initial check-in, version 0.5.0
This commit is contained in:
Rich Felker
commit
0b44a0315b
1021 changed files with 45711 additions and 0 deletions
14
src/math/__fpclassify.c
Normal file
14
src/math/__fpclassify.c
Normal file
|
|
@ -0,0 +1,14 @@
|
|||
#include <stdint.h>
|
||||
#include <math.h>
|
||||
|
||||
int __fpclassify(double __x)
|
||||
{
|
||||
union {
|
||||
double __d;
|
||||
__uint64_t __i;
|
||||
} __y = { __x };
|
||||
int __ee = __y.__i>>52 & 0x7ff;
|
||||
if (!__ee) return __y.__i<<1 ? FP_SUBNORMAL : FP_ZERO;
|
||||
if (__ee==0x7ff) return __y.__i<<12 ? FP_NAN : FP_INFINITE;
|
||||
return FP_NORMAL;
|
||||
}
|
||||
14
src/math/__fpclassifyf.c
Normal file
14
src/math/__fpclassifyf.c
Normal file
|
|
@ -0,0 +1,14 @@
|
|||
#include <stdint.h>
|
||||
#include <math.h>
|
||||
|
||||
int __fpclassifyf(float __x)
|
||||
{
|
||||
union {
|
||||
float __f;
|
||||
__uint32_t __i;
|
||||
} __y = { __x };
|
||||
int __ee = __y.__i>>23 & 0xff;
|
||||
if (!__ee) return __y.__i<<1 ? FP_SUBNORMAL : FP_ZERO;
|
||||
if (__ee==0xff) return __y.__i<<9 ? FP_NAN : FP_INFINITE;
|
||||
return FP_NORMAL;
|
||||
}
|
||||
16
src/math/__fpclassifyl.c
Normal file
16
src/math/__fpclassifyl.c
Normal file
|
|
@ -0,0 +1,16 @@
|
|||
#include <stdint.h>
|
||||
#include <math.h>
|
||||
|
||||
/* FIXME: move this to arch-specific file */
|
||||
int __fpclassifyl(long double __x)
|
||||
{
|
||||
union {
|
||||
long double __ld;
|
||||
__uint16_t __hw[5];
|
||||
__uint64_t __m;
|
||||
} __y = { __x };
|
||||
int __ee = __y.__hw[4]&0x7fff;
|
||||
if (!__ee) return __y.__m ? FP_SUBNORMAL : FP_ZERO;
|
||||
if (__ee==0x7fff) return __y.__m ? FP_NAN : FP_INFINITE;
|
||||
return FP_NORMAL;
|
||||
}
|
||||
99
src/math/e_acos.c
Normal file
99
src/math/e_acos.c
Normal file
|
|
@ -0,0 +1,99 @@
|
|||
/* @(#)e_acos.c 1.3 95/01/18 */
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunSoft, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/* acos(x)
|
||||
* Method :
|
||||
* acos(x) = pi/2 - asin(x)
|
||||
* acos(-x) = pi/2 + asin(x)
|
||||
* For |x|<=0.5
|
||||
* acos(x) = pi/2 - (x + x*x^2*R(x^2)) (see asin.c)
|
||||
* For x>0.5
|
||||
* acos(x) = pi/2 - (pi/2 - 2asin(sqrt((1-x)/2)))
|
||||
* = 2asin(sqrt((1-x)/2))
|
||||
* = 2s + 2s*z*R(z) ...z=(1-x)/2, s=sqrt(z)
|
||||
* = 2f + (2c + 2s*z*R(z))
|
||||
* where f=hi part of s, and c = (z-f*f)/(s+f) is the correction term
|
||||
* for f so that f+c ~ sqrt(z).
|
||||
* For x<-0.5
|
||||
* acos(x) = pi - 2asin(sqrt((1-|x|)/2))
|
||||
* = pi - 0.5*(s+s*z*R(z)), where z=(1-|x|)/2,s=sqrt(z)
|
||||
*
|
||||
* Special cases:
|
||||
* if x is NaN, return x itself;
|
||||
* if |x|>1, return NaN with invalid signal.
|
||||
*
|
||||
* Function needed: sqrt
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const double
|
||||
one= 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
|
||||
pi = 3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
|
||||
pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
|
||||
pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
|
||||
pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
|
||||
pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
|
||||
pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
|
||||
pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
|
||||
pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
|
||||
pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
|
||||
qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
|
||||
qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
|
||||
qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
|
||||
qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
|
||||
|
||||
double
|
||||
acos(double x)
|
||||
{
|
||||
double z,p,q,r,w,s,c,df;
|
||||
int32_t hx,ix;
|
||||
GET_HIGH_WORD(hx,x);
|
||||
ix = hx&0x7fffffff;
|
||||
if(ix>=0x3ff00000) { /* |x| >= 1 */
|
||||
uint32_t lx;
|
||||
GET_LOW_WORD(lx,x);
|
||||
if(((ix-0x3ff00000)|lx)==0) { /* |x|==1 */
|
||||
if(hx>0) return 0.0; /* acos(1) = 0 */
|
||||
else return pi+2.0*pio2_lo; /* acos(-1)= pi */
|
||||
}
|
||||
return (x-x)/(x-x); /* acos(|x|>1) is NaN */
|
||||
}
|
||||
if(ix<0x3fe00000) { /* |x| < 0.5 */
|
||||
if(ix<=0x3c600000) return pio2_hi+pio2_lo;/*if|x|<2**-57*/
|
||||
z = x*x;
|
||||
p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5)))));
|
||||
q = one+z*(qS1+z*(qS2+z*(qS3+z*qS4)));
|
||||
r = p/q;
|
||||
return pio2_hi - (x - (pio2_lo-x*r));
|
||||
} else if (hx<0) { /* x < -0.5 */
|
||||
z = (one+x)*0.5;
|
||||
p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5)))));
|
||||
q = one+z*(qS1+z*(qS2+z*(qS3+z*qS4)));
|
||||
s = sqrt(z);
|
||||
r = p/q;
|
||||
w = r*s-pio2_lo;
|
||||
return pi - 2.0*(s+w);
|
||||
} else { /* x > 0.5 */
|
||||
z = (one-x)*0.5;
|
||||
s = sqrt(z);
|
||||
df = s;
|
||||
SET_LOW_WORD(df,0);
|
||||
c = (z-df*df)/(s+df);
|
||||
p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5)))));
|
||||
q = one+z*(qS1+z*(qS2+z*(qS3+z*qS4)));
|
||||
r = p/q;
|
||||
w = r*s+c;
|
||||
return 2.0*(df+w);
|
||||
}
|
||||
}
|
||||
77
src/math/e_acosf.c
Normal file
77
src/math/e_acosf.c
Normal file
|
|
@ -0,0 +1,77 @@
|
|||
/* e_acosf.c -- float version of e_acos.c.
|
||||
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
||||
*/
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const float
|
||||
one = 1.0000000000e+00, /* 0x3F800000 */
|
||||
pi = 3.1415925026e+00, /* 0x40490fda */
|
||||
pio2_hi = 1.5707962513e+00, /* 0x3fc90fda */
|
||||
pio2_lo = 7.5497894159e-08, /* 0x33a22168 */
|
||||
pS0 = 1.6666667163e-01, /* 0x3e2aaaab */
|
||||
pS1 = -3.2556581497e-01, /* 0xbea6b090 */
|
||||
pS2 = 2.0121252537e-01, /* 0x3e4e0aa8 */
|
||||
pS3 = -4.0055535734e-02, /* 0xbd241146 */
|
||||
pS4 = 7.9153501429e-04, /* 0x3a4f7f04 */
|
||||
pS5 = 3.4793309169e-05, /* 0x3811ef08 */
|
||||
qS1 = -2.4033949375e+00, /* 0xc019d139 */
|
||||
qS2 = 2.0209457874e+00, /* 0x4001572d */
|
||||
qS3 = -6.8828397989e-01, /* 0xbf303361 */
|
||||
qS4 = 7.7038154006e-02; /* 0x3d9dc62e */
|
||||
|
||||
float
|
||||
acosf(float x)
|
||||
{
|
||||
float z,p,q,r,w,s,c,df;
|
||||
int32_t hx,ix;
|
||||
GET_FLOAT_WORD(hx,x);
|
||||
ix = hx&0x7fffffff;
|
||||
if(ix==0x3f800000) { /* |x|==1 */
|
||||
if(hx>0) return 0.0; /* acos(1) = 0 */
|
||||
else return pi+(float)2.0*pio2_lo; /* acos(-1)= pi */
|
||||
} else if(ix>0x3f800000) { /* |x| >= 1 */
|
||||
return (x-x)/(x-x); /* acos(|x|>1) is NaN */
|
||||
}
|
||||
if(ix<0x3f000000) { /* |x| < 0.5 */
|
||||
if(ix<=0x23000000) return pio2_hi+pio2_lo;/*if|x|<2**-57*/
|
||||
z = x*x;
|
||||
p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5)))));
|
||||
q = one+z*(qS1+z*(qS2+z*(qS3+z*qS4)));
|
||||
r = p/q;
|
||||
return pio2_hi - (x - (pio2_lo-x*r));
|
||||
} else if (hx<0) { /* x < -0.5 */
|
||||
z = (one+x)*(float)0.5;
|
||||
p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5)))));
|
||||
q = one+z*(qS1+z*(qS2+z*(qS3+z*qS4)));
|
||||
s = sqrtf(z);
|
||||
r = p/q;
|
||||
w = r*s-pio2_lo;
|
||||
return pi - (float)2.0*(s+w);
|
||||
} else { /* x > 0.5 */
|
||||
int32_t idf;
|
||||
z = (one-x)*(float)0.5;
|
||||
s = sqrtf(z);
|
||||
df = s;
|
||||
GET_FLOAT_WORD(idf,df);
|
||||
SET_FLOAT_WORD(df,idf&0xfffff000);
|
||||
c = (z-df*df)/(s+df);
|
||||
p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5)))));
|
||||
q = one+z*(qS1+z*(qS2+z*(qS3+z*qS4)));
|
||||
r = p/q;
|
||||
w = r*s+c;
|
||||
return (float)2.0*(df+w);
|
||||
}
|
||||
}
|
||||
59
src/math/e_acosh.c
Normal file
59
src/math/e_acosh.c
Normal file
|
|
@ -0,0 +1,59 @@
|
|||
|
||||
/* @(#)e_acosh.c 1.3 95/01/18 */
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunSoft, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*
|
||||
*/
|
||||
|
||||
/* acosh(x)
|
||||
* Method :
|
||||
* Based on
|
||||
* acosh(x) = log [ x + sqrt(x*x-1) ]
|
||||
* we have
|
||||
* acosh(x) := log(x)+ln2, if x is large; else
|
||||
* acosh(x) := log(2x-1/(sqrt(x*x-1)+x)) if x>2; else
|
||||
* acosh(x) := log1p(t+sqrt(2.0*t+t*t)); where t=x-1.
|
||||
*
|
||||
* Special cases:
|
||||
* acosh(x) is NaN with signal if x<1.
|
||||
* acosh(NaN) is NaN without signal.
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const double
|
||||
one = 1.0,
|
||||
ln2 = 6.93147180559945286227e-01; /* 0x3FE62E42, 0xFEFA39EF */
|
||||
|
||||
double
|
||||
acosh(double x)
|
||||
{
|
||||
double t;
|
||||
int32_t hx;
|
||||
uint32_t lx;
|
||||
EXTRACT_WORDS(hx,lx,x);
|
||||
if(hx<0x3ff00000) { /* x < 1 */
|
||||
return (x-x)/(x-x);
|
||||
} else if(hx >=0x41b00000) { /* x > 2**28 */
|
||||
if(hx >=0x7ff00000) { /* x is inf of NaN */
|
||||
return x+x;
|
||||
} else
|
||||
return log(x)+ln2; /* acosh(huge)=log(2x) */
|
||||
} else if(((hx-0x3ff00000)|lx)==0) {
|
||||
return 0.0; /* acosh(1) = 0 */
|
||||
} else if (hx > 0x40000000) { /* 2**28 > x > 2 */
|
||||
t=x*x;
|
||||
return log(2.0*x-one/(x+sqrt(t-one)));
|
||||
} else { /* 1<x<2 */
|
||||
t = x-one;
|
||||
return log1p(t+sqrt(2.0*t+t*t));
|
||||
}
|
||||
}
|
||||
45
src/math/e_acoshf.c
Normal file
45
src/math/e_acoshf.c
Normal file
|
|
@ -0,0 +1,45 @@
|
|||
/* e_acoshf.c -- float version of e_acosh.c.
|
||||
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
||||
*/
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const float
|
||||
one = 1.0,
|
||||
ln2 = 6.9314718246e-01; /* 0x3f317218 */
|
||||
|
||||
float
|
||||
acoshf(float x)
|
||||
{
|
||||
float t;
|
||||
int32_t hx;
|
||||
GET_FLOAT_WORD(hx,x);
|
||||
if(hx<0x3f800000) { /* x < 1 */
|
||||
return (x-x)/(x-x);
|
||||
} else if(hx >=0x4d800000) { /* x > 2**28 */
|
||||
if(hx >=0x7f800000) { /* x is inf of NaN */
|
||||
return x+x;
|
||||
} else
|
||||
return logf(x)+ln2; /* acosh(huge)=log(2x) */
|
||||
} else if (hx==0x3f800000) {
|
||||
return 0.0; /* acosh(1) = 0 */
|
||||
} else if (hx > 0x40000000) { /* 2**28 > x > 2 */
|
||||
t=x*x;
|
||||
return logf((float)2.0*x-one/(x+sqrtf(t-one)));
|
||||
} else { /* 1<x<2 */
|
||||
t = x-one;
|
||||
return log1pf(t+sqrtf((float)2.0*t+t*t));
|
||||
}
|
||||
}
|
||||
109
src/math/e_asin.c
Normal file
109
src/math/e_asin.c
Normal file
|
|
@ -0,0 +1,109 @@
|
|||
|
||||
/* @(#)e_asin.c 1.3 95/01/18 */
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunSoft, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/* asin(x)
|
||||
* Method :
|
||||
* Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
|
||||
* we approximate asin(x) on [0,0.5] by
|
||||
* asin(x) = x + x*x^2*R(x^2)
|
||||
* where
|
||||
* R(x^2) is a rational approximation of (asin(x)-x)/x^3
|
||||
* and its remez error is bounded by
|
||||
* |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
|
||||
*
|
||||
* For x in [0.5,1]
|
||||
* asin(x) = pi/2-2*asin(sqrt((1-x)/2))
|
||||
* Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
|
||||
* then for x>0.98
|
||||
* asin(x) = pi/2 - 2*(s+s*z*R(z))
|
||||
* = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
|
||||
* For x<=0.98, let pio4_hi = pio2_hi/2, then
|
||||
* f = hi part of s;
|
||||
* c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z)
|
||||
* and
|
||||
* asin(x) = pi/2 - 2*(s+s*z*R(z))
|
||||
* = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
|
||||
* = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
|
||||
*
|
||||
* Special cases:
|
||||
* if x is NaN, return x itself;
|
||||
* if |x|>1, return NaN with invalid signal.
|
||||
*
|
||||
*/
|
||||
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const double
|
||||
one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
|
||||
huge = 1.000e+300,
|
||||
pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
|
||||
pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
|
||||
pio4_hi = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
|
||||
/* coefficient for R(x^2) */
|
||||
pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
|
||||
pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
|
||||
pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
|
||||
pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
|
||||
pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
|
||||
pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
|
||||
qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
|
||||
qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
|
||||
qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
|
||||
qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
|
||||
|
||||
double
|
||||
asin(double x)
|
||||
{
|
||||
double t=0.0,w,p,q,c,r,s;
|
||||
int32_t hx,ix;
|
||||
GET_HIGH_WORD(hx,x);
|
||||
ix = hx&0x7fffffff;
|
||||
if(ix>= 0x3ff00000) { /* |x|>= 1 */
|
||||
uint32_t lx;
|
||||
GET_LOW_WORD(lx,x);
|
||||
if(((ix-0x3ff00000)|lx)==0)
|
||||
/* asin(1)=+-pi/2 with inexact */
|
||||
return x*pio2_hi+x*pio2_lo;
|
||||
return (x-x)/(x-x); /* asin(|x|>1) is NaN */
|
||||
} else if (ix<0x3fe00000) { /* |x|<0.5 */
|
||||
if(ix<0x3e400000) { /* if |x| < 2**-27 */
|
||||
if(huge+x>one) return x;/* return x with inexact if x!=0*/
|
||||
} else
|
||||
t = x*x;
|
||||
p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
|
||||
q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
|
||||
w = p/q;
|
||||
return x+x*w;
|
||||
}
|
||||
/* 1> |x|>= 0.5 */
|
||||
w = one-fabs(x);
|
||||
t = w*0.5;
|
||||
p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
|
||||
q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
|
||||
s = sqrt(t);
|
||||
if(ix>=0x3FEF3333) { /* if |x| > 0.975 */
|
||||
w = p/q;
|
||||
t = pio2_hi-(2.0*(s+s*w)-pio2_lo);
|
||||
} else {
|
||||
w = s;
|
||||
SET_LOW_WORD(w,0);
|
||||
c = (t-w*w)/(s+w);
|
||||
r = p/q;
|
||||
p = 2.0*s*r-(pio2_lo-2.0*c);
|
||||
q = pio4_hi-2.0*w;
|
||||
t = pio4_hi-(p-q);
|
||||
}
|
||||
if(hx>0) return t; else return -t;
|
||||
}
|
||||
80
src/math/e_asinf.c
Normal file
80
src/math/e_asinf.c
Normal file
|
|
@ -0,0 +1,80 @@
|
|||
/* e_asinf.c -- float version of e_asin.c.
|
||||
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
||||
*/
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const float
|
||||
one = 1.0000000000e+00, /* 0x3F800000 */
|
||||
huge = 1.000e+30,
|
||||
pio2_hi = 1.5707962513e+00, /* 0x3fc90fda */
|
||||
pio2_lo = 7.5497894159e-08, /* 0x33a22168 */
|
||||
pio4_hi = 7.8539818525e-01, /* 0x3f490fdb */
|
||||
/* coefficient for R(x^2) */
|
||||
pS0 = 1.6666667163e-01, /* 0x3e2aaaab */
|
||||
pS1 = -3.2556581497e-01, /* 0xbea6b090 */
|
||||
pS2 = 2.0121252537e-01, /* 0x3e4e0aa8 */
|
||||
pS3 = -4.0055535734e-02, /* 0xbd241146 */
|
||||
pS4 = 7.9153501429e-04, /* 0x3a4f7f04 */
|
||||
pS5 = 3.4793309169e-05, /* 0x3811ef08 */
|
||||
qS1 = -2.4033949375e+00, /* 0xc019d139 */
|
||||
qS2 = 2.0209457874e+00, /* 0x4001572d */
|
||||
qS3 = -6.8828397989e-01, /* 0xbf303361 */
|
||||
qS4 = 7.7038154006e-02; /* 0x3d9dc62e */
|
||||
|
||||
float
|
||||
asinf(float x)
|
||||
{
|
||||
float t=0.0,w,p,q,c,r,s;
|
||||
int32_t hx,ix;
|
||||
GET_FLOAT_WORD(hx,x);
|
||||
ix = hx&0x7fffffff;
|
||||
if(ix==0x3f800000) {
|
||||
/* asin(1)=+-pi/2 with inexact */
|
||||
return x*pio2_hi+x*pio2_lo;
|
||||
} else if(ix> 0x3f800000) { /* |x|>= 1 */
|
||||
return (x-x)/(x-x); /* asin(|x|>1) is NaN */
|
||||
} else if (ix<0x3f000000) { /* |x|<0.5 */
|
||||
if(ix<0x32000000) { /* if |x| < 2**-27 */
|
||||
if(huge+x>one) return x;/* return x with inexact if x!=0*/
|
||||
} else
|
||||
t = x*x;
|
||||
p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
|
||||
q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
|
||||
w = p/q;
|
||||
return x+x*w;
|
||||
}
|
||||
/* 1> |x|>= 0.5 */
|
||||
w = one-fabsf(x);
|
||||
t = w*(float)0.5;
|
||||
p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
|
||||
q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
|
||||
s = sqrtf(t);
|
||||
if(ix>=0x3F79999A) { /* if |x| > 0.975 */
|
||||
w = p/q;
|
||||
t = pio2_hi-((float)2.0*(s+s*w)-pio2_lo);
|
||||
} else {
|
||||
int32_t iw;
|
||||
w = s;
|
||||
GET_FLOAT_WORD(iw,w);
|
||||
SET_FLOAT_WORD(w,iw&0xfffff000);
|
||||
c = (t-w*w)/(s+w);
|
||||
r = p/q;
|
||||
p = (float)2.0*s*r-(pio2_lo-(float)2.0*c);
|
||||
q = pio4_hi-(float)2.0*w;
|
||||
t = pio4_hi-(p-q);
|
||||
}
|
||||
if(hx>0) return t; else return -t;
|
||||
}
|
||||
120
src/math/e_atan2.c
Normal file
120
src/math/e_atan2.c
Normal file
|
|
@ -0,0 +1,120 @@
|
|||
|
||||
/* @(#)e_atan2.c 1.3 95/01/18 */
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunSoft, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*
|
||||
*/
|
||||
|
||||
/* atan2(y,x)
|
||||
* Method :
|
||||
* 1. Reduce y to positive by atan2(y,x)=-atan2(-y,x).
|
||||
* 2. Reduce x to positive by (if x and y are unexceptional):
|
||||
* ARG (x+iy) = arctan(y/x) ... if x > 0,
|
||||
* ARG (x+iy) = pi - arctan[y/(-x)] ... if x < 0,
|
||||
*
|
||||
* Special cases:
|
||||
*
|
||||
* ATAN2((anything), NaN ) is NaN;
|
||||
* ATAN2(NAN , (anything) ) is NaN;
|
||||
* ATAN2(+-0, +(anything but NaN)) is +-0 ;
|
||||
* ATAN2(+-0, -(anything but NaN)) is +-pi ;
|
||||
* ATAN2(+-(anything but 0 and NaN), 0) is +-pi/2;
|
||||
* ATAN2(+-(anything but INF and NaN), +INF) is +-0 ;
|
||||
* ATAN2(+-(anything but INF and NaN), -INF) is +-pi;
|
||||
* ATAN2(+-INF,+INF ) is +-pi/4 ;
|
||||
* ATAN2(+-INF,-INF ) is +-3pi/4;
|
||||
* ATAN2(+-INF, (anything but,0,NaN, and INF)) is +-pi/2;
|
||||
*
|
||||
* Constants:
|
||||
* The hexadecimal values are the intended ones for the following
|
||||
* constants. The decimal values may be used, provided that the
|
||||
* compiler will convert from decimal to binary accurately enough
|
||||
* to produce the hexadecimal values shown.
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const double
|
||||
tiny = 1.0e-300,
|
||||
zero = 0.0,
|
||||
pi_o_4 = 7.8539816339744827900E-01, /* 0x3FE921FB, 0x54442D18 */
|
||||
pi_o_2 = 1.5707963267948965580E+00, /* 0x3FF921FB, 0x54442D18 */
|
||||
pi = 3.1415926535897931160E+00, /* 0x400921FB, 0x54442D18 */
|
||||
pi_lo = 1.2246467991473531772E-16; /* 0x3CA1A626, 0x33145C07 */
|
||||
|
||||
double
|
||||
atan2(double y, double x)
|
||||
{
|
||||
double z;
|
||||
int32_t k,m,hx,hy,ix,iy;
|
||||
uint32_t lx,ly;
|
||||
|
||||
EXTRACT_WORDS(hx,lx,x);
|
||||
ix = hx&0x7fffffff;
|
||||
EXTRACT_WORDS(hy,ly,y);
|
||||
iy = hy&0x7fffffff;
|
||||
if(((ix|((lx|-lx)>>31))>0x7ff00000)||
|
||||
((iy|((ly|-ly)>>31))>0x7ff00000)) /* x or y is NaN */
|
||||
return x+y;
|
||||
if(((hx-0x3ff00000)|lx)==0) return atan(y); /* x=1.0 */
|
||||
m = ((hy>>31)&1)|((hx>>30)&2); /* 2*sign(x)+sign(y) */
|
||||
|
||||
/* when y = 0 */
|
||||
if((iy|ly)==0) {
|
||||
switch(m) {
|
||||
case 0:
|
||||
case 1: return y; /* atan(+-0,+anything)=+-0 */
|
||||
case 2: return pi+tiny;/* atan(+0,-anything) = pi */
|
||||
case 3: return -pi-tiny;/* atan(-0,-anything) =-pi */
|
||||
}
|
||||
}
|
||||
/* when x = 0 */
|
||||
if((ix|lx)==0) return (hy<0)? -pi_o_2-tiny: pi_o_2+tiny;
|
||||
|
||||
/* when x is INF */
|
||||
if(ix==0x7ff00000) {
|
||||
if(iy==0x7ff00000) {
|
||||
switch(m) {
|
||||
case 0: return pi_o_4+tiny;/* atan(+INF,+INF) */
|
||||
case 1: return -pi_o_4-tiny;/* atan(-INF,+INF) */
|
||||
case 2: return 3.0*pi_o_4+tiny;/*atan(+INF,-INF)*/
|
||||
case 3: return -3.0*pi_o_4-tiny;/*atan(-INF,-INF)*/
|
||||
}
|
||||
} else {
|
||||
switch(m) {
|
||||
case 0: return zero ; /* atan(+...,+INF) */
|
||||
case 1: return -zero ; /* atan(-...,+INF) */
|
||||
case 2: return pi+tiny ; /* atan(+...,-INF) */
|
||||
case 3: return -pi-tiny ; /* atan(-...,-INF) */
|
||||
}
|
||||
}
|
||||
}
|
||||
/* when y is INF */
|
||||
if(iy==0x7ff00000) return (hy<0)? -pi_o_2-tiny: pi_o_2+tiny;
|
||||
|
||||
/* compute y/x */
|
||||
k = (iy-ix)>>20;
|
||||
if(k > 60) z=pi_o_2+0.5*pi_lo; /* |y/x| > 2**60 */
|
||||
else if(hx<0&&k<-60) z=0.0; /* |y|/x < -2**60 */
|
||||
else z=atan(fabs(y/x)); /* safe to do y/x */
|
||||
switch (m) {
|
||||
case 0: return z ; /* atan(+,+) */
|
||||
case 1: {
|
||||
uint32_t zh;
|
||||
GET_HIGH_WORD(zh,z);
|
||||
SET_HIGH_WORD(z,zh ^ 0x80000000);
|
||||
}
|
||||
return z ; /* atan(-,+) */
|
||||
case 2: return pi-(z-pi_lo);/* atan(+,-) */
|
||||
default: /* case 3 */
|
||||
return (z-pi_lo)-pi;/* atan(-,-) */
|
||||
}
|
||||
}
|
||||
93
src/math/e_atan2f.c
Normal file
93
src/math/e_atan2f.c
Normal file
|
|
@ -0,0 +1,93 @@
|
|||
/* e_atan2f.c -- float version of e_atan2.c.
|
||||
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
||||
*/
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const float
|
||||
tiny = 1.0e-30,
|
||||
zero = 0.0,
|
||||
pi_o_4 = 7.8539818525e-01, /* 0x3f490fdb */
|
||||
pi_o_2 = 1.5707963705e+00, /* 0x3fc90fdb */
|
||||
pi = 3.1415927410e+00, /* 0x40490fdb */
|
||||
pi_lo = -8.7422776573e-08; /* 0xb3bbbd2e */
|
||||
|
||||
float
|
||||
atan2f(float y, float x)
|
||||
{
|
||||
float z;
|
||||
int32_t k,m,hx,hy,ix,iy;
|
||||
|
||||
GET_FLOAT_WORD(hx,x);
|
||||
ix = hx&0x7fffffff;
|
||||
GET_FLOAT_WORD(hy,y);
|
||||
iy = hy&0x7fffffff;
|
||||
if((ix>0x7f800000)||
|
||||
(iy>0x7f800000)) /* x or y is NaN */
|
||||
return x+y;
|
||||
if(hx==0x3f800000) return atanf(y); /* x=1.0 */
|
||||
m = ((hy>>31)&1)|((hx>>30)&2); /* 2*sign(x)+sign(y) */
|
||||
|
||||
/* when y = 0 */
|
||||
if(iy==0) {
|
||||
switch(m) {
|
||||
case 0:
|
||||
case 1: return y; /* atan(+-0,+anything)=+-0 */
|
||||
case 2: return pi+tiny;/* atan(+0,-anything) = pi */
|
||||
case 3: return -pi-tiny;/* atan(-0,-anything) =-pi */
|
||||
}
|
||||
}
|
||||
/* when x = 0 */
|
||||
if(ix==0) return (hy<0)? -pi_o_2-tiny: pi_o_2+tiny;
|
||||
|
||||
/* when x is INF */
|
||||
if(ix==0x7f800000) {
|
||||
if(iy==0x7f800000) {
|
||||
switch(m) {
|
||||
case 0: return pi_o_4+tiny;/* atan(+INF,+INF) */
|
||||
case 1: return -pi_o_4-tiny;/* atan(-INF,+INF) */
|
||||
case 2: return (float)3.0*pi_o_4+tiny;/*atan(+INF,-INF)*/
|
||||
case 3: return (float)-3.0*pi_o_4-tiny;/*atan(-INF,-INF)*/
|
||||
}
|
||||
} else {
|
||||
switch(m) {
|
||||
case 0: return zero ; /* atan(+...,+INF) */
|
||||
case 1: return -zero ; /* atan(-...,+INF) */
|
||||
case 2: return pi+tiny ; /* atan(+...,-INF) */
|
||||
case 3: return -pi-tiny ; /* atan(-...,-INF) */
|
||||
}
|
||||
}
|
||||
}
|
||||
/* when y is INF */
|
||||
if(iy==0x7f800000) return (hy<0)? -pi_o_2-tiny: pi_o_2+tiny;
|
||||
|
||||
/* compute y/x */
|
||||
k = (iy-ix)>>23;
|
||||
if(k > 60) z=pi_o_2+(float)0.5*pi_lo; /* |y/x| > 2**60 */
|
||||
else if(hx<0&&k<-60) z=0.0; /* |y|/x < -2**60 */
|
||||
else z=atanf(fabsf(y/x)); /* safe to do y/x */
|
||||
switch (m) {
|
||||
case 0: return z ; /* atan(+,+) */
|
||||
case 1: {
|
||||
uint32_t zh;
|
||||
GET_FLOAT_WORD(zh,z);
|
||||
SET_FLOAT_WORD(z,zh ^ 0x80000000);
|
||||
}
|
||||
return z ; /* atan(-,+) */
|
||||
case 2: return pi-(z-pi_lo);/* atan(+,-) */
|
||||
default: /* case 3 */
|
||||
return (z-pi_lo)-pi;/* atan(-,-) */
|
||||
}
|
||||
}
|
||||
59
src/math/e_atanh.c
Normal file
59
src/math/e_atanh.c
Normal file
|
|
@ -0,0 +1,59 @@
|
|||
|
||||
/* @(#)e_atanh.c 1.3 95/01/18 */
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunSoft, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*
|
||||
*/
|
||||
|
||||
/* atanh(x)
|
||||
* Method :
|
||||
* 1.Reduced x to positive by atanh(-x) = -atanh(x)
|
||||
* 2.For x>=0.5
|
||||
* 1 2x x
|
||||
* atanh(x) = --- * log(1 + -------) = 0.5 * log1p(2 * --------)
|
||||
* 2 1 - x 1 - x
|
||||
*
|
||||
* For x<0.5
|
||||
* atanh(x) = 0.5*log1p(2x+2x*x/(1-x))
|
||||
*
|
||||
* Special cases:
|
||||
* atanh(x) is NaN if |x| > 1 with signal;
|
||||
* atanh(NaN) is that NaN with no signal;
|
||||
* atanh(+-1) is +-INF with signal.
|
||||
*
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const double one = 1.0, huge = 1e300;
|
||||
static const double zero = 0.0;
|
||||
|
||||
double
|
||||
atanh(double x)
|
||||
{
|
||||
double t;
|
||||
int32_t hx,ix;
|
||||
uint32_t lx;
|
||||
EXTRACT_WORDS(hx,lx,x);
|
||||
ix = hx&0x7fffffff;
|
||||
if ((ix|((lx|(-lx))>>31))>0x3ff00000) /* |x|>1 */
|
||||
return (x-x)/(x-x);
|
||||
if(ix==0x3ff00000)
|
||||
return x/zero;
|
||||
if(ix<0x3e300000&&(huge+x)>zero) return x; /* x<2**-28 */
|
||||
SET_HIGH_WORD(x,ix);
|
||||
if(ix<0x3fe00000) { /* x < 0.5 */
|
||||
t = x+x;
|
||||
t = 0.5*log1p(t+t*x/(one-x));
|
||||
} else
|
||||
t = 0.5*log1p((x+x)/(one-x));
|
||||
if(hx>=0) return t; else return -t;
|
||||
}
|
||||
42
src/math/e_atanhf.c
Normal file
42
src/math/e_atanhf.c
Normal file
|
|
@ -0,0 +1,42 @@
|
|||
/* e_atanhf.c -- float version of e_atanh.c.
|
||||
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
||||
*/
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const float one = 1.0, huge = 1e30;
|
||||
|
||||
static const float zero = 0.0;
|
||||
|
||||
float
|
||||
atanhf(float x)
|
||||
{
|
||||
float t;
|
||||
int32_t hx,ix;
|
||||
GET_FLOAT_WORD(hx,x);
|
||||
ix = hx&0x7fffffff;
|
||||
if (ix>0x3f800000) /* |x|>1 */
|
||||
return (x-x)/(x-x);
|
||||
if(ix==0x3f800000)
|
||||
return x/zero;
|
||||
if(ix<0x31800000&&(huge+x)>zero) return x; /* x<2**-28 */
|
||||
SET_FLOAT_WORD(x,ix);
|
||||
if(ix<0x3f000000) { /* x < 0.5 */
|
||||
t = x+x;
|
||||
t = (float)0.5*log1pf(t+t*x/(one-x));
|
||||
} else
|
||||
t = (float)0.5*log1pf((x+x)/(one-x));
|
||||
if(hx>=0) return t; else return -t;
|
||||
}
|
||||
82
src/math/e_cosh.c
Normal file
82
src/math/e_cosh.c
Normal file
|
|
@ -0,0 +1,82 @@
|
|||
|
||||
/* @(#)e_cosh.c 1.3 95/01/18 */
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunSoft, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/* cosh(x)
|
||||
* Method :
|
||||
* mathematically cosh(x) if defined to be (exp(x)+exp(-x))/2
|
||||
* 1. Replace x by |x| (cosh(x) = cosh(-x)).
|
||||
* 2.
|
||||
* [ exp(x) - 1 ]^2
|
||||
* 0 <= x <= ln2/2 : cosh(x) := 1 + -------------------
|
||||
* 2*exp(x)
|
||||
*
|
||||
* exp(x) + 1/exp(x)
|
||||
* ln2/2 <= x <= 22 : cosh(x) := -------------------
|
||||
* 2
|
||||
* 22 <= x <= lnovft : cosh(x) := exp(x)/2
|
||||
* lnovft <= x <= ln2ovft: cosh(x) := exp(x/2)/2 * exp(x/2)
|
||||
* ln2ovft < x : cosh(x) := huge*huge (overflow)
|
||||
*
|
||||
* Special cases:
|
||||
* cosh(x) is |x| if x is +INF, -INF, or NaN.
|
||||
* only cosh(0)=1 is exact for finite x.
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const double one = 1.0, half=0.5, huge = 1.0e300;
|
||||
|
||||
double
|
||||
cosh(double x)
|
||||
{
|
||||
double t,w;
|
||||
int32_t ix;
|
||||
uint32_t lx;
|
||||
|
||||
/* High word of |x|. */
|
||||
GET_HIGH_WORD(ix,x);
|
||||
ix &= 0x7fffffff;
|
||||
|
||||
/* x is INF or NaN */
|
||||
if(ix>=0x7ff00000) return x*x;
|
||||
|
||||
/* |x| in [0,0.5*ln2], return 1+expm1(|x|)^2/(2*exp(|x|)) */
|
||||
if(ix<0x3fd62e43) {
|
||||
t = expm1(fabs(x));
|
||||
w = one+t;
|
||||
if (ix<0x3c800000) return w; /* cosh(tiny) = 1 */
|
||||
return one+(t*t)/(w+w);
|
||||
}
|
||||
|
||||
/* |x| in [0.5*ln2,22], return (exp(|x|)+1/exp(|x|)/2; */
|
||||
if (ix < 0x40360000) {
|
||||
t = exp(fabs(x));
|
||||
return half*t+half/t;
|
||||
}
|
||||
|
||||
/* |x| in [22, log(maxdouble)] return half*exp(|x|) */
|
||||
if (ix < 0x40862E42) return half*exp(fabs(x));
|
||||
|
||||
/* |x| in [log(maxdouble), overflowthresold] */
|
||||
GET_LOW_WORD(lx,x);
|
||||
if (ix<0x408633CE ||
|
||||
((ix==0x408633ce)&&(lx<=(uint32_t)0x8fb9f87d))) {
|
||||
w = exp(half*fabs(x));
|
||||
t = half*w;
|
||||
return t*w;
|
||||
}
|
||||
|
||||
/* |x| > overflowthresold, cosh(x) overflow */
|
||||
return huge*huge;
|
||||
}
|
||||
59
src/math/e_coshf.c
Normal file
59
src/math/e_coshf.c
Normal file
|
|
@ -0,0 +1,59 @@
|
|||
/* e_coshf.c -- float version of e_cosh.c.
|
||||
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
||||
*/
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const float one = 1.0, half=0.5, huge = 1.0e30;
|
||||
|
||||
float
|
||||
coshf(float x)
|
||||
{
|
||||
float t,w;
|
||||
int32_t ix;
|
||||
|
||||
GET_FLOAT_WORD(ix,x);
|
||||
ix &= 0x7fffffff;
|
||||
|
||||
/* x is INF or NaN */
|
||||
if(ix>=0x7f800000) return x*x;
|
||||
|
||||
/* |x| in [0,0.5*ln2], return 1+expm1(|x|)^2/(2*exp(|x|)) */
|
||||
if(ix<0x3eb17218) {
|
||||
t = expm1f(fabsf(x));
|
||||
w = one+t;
|
||||
if (ix<0x24000000) return w; /* cosh(tiny) = 1 */
|
||||
return one+(t*t)/(w+w);
|
||||
}
|
||||
|
||||
/* |x| in [0.5*ln2,22], return (exp(|x|)+1/exp(|x|)/2; */
|
||||
if (ix < 0x41b00000) {
|
||||
t = expf(fabsf(x));
|
||||
return half*t+half/t;
|
||||
}
|
||||
|
||||
/* |x| in [22, log(maxdouble)] return half*exp(|x|) */
|
||||
if (ix < 0x42b17180) return half*expf(fabsf(x));
|
||||
|
||||
/* |x| in [log(maxdouble), overflowthresold] */
|
||||
if (ix<=0x42b2d4fc) {
|
||||
w = expf(half*fabsf(x));
|
||||
t = half*w;
|
||||
return t*w;
|
||||
}
|
||||
|
||||
/* |x| > overflowthresold, cosh(x) overflow */
|
||||
return huge*huge;
|
||||
}
|
||||
155
src/math/e_exp.c
Normal file
155
src/math/e_exp.c
Normal file
|
|
@ -0,0 +1,155 @@
|
|||
|
||||
/* @(#)e_exp.c 1.6 04/04/22 */
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/* exp(x)
|
||||
* Returns the exponential of x.
|
||||
*
|
||||
* Method
|
||||
* 1. Argument reduction:
|
||||
* Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
|
||||
* Given x, find r and integer k such that
|
||||
*
|
||||
* x = k*ln2 + r, |r| <= 0.5*ln2.
|
||||
*
|
||||
* Here r will be represented as r = hi-lo for better
|
||||
* accuracy.
|
||||
*
|
||||
* 2. Approximation of exp(r) by a special rational function on
|
||||
* the interval [0,0.34658]:
|
||||
* Write
|
||||
* R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
|
||||
* We use a special Remes algorithm on [0,0.34658] to generate
|
||||
* a polynomial of degree 5 to approximate R. The maximum error
|
||||
* of this polynomial approximation is bounded by 2**-59. In
|
||||
* other words,
|
||||
* R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
|
||||
* (where z=r*r, and the values of P1 to P5 are listed below)
|
||||
* and
|
||||
* | 5 | -59
|
||||
* | 2.0+P1*z+...+P5*z - R(z) | <= 2
|
||||
* | |
|
||||
* The computation of exp(r) thus becomes
|
||||
* 2*r
|
||||
* exp(r) = 1 + -------
|
||||
* R - r
|
||||
* r*R1(r)
|
||||
* = 1 + r + ----------- (for better accuracy)
|
||||
* 2 - R1(r)
|
||||
* where
|
||||
* 2 4 10
|
||||
* R1(r) = r - (P1*r + P2*r + ... + P5*r ).
|
||||
*
|
||||
* 3. Scale back to obtain exp(x):
|
||||
* From step 1, we have
|
||||
* exp(x) = 2^k * exp(r)
|
||||
*
|
||||
* Special cases:
|
||||
* exp(INF) is INF, exp(NaN) is NaN;
|
||||
* exp(-INF) is 0, and
|
||||
* for finite argument, only exp(0)=1 is exact.
|
||||
*
|
||||
* Accuracy:
|
||||
* according to an error analysis, the error is always less than
|
||||
* 1 ulp (unit in the last place).
|
||||
*
|
||||
* Misc. info.
|
||||
* For IEEE double
|
||||
* if x > 7.09782712893383973096e+02 then exp(x) overflow
|
||||
* if x < -7.45133219101941108420e+02 then exp(x) underflow
|
||||
*
|
||||
* Constants:
|
||||
* The hexadecimal values are the intended ones for the following
|
||||
* constants. The decimal values may be used, provided that the
|
||||
* compiler will convert from decimal to binary accurately enough
|
||||
* to produce the hexadecimal values shown.
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const double
|
||||
one = 1.0,
|
||||
halF[2] = {0.5,-0.5,},
|
||||
huge = 1.0e+300,
|
||||
twom1000= 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/
|
||||
o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
|
||||
u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */
|
||||
ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
|
||||
-6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
|
||||
ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
|
||||
-1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
|
||||
invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
|
||||
P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
|
||||
P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
|
||||
P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
|
||||
P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
|
||||
P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
|
||||
|
||||
|
||||
double
|
||||
exp(double x) /* default IEEE double exp */
|
||||
{
|
||||
double y,hi=0.0,lo=0.0,c,t;
|
||||
int32_t k=0,xsb;
|
||||
uint32_t hx;
|
||||
|
||||
GET_HIGH_WORD(hx,x);
|
||||
xsb = (hx>>31)&1; /* sign bit of x */
|
||||
hx &= 0x7fffffff; /* high word of |x| */
|
||||
|
||||
/* filter out non-finite argument */
|
||||
if(hx >= 0x40862E42) { /* if |x|>=709.78... */
|
||||
if(hx>=0x7ff00000) {
|
||||
uint32_t lx;
|
||||
GET_LOW_WORD(lx,x);
|
||||
if(((hx&0xfffff)|lx)!=0)
|
||||
return x+x; /* NaN */
|
||||
else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */
|
||||
}
|
||||
if(x > o_threshold) return huge*huge; /* overflow */
|
||||
if(x < u_threshold) return twom1000*twom1000; /* underflow */
|
||||
}
|
||||
|
||||
/* argument reduction */
|
||||
if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
|
||||
if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
|
||||
hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
|
||||
} else {
|
||||
k = (int)(invln2*x+halF[xsb]);
|
||||
t = k;
|
||||
hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */
|
||||
lo = t*ln2LO[0];
|
||||
}
|
||||
x = hi - lo;
|
||||
}
|
||||
else if(hx < 0x3e300000) { /* when |x|<2**-28 */
|
||||
if(huge+x>one) return one+x;/* trigger inexact */
|
||||
}
|
||||
else k = 0;
|
||||
|
||||
/* x is now in primary range */
|
||||
t = x*x;
|
||||
c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
|
||||
if(k==0) return one-((x*c)/(c-2.0)-x);
|
||||
else y = one-((lo-(x*c)/(2.0-c))-hi);
|
||||
if(k >= -1021) {
|
||||
uint32_t hy;
|
||||
GET_HIGH_WORD(hy,y);
|
||||
SET_HIGH_WORD(y,hy+(k<<20)); /* add k to y's exponent */
|
||||
return y;
|
||||
} else {
|
||||
uint32_t hy;
|
||||
GET_HIGH_WORD(hy,y);
|
||||
SET_HIGH_WORD(y,hy+((k+1000)<<20)); /* add k to y's exponent */
|
||||
return y*twom1000;
|
||||
}
|
||||
}
|
||||
91
src/math/e_expf.c
Normal file
91
src/math/e_expf.c
Normal file
|
|
@ -0,0 +1,91 @@
|
|||
/* e_expf.c -- float version of e_exp.c.
|
||||
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
||||
*/
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const float
|
||||
one = 1.0,
|
||||
halF[2] = {0.5,-0.5,},
|
||||
huge = 1.0e+30,
|
||||
twom100 = 7.8886090522e-31, /* 2**-100=0x0d800000 */
|
||||
o_threshold= 8.8721679688e+01, /* 0x42b17180 */
|
||||
u_threshold= -1.0397208405e+02, /* 0xc2cff1b5 */
|
||||
ln2HI[2] ={ 6.9313812256e-01, /* 0x3f317180 */
|
||||
-6.9313812256e-01,}, /* 0xbf317180 */
|
||||
ln2LO[2] ={ 9.0580006145e-06, /* 0x3717f7d1 */
|
||||
-9.0580006145e-06,}, /* 0xb717f7d1 */
|
||||
invln2 = 1.4426950216e+00, /* 0x3fb8aa3b */
|
||||
P1 = 1.6666667163e-01, /* 0x3e2aaaab */
|
||||
P2 = -2.7777778450e-03, /* 0xbb360b61 */
|
||||
P3 = 6.6137559770e-05, /* 0x388ab355 */
|
||||
P4 = -1.6533901999e-06, /* 0xb5ddea0e */
|
||||
P5 = 4.1381369442e-08; /* 0x3331bb4c */
|
||||
|
||||
float
|
||||
expf(float x) /* default IEEE double exp */
|
||||
{
|
||||
float y,hi=0.0,lo=0.0,c,t;
|
||||
int32_t k=0,xsb;
|
||||
uint32_t hx;
|
||||
|
||||
GET_FLOAT_WORD(hx,x);
|
||||
xsb = (hx>>31)&1; /* sign bit of x */
|
||||
hx &= 0x7fffffff; /* high word of |x| */
|
||||
|
||||
/* filter out non-finite argument */
|
||||
if(hx >= 0x42b17218) { /* if |x|>=88.721... */
|
||||
if(hx>0x7f800000)
|
||||
return x+x; /* NaN */
|
||||
if(hx==0x7f800000)
|
||||
return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */
|
||||
if(x > o_threshold) return huge*huge; /* overflow */
|
||||
if(x < u_threshold) return twom100*twom100; /* underflow */
|
||||
}
|
||||
|
||||
/* argument reduction */
|
||||
if(hx > 0x3eb17218) { /* if |x| > 0.5 ln2 */
|
||||
if(hx < 0x3F851592) { /* and |x| < 1.5 ln2 */
|
||||
hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
|
||||
} else {
|
||||
k = invln2*x+halF[xsb];
|
||||
t = k;
|
||||
hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */
|
||||
lo = t*ln2LO[0];
|
||||
}
|
||||
x = hi - lo;
|
||||
}
|
||||
else if(hx < 0x31800000) { /* when |x|<2**-28 */
|
||||
if(huge+x>one) return one+x;/* trigger inexact */
|
||||
}
|
||||
else k = 0;
|
||||
|
||||
/* x is now in primary range */
|
||||
t = x*x;
|
||||
c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
|
||||
if(k==0) return one-((x*c)/(c-(float)2.0)-x);
|
||||
else y = one-((lo-(x*c)/((float)2.0-c))-hi);
|
||||
if(k >= -125) {
|
||||
uint32_t hy;
|
||||
GET_FLOAT_WORD(hy,y);
|
||||
SET_FLOAT_WORD(y,hy+(k<<23)); /* add k to y's exponent */
|
||||
return y;
|
||||
} else {
|
||||
uint32_t hy;
|
||||
GET_FLOAT_WORD(hy,y);
|
||||
SET_FLOAT_WORD(y,hy+((k+100)<<23)); /* add k to y's exponent */
|
||||
return y*twom100;
|
||||
}
|
||||
}
|
||||
129
src/math/e_fmod.c
Normal file
129
src/math/e_fmod.c
Normal file
|
|
@ -0,0 +1,129 @@
|
|||
|
||||
/* @(#)e_fmod.c 1.3 95/01/18 */
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunSoft, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/*
|
||||
* fmod(x,y)
|
||||
* Return x mod y in exact arithmetic
|
||||
* Method: shift and subtract
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const double one = 1.0, Zero[] = {0.0, -0.0,};
|
||||
|
||||
double
|
||||
fmod(double x, double y)
|
||||
{
|
||||
int32_t n,hx,hy,hz,ix,iy,sx,i;
|
||||
uint32_t lx,ly,lz;
|
||||
|
||||
EXTRACT_WORDS(hx,lx,x);
|
||||
EXTRACT_WORDS(hy,ly,y);
|
||||
sx = hx&0x80000000; /* sign of x */
|
||||
hx ^=sx; /* |x| */
|
||||
hy &= 0x7fffffff; /* |y| */
|
||||
|
||||
/* purge off exception values */
|
||||
if((hy|ly)==0||(hx>=0x7ff00000)|| /* y=0,or x not finite */
|
||||
((hy|((ly|-ly)>>31))>0x7ff00000)) /* or y is NaN */
|
||||
return (x*y)/(x*y);
|
||||
if(hx<=hy) {
|
||||
if((hx<hy)||(lx<ly)) return x; /* |x|<|y| return x */
|
||||
if(lx==ly)
|
||||
return Zero[(uint32_t)sx>>31]; /* |x|=|y| return x*0*/
|
||||
}
|
||||
|
||||
/* determine ix = ilogb(x) */
|
||||
if(hx<0x00100000) { /* subnormal x */
|
||||
if(hx==0) {
|
||||
for (ix = -1043, i=lx; i>0; i<<=1) ix -=1;
|
||||
} else {
|
||||
for (ix = -1022,i=(hx<<11); i>0; i<<=1) ix -=1;
|
||||
}
|
||||
} else ix = (hx>>20)-1023;
|
||||
|
||||
/* determine iy = ilogb(y) */
|
||||
if(hy<0x00100000) { /* subnormal y */
|
||||
if(hy==0) {
|
||||
for (iy = -1043, i=ly; i>0; i<<=1) iy -=1;
|
||||
} else {
|
||||
for (iy = -1022,i=(hy<<11); i>0; i<<=1) iy -=1;
|
||||
}
|
||||
} else iy = (hy>>20)-1023;
|
||||
|
||||
/* set up {hx,lx}, {hy,ly} and align y to x */
|
||||
if(ix >= -1022)
|
||||
hx = 0x00100000|(0x000fffff&hx);
|
||||
else { /* subnormal x, shift x to normal */
|
||||
n = -1022-ix;
|
||||
if(n<=31) {
|
||||
hx = (hx<<n)|(lx>>(32-n));
|
||||
lx <<= n;
|
||||
} else {
|
||||
hx = lx<<(n-32);
|
||||
lx = 0;
|
||||
}
|
||||
}
|
||||
if(iy >= -1022)
|
||||
hy = 0x00100000|(0x000fffff&hy);
|
||||
else { /* subnormal y, shift y to normal */
|
||||
n = -1022-iy;
|
||||
if(n<=31) {
|
||||
hy = (hy<<n)|(ly>>(32-n));
|
||||
ly <<= n;
|
||||
} else {
|
||||
hy = ly<<(n-32);
|
||||
ly = 0;
|
||||
}
|
||||
}
|
||||
|
||||
/* fix point fmod */
|
||||
n = ix - iy;
|
||||
while(n--) {
|
||||
hz=hx-hy;lz=lx-ly; if(lx<ly) hz -= 1;
|
||||
if(hz<0){hx = hx+hx+(lx>>31); lx = lx+lx;}
|
||||
else {
|
||||
if((hz|lz)==0) /* return sign(x)*0 */
|
||||
return Zero[(uint32_t)sx>>31];
|
||||
hx = hz+hz+(lz>>31); lx = lz+lz;
|
||||
}
|
||||
}
|
||||
hz=hx-hy;lz=lx-ly; if(lx<ly) hz -= 1;
|
||||
if(hz>=0) {hx=hz;lx=lz;}
|
||||
|
||||
/* convert back to floating value and restore the sign */
|
||||
if((hx|lx)==0) /* return sign(x)*0 */
|
||||
return Zero[(uint32_t)sx>>31];
|
||||
while(hx<0x00100000) { /* normalize x */
|
||||
hx = hx+hx+(lx>>31); lx = lx+lx;
|
||||
iy -= 1;
|
||||
}
|
||||
if(iy>= -1022) { /* normalize output */
|
||||
hx = ((hx-0x00100000)|((iy+1023)<<20));
|
||||
INSERT_WORDS(x,hx|sx,lx);
|
||||
} else { /* subnormal output */
|
||||
n = -1022 - iy;
|
||||
if(n<=20) {
|
||||
lx = (lx>>n)|((uint32_t)hx<<(32-n));
|
||||
hx >>= n;
|
||||
} else if (n<=31) {
|
||||
lx = (hx<<(32-n))|(lx>>n); hx = sx;
|
||||
} else {
|
||||
lx = hx>>(n-32); hx = sx;
|
||||
}
|
||||
INSERT_WORDS(x,hx|sx,lx);
|
||||
x *= one; /* create necessary signal */
|
||||
}
|
||||
return x; /* exact output */
|
||||
}
|
||||
101
src/math/e_fmodf.c
Normal file
101
src/math/e_fmodf.c
Normal file
|
|
@ -0,0 +1,101 @@
|
|||
/* e_fmodf.c -- float version of e_fmod.c.
|
||||
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
||||
*/
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/*
|
||||
* fmodf(x,y)
|
||||
* Return x mod y in exact arithmetic
|
||||
* Method: shift and subtract
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const float one = 1.0, Zero[] = {0.0, -0.0,};
|
||||
|
||||
float
|
||||
fmodf(float x, float y)
|
||||
{
|
||||
int32_t n,hx,hy,hz,ix,iy,sx,i;
|
||||
|
||||
GET_FLOAT_WORD(hx,x);
|
||||
GET_FLOAT_WORD(hy,y);
|
||||
sx = hx&0x80000000; /* sign of x */
|
||||
hx ^=sx; /* |x| */
|
||||
hy &= 0x7fffffff; /* |y| */
|
||||
|
||||
/* purge off exception values */
|
||||
if(hy==0||(hx>=0x7f800000)|| /* y=0,or x not finite */
|
||||
(hy>0x7f800000)) /* or y is NaN */
|
||||
return (x*y)/(x*y);
|
||||
if(hx<hy) return x; /* |x|<|y| return x */
|
||||
if(hx==hy)
|
||||
return Zero[(uint32_t)sx>>31]; /* |x|=|y| return x*0*/
|
||||
|
||||
/* determine ix = ilogb(x) */
|
||||
if(hx<0x00800000) { /* subnormal x */
|
||||
for (ix = -126,i=(hx<<8); i>0; i<<=1) ix -=1;
|
||||
} else ix = (hx>>23)-127;
|
||||
|
||||
/* determine iy = ilogb(y) */
|
||||
if(hy<0x00800000) { /* subnormal y */
|
||||
for (iy = -126,i=(hy<<8); i>=0; i<<=1) iy -=1;
|
||||
} else iy = (hy>>23)-127;
|
||||
|
||||
/* set up {hx,lx}, {hy,ly} and align y to x */
|
||||
if(ix >= -126)
|
||||
hx = 0x00800000|(0x007fffff&hx);
|
||||
else { /* subnormal x, shift x to normal */
|
||||
n = -126-ix;
|
||||
hx = hx<<n;
|
||||
}
|
||||
if(iy >= -126)
|
||||
hy = 0x00800000|(0x007fffff&hy);
|
||||
else { /* subnormal y, shift y to normal */
|
||||
n = -126-iy;
|
||||
hy = hy<<n;
|
||||
}
|
||||
|
||||
/* fix point fmod */
|
||||
n = ix - iy;
|
||||
while(n--) {
|
||||
hz=hx-hy;
|
||||
if(hz<0){hx = hx+hx;}
|
||||
else {
|
||||
if(hz==0) /* return sign(x)*0 */
|
||||
return Zero[(uint32_t)sx>>31];
|
||||
hx = hz+hz;
|
||||
}
|
||||
}
|
||||
hz=hx-hy;
|
||||
if(hz>=0) {hx=hz;}
|
||||
|
||||
/* convert back to floating value and restore the sign */
|
||||
if(hx==0) /* return sign(x)*0 */
|
||||
return Zero[(uint32_t)sx>>31];
|
||||
while(hx<0x00800000) { /* normalize x */
|
||||
hx = hx+hx;
|
||||
iy -= 1;
|
||||
}
|
||||
if(iy>= -126) { /* normalize output */
|
||||
hx = ((hx-0x00800000)|((iy+127)<<23));
|
||||
SET_FLOAT_WORD(x,hx|sx);
|
||||
} else { /* subnormal output */
|
||||
n = -126 - iy;
|
||||
hx >>= n;
|
||||
SET_FLOAT_WORD(x,hx|sx);
|
||||
x *= one; /* create necessary signal */
|
||||
}
|
||||
return x; /* exact output */
|
||||
}
|
||||
121
src/math/e_hypot.c
Normal file
121
src/math/e_hypot.c
Normal file
|
|
@ -0,0 +1,121 @@
|
|||
|
||||
/* @(#)e_hypot.c 1.3 95/01/18 */
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunSoft, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/* hypot(x,y)
|
||||
*
|
||||
* Method :
|
||||
* If (assume round-to-nearest) z=x*x+y*y
|
||||
* has error less than sqrt(2)/2 ulp, than
|
||||
* sqrt(z) has error less than 1 ulp (exercise).
|
||||
*
|
||||
* So, compute sqrt(x*x+y*y) with some care as
|
||||
* follows to get the error below 1 ulp:
|
||||
*
|
||||
* Assume x>y>0;
|
||||
* (if possible, set rounding to round-to-nearest)
|
||||
* 1. if x > 2y use
|
||||
* x1*x1+(y*y+(x2*(x+x1))) for x*x+y*y
|
||||
* where x1 = x with lower 32 bits cleared, x2 = x-x1; else
|
||||
* 2. if x <= 2y use
|
||||
* t1*y1+((x-y)*(x-y)+(t1*y2+t2*y))
|
||||
* where t1 = 2x with lower 32 bits cleared, t2 = 2x-t1,
|
||||
* y1= y with lower 32 bits chopped, y2 = y-y1.
|
||||
*
|
||||
* NOTE: scaling may be necessary if some argument is too
|
||||
* large or too tiny
|
||||
*
|
||||
* Special cases:
|
||||
* hypot(x,y) is INF if x or y is +INF or -INF; else
|
||||
* hypot(x,y) is NAN if x or y is NAN.
|
||||
*
|
||||
* Accuracy:
|
||||
* hypot(x,y) returns sqrt(x^2+y^2) with error less
|
||||
* than 1 ulps (units in the last place)
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
double
|
||||
hypot(double x, double y)
|
||||
{
|
||||
double a=x,b=y,t1,t2,y1,y2,w;
|
||||
int32_t j,k,ha,hb;
|
||||
|
||||
GET_HIGH_WORD(ha,x);
|
||||
ha &= 0x7fffffff;
|
||||
GET_HIGH_WORD(hb,y);
|
||||
hb &= 0x7fffffff;
|
||||
if(hb > ha) {a=y;b=x;j=ha; ha=hb;hb=j;} else {a=x;b=y;}
|
||||
SET_HIGH_WORD(a,ha); /* a <- |a| */
|
||||
SET_HIGH_WORD(b,hb); /* b <- |b| */
|
||||
if((ha-hb)>0x3c00000) {return a+b;} /* x/y > 2**60 */
|
||||
k=0;
|
||||
if(ha > 0x5f300000) { /* a>2**500 */
|
||||
if(ha >= 0x7ff00000) { /* Inf or NaN */
|
||||
uint32_t low;
|
||||
w = a+b; /* for sNaN */
|
||||
GET_LOW_WORD(low,a);
|
||||
if(((ha&0xfffff)|low)==0) w = a;
|
||||
GET_LOW_WORD(low,b);
|
||||
if(((hb^0x7ff00000)|low)==0) w = b;
|
||||
return w;
|
||||
}
|
||||
/* scale a and b by 2**-600 */
|
||||
ha -= 0x25800000; hb -= 0x25800000; k += 600;
|
||||
SET_HIGH_WORD(a,ha);
|
||||
SET_HIGH_WORD(b,hb);
|
||||
}
|
||||
if(hb < 0x20b00000) { /* b < 2**-500 */
|
||||
if(hb <= 0x000fffff) { /* subnormal b or 0 */
|
||||
uint32_t low;
|
||||
GET_LOW_WORD(low,b);
|
||||
if((hb|low)==0) return a;
|
||||
t1=0;
|
||||
SET_HIGH_WORD(t1,0x7fd00000); /* t1=2^1022 */
|
||||
b *= t1;
|
||||
a *= t1;
|
||||
k -= 1022;
|
||||
} else { /* scale a and b by 2^600 */
|
||||
ha += 0x25800000; /* a *= 2^600 */
|
||||
hb += 0x25800000; /* b *= 2^600 */
|
||||
k -= 600;
|
||||
SET_HIGH_WORD(a,ha);
|
||||
SET_HIGH_WORD(b,hb);
|
||||
}
|
||||
}
|
||||
/* medium size a and b */
|
||||
w = a-b;
|
||||
if (w>b) {
|
||||
t1 = 0;
|
||||
SET_HIGH_WORD(t1,ha);
|
||||
t2 = a-t1;
|
||||
w = sqrt(t1*t1-(b*(-b)-t2*(a+t1)));
|
||||
} else {
|
||||
a = a+a;
|
||||
y1 = 0;
|
||||
SET_HIGH_WORD(y1,hb);
|
||||
y2 = b - y1;
|
||||
t1 = 0;
|
||||
SET_HIGH_WORD(t1,ha+0x00100000);
|
||||
t2 = a - t1;
|
||||
w = sqrt(t1*y1-(w*(-w)-(t1*y2+t2*b)));
|
||||
}
|
||||
if(k!=0) {
|
||||
uint32_t high;
|
||||
t1 = 1.0;
|
||||
GET_HIGH_WORD(high,t1);
|
||||
SET_HIGH_WORD(t1,high+(k<<20));
|
||||
return t1*w;
|
||||
} else return w;
|
||||
}
|
||||
79
src/math/e_hypotf.c
Normal file
79
src/math/e_hypotf.c
Normal file
|
|
@ -0,0 +1,79 @@
|
|||
/* e_hypotf.c -- float version of e_hypot.c.
|
||||
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
||||
*/
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
float
|
||||
hypotf(float x, float y)
|
||||
{
|
||||
float a=x,b=y,t1,t2,y1,y2,w;
|
||||
int32_t j,k,ha,hb;
|
||||
|
||||
GET_FLOAT_WORD(ha,x);
|
||||
ha &= 0x7fffffff;
|
||||
GET_FLOAT_WORD(hb,y);
|
||||
hb &= 0x7fffffff;
|
||||
if(hb > ha) {a=y;b=x;j=ha; ha=hb;hb=j;} else {a=x;b=y;}
|
||||
SET_FLOAT_WORD(a,ha); /* a <- |a| */
|
||||
SET_FLOAT_WORD(b,hb); /* b <- |b| */
|
||||
if((ha-hb)>0xf000000) {return a+b;} /* x/y > 2**30 */
|
||||
k=0;
|
||||
if(ha > 0x58800000) { /* a>2**50 */
|
||||
if(ha >= 0x7f800000) { /* Inf or NaN */
|
||||
w = a+b; /* for sNaN */
|
||||
if(ha == 0x7f800000) w = a;
|
||||
if(hb == 0x7f800000) w = b;
|
||||
return w;
|
||||
}
|
||||
/* scale a and b by 2**-68 */
|
||||
ha -= 0x22000000; hb -= 0x22000000; k += 68;
|
||||
SET_FLOAT_WORD(a,ha);
|
||||
SET_FLOAT_WORD(b,hb);
|
||||
}
|
||||
if(hb < 0x26800000) { /* b < 2**-50 */
|
||||
if(hb <= 0x007fffff) { /* subnormal b or 0 */
|
||||
if(hb==0) return a;
|
||||
SET_FLOAT_WORD(t1,0x7e800000); /* t1=2^126 */
|
||||
b *= t1;
|
||||
a *= t1;
|
||||
k -= 126;
|
||||
} else { /* scale a and b by 2^68 */
|
||||
ha += 0x22000000; /* a *= 2^68 */
|
||||
hb += 0x22000000; /* b *= 2^68 */
|
||||
k -= 68;
|
||||
SET_FLOAT_WORD(a,ha);
|
||||
SET_FLOAT_WORD(b,hb);
|
||||
}
|
||||
}
|
||||
/* medium size a and b */
|
||||
w = a-b;
|
||||
if (w>b) {
|
||||
SET_FLOAT_WORD(t1,ha&0xfffff000);
|
||||
t2 = a-t1;
|
||||
w = sqrtf(t1*t1-(b*(-b)-t2*(a+t1)));
|
||||
} else {
|
||||
a = a+a;
|
||||
SET_FLOAT_WORD(y1,hb&0xfffff000);
|
||||
y2 = b - y1;
|
||||
SET_FLOAT_WORD(t1,ha+0x00800000);
|
||||
t2 = a - t1;
|
||||
w = sqrtf(t1*y1-(w*(-w)-(t1*y2+t2*b)));
|
||||
}
|
||||
if(k!=0) {
|
||||
SET_FLOAT_WORD(t1,0x3f800000+(k<<23));
|
||||
return t1*w;
|
||||
} else return w;
|
||||
}
|
||||
131
src/math/e_log.c
Normal file
131
src/math/e_log.c
Normal file
|
|
@ -0,0 +1,131 @@
|
|||
|
||||
/* @(#)e_log.c 1.3 95/01/18 */
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunSoft, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/* log(x)
|
||||
* Return the logrithm of x
|
||||
*
|
||||
* Method :
|
||||
* 1. Argument Reduction: find k and f such that
|
||||
* x = 2^k * (1+f),
|
||||
* where sqrt(2)/2 < 1+f < sqrt(2) .
|
||||
*
|
||||
* 2. Approximation of log(1+f).
|
||||
* Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
|
||||
* = 2s + 2/3 s**3 + 2/5 s**5 + .....,
|
||||
* = 2s + s*R
|
||||
* We use a special Reme algorithm on [0,0.1716] to generate
|
||||
* a polynomial of degree 14 to approximate R The maximum error
|
||||
* of this polynomial approximation is bounded by 2**-58.45. In
|
||||
* other words,
|
||||
* 2 4 6 8 10 12 14
|
||||
* R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
|
||||
* (the values of Lg1 to Lg7 are listed in the program)
|
||||
* and
|
||||
* | 2 14 | -58.45
|
||||
* | Lg1*s +...+Lg7*s - R(z) | <= 2
|
||||
* | |
|
||||
* Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
|
||||
* In order to guarantee error in log below 1ulp, we compute log
|
||||
* by
|
||||
* log(1+f) = f - s*(f - R) (if f is not too large)
|
||||
* log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
|
||||
*
|
||||
* 3. Finally, log(x) = k*ln2 + log(1+f).
|
||||
* = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
|
||||
* Here ln2 is split into two floating point number:
|
||||
* ln2_hi + ln2_lo,
|
||||
* where n*ln2_hi is always exact for |n| < 2000.
|
||||
*
|
||||
* Special cases:
|
||||
* log(x) is NaN with signal if x < 0 (including -INF) ;
|
||||
* log(+INF) is +INF; log(0) is -INF with signal;
|
||||
* log(NaN) is that NaN with no signal.
|
||||
*
|
||||
* Accuracy:
|
||||
* according to an error analysis, the error is always less than
|
||||
* 1 ulp (unit in the last place).
|
||||
*
|
||||
* Constants:
|
||||
* The hexadecimal values are the intended ones for the following
|
||||
* constants. The decimal values may be used, provided that the
|
||||
* compiler will convert from decimal to binary accurately enough
|
||||
* to produce the hexadecimal values shown.
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const double
|
||||
ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
|
||||
ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
|
||||
two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
|
||||
Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
|
||||
Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
|
||||
Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
|
||||
Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
|
||||
Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
|
||||
Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
|
||||
Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
|
||||
|
||||
static const double zero = 0.0;
|
||||
|
||||
double
|
||||
log(double x)
|
||||
{
|
||||
double hfsq,f,s,z,R,w,t1,t2,dk;
|
||||
int32_t k,hx,i,j;
|
||||
uint32_t lx;
|
||||
|
||||
EXTRACT_WORDS(hx,lx,x);
|
||||
|
||||
k=0;
|
||||
if (hx < 0x00100000) { /* x < 2**-1022 */
|
||||
if (((hx&0x7fffffff)|lx)==0)
|
||||
return -two54/zero; /* log(+-0)=-inf */
|
||||
if (hx<0) return (x-x)/zero; /* log(-#) = NaN */
|
||||
k -= 54; x *= two54; /* subnormal number, scale up x */
|
||||
GET_HIGH_WORD(hx,x);
|
||||
}
|
||||
if (hx >= 0x7ff00000) return x+x;
|
||||
k += (hx>>20)-1023;
|
||||
hx &= 0x000fffff;
|
||||
i = (hx+0x95f64)&0x100000;
|
||||
SET_HIGH_WORD(x,hx|(i^0x3ff00000)); /* normalize x or x/2 */
|
||||
k += (i>>20);
|
||||
f = x-1.0;
|
||||
if((0x000fffff&(2+hx))<3) { /* |f| < 2**-20 */
|
||||
if(f==zero) { if(k==0) return zero; else {dk=(double)k;
|
||||
return dk*ln2_hi+dk*ln2_lo;} }
|
||||
R = f*f*(0.5-0.33333333333333333*f);
|
||||
if(k==0) return f-R; else {dk=(double)k;
|
||||
return dk*ln2_hi-((R-dk*ln2_lo)-f);}
|
||||
}
|
||||
s = f/(2.0+f);
|
||||
dk = (double)k;
|
||||
z = s*s;
|
||||
i = hx-0x6147a;
|
||||
w = z*z;
|
||||
j = 0x6b851-hx;
|
||||
t1= w*(Lg2+w*(Lg4+w*Lg6));
|
||||
t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
|
||||
i |= j;
|
||||
R = t2+t1;
|
||||
if(i>0) {
|
||||
hfsq=0.5*f*f;
|
||||
if(k==0) return f-(hfsq-s*(hfsq+R)); else
|
||||
return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
|
||||
} else {
|
||||
if(k==0) return f-s*(f-R); else
|
||||
return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
|
||||
}
|
||||
}
|
||||
83
src/math/e_log10.c
Normal file
83
src/math/e_log10.c
Normal file
|
|
@ -0,0 +1,83 @@
|
|||
|
||||
/* @(#)e_log10.c 1.3 95/01/18 */
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunSoft, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/* log10(x)
|
||||
* Return the base 10 logarithm of x
|
||||
*
|
||||
* Method :
|
||||
* Let log10_2hi = leading 40 bits of log10(2) and
|
||||
* log10_2lo = log10(2) - log10_2hi,
|
||||
* ivln10 = 1/log(10) rounded.
|
||||
* Then
|
||||
* n = ilogb(x),
|
||||
* if(n<0) n = n+1;
|
||||
* x = scalbn(x,-n);
|
||||
* log10(x) := n*log10_2hi + (n*log10_2lo + ivln10*log(x))
|
||||
*
|
||||
* Note 1:
|
||||
* To guarantee log10(10**n)=n, where 10**n is normal, the rounding
|
||||
* mode must set to Round-to-Nearest.
|
||||
* Note 2:
|
||||
* [1/log(10)] rounded to 53 bits has error .198 ulps;
|
||||
* log10 is monotonic at all binary break points.
|
||||
*
|
||||
* Special cases:
|
||||
* log10(x) is NaN with signal if x < 0;
|
||||
* log10(+INF) is +INF with no signal; log10(0) is -INF with signal;
|
||||
* log10(NaN) is that NaN with no signal;
|
||||
* log10(10**N) = N for N=0,1,...,22.
|
||||
*
|
||||
* Constants:
|
||||
* The hexadecimal values are the intended ones for the following constants.
|
||||
* The decimal values may be used, provided that the compiler will convert
|
||||
* from decimal to binary accurately enough to produce the hexadecimal values
|
||||
* shown.
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const double
|
||||
two54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */
|
||||
ivln10 = 4.34294481903251816668e-01, /* 0x3FDBCB7B, 0x1526E50E */
|
||||
log10_2hi = 3.01029995663611771306e-01, /* 0x3FD34413, 0x509F6000 */
|
||||
log10_2lo = 3.69423907715893078616e-13; /* 0x3D59FEF3, 0x11F12B36 */
|
||||
|
||||
static const double zero = 0.0;
|
||||
|
||||
double
|
||||
log10(double x)
|
||||
{
|
||||
double y,z;
|
||||
int32_t i,k,hx;
|
||||
uint32_t lx;
|
||||
|
||||
EXTRACT_WORDS(hx,lx,x);
|
||||
|
||||
k=0;
|
||||
if (hx < 0x00100000) { /* x < 2**-1022 */
|
||||
if (((hx&0x7fffffff)|lx)==0)
|
||||
return -two54/zero; /* log(+-0)=-inf */
|
||||
if (hx<0) return (x-x)/zero; /* log(-#) = NaN */
|
||||
k -= 54; x *= two54; /* subnormal number, scale up x */
|
||||
GET_HIGH_WORD(hx,x);
|
||||
}
|
||||
if (hx >= 0x7ff00000) return x+x;
|
||||
k += (hx>>20)-1023;
|
||||
i = ((uint32_t)k&0x80000000)>>31;
|
||||
hx = (hx&0x000fffff)|((0x3ff-i)<<20);
|
||||
y = (double)(k+i);
|
||||
SET_HIGH_WORD(x,hx);
|
||||
z = y*log10_2lo + ivln10*log(x);
|
||||
return z+y*log10_2hi;
|
||||
}
|
||||
51
src/math/e_log10f.c
Normal file
51
src/math/e_log10f.c
Normal file
|
|
@ -0,0 +1,51 @@
|
|||
/* e_log10f.c -- float version of e_log10.c.
|
||||
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
||||
*/
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const float
|
||||
two25 = 3.3554432000e+07, /* 0x4c000000 */
|
||||
ivln10 = 4.3429449201e-01, /* 0x3ede5bd9 */
|
||||
log10_2hi = 3.0102920532e-01, /* 0x3e9a2080 */
|
||||
log10_2lo = 7.9034151668e-07; /* 0x355427db */
|
||||
|
||||
static const float zero = 0.0;
|
||||
|
||||
float
|
||||
log10f(float x)
|
||||
{
|
||||
float y,z;
|
||||
int32_t i,k,hx;
|
||||
|
||||
GET_FLOAT_WORD(hx,x);
|
||||
|
||||
k=0;
|
||||
if (hx < 0x00800000) { /* x < 2**-126 */
|
||||
if ((hx&0x7fffffff)==0)
|
||||
return -two25/zero; /* log(+-0)=-inf */
|
||||
if (hx<0) return (x-x)/zero; /* log(-#) = NaN */
|
||||
k -= 25; x *= two25; /* subnormal number, scale up x */
|
||||
GET_FLOAT_WORD(hx,x);
|
||||
}
|
||||
if (hx >= 0x7f800000) return x+x;
|
||||
k += (hx>>23)-127;
|
||||
i = ((uint32_t)k&0x80000000)>>31;
|
||||
hx = (hx&0x007fffff)|((0x7f-i)<<23);
|
||||
y = (float)(k+i);
|
||||
SET_FLOAT_WORD(x,hx);
|
||||
z = y*log10_2lo + ivln10*logf(x);
|
||||
return z+y*log10_2hi;
|
||||
}
|
||||
81
src/math/e_logf.c
Normal file
81
src/math/e_logf.c
Normal file
|
|
@ -0,0 +1,81 @@
|
|||
/* e_logf.c -- float version of e_log.c.
|
||||
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
||||
*/
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const float
|
||||
ln2_hi = 6.9313812256e-01, /* 0x3f317180 */
|
||||
ln2_lo = 9.0580006145e-06, /* 0x3717f7d1 */
|
||||
two25 = 3.355443200e+07, /* 0x4c000000 */
|
||||
Lg1 = 6.6666668653e-01, /* 3F2AAAAB */
|
||||
Lg2 = 4.0000000596e-01, /* 3ECCCCCD */
|
||||
Lg3 = 2.8571429849e-01, /* 3E924925 */
|
||||
Lg4 = 2.2222198546e-01, /* 3E638E29 */
|
||||
Lg5 = 1.8183572590e-01, /* 3E3A3325 */
|
||||
Lg6 = 1.5313838422e-01, /* 3E1CD04F */
|
||||
Lg7 = 1.4798198640e-01; /* 3E178897 */
|
||||
|
||||
static const float zero = 0.0;
|
||||
|
||||
float
|
||||
logf(float x)
|
||||
{
|
||||
float hfsq,f,s,z,R,w,t1,t2,dk;
|
||||
int32_t k,ix,i,j;
|
||||
|
||||
GET_FLOAT_WORD(ix,x);
|
||||
|
||||
k=0;
|
||||
if (ix < 0x00800000) { /* x < 2**-126 */
|
||||
if ((ix&0x7fffffff)==0)
|
||||
return -two25/zero; /* log(+-0)=-inf */
|
||||
if (ix<0) return (x-x)/zero; /* log(-#) = NaN */
|
||||
k -= 25; x *= two25; /* subnormal number, scale up x */
|
||||
GET_FLOAT_WORD(ix,x);
|
||||
}
|
||||
if (ix >= 0x7f800000) return x+x;
|
||||
k += (ix>>23)-127;
|
||||
ix &= 0x007fffff;
|
||||
i = (ix+(0x95f64<<3))&0x800000;
|
||||
SET_FLOAT_WORD(x,ix|(i^0x3f800000)); /* normalize x or x/2 */
|
||||
k += (i>>23);
|
||||
f = x-(float)1.0;
|
||||
if((0x007fffff&(15+ix))<16) { /* |f| < 2**-20 */
|
||||
if(f==zero) { if(k==0) return zero; else {dk=(float)k;
|
||||
return dk*ln2_hi+dk*ln2_lo;} }
|
||||
R = f*f*((float)0.5-(float)0.33333333333333333*f);
|
||||
if(k==0) return f-R; else {dk=(float)k;
|
||||
return dk*ln2_hi-((R-dk*ln2_lo)-f);}
|
||||
}
|
||||
s = f/((float)2.0+f);
|
||||
dk = (float)k;
|
||||
z = s*s;
|
||||
i = ix-(0x6147a<<3);
|
||||
w = z*z;
|
||||
j = (0x6b851<<3)-ix;
|
||||
t1= w*(Lg2+w*(Lg4+w*Lg6));
|
||||
t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
|
||||
i |= j;
|
||||
R = t2+t1;
|
||||
if(i>0) {
|
||||
hfsq=(float)0.5*f*f;
|
||||
if(k==0) return f-(hfsq-s*(hfsq+R)); else
|
||||
return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
|
||||
} else {
|
||||
if(k==0) return f-s*(f-R); else
|
||||
return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
|
||||
}
|
||||
}
|
||||
300
src/math/e_pow.c
Normal file
300
src/math/e_pow.c
Normal file
|
|
@ -0,0 +1,300 @@
|
|||
/* @(#)e_pow.c 1.5 04/04/22 SMI */
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/* pow(x,y) return x**y
|
||||
*
|
||||
* n
|
||||
* Method: Let x = 2 * (1+f)
|
||||
* 1. Compute and return log2(x) in two pieces:
|
||||
* log2(x) = w1 + w2,
|
||||
* where w1 has 53-24 = 29 bit trailing zeros.
|
||||
* 2. Perform y*log2(x) = n+y' by simulating muti-precision
|
||||
* arithmetic, where |y'|<=0.5.
|
||||
* 3. Return x**y = 2**n*exp(y'*log2)
|
||||
*
|
||||
* Special cases:
|
||||
* 1. (anything) ** 0 is 1
|
||||
* 2. (anything) ** 1 is itself
|
||||
* 3. (anything) ** NAN is NAN
|
||||
* 4. NAN ** (anything except 0) is NAN
|
||||
* 5. +-(|x| > 1) ** +INF is +INF
|
||||
* 6. +-(|x| > 1) ** -INF is +0
|
||||
* 7. +-(|x| < 1) ** +INF is +0
|
||||
* 8. +-(|x| < 1) ** -INF is +INF
|
||||
* 9. +-1 ** +-INF is NAN
|
||||
* 10. +0 ** (+anything except 0, NAN) is +0
|
||||
* 11. -0 ** (+anything except 0, NAN, odd integer) is +0
|
||||
* 12. +0 ** (-anything except 0, NAN) is +INF
|
||||
* 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
|
||||
* 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
|
||||
* 15. +INF ** (+anything except 0,NAN) is +INF
|
||||
* 16. +INF ** (-anything except 0,NAN) is +0
|
||||
* 17. -INF ** (anything) = -0 ** (-anything)
|
||||
* 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
|
||||
* 19. (-anything except 0 and inf) ** (non-integer) is NAN
|
||||
*
|
||||
* Accuracy:
|
||||
* pow(x,y) returns x**y nearly rounded. In particular
|
||||
* pow(integer,integer)
|
||||
* always returns the correct integer provided it is
|
||||
* representable.
|
||||
*
|
||||
* Constants :
|
||||
* The hexadecimal values are the intended ones for the following
|
||||
* constants. The decimal values may be used, provided that the
|
||||
* compiler will convert from decimal to binary accurately enough
|
||||
* to produce the hexadecimal values shown.
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const double
|
||||
bp[] = {1.0, 1.5,},
|
||||
dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
|
||||
dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
|
||||
zero = 0.0,
|
||||
one = 1.0,
|
||||
two = 2.0,
|
||||
two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */
|
||||
huge = 1.0e300,
|
||||
tiny = 1.0e-300,
|
||||
/* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
|
||||
L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
|
||||
L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
|
||||
L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
|
||||
L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
|
||||
L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
|
||||
L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
|
||||
P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
|
||||
P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
|
||||
P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
|
||||
P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
|
||||
P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
|
||||
lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
|
||||
lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
|
||||
lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
|
||||
ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
|
||||
cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
|
||||
cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
|
||||
cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
|
||||
ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
|
||||
ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
|
||||
ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
|
||||
|
||||
double
|
||||
pow(double x, double y)
|
||||
{
|
||||
double z,ax,z_h,z_l,p_h,p_l;
|
||||
double y1,t1,t2,r,s,t,u,v,w;
|
||||
int32_t i,j,k,yisint,n;
|
||||
int32_t hx,hy,ix,iy;
|
||||
uint32_t lx,ly;
|
||||
|
||||
EXTRACT_WORDS(hx,lx,x);
|
||||
EXTRACT_WORDS(hy,ly,y);
|
||||
ix = hx&0x7fffffff; iy = hy&0x7fffffff;
|
||||
|
||||
/* y==zero: x**0 = 1 */
|
||||
if((iy|ly)==0) return one;
|
||||
|
||||
/* +-NaN return x+y */
|
||||
if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
|
||||
iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
|
||||
return x+y;
|
||||
|
||||
/* determine if y is an odd int when x < 0
|
||||
* yisint = 0 ... y is not an integer
|
||||
* yisint = 1 ... y is an odd int
|
||||
* yisint = 2 ... y is an even int
|
||||
*/
|
||||
yisint = 0;
|
||||
if(hx<0) {
|
||||
if(iy>=0x43400000) yisint = 2; /* even integer y */
|
||||
else if(iy>=0x3ff00000) {
|
||||
k = (iy>>20)-0x3ff; /* exponent */
|
||||
if(k>20) {
|
||||
j = ly>>(52-k);
|
||||
if((j<<(52-k))==ly) yisint = 2-(j&1);
|
||||
} else if(ly==0) {
|
||||
j = iy>>(20-k);
|
||||
if((j<<(20-k))==iy) yisint = 2-(j&1);
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
/* special value of y */
|
||||
if(ly==0) {
|
||||
if (iy==0x7ff00000) { /* y is +-inf */
|
||||
if(((ix-0x3ff00000)|lx)==0)
|
||||
return y - y; /* inf**+-1 is NaN */
|
||||
else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
|
||||
return (hy>=0)? y: zero;
|
||||
else /* (|x|<1)**-,+inf = inf,0 */
|
||||
return (hy<0)?-y: zero;
|
||||
}
|
||||
if(iy==0x3ff00000) { /* y is +-1 */
|
||||
if(hy<0) return one/x; else return x;
|
||||
}
|
||||
if(hy==0x40000000) return x*x; /* y is 2 */
|
||||
if(hy==0x3fe00000) { /* y is 0.5 */
|
||||
if(hx>=0) /* x >= +0 */
|
||||
return sqrt(x);
|
||||
}
|
||||
}
|
||||
|
||||
ax = fabs(x);
|
||||
/* special value of x */
|
||||
if(lx==0) {
|
||||
if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
|
||||
z = ax; /*x is +-0,+-inf,+-1*/
|
||||
if(hy<0) z = one/z; /* z = (1/|x|) */
|
||||
if(hx<0) {
|
||||
if(((ix-0x3ff00000)|yisint)==0) {
|
||||
z = (z-z)/(z-z); /* (-1)**non-int is NaN */
|
||||
} else if(yisint==1)
|
||||
z = -z; /* (x<0)**odd = -(|x|**odd) */
|
||||
}
|
||||
return z;
|
||||
}
|
||||
}
|
||||
|
||||
/* CYGNUS LOCAL + fdlibm-5.3 fix: This used to be
|
||||
n = (hx>>31)+1;
|
||||
but ANSI C says a right shift of a signed negative quantity is
|
||||
implementation defined. */
|
||||
n = ((uint32_t)hx>>31)-1;
|
||||
|
||||
/* (x<0)**(non-int) is NaN */
|
||||
if((n|yisint)==0) return (x-x)/(x-x);
|
||||
|
||||
s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
|
||||
if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */
|
||||
|
||||
/* |y| is huge */
|
||||
if(iy>0x41e00000) { /* if |y| > 2**31 */
|
||||
if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */
|
||||
if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
|
||||
if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
|
||||
}
|
||||
/* over/underflow if x is not close to one */
|
||||
if(ix<0x3fefffff) return (hy<0)? s*huge*huge:s*tiny*tiny;
|
||||
if(ix>0x3ff00000) return (hy>0)? s*huge*huge:s*tiny*tiny;
|
||||
/* now |1-x| is tiny <= 2**-20, suffice to compute
|
||||
log(x) by x-x^2/2+x^3/3-x^4/4 */
|
||||
t = ax-one; /* t has 20 trailing zeros */
|
||||
w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
|
||||
u = ivln2_h*t; /* ivln2_h has 21 sig. bits */
|
||||
v = t*ivln2_l-w*ivln2;
|
||||
t1 = u+v;
|
||||
SET_LOW_WORD(t1,0);
|
||||
t2 = v-(t1-u);
|
||||
} else {
|
||||
double ss,s2,s_h,s_l,t_h,t_l;
|
||||
n = 0;
|
||||
/* take care subnormal number */
|
||||
if(ix<0x00100000)
|
||||
{ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); }
|
||||
n += ((ix)>>20)-0x3ff;
|
||||
j = ix&0x000fffff;
|
||||
/* determine interval */
|
||||
ix = j|0x3ff00000; /* normalize ix */
|
||||
if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */
|
||||
else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */
|
||||
else {k=0;n+=1;ix -= 0x00100000;}
|
||||
SET_HIGH_WORD(ax,ix);
|
||||
|
||||
/* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
|
||||
u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
|
||||
v = one/(ax+bp[k]);
|
||||
ss = u*v;
|
||||
s_h = ss;
|
||||
SET_LOW_WORD(s_h,0);
|
||||
/* t_h=ax+bp[k] High */
|
||||
t_h = zero;
|
||||
SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18));
|
||||
t_l = ax - (t_h-bp[k]);
|
||||
s_l = v*((u-s_h*t_h)-s_h*t_l);
|
||||
/* compute log(ax) */
|
||||
s2 = ss*ss;
|
||||
r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
|
||||
r += s_l*(s_h+ss);
|
||||
s2 = s_h*s_h;
|
||||
t_h = 3.0+s2+r;
|
||||
SET_LOW_WORD(t_h,0);
|
||||
t_l = r-((t_h-3.0)-s2);
|
||||
/* u+v = ss*(1+...) */
|
||||
u = s_h*t_h;
|
||||
v = s_l*t_h+t_l*ss;
|
||||
/* 2/(3log2)*(ss+...) */
|
||||
p_h = u+v;
|
||||
SET_LOW_WORD(p_h,0);
|
||||
p_l = v-(p_h-u);
|
||||
z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
|
||||
z_l = cp_l*p_h+p_l*cp+dp_l[k];
|
||||
/* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
|
||||
t = (double)n;
|
||||
t1 = (((z_h+z_l)+dp_h[k])+t);
|
||||
SET_LOW_WORD(t1,0);
|
||||
t2 = z_l-(((t1-t)-dp_h[k])-z_h);
|
||||
}
|
||||
|
||||
/* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
|
||||
y1 = y;
|
||||
SET_LOW_WORD(y1,0);
|
||||
p_l = (y-y1)*t1+y*t2;
|
||||
p_h = y1*t1;
|
||||
z = p_l+p_h;
|
||||
EXTRACT_WORDS(j,i,z);
|
||||
if (j>=0x40900000) { /* z >= 1024 */
|
||||
if(((j-0x40900000)|i)!=0) /* if z > 1024 */
|
||||
return s*huge*huge; /* overflow */
|
||||
else {
|
||||
if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */
|
||||
}
|
||||
} else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */
|
||||
if(((j-0xc090cc00)|i)!=0) /* z < -1075 */
|
||||
return s*tiny*tiny; /* underflow */
|
||||
else {
|
||||
if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */
|
||||
}
|
||||
}
|
||||
/*
|
||||
* compute 2**(p_h+p_l)
|
||||
*/
|
||||
i = j&0x7fffffff;
|
||||
k = (i>>20)-0x3ff;
|
||||
n = 0;
|
||||
if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */
|
||||
n = j+(0x00100000>>(k+1));
|
||||
k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */
|
||||
t = zero;
|
||||
SET_HIGH_WORD(t,n&~(0x000fffff>>k));
|
||||
n = ((n&0x000fffff)|0x00100000)>>(20-k);
|
||||
if(j<0) n = -n;
|
||||
p_h -= t;
|
||||
}
|
||||
t = p_l+p_h;
|
||||
SET_LOW_WORD(t,0);
|
||||
u = t*lg2_h;
|
||||
v = (p_l-(t-p_h))*lg2+t*lg2_l;
|
||||
z = u+v;
|
||||
w = v-(z-u);
|
||||
t = z*z;
|
||||
t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
|
||||
r = (z*t1)/(t1-two)-(w+z*w);
|
||||
z = one-(r-z);
|
||||
GET_HIGH_WORD(j,z);
|
||||
j += (n<<20);
|
||||
if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */
|
||||
else SET_HIGH_WORD(z,j);
|
||||
return s*z;
|
||||
}
|
||||
243
src/math/e_powf.c
Normal file
243
src/math/e_powf.c
Normal file
|
|
@ -0,0 +1,243 @@
|
|||
/* e_powf.c -- float version of e_pow.c.
|
||||
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
||||
*/
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const float
|
||||
bp[] = {1.0, 1.5,},
|
||||
dp_h[] = { 0.0, 5.84960938e-01,}, /* 0x3f15c000 */
|
||||
dp_l[] = { 0.0, 1.56322085e-06,}, /* 0x35d1cfdc */
|
||||
zero = 0.0,
|
||||
one = 1.0,
|
||||
two = 2.0,
|
||||
two24 = 16777216.0, /* 0x4b800000 */
|
||||
huge = 1.0e30,
|
||||
tiny = 1.0e-30,
|
||||
/* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
|
||||
L1 = 6.0000002384e-01, /* 0x3f19999a */
|
||||
L2 = 4.2857143283e-01, /* 0x3edb6db7 */
|
||||
L3 = 3.3333334327e-01, /* 0x3eaaaaab */
|
||||
L4 = 2.7272811532e-01, /* 0x3e8ba305 */
|
||||
L5 = 2.3066075146e-01, /* 0x3e6c3255 */
|
||||
L6 = 2.0697501302e-01, /* 0x3e53f142 */
|
||||
P1 = 1.6666667163e-01, /* 0x3e2aaaab */
|
||||
P2 = -2.7777778450e-03, /* 0xbb360b61 */
|
||||
P3 = 6.6137559770e-05, /* 0x388ab355 */
|
||||
P4 = -1.6533901999e-06, /* 0xb5ddea0e */
|
||||
P5 = 4.1381369442e-08, /* 0x3331bb4c */
|
||||
lg2 = 6.9314718246e-01, /* 0x3f317218 */
|
||||
lg2_h = 6.93145752e-01, /* 0x3f317200 */
|
||||
lg2_l = 1.42860654e-06, /* 0x35bfbe8c */
|
||||
ovt = 4.2995665694e-08, /* -(128-log2(ovfl+.5ulp)) */
|
||||
cp = 9.6179670095e-01, /* 0x3f76384f =2/(3ln2) */
|
||||
cp_h = 9.6179199219e-01, /* 0x3f763800 =head of cp */
|
||||
cp_l = 4.7017383622e-06, /* 0x369dc3a0 =tail of cp_h */
|
||||
ivln2 = 1.4426950216e+00, /* 0x3fb8aa3b =1/ln2 */
|
||||
ivln2_h = 1.4426879883e+00, /* 0x3fb8aa00 =16b 1/ln2*/
|
||||
ivln2_l = 7.0526075433e-06; /* 0x36eca570 =1/ln2 tail*/
|
||||
|
||||
float
|
||||
powf(float x, float y)
|
||||
{
|
||||
float z,ax,z_h,z_l,p_h,p_l;
|
||||
float y1,t1,t2,r,s,sn,t,u,v,w;
|
||||
int32_t i,j,k,yisint,n;
|
||||
int32_t hx,hy,ix,iy,is;
|
||||
|
||||
GET_FLOAT_WORD(hx,x);
|
||||
GET_FLOAT_WORD(hy,y);
|
||||
ix = hx&0x7fffffff; iy = hy&0x7fffffff;
|
||||
|
||||
/* y==zero: x**0 = 1 */
|
||||
if(iy==0) return one;
|
||||
|
||||
/* +-NaN return x+y */
|
||||
if(ix > 0x7f800000 ||
|
||||
iy > 0x7f800000)
|
||||
return x+y;
|
||||
|
||||
/* determine if y is an odd int when x < 0
|
||||
* yisint = 0 ... y is not an integer
|
||||
* yisint = 1 ... y is an odd int
|
||||
* yisint = 2 ... y is an even int
|
||||
*/
|
||||
yisint = 0;
|
||||
if(hx<0) {
|
||||
if(iy>=0x4b800000) yisint = 2; /* even integer y */
|
||||
else if(iy>=0x3f800000) {
|
||||
k = (iy>>23)-0x7f; /* exponent */
|
||||
j = iy>>(23-k);
|
||||
if((j<<(23-k))==iy) yisint = 2-(j&1);
|
||||
}
|
||||
}
|
||||
|
||||
/* special value of y */
|
||||
if (iy==0x7f800000) { /* y is +-inf */
|
||||
if (ix==0x3f800000)
|
||||
return y - y; /* inf**+-1 is NaN */
|
||||
else if (ix > 0x3f800000)/* (|x|>1)**+-inf = inf,0 */
|
||||
return (hy>=0)? y: zero;
|
||||
else /* (|x|<1)**-,+inf = inf,0 */
|
||||
return (hy<0)?-y: zero;
|
||||
}
|
||||
if(iy==0x3f800000) { /* y is +-1 */
|
||||
if(hy<0) return one/x; else return x;
|
||||
}
|
||||
if(hy==0x40000000) return x*x; /* y is 2 */
|
||||
if(hy==0x3f000000) { /* y is 0.5 */
|
||||
if(hx>=0) /* x >= +0 */
|
||||
return sqrtf(x);
|
||||
}
|
||||
|
||||
ax = fabsf(x);
|
||||
/* special value of x */
|
||||
if(ix==0x7f800000||ix==0||ix==0x3f800000){
|
||||
z = ax; /*x is +-0,+-inf,+-1*/
|
||||
if(hy<0) z = one/z; /* z = (1/|x|) */
|
||||
if(hx<0) {
|
||||
if(((ix-0x3f800000)|yisint)==0) {
|
||||
z = (z-z)/(z-z); /* (-1)**non-int is NaN */
|
||||
} else if(yisint==1)
|
||||
z = -z; /* (x<0)**odd = -(|x|**odd) */
|
||||
}
|
||||
return z;
|
||||
}
|
||||
|
||||
n = ((uint32_t)hx>>31)-1;
|
||||
|
||||
/* (x<0)**(non-int) is NaN */
|
||||
if((n|yisint)==0) return (x-x)/(x-x);
|
||||
|
||||
sn = one; /* s (sign of result -ve**odd) = -1 else = 1 */
|
||||
if((n|(yisint-1))==0) sn = -one;/* (-ve)**(odd int) */
|
||||
|
||||
/* |y| is huge */
|
||||
if(iy>0x4d000000) { /* if |y| > 2**27 */
|
||||
/* over/underflow if x is not close to one */
|
||||
if(ix<0x3f7ffff8) return (hy<0)? sn*huge*huge:sn*tiny*tiny;
|
||||
if(ix>0x3f800007) return (hy>0)? sn*huge*huge:sn*tiny*tiny;
|
||||
/* now |1-x| is tiny <= 2**-20, suffice to compute
|
||||
log(x) by x-x^2/2+x^3/3-x^4/4 */
|
||||
t = ax-1; /* t has 20 trailing zeros */
|
||||
w = (t*t)*((float)0.5-t*((float)0.333333333333-t*(float)0.25));
|
||||
u = ivln2_h*t; /* ivln2_h has 16 sig. bits */
|
||||
v = t*ivln2_l-w*ivln2;
|
||||
t1 = u+v;
|
||||
GET_FLOAT_WORD(is,t1);
|
||||
SET_FLOAT_WORD(t1,is&0xfffff000);
|
||||
t2 = v-(t1-u);
|
||||
} else {
|
||||
float s2,s_h,s_l,t_h,t_l;
|
||||
n = 0;
|
||||
/* take care subnormal number */
|
||||
if(ix<0x00800000)
|
||||
{ax *= two24; n -= 24; GET_FLOAT_WORD(ix,ax); }
|
||||
n += ((ix)>>23)-0x7f;
|
||||
j = ix&0x007fffff;
|
||||
/* determine interval */
|
||||
ix = j|0x3f800000; /* normalize ix */
|
||||
if(j<=0x1cc471) k=0; /* |x|<sqrt(3/2) */
|
||||
else if(j<0x5db3d7) k=1; /* |x|<sqrt(3) */
|
||||
else {k=0;n+=1;ix -= 0x00800000;}
|
||||
SET_FLOAT_WORD(ax,ix);
|
||||
|
||||
/* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
|
||||
u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
|
||||
v = one/(ax+bp[k]);
|
||||
s = u*v;
|
||||
s_h = s;
|
||||
GET_FLOAT_WORD(is,s_h);
|
||||
SET_FLOAT_WORD(s_h,is&0xfffff000);
|
||||
/* t_h=ax+bp[k] High */
|
||||
is = ((ix>>1)&0xfffff000)|0x20000000;
|
||||
SET_FLOAT_WORD(t_h,is+0x00400000+(k<<21));
|
||||
t_l = ax - (t_h-bp[k]);
|
||||
s_l = v*((u-s_h*t_h)-s_h*t_l);
|
||||
/* compute log(ax) */
|
||||
s2 = s*s;
|
||||
r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
|
||||
r += s_l*(s_h+s);
|
||||
s2 = s_h*s_h;
|
||||
t_h = (float)3.0+s2+r;
|
||||
GET_FLOAT_WORD(is,t_h);
|
||||
SET_FLOAT_WORD(t_h,is&0xfffff000);
|
||||
t_l = r-((t_h-(float)3.0)-s2);
|
||||
/* u+v = s*(1+...) */
|
||||
u = s_h*t_h;
|
||||
v = s_l*t_h+t_l*s;
|
||||
/* 2/(3log2)*(s+...) */
|
||||
p_h = u+v;
|
||||
GET_FLOAT_WORD(is,p_h);
|
||||
SET_FLOAT_WORD(p_h,is&0xfffff000);
|
||||
p_l = v-(p_h-u);
|
||||
z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
|
||||
z_l = cp_l*p_h+p_l*cp+dp_l[k];
|
||||
/* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
|
||||
t = (float)n;
|
||||
t1 = (((z_h+z_l)+dp_h[k])+t);
|
||||
GET_FLOAT_WORD(is,t1);
|
||||
SET_FLOAT_WORD(t1,is&0xfffff000);
|
||||
t2 = z_l-(((t1-t)-dp_h[k])-z_h);
|
||||
}
|
||||
|
||||
/* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
|
||||
GET_FLOAT_WORD(is,y);
|
||||
SET_FLOAT_WORD(y1,is&0xfffff000);
|
||||
p_l = (y-y1)*t1+y*t2;
|
||||
p_h = y1*t1;
|
||||
z = p_l+p_h;
|
||||
GET_FLOAT_WORD(j,z);
|
||||
if (j>0x43000000) /* if z > 128 */
|
||||
return sn*huge*huge; /* overflow */
|
||||
else if (j==0x43000000) { /* if z == 128 */
|
||||
if(p_l+ovt>z-p_h) return sn*huge*huge; /* overflow */
|
||||
}
|
||||
else if ((j&0x7fffffff)>0x43160000) /* z <= -150 */
|
||||
return sn*tiny*tiny; /* underflow */
|
||||
else if (j==0xc3160000){ /* z == -150 */
|
||||
if(p_l<=z-p_h) return sn*tiny*tiny; /* underflow */
|
||||
}
|
||||
/*
|
||||
* compute 2**(p_h+p_l)
|
||||
*/
|
||||
i = j&0x7fffffff;
|
||||
k = (i>>23)-0x7f;
|
||||
n = 0;
|
||||
if(i>0x3f000000) { /* if |z| > 0.5, set n = [z+0.5] */
|
||||
n = j+(0x00800000>>(k+1));
|
||||
k = ((n&0x7fffffff)>>23)-0x7f; /* new k for n */
|
||||
SET_FLOAT_WORD(t,n&~(0x007fffff>>k));
|
||||
n = ((n&0x007fffff)|0x00800000)>>(23-k);
|
||||
if(j<0) n = -n;
|
||||
p_h -= t;
|
||||
}
|
||||
t = p_l+p_h;
|
||||
GET_FLOAT_WORD(is,t);
|
||||
SET_FLOAT_WORD(t,is&0xffff8000);
|
||||
u = t*lg2_h;
|
||||
v = (p_l-(t-p_h))*lg2+t*lg2_l;
|
||||
z = u+v;
|
||||
w = v-(z-u);
|
||||
t = z*z;
|
||||
t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
|
||||
r = (z*t1)/(t1-two)-(w+z*w);
|
||||
z = one-(r-z);
|
||||
GET_FLOAT_WORD(j,z);
|
||||
j += (n<<23);
|
||||
if((j>>23)<=0) z = scalbnf(z,n); /* subnormal output */
|
||||
else SET_FLOAT_WORD(z,j);
|
||||
return sn*z;
|
||||
}
|
||||
163
src/math/e_rem_pio2.c
Normal file
163
src/math/e_rem_pio2.c
Normal file
|
|
@ -0,0 +1,163 @@
|
|||
|
||||
/* @(#)e_rem_pio2.c 1.4 95/01/18 */
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunSoft, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*
|
||||
*/
|
||||
|
||||
/* __ieee754_rem_pio2(x,y)
|
||||
*
|
||||
* return the remainder of x rem pi/2 in y[0]+y[1]
|
||||
* use __kernel_rem_pio2()
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
/*
|
||||
* Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi
|
||||
*/
|
||||
static const int32_t two_over_pi[] = {
|
||||
0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62,
|
||||
0x95993C, 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A,
|
||||
0x424DD2, 0xE00649, 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129,
|
||||
0xA73EE8, 0x8235F5, 0x2EBB44, 0x84E99C, 0x7026B4, 0x5F7E41,
|
||||
0x3991D6, 0x398353, 0x39F49C, 0x845F8B, 0xBDF928, 0x3B1FF8,
|
||||
0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D, 0x367ECF,
|
||||
0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5,
|
||||
0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08,
|
||||
0x560330, 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3,
|
||||
0x91615E, 0xE61B08, 0x659985, 0x5F14A0, 0x68408D, 0xFFD880,
|
||||
0x4D7327, 0x310606, 0x1556CA, 0x73A8C9, 0x60E27B, 0xC08C6B,
|
||||
};
|
||||
|
||||
static const int32_t npio2_hw[] = {
|
||||
0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C,
|
||||
0x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C,
|
||||
0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A,
|
||||
0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C,
|
||||
0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB,
|
||||
0x404858EB, 0x404921FB,
|
||||
};
|
||||
|
||||
/*
|
||||
* invpio2: 53 bits of 2/pi
|
||||
* pio2_1: first 33 bit of pi/2
|
||||
* pio2_1t: pi/2 - pio2_1
|
||||
* pio2_2: second 33 bit of pi/2
|
||||
* pio2_2t: pi/2 - (pio2_1+pio2_2)
|
||||
* pio2_3: third 33 bit of pi/2
|
||||
* pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
|
||||
*/
|
||||
|
||||
static const double
|
||||
zero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
|
||||
half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
|
||||
two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
|
||||
invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
|
||||
pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
|
||||
pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
|
||||
pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
|
||||
pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
|
||||
pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
|
||||
pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
|
||||
|
||||
int32_t __ieee754_rem_pio2(double x, double *y)
|
||||
{
|
||||
double z,w,t,r,fn;
|
||||
double tx[3];
|
||||
int32_t e0,i,j,nx,n,ix,hx;
|
||||
uint32_t low;
|
||||
|
||||
GET_HIGH_WORD(hx,x); /* high word of x */
|
||||
ix = hx&0x7fffffff;
|
||||
if(ix<=0x3fe921fb) /* |x| ~<= pi/4 , no need for reduction */
|
||||
{y[0] = x; y[1] = 0; return 0;}
|
||||
if(ix<0x4002d97c) { /* |x| < 3pi/4, special case with n=+-1 */
|
||||
if(hx>0) {
|
||||
z = x - pio2_1;
|
||||
if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */
|
||||
y[0] = z - pio2_1t;
|
||||
y[1] = (z-y[0])-pio2_1t;
|
||||
} else { /* near pi/2, use 33+33+53 bit pi */
|
||||
z -= pio2_2;
|
||||
y[0] = z - pio2_2t;
|
||||
y[1] = (z-y[0])-pio2_2t;
|
||||
}
|
||||
return 1;
|
||||
} else { /* negative x */
|
||||
z = x + pio2_1;
|
||||
if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */
|
||||
y[0] = z + pio2_1t;
|
||||
y[1] = (z-y[0])+pio2_1t;
|
||||
} else { /* near pi/2, use 33+33+53 bit pi */
|
||||
z += pio2_2;
|
||||
y[0] = z + pio2_2t;
|
||||
y[1] = (z-y[0])+pio2_2t;
|
||||
}
|
||||
return -1;
|
||||
}
|
||||
}
|
||||
if(ix<=0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */
|
||||
t = fabs(x);
|
||||
n = (int32_t) (t*invpio2+half);
|
||||
fn = (double)n;
|
||||
r = t-fn*pio2_1;
|
||||
w = fn*pio2_1t; /* 1st round good to 85 bit */
|
||||
if(n<32&&ix!=npio2_hw[n-1]) {
|
||||
y[0] = r-w; /* quick check no cancellation */
|
||||
} else {
|
||||
uint32_t high;
|
||||
j = ix>>20;
|
||||
y[0] = r-w;
|
||||
GET_HIGH_WORD(high,y[0]);
|
||||
i = j-((high>>20)&0x7ff);
|
||||
if(i>16) { /* 2nd iteration needed, good to 118 */
|
||||
t = r;
|
||||
w = fn*pio2_2;
|
||||
r = t-w;
|
||||
w = fn*pio2_2t-((t-r)-w);
|
||||
y[0] = r-w;
|
||||
GET_HIGH_WORD(high,y[0]);
|
||||
i = j-((high>>20)&0x7ff);
|
||||
if(i>49) { /* 3rd iteration need, 151 bits acc */
|
||||
t = r; /* will cover all possible cases */
|
||||
w = fn*pio2_3;
|
||||
r = t-w;
|
||||
w = fn*pio2_3t-((t-r)-w);
|
||||
y[0] = r-w;
|
||||
}
|
||||
}
|
||||
}
|
||||
y[1] = (r-y[0])-w;
|
||||
if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;}
|
||||
else return n;
|
||||
}
|
||||
/*
|
||||
* all other (large) arguments
|
||||
*/
|
||||
if(ix>=0x7ff00000) { /* x is inf or NaN */
|
||||
y[0]=y[1]=x-x; return 0;
|
||||
}
|
||||
/* set z = scalbn(|x|,ilogb(x)-23) */
|
||||
GET_LOW_WORD(low,x);
|
||||
e0 = (ix>>20)-1046; /* e0 = ilogb(z)-23; */
|
||||
INSERT_WORDS(z, ix - ((int32_t)(e0<<20)), low);
|
||||
for(i=0;i<2;i++) {
|
||||
tx[i] = (double)((int32_t)(z));
|
||||
z = (z-tx[i])*two24;
|
||||
}
|
||||
tx[2] = z;
|
||||
nx = 3;
|
||||
while(tx[nx-1]==zero) nx--; /* skip zero term */
|
||||
n = __kernel_rem_pio2(tx,y,e0,nx,2,two_over_pi);
|
||||
if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;}
|
||||
return n;
|
||||
}
|
||||
175
src/math/e_rem_pio2f.c
Normal file
175
src/math/e_rem_pio2f.c
Normal file
|
|
@ -0,0 +1,175 @@
|
|||
/* e_rem_pio2f.c -- float version of e_rem_pio2.c
|
||||
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
||||
*/
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/* __ieee754_rem_pio2f(x,y)
|
||||
*
|
||||
* return the remainder of x rem pi/2 in y[0]+y[1]
|
||||
* use __kernel_rem_pio2f()
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
/*
|
||||
* Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi
|
||||
*/
|
||||
static const int32_t two_over_pi[] = {
|
||||
0xA2, 0xF9, 0x83, 0x6E, 0x4E, 0x44, 0x15, 0x29, 0xFC,
|
||||
0x27, 0x57, 0xD1, 0xF5, 0x34, 0xDD, 0xC0, 0xDB, 0x62,
|
||||
0x95, 0x99, 0x3C, 0x43, 0x90, 0x41, 0xFE, 0x51, 0x63,
|
||||
0xAB, 0xDE, 0xBB, 0xC5, 0x61, 0xB7, 0x24, 0x6E, 0x3A,
|
||||
0x42, 0x4D, 0xD2, 0xE0, 0x06, 0x49, 0x2E, 0xEA, 0x09,
|
||||
0xD1, 0x92, 0x1C, 0xFE, 0x1D, 0xEB, 0x1C, 0xB1, 0x29,
|
||||
0xA7, 0x3E, 0xE8, 0x82, 0x35, 0xF5, 0x2E, 0xBB, 0x44,
|
||||
0x84, 0xE9, 0x9C, 0x70, 0x26, 0xB4, 0x5F, 0x7E, 0x41,
|
||||
0x39, 0x91, 0xD6, 0x39, 0x83, 0x53, 0x39, 0xF4, 0x9C,
|
||||
0x84, 0x5F, 0x8B, 0xBD, 0xF9, 0x28, 0x3B, 0x1F, 0xF8,
|
||||
0x97, 0xFF, 0xDE, 0x05, 0x98, 0x0F, 0xEF, 0x2F, 0x11,
|
||||
0x8B, 0x5A, 0x0A, 0x6D, 0x1F, 0x6D, 0x36, 0x7E, 0xCF,
|
||||
0x27, 0xCB, 0x09, 0xB7, 0x4F, 0x46, 0x3F, 0x66, 0x9E,
|
||||
0x5F, 0xEA, 0x2D, 0x75, 0x27, 0xBA, 0xC7, 0xEB, 0xE5,
|
||||
0xF1, 0x7B, 0x3D, 0x07, 0x39, 0xF7, 0x8A, 0x52, 0x92,
|
||||
0xEA, 0x6B, 0xFB, 0x5F, 0xB1, 0x1F, 0x8D, 0x5D, 0x08,
|
||||
0x56, 0x03, 0x30, 0x46, 0xFC, 0x7B, 0x6B, 0xAB, 0xF0,
|
||||
0xCF, 0xBC, 0x20, 0x9A, 0xF4, 0x36, 0x1D, 0xA9, 0xE3,
|
||||
0x91, 0x61, 0x5E, 0xE6, 0x1B, 0x08, 0x65, 0x99, 0x85,
|
||||
0x5F, 0x14, 0xA0, 0x68, 0x40, 0x8D, 0xFF, 0xD8, 0x80,
|
||||
0x4D, 0x73, 0x27, 0x31, 0x06, 0x06, 0x15, 0x56, 0xCA,
|
||||
0x73, 0xA8, 0xC9, 0x60, 0xE2, 0x7B, 0xC0, 0x8C, 0x6B,
|
||||
};
|
||||
|
||||
/* This array is like the one in e_rem_pio2.c, but the numbers are
|
||||
single precision and the last 8 bits are forced to 0. */
|
||||
static const int32_t npio2_hw[] = {
|
||||
0x3fc90f00, 0x40490f00, 0x4096cb00, 0x40c90f00, 0x40fb5300, 0x4116cb00,
|
||||
0x412fed00, 0x41490f00, 0x41623100, 0x417b5300, 0x418a3a00, 0x4196cb00,
|
||||
0x41a35c00, 0x41afed00, 0x41bc7e00, 0x41c90f00, 0x41d5a000, 0x41e23100,
|
||||
0x41eec200, 0x41fb5300, 0x4203f200, 0x420a3a00, 0x42108300, 0x4216cb00,
|
||||
0x421d1400, 0x42235c00, 0x4229a500, 0x422fed00, 0x42363600, 0x423c7e00,
|
||||
0x4242c700, 0x42490f00
|
||||
};
|
||||
|
||||
/*
|
||||
* invpio2: 24 bits of 2/pi
|
||||
* pio2_1: first 17 bit of pi/2
|
||||
* pio2_1t: pi/2 - pio2_1
|
||||
* pio2_2: second 17 bit of pi/2
|
||||
* pio2_2t: pi/2 - (pio2_1+pio2_2)
|
||||
* pio2_3: third 17 bit of pi/2
|
||||
* pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
|
||||
*/
|
||||
|
||||
static const float
|
||||
zero = 0.0000000000e+00, /* 0x00000000 */
|
||||
half = 5.0000000000e-01, /* 0x3f000000 */
|
||||
two8 = 2.5600000000e+02, /* 0x43800000 */
|
||||
invpio2 = 6.3661980629e-01, /* 0x3f22f984 */
|
||||
pio2_1 = 1.5707855225e+00, /* 0x3fc90f80 */
|
||||
pio2_1t = 1.0804334124e-05, /* 0x37354443 */
|
||||
pio2_2 = 1.0804273188e-05, /* 0x37354400 */
|
||||
pio2_2t = 6.0770999344e-11, /* 0x2e85a308 */
|
||||
pio2_3 = 6.0770943833e-11, /* 0x2e85a300 */
|
||||
pio2_3t = 6.1232342629e-17; /* 0x248d3132 */
|
||||
|
||||
int32_t __ieee754_rem_pio2f(float x, float *y)
|
||||
{
|
||||
float z,w,t,r,fn;
|
||||
float tx[3];
|
||||
int32_t e0,i,j,nx,n,ix,hx;
|
||||
|
||||
GET_FLOAT_WORD(hx,x);
|
||||
ix = hx&0x7fffffff;
|
||||
if(ix<=0x3f490fd8) /* |x| ~<= pi/4 , no need for reduction */
|
||||
{y[0] = x; y[1] = 0; return 0;}
|
||||
if(ix<0x4016cbe4) { /* |x| < 3pi/4, special case with n=+-1 */
|
||||
if(hx>0) {
|
||||
z = x - pio2_1;
|
||||
if((ix&0xfffffff0)!=0x3fc90fd0) { /* 24+24 bit pi OK */
|
||||
y[0] = z - pio2_1t;
|
||||
y[1] = (z-y[0])-pio2_1t;
|
||||
} else { /* near pi/2, use 24+24+24 bit pi */
|
||||
z -= pio2_2;
|
||||
y[0] = z - pio2_2t;
|
||||
y[1] = (z-y[0])-pio2_2t;
|
||||
}
|
||||
return 1;
|
||||
} else { /* negative x */
|
||||
z = x + pio2_1;
|
||||
if((ix&0xfffffff0)!=0x3fc90fd0) { /* 24+24 bit pi OK */
|
||||
y[0] = z + pio2_1t;
|
||||
y[1] = (z-y[0])+pio2_1t;
|
||||
} else { /* near pi/2, use 24+24+24 bit pi */
|
||||
z += pio2_2;
|
||||
y[0] = z + pio2_2t;
|
||||
y[1] = (z-y[0])+pio2_2t;
|
||||
}
|
||||
return -1;
|
||||
}
|
||||
}
|
||||
if(ix<=0x43490f80) { /* |x| ~<= 2^7*(pi/2), medium size */
|
||||
t = fabsf(x);
|
||||
n = (int32_t) (t*invpio2+half);
|
||||
fn = (float)n;
|
||||
r = t-fn*pio2_1;
|
||||
w = fn*pio2_1t; /* 1st round good to 40 bit */
|
||||
if(n<32&&(ix&0xffffff00)!=npio2_hw[n-1]) {
|
||||
y[0] = r-w; /* quick check no cancellation */
|
||||
} else {
|
||||
uint32_t high;
|
||||
j = ix>>23;
|
||||
y[0] = r-w;
|
||||
GET_FLOAT_WORD(high,y[0]);
|
||||
i = j-((high>>23)&0xff);
|
||||
if(i>8) { /* 2nd iteration needed, good to 57 */
|
||||
t = r;
|
||||
w = fn*pio2_2;
|
||||
r = t-w;
|
||||
w = fn*pio2_2t-((t-r)-w);
|
||||
y[0] = r-w;
|
||||
GET_FLOAT_WORD(high,y[0]);
|
||||
i = j-((high>>23)&0xff);
|
||||
if(i>25) { /* 3rd iteration need, 74 bits acc */
|
||||
t = r; /* will cover all possible cases */
|
||||
w = fn*pio2_3;
|
||||
r = t-w;
|
||||
w = fn*pio2_3t-((t-r)-w);
|
||||
y[0] = r-w;
|
||||
}
|
||||
}
|
||||
}
|
||||
y[1] = (r-y[0])-w;
|
||||
if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;}
|
||||
else return n;
|
||||
}
|
||||
/*
|
||||
* all other (large) arguments
|
||||
*/
|
||||
if(ix>=0x7f800000) { /* x is inf or NaN */
|
||||
y[0]=y[1]=x-x; return 0;
|
||||
}
|
||||
/* set z = scalbn(|x|,ilogb(x)-7) */
|
||||
e0 = (ix>>23)-134; /* e0 = ilogb(z)-7; */
|
||||
SET_FLOAT_WORD(z, ix - ((int32_t)(e0<<23)));
|
||||
for(i=0;i<2;i++) {
|
||||
tx[i] = (float)((int32_t)(z));
|
||||
z = (z-tx[i])*two8;
|
||||
}
|
||||
tx[2] = z;
|
||||
nx = 3;
|
||||
while(tx[nx-1]==zero) nx--; /* skip zero term */
|
||||
n = __kernel_rem_pio2f(tx,y,e0,nx,2,two_over_pi);
|
||||
if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;}
|
||||
return n;
|
||||
}
|
||||
69
src/math/e_remainder.c
Normal file
69
src/math/e_remainder.c
Normal file
|
|
@ -0,0 +1,69 @@
|
|||
|
||||
/* @(#)e_remainder.c 1.3 95/01/18 */
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunSoft, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/* remainder(x,p)
|
||||
* Return :
|
||||
* returns x REM p = x - [x/p]*p as if in infinite
|
||||
* precise arithmetic, where [x/p] is the (infinite bit)
|
||||
* integer nearest x/p (in half way case choose the even one).
|
||||
* Method :
|
||||
* Based on fmod() return x-[x/p]chopped*p exactlp.
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const double zero = 0.0;
|
||||
|
||||
|
||||
double
|
||||
remainder(double x, double p)
|
||||
{
|
||||
int32_t hx,hp;
|
||||
uint32_t sx,lx,lp;
|
||||
double p_half;
|
||||
|
||||
EXTRACT_WORDS(hx,lx,x);
|
||||
EXTRACT_WORDS(hp,lp,p);
|
||||
sx = hx&0x80000000;
|
||||
hp &= 0x7fffffff;
|
||||
hx &= 0x7fffffff;
|
||||
|
||||
/* purge off exception values */
|
||||
if((hp|lp)==0) return (x*p)/(x*p); /* p = 0 */
|
||||
if((hx>=0x7ff00000)|| /* x not finite */
|
||||
((hp>=0x7ff00000)&& /* p is NaN */
|
||||
(((hp-0x7ff00000)|lp)!=0)))
|
||||
return (x*p)/(x*p);
|
||||
|
||||
|
||||
if (hp<=0x7fdfffff) x = fmod(x,p+p); /* now x < 2p */
|
||||
if (((hx-hp)|(lx-lp))==0) return zero*x;
|
||||
x = fabs(x);
|
||||
p = fabs(p);
|
||||
if (hp<0x00200000) {
|
||||
if(x+x>p) {
|
||||
x-=p;
|
||||
if(x+x>=p) x -= p;
|
||||
}
|
||||
} else {
|
||||
p_half = 0.5*p;
|
||||
if(x>p_half) {
|
||||
x-=p;
|
||||
if(x>=p_half) x -= p;
|
||||
}
|
||||
}
|
||||
GET_HIGH_WORD(hx,x);
|
||||
SET_HIGH_WORD(x,hx^sx);
|
||||
return x;
|
||||
}
|
||||
61
src/math/e_remainderf.c
Normal file
61
src/math/e_remainderf.c
Normal file
|
|
@ -0,0 +1,61 @@
|
|||
/* e_remainderf.c -- float version of e_remainder.c.
|
||||
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
||||
*/
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const float zero = 0.0;
|
||||
|
||||
|
||||
float
|
||||
remainderf(float x, float p)
|
||||
{
|
||||
int32_t hx,hp;
|
||||
uint32_t sx;
|
||||
float p_half;
|
||||
|
||||
GET_FLOAT_WORD(hx,x);
|
||||
GET_FLOAT_WORD(hp,p);
|
||||
sx = hx&0x80000000;
|
||||
hp &= 0x7fffffff;
|
||||
hx &= 0x7fffffff;
|
||||
|
||||
/* purge off exception values */
|
||||
if(hp==0) return (x*p)/(x*p); /* p = 0 */
|
||||
if((hx>=0x7f800000)|| /* x not finite */
|
||||
((hp>0x7f800000))) /* p is NaN */
|
||||
return (x*p)/(x*p);
|
||||
|
||||
|
||||
if (hp<=0x7effffff) x = fmodf(x,p+p); /* now x < 2p */
|
||||
if ((hx-hp)==0) return zero*x;
|
||||
x = fabsf(x);
|
||||
p = fabsf(p);
|
||||
if (hp<0x01000000) {
|
||||
if(x+x>p) {
|
||||
x-=p;
|
||||
if(x+x>=p) x -= p;
|
||||
}
|
||||
} else {
|
||||
p_half = (float)0.5*p;
|
||||
if(x>p_half) {
|
||||
x-=p;
|
||||
if(x>=p_half) x -= p;
|
||||
}
|
||||
}
|
||||
GET_FLOAT_WORD(hx,x);
|
||||
SET_FLOAT_WORD(x,hx^sx);
|
||||
return x;
|
||||
}
|
||||
35
src/math/e_scalb.c
Normal file
35
src/math/e_scalb.c
Normal file
|
|
@ -0,0 +1,35 @@
|
|||
|
||||
/* @(#)e_scalb.c 1.3 95/01/18 */
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunSoft, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/*
|
||||
* scalb(x, fn) is provide for
|
||||
* passing various standard test suite. One
|
||||
* should use scalbn() instead.
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
double
|
||||
scalb(double x, double fn)
|
||||
{
|
||||
if (isnan(x)||isnan(fn)) return x*fn;
|
||||
if (!isfinite(fn)) {
|
||||
if(fn>0.0) return x*fn;
|
||||
else return x/(-fn);
|
||||
}
|
||||
if (rint(fn)!=fn) return (fn-fn)/(fn-fn);
|
||||
if ( fn > 65000.0) return scalbn(x, 65000);
|
||||
if (-fn > 65000.0) return scalbn(x,-65000);
|
||||
return scalbn(x,(int)fn);
|
||||
}
|
||||
31
src/math/e_scalbf.c
Normal file
31
src/math/e_scalbf.c
Normal file
|
|
@ -0,0 +1,31 @@
|
|||
/* e_scalbf.c -- float version of e_scalb.c.
|
||||
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
||||
*/
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
float
|
||||
scalbf(float x, float fn)
|
||||
{
|
||||
if (isnan(x)||isnan(fn)) return x*fn;
|
||||
if (!isfinite(fn)) {
|
||||
if(fn>(float)0.0) return x*fn;
|
||||
else return x/(-fn);
|
||||
}
|
||||
if (rintf(fn)!=fn) return (fn-fn)/(fn-fn);
|
||||
if ( fn > (float)65000.0) return scalbnf(x, 65000);
|
||||
if (-fn > (float)65000.0) return scalbnf(x,-65000);
|
||||
return scalbnf(x,(int)fn);
|
||||
}
|
||||
75
src/math/e_sinh.c
Normal file
75
src/math/e_sinh.c
Normal file
|
|
@ -0,0 +1,75 @@
|
|||
|
||||
/* @(#)e_sinh.c 1.3 95/01/18 */
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunSoft, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/* sinh(x)
|
||||
* Method :
|
||||
* mathematically sinh(x) if defined to be (exp(x)-exp(-x))/2
|
||||
* 1. Replace x by |x| (sinh(-x) = -sinh(x)).
|
||||
* 2.
|
||||
* E + E/(E+1)
|
||||
* 0 <= x <= 22 : sinh(x) := --------------, E=expm1(x)
|
||||
* 2
|
||||
*
|
||||
* 22 <= x <= lnovft : sinh(x) := exp(x)/2
|
||||
* lnovft <= x <= ln2ovft: sinh(x) := exp(x/2)/2 * exp(x/2)
|
||||
* ln2ovft < x : sinh(x) := x*shuge (overflow)
|
||||
*
|
||||
* Special cases:
|
||||
* sinh(x) is |x| if x is +INF, -INF, or NaN.
|
||||
* only sinh(0)=0 is exact for finite x.
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const double one = 1.0, shuge = 1.0e307;
|
||||
|
||||
double
|
||||
sinh(double x)
|
||||
{
|
||||
double t,w,h;
|
||||
int32_t ix,jx;
|
||||
uint32_t lx;
|
||||
|
||||
/* High word of |x|. */
|
||||
GET_HIGH_WORD(jx,x);
|
||||
ix = jx&0x7fffffff;
|
||||
|
||||
/* x is INF or NaN */
|
||||
if(ix>=0x7ff00000) return x+x;
|
||||
|
||||
h = 0.5;
|
||||
if (jx<0) h = -h;
|
||||
/* |x| in [0,22], return sign(x)*0.5*(E+E/(E+1))) */
|
||||
if (ix < 0x40360000) { /* |x|<22 */
|
||||
if (ix<0x3e300000) /* |x|<2**-28 */
|
||||
if(shuge+x>one) return x;/* sinh(tiny) = tiny with inexact */
|
||||
t = expm1(fabs(x));
|
||||
if(ix<0x3ff00000) return h*(2.0*t-t*t/(t+one));
|
||||
return h*(t+t/(t+one));
|
||||
}
|
||||
|
||||
/* |x| in [22, log(maxdouble)] return 0.5*exp(|x|) */
|
||||
if (ix < 0x40862E42) return h*exp(fabs(x));
|
||||
|
||||
/* |x| in [log(maxdouble), overflowthresold] */
|
||||
GET_LOW_WORD(lx,x);
|
||||
if (ix<0x408633CE || ((ix==0x408633ce)&&(lx<=(uint32_t)0x8fb9f87d))) {
|
||||
w = exp(0.5*fabs(x));
|
||||
t = h*w;
|
||||
return t*w;
|
||||
}
|
||||
|
||||
/* |x| > overflowthresold, sinh(x) overflow */
|
||||
return x*shuge;
|
||||
}
|
||||
56
src/math/e_sinhf.c
Normal file
56
src/math/e_sinhf.c
Normal file
|
|
@ -0,0 +1,56 @@
|
|||
/* e_sinhf.c -- float version of e_sinh.c.
|
||||
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
||||
*/
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const float one = 1.0, shuge = 1.0e37;
|
||||
|
||||
float
|
||||
sinhf(float x)
|
||||
{
|
||||
float t,w,h;
|
||||
int32_t ix,jx;
|
||||
|
||||
GET_FLOAT_WORD(jx,x);
|
||||
ix = jx&0x7fffffff;
|
||||
|
||||
/* x is INF or NaN */
|
||||
if(ix>=0x7f800000) return x+x;
|
||||
|
||||
h = 0.5;
|
||||
if (jx<0) h = -h;
|
||||
/* |x| in [0,22], return sign(x)*0.5*(E+E/(E+1))) */
|
||||
if (ix < 0x41b00000) { /* |x|<22 */
|
||||
if (ix<0x31800000) /* |x|<2**-28 */
|
||||
if(shuge+x>one) return x;/* sinh(tiny) = tiny with inexact */
|
||||
t = expm1f(fabsf(x));
|
||||
if(ix<0x3f800000) return h*((float)2.0*t-t*t/(t+one));
|
||||
return h*(t+t/(t+one));
|
||||
}
|
||||
|
||||
/* |x| in [22, log(maxdouble)] return 0.5*exp(|x|) */
|
||||
if (ix < 0x42b17180) return h*expf(fabsf(x));
|
||||
|
||||
/* |x| in [log(maxdouble), overflowthresold] */
|
||||
if (ix<=0x42b2d4fc) {
|
||||
w = expf((float)0.5*fabsf(x));
|
||||
t = h*w;
|
||||
return t*w;
|
||||
}
|
||||
|
||||
/* |x| > overflowthresold, sinh(x) overflow */
|
||||
return x*shuge;
|
||||
}
|
||||
442
src/math/e_sqrt.c
Normal file
442
src/math/e_sqrt.c
Normal file
|
|
@ -0,0 +1,442 @@
|
|||
|
||||
/* @(#)e_sqrt.c 1.3 95/01/18 */
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunSoft, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/* sqrt(x)
|
||||
* Return correctly rounded sqrt.
|
||||
* ------------------------------------------
|
||||
* | Use the hardware sqrt if you have one |
|
||||
* ------------------------------------------
|
||||
* Method:
|
||||
* Bit by bit method using integer arithmetic. (Slow, but portable)
|
||||
* 1. Normalization
|
||||
* Scale x to y in [1,4) with even powers of 2:
|
||||
* find an integer k such that 1 <= (y=x*2^(2k)) < 4, then
|
||||
* sqrt(x) = 2^k * sqrt(y)
|
||||
* 2. Bit by bit computation
|
||||
* Let q = sqrt(y) truncated to i bit after binary point (q = 1),
|
||||
* i 0
|
||||
* i+1 2
|
||||
* s = 2*q , and y = 2 * ( y - q ). (1)
|
||||
* i i i i
|
||||
*
|
||||
* To compute q from q , one checks whether
|
||||
* i+1 i
|
||||
*
|
||||
* -(i+1) 2
|
||||
* (q + 2 ) <= y. (2)
|
||||
* i
|
||||
* -(i+1)
|
||||
* If (2) is false, then q = q ; otherwise q = q + 2 .
|
||||
* i+1 i i+1 i
|
||||
*
|
||||
* With some algebric manipulation, it is not difficult to see
|
||||
* that (2) is equivalent to
|
||||
* -(i+1)
|
||||
* s + 2 <= y (3)
|
||||
* i i
|
||||
*
|
||||
* The advantage of (3) is that s and y can be computed by
|
||||
* i i
|
||||
* the following recurrence formula:
|
||||
* if (3) is false
|
||||
*
|
||||
* s = s , y = y ; (4)
|
||||
* i+1 i i+1 i
|
||||
*
|
||||
* otherwise,
|
||||
* -i -(i+1)
|
||||
* s = s + 2 , y = y - s - 2 (5)
|
||||
* i+1 i i+1 i i
|
||||
*
|
||||
* One may easily use induction to prove (4) and (5).
|
||||
* Note. Since the left hand side of (3) contain only i+2 bits,
|
||||
* it does not necessary to do a full (53-bit) comparison
|
||||
* in (3).
|
||||
* 3. Final rounding
|
||||
* After generating the 53 bits result, we compute one more bit.
|
||||
* Together with the remainder, we can decide whether the
|
||||
* result is exact, bigger than 1/2ulp, or less than 1/2ulp
|
||||
* (it will never equal to 1/2ulp).
|
||||
* The rounding mode can be detected by checking whether
|
||||
* huge + tiny is equal to huge, and whether huge - tiny is
|
||||
* equal to huge for some floating point number "huge" and "tiny".
|
||||
*
|
||||
* Special cases:
|
||||
* sqrt(+-0) = +-0 ... exact
|
||||
* sqrt(inf) = inf
|
||||
* sqrt(-ve) = NaN ... with invalid signal
|
||||
* sqrt(NaN) = NaN ... with invalid signal for signaling NaN
|
||||
*
|
||||
* Other methods : see the appended file at the end of the program below.
|
||||
*---------------
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const double one = 1.0, tiny=1.0e-300;
|
||||
|
||||
double
|
||||
sqrt(double x)
|
||||
{
|
||||
double z;
|
||||
int32_t sign = (int)0x80000000;
|
||||
int32_t ix0,s0,q,m,t,i;
|
||||
uint32_t r,t1,s1,ix1,q1;
|
||||
|
||||
EXTRACT_WORDS(ix0,ix1,x);
|
||||
|
||||
/* take care of Inf and NaN */
|
||||
if((ix0&0x7ff00000)==0x7ff00000) {
|
||||
return x*x+x; /* sqrt(NaN)=NaN, sqrt(+inf)=+inf
|
||||
sqrt(-inf)=sNaN */
|
||||
}
|
||||
/* take care of zero */
|
||||
if(ix0<=0) {
|
||||
if(((ix0&(~sign))|ix1)==0) return x;/* sqrt(+-0) = +-0 */
|
||||
else if(ix0<0)
|
||||
return (x-x)/(x-x); /* sqrt(-ve) = sNaN */
|
||||
}
|
||||
/* normalize x */
|
||||
m = (ix0>>20);
|
||||
if(m==0) { /* subnormal x */
|
||||
while(ix0==0) {
|
||||
m -= 21;
|
||||
ix0 |= (ix1>>11); ix1 <<= 21;
|
||||
}
|
||||
for(i=0;(ix0&0x00100000)==0;i++) ix0<<=1;
|
||||
m -= i-1;
|
||||
ix0 |= (ix1>>(32-i));
|
||||
ix1 <<= i;
|
||||
}
|
||||
m -= 1023; /* unbias exponent */
|
||||
ix0 = (ix0&0x000fffff)|0x00100000;
|
||||
if(m&1){ /* odd m, double x to make it even */
|
||||
ix0 += ix0 + ((ix1&sign)>>31);
|
||||
ix1 += ix1;
|
||||
}
|
||||
m >>= 1; /* m = [m/2] */
|
||||
|
||||
/* generate sqrt(x) bit by bit */
|
||||
ix0 += ix0 + ((ix1&sign)>>31);
|
||||
ix1 += ix1;
|
||||
q = q1 = s0 = s1 = 0; /* [q,q1] = sqrt(x) */
|
||||
r = 0x00200000; /* r = moving bit from right to left */
|
||||
|
||||
while(r!=0) {
|
||||
t = s0+r;
|
||||
if(t<=ix0) {
|
||||
s0 = t+r;
|
||||
ix0 -= t;
|
||||
q += r;
|
||||
}
|
||||
ix0 += ix0 + ((ix1&sign)>>31);
|
||||
ix1 += ix1;
|
||||
r>>=1;
|
||||
}
|
||||
|
||||
r = sign;
|
||||
while(r!=0) {
|
||||
t1 = s1+r;
|
||||
t = s0;
|
||||
if((t<ix0)||((t==ix0)&&(t1<=ix1))) {
|
||||
s1 = t1+r;
|
||||
if(((t1&sign)==sign)&&(s1&sign)==0) s0 += 1;
|
||||
ix0 -= t;
|
||||
if (ix1 < t1) ix0 -= 1;
|
||||
ix1 -= t1;
|
||||
q1 += r;
|
||||
}
|
||||
ix0 += ix0 + ((ix1&sign)>>31);
|
||||
ix1 += ix1;
|
||||
r>>=1;
|
||||
}
|
||||
|
||||
/* use floating add to find out rounding direction */
|
||||
if((ix0|ix1)!=0) {
|
||||
z = one-tiny; /* trigger inexact flag */
|
||||
if (z>=one) {
|
||||
z = one+tiny;
|
||||
if (q1==(uint32_t)0xffffffff) { q1=0; q += 1;}
|
||||
else if (z>one) {
|
||||
if (q1==(uint32_t)0xfffffffe) q+=1;
|
||||
q1+=2;
|
||||
} else
|
||||
q1 += (q1&1);
|
||||
}
|
||||
}
|
||||
ix0 = (q>>1)+0x3fe00000;
|
||||
ix1 = q1>>1;
|
||||
if ((q&1)==1) ix1 |= sign;
|
||||
ix0 += (m <<20);
|
||||
INSERT_WORDS(z,ix0,ix1);
|
||||
return z;
|
||||
}
|
||||
|
||||
/*
|
||||
Other methods (use floating-point arithmetic)
|
||||
-------------
|
||||
(This is a copy of a drafted paper by Prof W. Kahan
|
||||
and K.C. Ng, written in May, 1986)
|
||||
|
||||
Two algorithms are given here to implement sqrt(x)
|
||||
(IEEE double precision arithmetic) in software.
|
||||
Both supply sqrt(x) correctly rounded. The first algorithm (in
|
||||
Section A) uses newton iterations and involves four divisions.
|
||||
The second one uses reciproot iterations to avoid division, but
|
||||
requires more multiplications. Both algorithms need the ability
|
||||
to chop results of arithmetic operations instead of round them,
|
||||
and the INEXACT flag to indicate when an arithmetic operation
|
||||
is executed exactly with no roundoff error, all part of the
|
||||
standard (IEEE 754-1985). The ability to perform shift, add,
|
||||
subtract and logical AND operations upon 32-bit words is needed
|
||||
too, though not part of the standard.
|
||||
|
||||
A. sqrt(x) by Newton Iteration
|
||||
|
||||
(1) Initial approximation
|
||||
|
||||
Let x0 and x1 be the leading and the trailing 32-bit words of
|
||||
a floating point number x (in IEEE double format) respectively
|
||||
|
||||
1 11 52 ...widths
|
||||
------------------------------------------------------
|
||||
x: |s| e | f |
|
||||
------------------------------------------------------
|
||||
msb lsb msb lsb ...order
|
||||
|
||||
|
||||
------------------------ ------------------------
|
||||
x0: |s| e | f1 | x1: | f2 |
|
||||
------------------------ ------------------------
|
||||
|
||||
By performing shifts and subtracts on x0 and x1 (both regarded
|
||||
as integers), we obtain an 8-bit approximation of sqrt(x) as
|
||||
follows.
|
||||
|
||||
k := (x0>>1) + 0x1ff80000;
|
||||
y0 := k - T1[31&(k>>15)]. ... y ~ sqrt(x) to 8 bits
|
||||
Here k is a 32-bit integer and T1[] is an integer array containing
|
||||
correction terms. Now magically the floating value of y (y's
|
||||
leading 32-bit word is y0, the value of its trailing word is 0)
|
||||
approximates sqrt(x) to almost 8-bit.
|
||||
|
||||
Value of T1:
|
||||
static int T1[32]= {
|
||||
0, 1024, 3062, 5746, 9193, 13348, 18162, 23592,
|
||||
29598, 36145, 43202, 50740, 58733, 67158, 75992, 85215,
|
||||
83599, 71378, 60428, 50647, 41945, 34246, 27478, 21581,
|
||||
16499, 12183, 8588, 5674, 3403, 1742, 661, 130,};
|
||||
|
||||
(2) Iterative refinement
|
||||
|
||||
Apply Heron's rule three times to y, we have y approximates
|
||||
sqrt(x) to within 1 ulp (Unit in the Last Place):
|
||||
|
||||
y := (y+x/y)/2 ... almost 17 sig. bits
|
||||
y := (y+x/y)/2 ... almost 35 sig. bits
|
||||
y := y-(y-x/y)/2 ... within 1 ulp
|
||||
|
||||
|
||||
Remark 1.
|
||||
Another way to improve y to within 1 ulp is:
|
||||
|
||||
y := (y+x/y) ... almost 17 sig. bits to 2*sqrt(x)
|
||||
y := y - 0x00100006 ... almost 18 sig. bits to sqrt(x)
|
||||
|
||||
2
|
||||
(x-y )*y
|
||||
y := y + 2* ---------- ...within 1 ulp
|
||||
2
|
||||
3y + x
|
||||
|
||||
|
||||
This formula has one division fewer than the one above; however,
|
||||
it requires more multiplications and additions. Also x must be
|
||||
scaled in advance to avoid spurious overflow in evaluating the
|
||||
expression 3y*y+x. Hence it is not recommended uless division
|
||||
is slow. If division is very slow, then one should use the
|
||||
reciproot algorithm given in section B.
|
||||
|
||||
(3) Final adjustment
|
||||
|
||||
By twiddling y's last bit it is possible to force y to be
|
||||
correctly rounded according to the prevailing rounding mode
|
||||
as follows. Let r and i be copies of the rounding mode and
|
||||
inexact flag before entering the square root program. Also we
|
||||
use the expression y+-ulp for the next representable floating
|
||||
numbers (up and down) of y. Note that y+-ulp = either fixed
|
||||
point y+-1, or multiply y by nextafter(1,+-inf) in chopped
|
||||
mode.
|
||||
|
||||
I := FALSE; ... reset INEXACT flag I
|
||||
R := RZ; ... set rounding mode to round-toward-zero
|
||||
z := x/y; ... chopped quotient, possibly inexact
|
||||
If(not I) then { ... if the quotient is exact
|
||||
if(z=y) {
|
||||
I := i; ... restore inexact flag
|
||||
R := r; ... restore rounded mode
|
||||
return sqrt(x):=y.
|
||||
} else {
|
||||
z := z - ulp; ... special rounding
|
||||
}
|
||||
}
|
||||
i := TRUE; ... sqrt(x) is inexact
|
||||
If (r=RN) then z=z+ulp ... rounded-to-nearest
|
||||
If (r=RP) then { ... round-toward-+inf
|
||||
y = y+ulp; z=z+ulp;
|
||||
}
|
||||
y := y+z; ... chopped sum
|
||||
y0:=y0-0x00100000; ... y := y/2 is correctly rounded.
|
||||
I := i; ... restore inexact flag
|
||||
R := r; ... restore rounded mode
|
||||
return sqrt(x):=y.
|
||||
|
||||
(4) Special cases
|
||||
|
||||
Square root of +inf, +-0, or NaN is itself;
|
||||
Square root of a negative number is NaN with invalid signal.
|
||||
|
||||
|
||||
B. sqrt(x) by Reciproot Iteration
|
||||
|
||||
(1) Initial approximation
|
||||
|
||||
Let x0 and x1 be the leading and the trailing 32-bit words of
|
||||
a floating point number x (in IEEE double format) respectively
|
||||
(see section A). By performing shifs and subtracts on x0 and y0,
|
||||
we obtain a 7.8-bit approximation of 1/sqrt(x) as follows.
|
||||
|
||||
k := 0x5fe80000 - (x0>>1);
|
||||
y0:= k - T2[63&(k>>14)]. ... y ~ 1/sqrt(x) to 7.8 bits
|
||||
|
||||
Here k is a 32-bit integer and T2[] is an integer array
|
||||
containing correction terms. Now magically the floating
|
||||
value of y (y's leading 32-bit word is y0, the value of
|
||||
its trailing word y1 is set to zero) approximates 1/sqrt(x)
|
||||
to almost 7.8-bit.
|
||||
|
||||
Value of T2:
|
||||
static int T2[64]= {
|
||||
0x1500, 0x2ef8, 0x4d67, 0x6b02, 0x87be, 0xa395, 0xbe7a, 0xd866,
|
||||
0xf14a, 0x1091b,0x11fcd,0x13552,0x14999,0x15c98,0x16e34,0x17e5f,
|
||||
0x18d03,0x19a01,0x1a545,0x1ae8a,0x1b5c4,0x1bb01,0x1bfde,0x1c28d,
|
||||
0x1c2de,0x1c0db,0x1ba73,0x1b11c,0x1a4b5,0x1953d,0x18266,0x16be0,
|
||||
0x1683e,0x179d8,0x18a4d,0x19992,0x1a789,0x1b445,0x1bf61,0x1c989,
|
||||
0x1d16d,0x1d77b,0x1dddf,0x1e2ad,0x1e5bf,0x1e6e8,0x1e654,0x1e3cd,
|
||||
0x1df2a,0x1d635,0x1cb16,0x1be2c,0x1ae4e,0x19bde,0x1868e,0x16e2e,
|
||||
0x1527f,0x1334a,0x11051,0xe951, 0xbe01, 0x8e0d, 0x5924, 0x1edd,};
|
||||
|
||||
(2) Iterative refinement
|
||||
|
||||
Apply Reciproot iteration three times to y and multiply the
|
||||
result by x to get an approximation z that matches sqrt(x)
|
||||
to about 1 ulp. To be exact, we will have
|
||||
-1ulp < sqrt(x)-z<1.0625ulp.
|
||||
|
||||
... set rounding mode to Round-to-nearest
|
||||
y := y*(1.5-0.5*x*y*y) ... almost 15 sig. bits to 1/sqrt(x)
|
||||
y := y*((1.5-2^-30)+0.5*x*y*y)... about 29 sig. bits to 1/sqrt(x)
|
||||
... special arrangement for better accuracy
|
||||
z := x*y ... 29 bits to sqrt(x), with z*y<1
|
||||
z := z + 0.5*z*(1-z*y) ... about 1 ulp to sqrt(x)
|
||||
|
||||
Remark 2. The constant 1.5-2^-30 is chosen to bias the error so that
|
||||
(a) the term z*y in the final iteration is always less than 1;
|
||||
(b) the error in the final result is biased upward so that
|
||||
-1 ulp < sqrt(x) - z < 1.0625 ulp
|
||||
instead of |sqrt(x)-z|<1.03125ulp.
|
||||
|
||||
(3) Final adjustment
|
||||
|
||||
By twiddling y's last bit it is possible to force y to be
|
||||
correctly rounded according to the prevailing rounding mode
|
||||
as follows. Let r and i be copies of the rounding mode and
|
||||
inexact flag before entering the square root program. Also we
|
||||
use the expression y+-ulp for the next representable floating
|
||||
numbers (up and down) of y. Note that y+-ulp = either fixed
|
||||
point y+-1, or multiply y by nextafter(1,+-inf) in chopped
|
||||
mode.
|
||||
|
||||
R := RZ; ... set rounding mode to round-toward-zero
|
||||
switch(r) {
|
||||
case RN: ... round-to-nearest
|
||||
if(x<= z*(z-ulp)...chopped) z = z - ulp; else
|
||||
if(x<= z*(z+ulp)...chopped) z = z; else z = z+ulp;
|
||||
break;
|
||||
case RZ:case RM: ... round-to-zero or round-to--inf
|
||||
R:=RP; ... reset rounding mod to round-to-+inf
|
||||
if(x<z*z ... rounded up) z = z - ulp; else
|
||||
if(x>=(z+ulp)*(z+ulp) ...rounded up) z = z+ulp;
|
||||
break;
|
||||
case RP: ... round-to-+inf
|
||||
if(x>(z+ulp)*(z+ulp)...chopped) z = z+2*ulp; else
|
||||
if(x>z*z ...chopped) z = z+ulp;
|
||||
break;
|
||||
}
|
||||
|
||||
Remark 3. The above comparisons can be done in fixed point. For
|
||||
example, to compare x and w=z*z chopped, it suffices to compare
|
||||
x1 and w1 (the trailing parts of x and w), regarding them as
|
||||
two's complement integers.
|
||||
|
||||
...Is z an exact square root?
|
||||
To determine whether z is an exact square root of x, let z1 be the
|
||||
trailing part of z, and also let x0 and x1 be the leading and
|
||||
trailing parts of x.
|
||||
|
||||
If ((z1&0x03ffffff)!=0) ... not exact if trailing 26 bits of z!=0
|
||||
I := 1; ... Raise Inexact flag: z is not exact
|
||||
else {
|
||||
j := 1 - [(x0>>20)&1] ... j = logb(x) mod 2
|
||||
k := z1 >> 26; ... get z's 25-th and 26-th
|
||||
fraction bits
|
||||
I := i or (k&j) or ((k&(j+j+1))!=(x1&3));
|
||||
}
|
||||
R:= r ... restore rounded mode
|
||||
return sqrt(x):=z.
|
||||
|
||||
If multiplication is cheaper then the foregoing red tape, the
|
||||
Inexact flag can be evaluated by
|
||||
|
||||
I := i;
|
||||
I := (z*z!=x) or I.
|
||||
|
||||
Note that z*z can overwrite I; this value must be sensed if it is
|
||||
True.
|
||||
|
||||
Remark 4. If z*z = x exactly, then bit 25 to bit 0 of z1 must be
|
||||
zero.
|
||||
|
||||
--------------------
|
||||
z1: | f2 |
|
||||
--------------------
|
||||
bit 31 bit 0
|
||||
|
||||
Further more, bit 27 and 26 of z1, bit 0 and 1 of x1, and the odd
|
||||
or even of logb(x) have the following relations:
|
||||
|
||||
-------------------------------------------------
|
||||
bit 27,26 of z1 bit 1,0 of x1 logb(x)
|
||||
-------------------------------------------------
|
||||
00 00 odd and even
|
||||
01 01 even
|
||||
10 10 odd
|
||||
10 00 even
|
||||
11 01 even
|
||||
-------------------------------------------------
|
||||
|
||||
(4) Special cases (see (4) of Section A).
|
||||
|
||||
*/
|
||||
|
||||
85
src/math/e_sqrtf.c
Normal file
85
src/math/e_sqrtf.c
Normal file
|
|
@ -0,0 +1,85 @@
|
|||
/* e_sqrtf.c -- float version of e_sqrt.c.
|
||||
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
||||
*/
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const float one = 1.0, tiny=1.0e-30;
|
||||
|
||||
float
|
||||
sqrtf(float x)
|
||||
{
|
||||
float z;
|
||||
int32_t sign = (int)0x80000000;
|
||||
int32_t ix,s,q,m,t,i;
|
||||
uint32_t r;
|
||||
|
||||
GET_FLOAT_WORD(ix,x);
|
||||
|
||||
/* take care of Inf and NaN */
|
||||
if((ix&0x7f800000)==0x7f800000) {
|
||||
return x*x+x; /* sqrt(NaN)=NaN, sqrt(+inf)=+inf
|
||||
sqrt(-inf)=sNaN */
|
||||
}
|
||||
/* take care of zero */
|
||||
if(ix<=0) {
|
||||
if((ix&(~sign))==0) return x;/* sqrt(+-0) = +-0 */
|
||||
else if(ix<0)
|
||||
return (x-x)/(x-x); /* sqrt(-ve) = sNaN */
|
||||
}
|
||||
/* normalize x */
|
||||
m = (ix>>23);
|
||||
if(m==0) { /* subnormal x */
|
||||
for(i=0;(ix&0x00800000)==0;i++) ix<<=1;
|
||||
m -= i-1;
|
||||
}
|
||||
m -= 127; /* unbias exponent */
|
||||
ix = (ix&0x007fffff)|0x00800000;
|
||||
if(m&1) /* odd m, double x to make it even */
|
||||
ix += ix;
|
||||
m >>= 1; /* m = [m/2] */
|
||||
|
||||
/* generate sqrt(x) bit by bit */
|
||||
ix += ix;
|
||||
q = s = 0; /* q = sqrt(x) */
|
||||
r = 0x01000000; /* r = moving bit from right to left */
|
||||
|
||||
while(r!=0) {
|
||||
t = s+r;
|
||||
if(t<=ix) {
|
||||
s = t+r;
|
||||
ix -= t;
|
||||
q += r;
|
||||
}
|
||||
ix += ix;
|
||||
r>>=1;
|
||||
}
|
||||
|
||||
/* use floating add to find out rounding direction */
|
||||
if(ix!=0) {
|
||||
z = one-tiny; /* trigger inexact flag */
|
||||
if (z>=one) {
|
||||
z = one+tiny;
|
||||
if (z>one)
|
||||
q += 2;
|
||||
else
|
||||
q += (q&1);
|
||||
}
|
||||
}
|
||||
ix = (q>>1)+0x3f000000;
|
||||
ix += (m <<23);
|
||||
SET_FLOAT_WORD(z,ix);
|
||||
return z;
|
||||
}
|
||||
36
src/math/i386/e_exp.s
Normal file
36
src/math/i386/e_exp.s
Normal file
|
|
@ -0,0 +1,36 @@
|
|||
.global expf
|
||||
expf:
|
||||
mov 4(%esp),%eax
|
||||
flds 4(%esp)
|
||||
shr $23,%eax
|
||||
inc %al
|
||||
jz 1f
|
||||
jmp 0f
|
||||
|
||||
.global exp
|
||||
exp:
|
||||
mov 8(%esp),%eax
|
||||
fldl 4(%esp)
|
||||
shl %eax
|
||||
cmp $0xffe00000,%eax
|
||||
jae 1f
|
||||
|
||||
0: fldl2e
|
||||
fmulp
|
||||
fst %st(1)
|
||||
frndint
|
||||
fst %st(2)
|
||||
fsubrp
|
||||
f2xm1
|
||||
fld1
|
||||
faddp
|
||||
fscale
|
||||
fstp %st(1)
|
||||
ret
|
||||
|
||||
1: fsts 4(%esp)
|
||||
cmpl $0xff800000,4(%esp)
|
||||
jnz 1f
|
||||
fstp %st(0)
|
||||
fldz
|
||||
1: ret
|
||||
1
src/math/i386/e_expf.s
Normal file
1
src/math/i386/e_expf.s
Normal file
|
|
@ -0,0 +1 @@
|
|||
|
||||
6
src/math/i386/e_log.s
Normal file
6
src/math/i386/e_log.s
Normal file
|
|
@ -0,0 +1,6 @@
|
|||
.global log
|
||||
log:
|
||||
fldln2
|
||||
fldl 4(%esp)
|
||||
fyl2x
|
||||
ret
|
||||
6
src/math/i386/e_log10.s
Normal file
6
src/math/i386/e_log10.s
Normal file
|
|
@ -0,0 +1,6 @@
|
|||
.global log10
|
||||
log10:
|
||||
fldlg2
|
||||
fldl 4(%esp)
|
||||
fyl2x
|
||||
ret
|
||||
6
src/math/i386/e_log10f.s
Normal file
6
src/math/i386/e_log10f.s
Normal file
|
|
@ -0,0 +1,6 @@
|
|||
.global log10f
|
||||
log10f:
|
||||
fldlg2
|
||||
flds 4(%esp)
|
||||
fyl2x
|
||||
ret
|
||||
6
src/math/i386/e_logf.s
Normal file
6
src/math/i386/e_logf.s
Normal file
|
|
@ -0,0 +1,6 @@
|
|||
.global logf
|
||||
logf:
|
||||
fldln2
|
||||
flds 4(%esp)
|
||||
fyl2x
|
||||
ret
|
||||
16
src/math/i386/e_remainder.s
Normal file
16
src/math/i386/e_remainder.s
Normal file
|
|
@ -0,0 +1,16 @@
|
|||
.global remainderf
|
||||
remainderf:
|
||||
flds 8(%esp)
|
||||
flds 4(%esp)
|
||||
jmp 1f
|
||||
|
||||
.global remainder
|
||||
remainder:
|
||||
fldl 12(%esp)
|
||||
fldl 4(%esp)
|
||||
1: fprem1
|
||||
fstsw %ax
|
||||
sahf
|
||||
jp 1b
|
||||
fstp %st(1)
|
||||
ret
|
||||
0
src/math/i386/e_remainderf.s
Normal file
0
src/math/i386/e_remainderf.s
Normal file
4
src/math/i386/e_sqrt.s
Normal file
4
src/math/i386/e_sqrt.s
Normal file
|
|
@ -0,0 +1,4 @@
|
|||
.global sqrt
|
||||
sqrt: fldl 4(%esp)
|
||||
fsqrt
|
||||
ret
|
||||
4
src/math/i386/e_sqrtf.s
Normal file
4
src/math/i386/e_sqrtf.s
Normal file
|
|
@ -0,0 +1,4 @@
|
|||
.global sqrtf
|
||||
sqrtf: flds 4(%esp)
|
||||
fsqrt
|
||||
ret
|
||||
0
src/math/i386/s_ceil.s
Normal file
0
src/math/i386/s_ceil.s
Normal file
0
src/math/i386/s_ceilf.s
Normal file
0
src/math/i386/s_ceilf.s
Normal file
5
src/math/i386/s_fabs.s
Normal file
5
src/math/i386/s_fabs.s
Normal file
|
|
@ -0,0 +1,5 @@
|
|||
.global fabs
|
||||
fabs:
|
||||
fldl 4(%esp)
|
||||
fabs
|
||||
ret
|
||||
5
src/math/i386/s_fabsf.s
Normal file
5
src/math/i386/s_fabsf.s
Normal file
|
|
@ -0,0 +1,5 @@
|
|||
.global fabsf
|
||||
fabsf:
|
||||
flds 4(%esp)
|
||||
fabs
|
||||
ret
|
||||
0
src/math/i386/s_floor.s
Normal file
0
src/math/i386/s_floor.s
Normal file
0
src/math/i386/s_floorf.s
Normal file
0
src/math/i386/s_floorf.s
Normal file
0
src/math/i386/s_ldexp.s
Normal file
0
src/math/i386/s_ldexp.s
Normal file
0
src/math/i386/s_ldexpf.s
Normal file
0
src/math/i386/s_ldexpf.s
Normal file
5
src/math/i386/s_rint.s
Normal file
5
src/math/i386/s_rint.s
Normal file
|
|
@ -0,0 +1,5 @@
|
|||
.global rint
|
||||
rint:
|
||||
fldl 4(%esp)
|
||||
frndint
|
||||
ret
|
||||
5
src/math/i386/s_rintf.s
Normal file
5
src/math/i386/s_rintf.s
Normal file
|
|
@ -0,0 +1,5 @@
|
|||
.global rintf
|
||||
rintf:
|
||||
flds 4(%esp)
|
||||
frndint
|
||||
ret
|
||||
11
src/math/i386/s_scalbln.s
Normal file
11
src/math/i386/s_scalbln.s
Normal file
|
|
@ -0,0 +1,11 @@
|
|||
.global ldexp
|
||||
.global scalbn
|
||||
.global scalbln
|
||||
ldexp:
|
||||
scalbn:
|
||||
scalbln:
|
||||
fildl 12(%esp)
|
||||
fldl 4(%esp)
|
||||
fscale
|
||||
fstp %st(1)
|
||||
ret
|
||||
11
src/math/i386/s_scalblnf.s
Normal file
11
src/math/i386/s_scalblnf.s
Normal file
|
|
@ -0,0 +1,11 @@
|
|||
.global ldexpf
|
||||
.global scalbnf
|
||||
.global scalblnf
|
||||
ldexpf:
|
||||
scalbnf:
|
||||
scalblnf:
|
||||
fildl 8(%esp)
|
||||
flds 4(%esp)
|
||||
fscale
|
||||
fstp %st(1)
|
||||
ret
|
||||
36
src/math/i386/s_trunc.s
Normal file
36
src/math/i386/s_trunc.s
Normal file
|
|
@ -0,0 +1,36 @@
|
|||
.global ceilf
|
||||
ceilf: flds 4(%esp)
|
||||
jmp 1f
|
||||
|
||||
.global ceil
|
||||
ceil: fldl 4(%esp)
|
||||
1: mov $0x08fb,%edx
|
||||
jmp 0f
|
||||
|
||||
.global floorf
|
||||
floorf: flds 4(%esp)
|
||||
jmp 1f
|
||||
|
||||
.global floor
|
||||
floor: fldl 4(%esp)
|
||||
1: mov $0x04f7,%edx
|
||||
jmp 0f
|
||||
|
||||
.global truncf
|
||||
truncf: flds 4(%esp)
|
||||
jmp 1f
|
||||
|
||||
.global trunc
|
||||
trunc: fldl 4(%esp)
|
||||
1: mov $0x0cff,%edx
|
||||
|
||||
0: fstcw 4(%esp)
|
||||
mov 5(%esp),%ah
|
||||
or %dh,%ah
|
||||
and %dl,%ah
|
||||
xchg %ah,5(%esp)
|
||||
fldcw 4(%esp)
|
||||
frndint
|
||||
mov %ah,5(%esp)
|
||||
fldcw 4(%esp)
|
||||
ret
|
||||
0
src/math/i386/s_truncf.s
Normal file
0
src/math/i386/s_truncf.s
Normal file
85
src/math/k_cos.c
Normal file
85
src/math/k_cos.c
Normal file
|
|
@ -0,0 +1,85 @@
|
|||
|
||||
/* @(#)k_cos.c 1.3 95/01/18 */
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunSoft, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/*
|
||||
* __kernel_cos( x, y )
|
||||
* kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
|
||||
* Input x is assumed to be bounded by ~pi/4 in magnitude.
|
||||
* Input y is the tail of x.
|
||||
*
|
||||
* Algorithm
|
||||
* 1. Since cos(-x) = cos(x), we need only to consider positive x.
|
||||
* 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
|
||||
* 3. cos(x) is approximated by a polynomial of degree 14 on
|
||||
* [0,pi/4]
|
||||
* 4 14
|
||||
* cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
|
||||
* where the remez error is
|
||||
*
|
||||
* | 2 4 6 8 10 12 14 | -58
|
||||
* |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2
|
||||
* | |
|
||||
*
|
||||
* 4 6 8 10 12 14
|
||||
* 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then
|
||||
* cos(x) = 1 - x*x/2 + r
|
||||
* since cos(x+y) ~ cos(x) - sin(x)*y
|
||||
* ~ cos(x) - x*y,
|
||||
* a correction term is necessary in cos(x) and hence
|
||||
* cos(x+y) = 1 - (x*x/2 - (r - x*y))
|
||||
* For better accuracy when x > 0.3, let qx = |x|/4 with
|
||||
* the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125.
|
||||
* Then
|
||||
* cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)).
|
||||
* Note that 1-qx and (x*x/2-qx) is EXACT here, and the
|
||||
* magnitude of the latter is at least a quarter of x*x/2,
|
||||
* thus, reducing the rounding error in the subtraction.
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const double
|
||||
one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
|
||||
C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */
|
||||
C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */
|
||||
C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */
|
||||
C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */
|
||||
C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */
|
||||
C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */
|
||||
|
||||
double
|
||||
__kernel_cos(double x, double y)
|
||||
{
|
||||
double a,hz,z,r,qx;
|
||||
int32_t ix;
|
||||
GET_HIGH_WORD(ix,x);
|
||||
ix &= 0x7fffffff; /* ix = |x|'s high word*/
|
||||
if(ix<0x3e400000) { /* if x < 2**27 */
|
||||
if(((int)x)==0) return one; /* generate inexact */
|
||||
}
|
||||
z = x*x;
|
||||
r = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6)))));
|
||||
if(ix < 0x3FD33333) /* if |x| < 0.3 */
|
||||
return one - (0.5*z - (z*r - x*y));
|
||||
else {
|
||||
if(ix > 0x3fe90000) { /* x > 0.78125 */
|
||||
qx = 0.28125;
|
||||
} else {
|
||||
INSERT_WORDS(qx,ix-0x00200000,0); /* x/4 */
|
||||
}
|
||||
hz = 0.5*z-qx;
|
||||
a = one-qx;
|
||||
return a - (hz - (z*r-x*y));
|
||||
}
|
||||
}
|
||||
52
src/math/k_cosf.c
Normal file
52
src/math/k_cosf.c
Normal file
|
|
@ -0,0 +1,52 @@
|
|||
/* k_cosf.c -- float version of k_cos.c
|
||||
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
||||
*/
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const float
|
||||
one = 1.0000000000e+00, /* 0x3f800000 */
|
||||
C1 = 4.1666667908e-02, /* 0x3d2aaaab */
|
||||
C2 = -1.3888889225e-03, /* 0xbab60b61 */
|
||||
C3 = 2.4801587642e-05, /* 0x37d00d01 */
|
||||
C4 = -2.7557314297e-07, /* 0xb493f27c */
|
||||
C5 = 2.0875723372e-09, /* 0x310f74f6 */
|
||||
C6 = -1.1359647598e-11; /* 0xad47d74e */
|
||||
|
||||
float
|
||||
__kernel_cosf(float x, float y)
|
||||
{
|
||||
float a,hz,z,r,qx;
|
||||
int32_t ix;
|
||||
GET_FLOAT_WORD(ix,x);
|
||||
ix &= 0x7fffffff; /* ix = |x|'s high word*/
|
||||
if(ix<0x32000000) { /* if x < 2**27 */
|
||||
if(((int)x)==0) return one; /* generate inexact */
|
||||
}
|
||||
z = x*x;
|
||||
r = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6)))));
|
||||
if(ix < 0x3e99999a) /* if |x| < 0.3 */
|
||||
return one - ((float)0.5*z - (z*r - x*y));
|
||||
else {
|
||||
if(ix > 0x3f480000) { /* x > 0.78125 */
|
||||
qx = (float)0.28125;
|
||||
} else {
|
||||
SET_FLOAT_WORD(qx,ix-0x01000000); /* x/4 */
|
||||
}
|
||||
hz = (float)0.5*z-qx;
|
||||
a = one-qx;
|
||||
return a - (hz - (z*r-x*y));
|
||||
}
|
||||
}
|
||||
300
src/math/k_rem_pio2.c
Normal file
300
src/math/k_rem_pio2.c
Normal file
|
|
@ -0,0 +1,300 @@
|
|||
|
||||
/* @(#)k_rem_pio2.c 1.3 95/01/18 */
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunSoft, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/*
|
||||
* __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
|
||||
* double x[],y[]; int e0,nx,prec; int ipio2[];
|
||||
*
|
||||
* __kernel_rem_pio2 return the last three digits of N with
|
||||
* y = x - N*pi/2
|
||||
* so that |y| < pi/2.
|
||||
*
|
||||
* The method is to compute the integer (mod 8) and fraction parts of
|
||||
* (2/pi)*x without doing the full multiplication. In general we
|
||||
* skip the part of the product that are known to be a huge integer (
|
||||
* more accurately, = 0 mod 8 ). Thus the number of operations are
|
||||
* independent of the exponent of the input.
|
||||
*
|
||||
* (2/pi) is represented by an array of 24-bit integers in ipio2[].
|
||||
*
|
||||
* Input parameters:
|
||||
* x[] The input value (must be positive) is broken into nx
|
||||
* pieces of 24-bit integers in double precision format.
|
||||
* x[i] will be the i-th 24 bit of x. The scaled exponent
|
||||
* of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
|
||||
* match x's up to 24 bits.
|
||||
*
|
||||
* Example of breaking a double positive z into x[0]+x[1]+x[2]:
|
||||
* e0 = ilogb(z)-23
|
||||
* z = scalbn(z,-e0)
|
||||
* for i = 0,1,2
|
||||
* x[i] = floor(z)
|
||||
* z = (z-x[i])*2**24
|
||||
*
|
||||
*
|
||||
* y[] ouput result in an array of double precision numbers.
|
||||
* The dimension of y[] is:
|
||||
* 24-bit precision 1
|
||||
* 53-bit precision 2
|
||||
* 64-bit precision 2
|
||||
* 113-bit precision 3
|
||||
* The actual value is the sum of them. Thus for 113-bit
|
||||
* precison, one may have to do something like:
|
||||
*
|
||||
* long double t,w,r_head, r_tail;
|
||||
* t = (long double)y[2] + (long double)y[1];
|
||||
* w = (long double)y[0];
|
||||
* r_head = t+w;
|
||||
* r_tail = w - (r_head - t);
|
||||
*
|
||||
* e0 The exponent of x[0]
|
||||
*
|
||||
* nx dimension of x[]
|
||||
*
|
||||
* prec an integer indicating the precision:
|
||||
* 0 24 bits (single)
|
||||
* 1 53 bits (double)
|
||||
* 2 64 bits (extended)
|
||||
* 3 113 bits (quad)
|
||||
*
|
||||
* ipio2[]
|
||||
* integer array, contains the (24*i)-th to (24*i+23)-th
|
||||
* bit of 2/pi after binary point. The corresponding
|
||||
* floating value is
|
||||
*
|
||||
* ipio2[i] * 2^(-24(i+1)).
|
||||
*
|
||||
* External function:
|
||||
* double scalbn(), floor();
|
||||
*
|
||||
*
|
||||
* Here is the description of some local variables:
|
||||
*
|
||||
* jk jk+1 is the initial number of terms of ipio2[] needed
|
||||
* in the computation. The recommended value is 2,3,4,
|
||||
* 6 for single, double, extended,and quad.
|
||||
*
|
||||
* jz local integer variable indicating the number of
|
||||
* terms of ipio2[] used.
|
||||
*
|
||||
* jx nx - 1
|
||||
*
|
||||
* jv index for pointing to the suitable ipio2[] for the
|
||||
* computation. In general, we want
|
||||
* ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
|
||||
* is an integer. Thus
|
||||
* e0-3-24*jv >= 0 or (e0-3)/24 >= jv
|
||||
* Hence jv = max(0,(e0-3)/24).
|
||||
*
|
||||
* jp jp+1 is the number of terms in PIo2[] needed, jp = jk.
|
||||
*
|
||||
* q[] double array with integral value, representing the
|
||||
* 24-bits chunk of the product of x and 2/pi.
|
||||
*
|
||||
* q0 the corresponding exponent of q[0]. Note that the
|
||||
* exponent for q[i] would be q0-24*i.
|
||||
*
|
||||
* PIo2[] double precision array, obtained by cutting pi/2
|
||||
* into 24 bits chunks.
|
||||
*
|
||||
* f[] ipio2[] in floating point
|
||||
*
|
||||
* iq[] integer array by breaking up q[] in 24-bits chunk.
|
||||
*
|
||||
* fq[] final product of x*(2/pi) in fq[0],..,fq[jk]
|
||||
*
|
||||
* ih integer. If >0 it indicates q[] is >= 0.5, hence
|
||||
* it also indicates the *sign* of the result.
|
||||
*
|
||||
*/
|
||||
|
||||
|
||||
/*
|
||||
* Constants:
|
||||
* The hexadecimal values are the intended ones for the following
|
||||
* constants. The decimal values may be used, provided that the
|
||||
* compiler will convert from decimal to binary accurately enough
|
||||
* to produce the hexadecimal values shown.
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const int init_jk[] = {2,3,4,6}; /* initial value for jk */
|
||||
|
||||
static const double PIo2[] = {
|
||||
1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
|
||||
7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
|
||||
5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
|
||||
3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
|
||||
1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
|
||||
1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
|
||||
2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
|
||||
2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
|
||||
};
|
||||
|
||||
static const double
|
||||
zero = 0.0,
|
||||
one = 1.0,
|
||||
two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
|
||||
twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
|
||||
|
||||
int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int32_t *ipio2)
|
||||
{
|
||||
int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;
|
||||
double z,fw,f[20],fq[20],q[20];
|
||||
|
||||
/* initialize jk*/
|
||||
jk = init_jk[prec];
|
||||
jp = jk;
|
||||
|
||||
/* determine jx,jv,q0, note that 3>q0 */
|
||||
jx = nx-1;
|
||||
jv = (e0-3)/24; if(jv<0) jv=0;
|
||||
q0 = e0-24*(jv+1);
|
||||
|
||||
/* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
|
||||
j = jv-jx; m = jx+jk;
|
||||
for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j];
|
||||
|
||||
/* compute q[0],q[1],...q[jk] */
|
||||
for (i=0;i<=jk;i++) {
|
||||
for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw;
|
||||
}
|
||||
|
||||
jz = jk;
|
||||
recompute:
|
||||
/* distill q[] into iq[] reversingly */
|
||||
for(i=0,j=jz,z=q[jz];j>0;i++,j--) {
|
||||
fw = (double)((int32_t)(twon24* z));
|
||||
iq[i] = (int32_t)(z-two24*fw);
|
||||
z = q[j-1]+fw;
|
||||
}
|
||||
|
||||
/* compute n */
|
||||
z = scalbn(z,q0); /* actual value of z */
|
||||
z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */
|
||||
n = (int32_t) z;
|
||||
z -= (double)n;
|
||||
ih = 0;
|
||||
if(q0>0) { /* need iq[jz-1] to determine n */
|
||||
i = (iq[jz-1]>>(24-q0)); n += i;
|
||||
iq[jz-1] -= i<<(24-q0);
|
||||
ih = iq[jz-1]>>(23-q0);
|
||||
}
|
||||
else if(q0==0) ih = iq[jz-1]>>23;
|
||||
else if(z>=0.5) ih=2;
|
||||
|
||||
if(ih>0) { /* q > 0.5 */
|
||||
n += 1; carry = 0;
|
||||
for(i=0;i<jz ;i++) { /* compute 1-q */
|
||||
j = iq[i];
|
||||
if(carry==0) {
|
||||
if(j!=0) {
|
||||
carry = 1; iq[i] = 0x1000000- j;
|
||||
}
|
||||
} else iq[i] = 0xffffff - j;
|
||||
}
|
||||
if(q0>0) { /* rare case: chance is 1 in 12 */
|
||||
switch(q0) {
|
||||
case 1:
|
||||
iq[jz-1] &= 0x7fffff; break;
|
||||
case 2:
|
||||
iq[jz-1] &= 0x3fffff; break;
|
||||
}
|
||||
}
|
||||
if(ih==2) {
|
||||
z = one - z;
|
||||
if(carry!=0) z -= scalbn(one,q0);
|
||||
}
|
||||
}
|
||||
|
||||
/* check if recomputation is needed */
|
||||
if(z==zero) {
|
||||
j = 0;
|
||||
for (i=jz-1;i>=jk;i--) j |= iq[i];
|
||||
if(j==0) { /* need recomputation */
|
||||
for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */
|
||||
|
||||
for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */
|
||||
f[jx+i] = (double) ipio2[jv+i];
|
||||
for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
|
||||
q[i] = fw;
|
||||
}
|
||||
jz += k;
|
||||
goto recompute;
|
||||
}
|
||||
}
|
||||
|
||||
/* chop off zero terms */
|
||||
if(z==0.0) {
|
||||
jz -= 1; q0 -= 24;
|
||||
while(iq[jz]==0) { jz--; q0-=24;}
|
||||
} else { /* break z into 24-bit if necessary */
|
||||
z = scalbn(z,-q0);
|
||||
if(z>=two24) {
|
||||
fw = (double)((int32_t)(twon24*z));
|
||||
iq[jz] = (int32_t)(z-two24*fw);
|
||||
jz += 1; q0 += 24;
|
||||
iq[jz] = (int32_t) fw;
|
||||
} else iq[jz] = (int32_t) z ;
|
||||
}
|
||||
|
||||
/* convert integer "bit" chunk to floating-point value */
|
||||
fw = scalbn(one,q0);
|
||||
for(i=jz;i>=0;i--) {
|
||||
q[i] = fw*(double)iq[i]; fw*=twon24;
|
||||
}
|
||||
|
||||
/* compute PIo2[0,...,jp]*q[jz,...,0] */
|
||||
for(i=jz;i>=0;i--) {
|
||||
for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
|
||||
fq[jz-i] = fw;
|
||||
}
|
||||
|
||||
/* compress fq[] into y[] */
|
||||
switch(prec) {
|
||||
case 0:
|
||||
fw = 0.0;
|
||||
for (i=jz;i>=0;i--) fw += fq[i];
|
||||
y[0] = (ih==0)? fw: -fw;
|
||||
break;
|
||||
case 1:
|
||||
case 2:
|
||||
fw = 0.0;
|
||||
for (i=jz;i>=0;i--) fw += fq[i];
|
||||
y[0] = (ih==0)? fw: -fw;
|
||||
fw = fq[0]-fw;
|
||||
for (i=1;i<=jz;i++) fw += fq[i];
|
||||
y[1] = (ih==0)? fw: -fw;
|
||||
break;
|
||||
case 3: /* painful */
|
||||
for (i=jz;i>0;i--) {
|
||||
fw = fq[i-1]+fq[i];
|
||||
fq[i] += fq[i-1]-fw;
|
||||
fq[i-1] = fw;
|
||||
}
|
||||
for (i=jz;i>1;i--) {
|
||||
fw = fq[i-1]+fq[i];
|
||||
fq[i] += fq[i-1]-fw;
|
||||
fq[i-1] = fw;
|
||||
}
|
||||
for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];
|
||||
if(ih==0) {
|
||||
y[0] = fq[0]; y[1] = fq[1]; y[2] = fw;
|
||||
} else {
|
||||
y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
|
||||
}
|
||||
}
|
||||
return n&7;
|
||||
}
|
||||
192
src/math/k_rem_pio2f.c
Normal file
192
src/math/k_rem_pio2f.c
Normal file
|
|
@ -0,0 +1,192 @@
|
|||
/* k_rem_pio2f.c -- float version of k_rem_pio2.c
|
||||
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
||||
*/
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
/* In the float version, the input parameter x contains 8 bit
|
||||
integers, not 24 bit integers. 113 bit precision is not supported. */
|
||||
|
||||
static const int init_jk[] = {4,7,9}; /* initial value for jk */
|
||||
|
||||
static const float PIo2[] = {
|
||||
1.5703125000e+00, /* 0x3fc90000 */
|
||||
4.5776367188e-04, /* 0x39f00000 */
|
||||
2.5987625122e-05, /* 0x37da0000 */
|
||||
7.5437128544e-08, /* 0x33a20000 */
|
||||
6.0026650317e-11, /* 0x2e840000 */
|
||||
7.3896444519e-13, /* 0x2b500000 */
|
||||
5.3845816694e-15, /* 0x27c20000 */
|
||||
5.6378512969e-18, /* 0x22d00000 */
|
||||
8.3009228831e-20, /* 0x1fc40000 */
|
||||
3.2756352257e-22, /* 0x1bc60000 */
|
||||
6.3331015649e-25, /* 0x17440000 */
|
||||
};
|
||||
|
||||
static const float
|
||||
zero = 0.0,
|
||||
one = 1.0,
|
||||
two8 = 2.5600000000e+02, /* 0x43800000 */
|
||||
twon8 = 3.9062500000e-03; /* 0x3b800000 */
|
||||
|
||||
int __kernel_rem_pio2f(float *x, float *y, int e0, int nx, int prec, const int32_t *ipio2)
|
||||
{
|
||||
int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;
|
||||
float z,fw,f[20],fq[20],q[20];
|
||||
|
||||
/* initialize jk*/
|
||||
jk = init_jk[prec];
|
||||
jp = jk;
|
||||
|
||||
/* determine jx,jv,q0, note that 3>q0 */
|
||||
jx = nx-1;
|
||||
jv = (e0-3)/8; if(jv<0) jv=0;
|
||||
q0 = e0-8*(jv+1);
|
||||
|
||||
/* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
|
||||
j = jv-jx; m = jx+jk;
|
||||
for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (float) ipio2[j];
|
||||
|
||||
/* compute q[0],q[1],...q[jk] */
|
||||
for (i=0;i<=jk;i++) {
|
||||
for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw;
|
||||
}
|
||||
|
||||
jz = jk;
|
||||
recompute:
|
||||
/* distill q[] into iq[] reversingly */
|
||||
for(i=0,j=jz,z=q[jz];j>0;i++,j--) {
|
||||
fw = (float)((int32_t)(twon8* z));
|
||||
iq[i] = (int32_t)(z-two8*fw);
|
||||
z = q[j-1]+fw;
|
||||
}
|
||||
|
||||
/* compute n */
|
||||
z = scalbnf(z,q0); /* actual value of z */
|
||||
z -= (float)8.0*floorf(z*(float)0.125); /* trim off integer >= 8 */
|
||||
n = (int32_t) z;
|
||||
z -= (float)n;
|
||||
ih = 0;
|
||||
if(q0>0) { /* need iq[jz-1] to determine n */
|
||||
i = (iq[jz-1]>>(8-q0)); n += i;
|
||||
iq[jz-1] -= i<<(8-q0);
|
||||
ih = iq[jz-1]>>(7-q0);
|
||||
}
|
||||
else if(q0==0) ih = iq[jz-1]>>7;
|
||||
else if(z>=(float)0.5) ih=2;
|
||||
|
||||
if(ih>0) { /* q > 0.5 */
|
||||
n += 1; carry = 0;
|
||||
for(i=0;i<jz ;i++) { /* compute 1-q */
|
||||
j = iq[i];
|
||||
if(carry==0) {
|
||||
if(j!=0) {
|
||||
carry = 1; iq[i] = 0x100- j;
|
||||
}
|
||||
} else iq[i] = 0xff - j;
|
||||
}
|
||||
if(q0>0) { /* rare case: chance is 1 in 12 */
|
||||
switch(q0) {
|
||||
case 1:
|
||||
iq[jz-1] &= 0x7f; break;
|
||||
case 2:
|
||||
iq[jz-1] &= 0x3f; break;
|
||||
}
|
||||
}
|
||||
if(ih==2) {
|
||||
z = one - z;
|
||||
if(carry!=0) z -= scalbnf(one,q0);
|
||||
}
|
||||
}
|
||||
|
||||
/* check if recomputation is needed */
|
||||
if(z==zero) {
|
||||
j = 0;
|
||||
for (i=jz-1;i>=jk;i--) j |= iq[i];
|
||||
if(j==0) { /* need recomputation */
|
||||
for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */
|
||||
|
||||
for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */
|
||||
f[jx+i] = (float) ipio2[jv+i];
|
||||
for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
|
||||
q[i] = fw;
|
||||
}
|
||||
jz += k;
|
||||
goto recompute;
|
||||
}
|
||||
}
|
||||
|
||||
/* chop off zero terms */
|
||||
if(z==(float)0.0) {
|
||||
jz -= 1; q0 -= 8;
|
||||
while(iq[jz]==0) { jz--; q0-=8;}
|
||||
} else { /* break z into 8-bit if necessary */
|
||||
z = scalbnf(z,-q0);
|
||||
if(z>=two8) {
|
||||
fw = (float)((int32_t)(twon8*z));
|
||||
iq[jz] = (int32_t)(z-two8*fw);
|
||||
jz += 1; q0 += 8;
|
||||
iq[jz] = (int32_t) fw;
|
||||
} else iq[jz] = (int32_t) z ;
|
||||
}
|
||||
|
||||
/* convert integer "bit" chunk to floating-point value */
|
||||
fw = scalbnf(one,q0);
|
||||
for(i=jz;i>=0;i--) {
|
||||
q[i] = fw*(float)iq[i]; fw*=twon8;
|
||||
}
|
||||
|
||||
/* compute PIo2[0,...,jp]*q[jz,...,0] */
|
||||
for(i=jz;i>=0;i--) {
|
||||
for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
|
||||
fq[jz-i] = fw;
|
||||
}
|
||||
|
||||
/* compress fq[] into y[] */
|
||||
switch(prec) {
|
||||
case 0:
|
||||
fw = 0.0;
|
||||
for (i=jz;i>=0;i--) fw += fq[i];
|
||||
y[0] = (ih==0)? fw: -fw;
|
||||
break;
|
||||
case 1:
|
||||
case 2:
|
||||
fw = 0.0;
|
||||
for (i=jz;i>=0;i--) fw += fq[i];
|
||||
y[0] = (ih==0)? fw: -fw;
|
||||
fw = fq[0]-fw;
|
||||
for (i=1;i<=jz;i++) fw += fq[i];
|
||||
y[1] = (ih==0)? fw: -fw;
|
||||
break;
|
||||
case 3: /* painful */
|
||||
for (i=jz;i>0;i--) {
|
||||
fw = fq[i-1]+fq[i];
|
||||
fq[i] += fq[i-1]-fw;
|
||||
fq[i-1] = fw;
|
||||
}
|
||||
for (i=jz;i>1;i--) {
|
||||
fw = fq[i-1]+fq[i];
|
||||
fq[i] += fq[i-1]-fw;
|
||||
fq[i-1] = fw;
|
||||
}
|
||||
for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];
|
||||
if(ih==0) {
|
||||
y[0] = fq[0]; y[1] = fq[1]; y[2] = fw;
|
||||
} else {
|
||||
y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
|
||||
}
|
||||
}
|
||||
return n&7;
|
||||
}
|
||||
68
src/math/k_sin.c
Normal file
68
src/math/k_sin.c
Normal file
|
|
@ -0,0 +1,68 @@
|
|||
|
||||
/* @(#)k_sin.c 1.3 95/01/18 */
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunSoft, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/* __kernel_sin( x, y, iy)
|
||||
* kernel sin function on [-pi/4, pi/4], pi/4 ~ 0.7854
|
||||
* Input x is assumed to be bounded by ~pi/4 in magnitude.
|
||||
* Input y is the tail of x.
|
||||
* Input iy indicates whether y is 0. (if iy=0, y assume to be 0).
|
||||
*
|
||||
* Algorithm
|
||||
* 1. Since sin(-x) = -sin(x), we need only to consider positive x.
|
||||
* 2. if x < 2^-27 (hx<0x3e400000 0), return x with inexact if x!=0.
|
||||
* 3. sin(x) is approximated by a polynomial of degree 13 on
|
||||
* [0,pi/4]
|
||||
* 3 13
|
||||
* sin(x) ~ x + S1*x + ... + S6*x
|
||||
* where
|
||||
*
|
||||
* |sin(x) 2 4 6 8 10 12 | -58
|
||||
* |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x +S6*x )| <= 2
|
||||
* | x |
|
||||
*
|
||||
* 4. sin(x+y) = sin(x) + sin'(x')*y
|
||||
* ~ sin(x) + (1-x*x/2)*y
|
||||
* For better accuracy, let
|
||||
* 3 2 2 2 2
|
||||
* r = x *(S2+x *(S3+x *(S4+x *(S5+x *S6))))
|
||||
* then 3 2
|
||||
* sin(x) = x + (S1*x + (x *(r-y/2)+y))
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const double
|
||||
half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
|
||||
S1 = -1.66666666666666324348e-01, /* 0xBFC55555, 0x55555549 */
|
||||
S2 = 8.33333333332248946124e-03, /* 0x3F811111, 0x1110F8A6 */
|
||||
S3 = -1.98412698298579493134e-04, /* 0xBF2A01A0, 0x19C161D5 */
|
||||
S4 = 2.75573137070700676789e-06, /* 0x3EC71DE3, 0x57B1FE7D */
|
||||
S5 = -2.50507602534068634195e-08, /* 0xBE5AE5E6, 0x8A2B9CEB */
|
||||
S6 = 1.58969099521155010221e-10; /* 0x3DE5D93A, 0x5ACFD57C */
|
||||
|
||||
double
|
||||
__kernel_sin(double x, double y, int iy)
|
||||
{
|
||||
double z,r,v;
|
||||
int32_t ix;
|
||||
GET_HIGH_WORD(ix,x);
|
||||
ix &= 0x7fffffff; /* high word of x */
|
||||
if(ix<0x3e400000) /* |x| < 2**-27 */
|
||||
{if((int)x==0) return x;} /* generate inexact */
|
||||
z = x*x;
|
||||
v = z*x;
|
||||
r = S2+z*(S3+z*(S4+z*(S5+z*S6)));
|
||||
if(iy==0) return x+v*(S1+z*r);
|
||||
else return x-((z*(half*y-v*r)-y)-v*S1);
|
||||
}
|
||||
42
src/math/k_sinf.c
Normal file
42
src/math/k_sinf.c
Normal file
|
|
@ -0,0 +1,42 @@
|
|||
/* k_sinf.c -- float version of k_sin.c
|
||||
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
||||
*/
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const float
|
||||
half = 5.0000000000e-01,/* 0x3f000000 */
|
||||
S1 = -1.6666667163e-01, /* 0xbe2aaaab */
|
||||
S2 = 8.3333337680e-03, /* 0x3c088889 */
|
||||
S3 = -1.9841270114e-04, /* 0xb9500d01 */
|
||||
S4 = 2.7557314297e-06, /* 0x3638ef1b */
|
||||
S5 = -2.5050759689e-08, /* 0xb2d72f34 */
|
||||
S6 = 1.5896910177e-10; /* 0x2f2ec9d3 */
|
||||
|
||||
float
|
||||
__kernel_sinf(float x, float y, int iy)
|
||||
{
|
||||
float z,r,v;
|
||||
int32_t ix;
|
||||
GET_FLOAT_WORD(ix,x);
|
||||
ix &= 0x7fffffff; /* high word of x */
|
||||
if(ix<0x32000000) /* |x| < 2**-27 */
|
||||
{if((int)x==0) return x;} /* generate inexact */
|
||||
z = x*x;
|
||||
v = z*x;
|
||||
r = S2+z*(S3+z*(S4+z*(S5+z*S6)));
|
||||
if(iy==0) return x+v*(S1+z*r);
|
||||
else return x-((z*(half*y-v*r)-y)-v*S1);
|
||||
}
|
||||
149
src/math/k_tan.c
Normal file
149
src/math/k_tan.c
Normal file
|
|
@ -0,0 +1,149 @@
|
|||
/* @(#)k_tan.c 1.5 04/04/22 SMI */
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright 2004 Sun Microsystems, Inc. All Rights Reserved.
|
||||
*
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/* __kernel_tan( x, y, k )
|
||||
* kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
|
||||
* Input x is assumed to be bounded by ~pi/4 in magnitude.
|
||||
* Input y is the tail of x.
|
||||
* Input k indicates whether tan (if k = 1) or -1/tan (if k = -1) is returned.
|
||||
*
|
||||
* Algorithm
|
||||
* 1. Since tan(-x) = -tan(x), we need only to consider positive x.
|
||||
* 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
|
||||
* 3. tan(x) is approximated by a odd polynomial of degree 27 on
|
||||
* [0,0.67434]
|
||||
* 3 27
|
||||
* tan(x) ~ x + T1*x + ... + T13*x
|
||||
* where
|
||||
*
|
||||
* |tan(x) 2 4 26 | -59.2
|
||||
* |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2
|
||||
* | x |
|
||||
*
|
||||
* Note: tan(x+y) = tan(x) + tan'(x)*y
|
||||
* ~ tan(x) + (1+x*x)*y
|
||||
* Therefore, for better accuracy in computing tan(x+y), let
|
||||
* 3 2 2 2 2
|
||||
* r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
|
||||
* then
|
||||
* 3 2
|
||||
* tan(x+y) = x + (T1*x + (x *(r+y)+y))
|
||||
*
|
||||
* 4. For x in [0.67434,pi/4], let y = pi/4 - x, then
|
||||
* tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
|
||||
* = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
static const double xxx[] = {
|
||||
3.33333333333334091986e-01, /* 3FD55555, 55555563 */
|
||||
1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */
|
||||
5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */
|
||||
2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */
|
||||
8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */
|
||||
3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */
|
||||
1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */
|
||||
5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */
|
||||
2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */
|
||||
7.81794442939557092300e-05, /* 3F147E88, A03792A6 */
|
||||
7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */
|
||||
-1.85586374855275456654e-05, /* BEF375CB, DB605373 */
|
||||
2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */
|
||||
/* one */ 1.00000000000000000000e+00, /* 3FF00000, 00000000 */
|
||||
/* pio4 */ 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */
|
||||
/* pio4lo */ 3.06161699786838301793e-17 /* 3C81A626, 33145C07 */
|
||||
};
|
||||
#define one xxx[13]
|
||||
#define pio4 xxx[14]
|
||||
#define pio4lo xxx[15]
|
||||
#define T xxx
|
||||
/* INDENT ON */
|
||||
|
||||
double
|
||||
__kernel_tan(double x, double y, int iy) {
|
||||
double z, r, v, w, s;
|
||||
int32_t ix, hx;
|
||||
|
||||
GET_HIGH_WORD(hx,x);
|
||||
ix = hx & 0x7fffffff; /* high word of |x| */
|
||||
if (ix < 0x3e300000) { /* x < 2**-28 */
|
||||
if ((int) x == 0) { /* generate inexact */
|
||||
uint32_t low;
|
||||
GET_LOW_WORD(low,x);
|
||||
if (((ix | low) | (iy + 1)) == 0)
|
||||
return one / fabs(x);
|
||||
else {
|
||||
if (iy == 1)
|
||||
return x;
|
||||
else { /* compute -1 / (x+y) carefully */
|
||||
double a, t;
|
||||
|
||||
z = w = x + y;
|
||||
SET_LOW_WORD(z, 0);
|
||||
v = y - (z - x);
|
||||
t = a = -one / w;
|
||||
SET_LOW_WORD(t, 0);
|
||||
s = one + t * z;
|
||||
return t + a * (s + t * v);
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
if (ix >= 0x3FE59428) { /* |x| >= 0.6744 */
|
||||
if (hx < 0) {
|
||||
x = -x;
|
||||
y = -y;
|
||||
}
|
||||
z = pio4 - x;
|
||||
w = pio4lo - y;
|
||||
x = z + w;
|
||||
y = 0.0;
|
||||
}
|
||||
z = x * x;
|
||||
w = z * z;
|
||||
/*
|
||||
* Break x^5*(T[1]+x^2*T[2]+...) into
|
||||
* x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
|
||||
* x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
|
||||
*/
|
||||
r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] +
|
||||
w * T[11]))));
|
||||
v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] +
|
||||
w * T[12])))));
|
||||
s = z * x;
|
||||
r = y + z * (s * (r + v) + y);
|
||||
r += T[0] * s;
|
||||
w = x + r;
|
||||
if (ix >= 0x3FE59428) {
|
||||
v = (double) iy;
|
||||
return (double) (1 - ((hx >> 30) & 2)) *
|
||||
(v - 2.0 * (x - (w * w / (w + v) - r)));
|
||||
}
|
||||
if (iy == 1)
|
||||
return w;
|
||||
else {
|
||||
/*
|
||||
* if allow error up to 2 ulp, simply return
|
||||
* -1.0 / (x+r) here
|
||||
*/
|
||||
/* compute -1.0 / (x+r) accurately */
|
||||
double a, t;
|
||||
z = w;
|
||||
SET_LOW_WORD(z,0);
|
||||
v = r - (z - x); /* z+v = r+x */
|
||||
t = a = -1.0 / w; /* a = -1.0/w */
|
||||
SET_LOW_WORD(t,0);
|
||||
s = 1.0 + t * z;
|
||||
return t + a * (s + t * v);
|
||||
}
|
||||
}
|
||||
105
src/math/k_tanf.c
Normal file
105
src/math/k_tanf.c
Normal file
|
|
@ -0,0 +1,105 @@
|
|||
/* k_tanf.c -- float version of k_tan.c
|
||||
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
||||
*/
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright 2004 Sun Microsystems, Inc. All Rights Reserved.
|
||||
*
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
static const float
|
||||
one = 1.0000000000e+00, /* 0x3f800000 */
|
||||
pio4 = 7.8539812565e-01, /* 0x3f490fda */
|
||||
pio4lo= 3.7748947079e-08, /* 0x33222168 */
|
||||
T[] = {
|
||||
3.3333334327e-01, /* 0x3eaaaaab */
|
||||
1.3333334029e-01, /* 0x3e088889 */
|
||||
5.3968254477e-02, /* 0x3d5d0dd1 */
|
||||
2.1869488060e-02, /* 0x3cb327a4 */
|
||||
8.8632395491e-03, /* 0x3c11371f */
|
||||
3.5920790397e-03, /* 0x3b6b6916 */
|
||||
1.4562094584e-03, /* 0x3abede48 */
|
||||
5.8804126456e-04, /* 0x3a1a26c8 */
|
||||
2.4646313977e-04, /* 0x398137b9 */
|
||||
7.8179444245e-05, /* 0x38a3f445 */
|
||||
7.1407252108e-05, /* 0x3895c07a */
|
||||
-1.8558637748e-05, /* 0xb79bae5f */
|
||||
2.5907305826e-05, /* 0x37d95384 */
|
||||
};
|
||||
|
||||
float
|
||||
__kernel_tanf(float x, float y, int iy)
|
||||
{
|
||||
float z,r,v,w,s;
|
||||
int32_t ix,hx;
|
||||
GET_FLOAT_WORD(hx,x);
|
||||
ix = hx&0x7fffffff; /* high word of |x| */
|
||||
if(ix<0x31800000) { /* x < 2**-28 */
|
||||
if ((int) x == 0) { /* generate inexact */
|
||||
if ((ix | (iy + 1)) == 0)
|
||||
return one / fabsf(x);
|
||||
else {
|
||||
if (iy == 1)
|
||||
return x;
|
||||
else { /* compute -1 / (x+y) carefully */
|
||||
double a, t;
|
||||
|
||||
z = w = x + y;
|
||||
GET_FLOAT_WORD(ix, z);
|
||||
SET_FLOAT_WORD(z, ix & 0xfffff000);
|
||||
v = y - (z - x);
|
||||
t = a = -one / w;
|
||||
GET_FLOAT_WORD(ix, t);
|
||||
SET_FLOAT_WORD(t, ix & 0xfffff000);
|
||||
s = one + t * z;
|
||||
return t + a * (s + t * v);
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
if(ix>=0x3f2ca140) { /* |x|>=0.6744 */
|
||||
if(hx<0) {x = -x; y = -y;}
|
||||
z = pio4-x;
|
||||
w = pio4lo-y;
|
||||
x = z+w; y = 0.0;
|
||||
}
|
||||
z = x*x;
|
||||
w = z*z;
|
||||
/* Break x^5*(T[1]+x^2*T[2]+...) into
|
||||
* x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
|
||||
* x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
|
||||
*/
|
||||
r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11]))));
|
||||
v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12])))));
|
||||
s = z*x;
|
||||
r = y + z*(s*(r+v)+y);
|
||||
r += T[0]*s;
|
||||
w = x+r;
|
||||
if(ix>=0x3f2ca140) {
|
||||
v = (float)iy;
|
||||
return (float)(1-((hx>>30)&2))*(v-(float)2.0*(x-(w*w/(w+v)-r)));
|
||||
}
|
||||
if(iy==1) return w;
|
||||
else { /* if allow error up to 2 ulp,
|
||||
simply return -1.0/(x+r) here */
|
||||
/* compute -1.0/(x+r) accurately */
|
||||
float a,t;
|
||||
int32_t i;
|
||||
z = w;
|
||||
GET_FLOAT_WORD(i,z);
|
||||
SET_FLOAT_WORD(z,i&0xfffff000);
|
||||
v = r-(z - x); /* z+v = r+x */
|
||||
t = a = -(float)1.0/w; /* a = -1.0/w */
|
||||
GET_FLOAT_WORD(i,t);
|
||||
SET_FLOAT_WORD(t,i&0xfffff000);
|
||||
s = (float)1.0+t*z;
|
||||
return t+a*(s+t*v);
|
||||
}
|
||||
}
|
||||
143
src/math/math_private.h
Normal file
143
src/math/math_private.h
Normal file
|
|
@ -0,0 +1,143 @@
|
|||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
#ifndef _MATH_PRIVATE_H_
|
||||
#define _MATH_PRIVATE_H_
|
||||
|
||||
#include <inttypes.h>
|
||||
|
||||
/*
|
||||
* The original fdlibm code used statements like:
|
||||
* n0 = ((*(int*)&one)>>29)^1; * index of high word *
|
||||
* ix0 = *(n0+(int*)&x); * high word of x *
|
||||
* ix1 = *((1-n0)+(int*)&x); * low word of x *
|
||||
* to dig two 32 bit words out of the 64 bit IEEE floating point
|
||||
* value. That is non-ANSI, and, moreover, the gcc instruction
|
||||
* scheduler gets it wrong. We instead use the following macros.
|
||||
* Unlike the original code, we determine the endianness at compile
|
||||
* time, not at run time; I don't see much benefit to selecting
|
||||
* endianness at run time.
|
||||
*/
|
||||
|
||||
/*
|
||||
* A union which permits us to convert between a double and two 32 bit
|
||||
* ints.
|
||||
*/
|
||||
|
||||
typedef union
|
||||
{
|
||||
double value;
|
||||
uint64_t words;
|
||||
} ieee_double_shape_type;
|
||||
|
||||
/* Get two 32 bit ints from a double. */
|
||||
|
||||
#define EXTRACT_WORDS(ix0,ix1,d) \
|
||||
do { \
|
||||
ieee_double_shape_type ew_u; \
|
||||
ew_u.value = (d); \
|
||||
(ix0) = ew_u.words >> 32; \
|
||||
(ix1) = (uint32_t)ew_u.words; \
|
||||
} while (0)
|
||||
|
||||
/* Get the more significant 32 bit int from a double. */
|
||||
|
||||
#define GET_HIGH_WORD(i,d) \
|
||||
do { \
|
||||
ieee_double_shape_type gh_u; \
|
||||
gh_u.value = (d); \
|
||||
(i) = gh_u.words >> 32; \
|
||||
} while (0)
|
||||
|
||||
/* Get the less significant 32 bit int from a double. */
|
||||
|
||||
#define GET_LOW_WORD(i,d) \
|
||||
do { \
|
||||
ieee_double_shape_type gl_u; \
|
||||
gl_u.value = (d); \
|
||||
(i) = (uint32_t)gl_u.words; \
|
||||
} while (0)
|
||||
|
||||
/* Set a double from two 32 bit ints. */
|
||||
|
||||
#define INSERT_WORDS(d,ix0,ix1) \
|
||||
do { \
|
||||
ieee_double_shape_type iw_u; \
|
||||
iw_u.words = ((uint64_t)(ix0) << 32) | (ix1); \
|
||||
(d) = iw_u.value; \
|
||||
} while (0)
|
||||
|
||||
/* Set the more significant 32 bits of a double from an int. */
|
||||
|
||||
#define SET_HIGH_WORD(d,v) \
|
||||
do { \
|
||||
ieee_double_shape_type sh_u; \
|
||||
sh_u.value = (d); \
|
||||
sh_u.words &= 0xffffffff; \
|
||||
sh_u.words |= ((uint64_t)(v) << 32); \
|
||||
(d) = sh_u.value; \
|
||||
} while (0)
|
||||
|
||||
/* Set the less significant 32 bits of a double from an int. */
|
||||
|
||||
#define SET_LOW_WORD(d,v) \
|
||||
do { \
|
||||
ieee_double_shape_type sl_u; \
|
||||
sl_u.value = (d); \
|
||||
sl_u.words &= 0xffffffff00000000ull; \
|
||||
sl_u.words |= (uint32_t)(v); \
|
||||
(d) = sl_u.value; \
|
||||
} while (0)
|
||||
|
||||
/*
|
||||
* A union which permits us to convert between a float and a 32 bit
|
||||
* int.
|
||||
*/
|
||||
|
||||
typedef union
|
||||
{
|
||||
float value;
|
||||
uint32_t word;
|
||||
} ieee_float_shape_type;
|
||||
|
||||
/* Get a 32 bit int from a float. */
|
||||
|
||||
#define GET_FLOAT_WORD(i,d) \
|
||||
do { \
|
||||
ieee_float_shape_type gf_u; \
|
||||
gf_u.value = (d); \
|
||||
(i) = gf_u.word; \
|
||||
} while (0)
|
||||
|
||||
/* Set a float from a 32 bit int. */
|
||||
|
||||
#define SET_FLOAT_WORD(d,i) \
|
||||
do { \
|
||||
ieee_float_shape_type sf_u; \
|
||||
sf_u.word = (i); \
|
||||
(d) = sf_u.value; \
|
||||
} while (0)
|
||||
|
||||
/* fdlibm kernel function */
|
||||
int __ieee754_rem_pio2(double,double*);
|
||||
double __kernel_sin(double,double,int);
|
||||
double __kernel_cos(double,double);
|
||||
double __kernel_tan(double,double,int);
|
||||
int __kernel_rem_pio2(double*,double*,int,int,int,const int*);
|
||||
|
||||
/* float versions of fdlibm kernel functions */
|
||||
int __ieee754_rem_pio2f(float,float*);
|
||||
float __kernel_sinf(float,float,int);
|
||||
float __kernel_cosf(float,float);
|
||||
float __kernel_tanf(float,float,int);
|
||||
int __kernel_rem_pio2f(float*,float*,int,int,int,const int*);
|
||||
|
||||
#endif /* !_MATH_PRIVATE_H_ */
|
||||
53
src/math/s_asinh.c
Normal file
53
src/math/s_asinh.c
Normal file
|
|
@ -0,0 +1,53 @@
|
|||
/* @(#)s_asinh.c 5.1 93/09/24 */
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/* asinh(x)
|
||||
* Method :
|
||||
* Based on
|
||||
* asinh(x) = sign(x) * log [ |x| + sqrt(x*x+1) ]
|
||||
* we have
|
||||
* asinh(x) := x if 1+x*x=1,
|
||||
* := sign(x)*(log(x)+ln2)) for large |x|, else
|
||||
* := sign(x)*log(2|x|+1/(|x|+sqrt(x*x+1))) if|x|>2, else
|
||||
* := sign(x)*log1p(|x| + x^2/(1 + sqrt(1+x^2)))
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const double
|
||||
one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
|
||||
ln2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
|
||||
huge= 1.00000000000000000000e+300;
|
||||
|
||||
double
|
||||
asinh(double x)
|
||||
{
|
||||
double t,w;
|
||||
int32_t hx,ix;
|
||||
GET_HIGH_WORD(hx,x);
|
||||
ix = hx&0x7fffffff;
|
||||
if(ix>=0x7ff00000) return x+x; /* x is inf or NaN */
|
||||
if(ix< 0x3e300000) { /* |x|<2**-28 */
|
||||
if(huge+x>one) return x; /* return x inexact except 0 */
|
||||
}
|
||||
if(ix>0x41b00000) { /* |x| > 2**28 */
|
||||
w = log(fabs(x))+ln2;
|
||||
} else if (ix>0x40000000) { /* 2**28 > |x| > 2.0 */
|
||||
t = fabs(x);
|
||||
w = log(2.0*t+one/(sqrt(x*x+one)+t));
|
||||
} else { /* 2.0 > |x| > 2**-28 */
|
||||
t = x*x;
|
||||
w =log1p(fabs(x)+t/(one+sqrt(one+t)));
|
||||
}
|
||||
if(hx>0) return w; else return -w;
|
||||
}
|
||||
45
src/math/s_asinhf.c
Normal file
45
src/math/s_asinhf.c
Normal file
|
|
@ -0,0 +1,45 @@
|
|||
/* s_asinhf.c -- float version of s_asinh.c.
|
||||
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
||||
*/
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const float
|
||||
one = 1.0000000000e+00, /* 0x3F800000 */
|
||||
ln2 = 6.9314718246e-01, /* 0x3f317218 */
|
||||
huge= 1.0000000000e+30;
|
||||
|
||||
float
|
||||
asinhf(float x)
|
||||
{
|
||||
float t,w;
|
||||
int32_t hx,ix;
|
||||
GET_FLOAT_WORD(hx,x);
|
||||
ix = hx&0x7fffffff;
|
||||
if(ix>=0x7f800000) return x+x; /* x is inf or NaN */
|
||||
if(ix< 0x31800000) { /* |x|<2**-28 */
|
||||
if(huge+x>one) return x; /* return x inexact except 0 */
|
||||
}
|
||||
if(ix>0x4d800000) { /* |x| > 2**28 */
|
||||
w = logf(fabsf(x))+ln2;
|
||||
} else if (ix>0x40000000) { /* 2**28 > |x| > 2.0 */
|
||||
t = fabsf(x);
|
||||
w = logf((float)2.0*t+one/(sqrtf(x*x+one)+t));
|
||||
} else { /* 2.0 > |x| > 2**-28 */
|
||||
t = x*x;
|
||||
w =log1pf(fabsf(x)+t/(one+sqrtf(one+t)));
|
||||
}
|
||||
if(hx>0) return w; else return -w;
|
||||
}
|
||||
115
src/math/s_atan.c
Normal file
115
src/math/s_atan.c
Normal file
|
|
@ -0,0 +1,115 @@
|
|||
/* @(#)s_atan.c 5.1 93/09/24 */
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/* atan(x)
|
||||
* Method
|
||||
* 1. Reduce x to positive by atan(x) = -atan(-x).
|
||||
* 2. According to the integer k=4t+0.25 chopped, t=x, the argument
|
||||
* is further reduced to one of the following intervals and the
|
||||
* arctangent of t is evaluated by the corresponding formula:
|
||||
*
|
||||
* [0,7/16] atan(x) = t-t^3*(a1+t^2*(a2+...(a10+t^2*a11)...)
|
||||
* [7/16,11/16] atan(x) = atan(1/2) + atan( (t-0.5)/(1+t/2) )
|
||||
* [11/16.19/16] atan(x) = atan( 1 ) + atan( (t-1)/(1+t) )
|
||||
* [19/16,39/16] atan(x) = atan(3/2) + atan( (t-1.5)/(1+1.5t) )
|
||||
* [39/16,INF] atan(x) = atan(INF) + atan( -1/t )
|
||||
*
|
||||
* Constants:
|
||||
* The hexadecimal values are the intended ones for the following
|
||||
* constants. The decimal values may be used, provided that the
|
||||
* compiler will convert from decimal to binary accurately enough
|
||||
* to produce the hexadecimal values shown.
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const double atanhi[] = {
|
||||
4.63647609000806093515e-01, /* atan(0.5)hi 0x3FDDAC67, 0x0561BB4F */
|
||||
7.85398163397448278999e-01, /* atan(1.0)hi 0x3FE921FB, 0x54442D18 */
|
||||
9.82793723247329054082e-01, /* atan(1.5)hi 0x3FEF730B, 0xD281F69B */
|
||||
1.57079632679489655800e+00, /* atan(inf)hi 0x3FF921FB, 0x54442D18 */
|
||||
};
|
||||
|
||||
static const double atanlo[] = {
|
||||
2.26987774529616870924e-17, /* atan(0.5)lo 0x3C7A2B7F, 0x222F65E2 */
|
||||
3.06161699786838301793e-17, /* atan(1.0)lo 0x3C81A626, 0x33145C07 */
|
||||
1.39033110312309984516e-17, /* atan(1.5)lo 0x3C700788, 0x7AF0CBBD */
|
||||
6.12323399573676603587e-17, /* atan(inf)lo 0x3C91A626, 0x33145C07 */
|
||||
};
|
||||
|
||||
static const double aT[] = {
|
||||
3.33333333333329318027e-01, /* 0x3FD55555, 0x5555550D */
|
||||
-1.99999999998764832476e-01, /* 0xBFC99999, 0x9998EBC4 */
|
||||
1.42857142725034663711e-01, /* 0x3FC24924, 0x920083FF */
|
||||
-1.11111104054623557880e-01, /* 0xBFBC71C6, 0xFE231671 */
|
||||
9.09088713343650656196e-02, /* 0x3FB745CD, 0xC54C206E */
|
||||
-7.69187620504482999495e-02, /* 0xBFB3B0F2, 0xAF749A6D */
|
||||
6.66107313738753120669e-02, /* 0x3FB10D66, 0xA0D03D51 */
|
||||
-5.83357013379057348645e-02, /* 0xBFADDE2D, 0x52DEFD9A */
|
||||
4.97687799461593236017e-02, /* 0x3FA97B4B, 0x24760DEB */
|
||||
-3.65315727442169155270e-02, /* 0xBFA2B444, 0x2C6A6C2F */
|
||||
1.62858201153657823623e-02, /* 0x3F90AD3A, 0xE322DA11 */
|
||||
};
|
||||
|
||||
static const double
|
||||
one = 1.0,
|
||||
huge = 1.0e300;
|
||||
|
||||
double
|
||||
atan(double x)
|
||||
{
|
||||
double w,s1,s2,z;
|
||||
int32_t ix,hx,id;
|
||||
|
||||
GET_HIGH_WORD(hx,x);
|
||||
ix = hx&0x7fffffff;
|
||||
if(ix>=0x44100000) { /* if |x| >= 2^66 */
|
||||
uint32_t low;
|
||||
GET_LOW_WORD(low,x);
|
||||
if(ix>0x7ff00000||
|
||||
(ix==0x7ff00000&&(low!=0)))
|
||||
return x+x; /* NaN */
|
||||
if(hx>0) return atanhi[3]+atanlo[3];
|
||||
else return -atanhi[3]-atanlo[3];
|
||||
} if (ix < 0x3fdc0000) { /* |x| < 0.4375 */
|
||||
if (ix < 0x3e200000) { /* |x| < 2^-29 */
|
||||
if(huge+x>one) return x; /* raise inexact */
|
||||
}
|
||||
id = -1;
|
||||
} else {
|
||||
x = fabs(x);
|
||||
if (ix < 0x3ff30000) { /* |x| < 1.1875 */
|
||||
if (ix < 0x3fe60000) { /* 7/16 <=|x|<11/16 */
|
||||
id = 0; x = (2.0*x-one)/(2.0+x);
|
||||
} else { /* 11/16<=|x|< 19/16 */
|
||||
id = 1; x = (x-one)/(x+one);
|
||||
}
|
||||
} else {
|
||||
if (ix < 0x40038000) { /* |x| < 2.4375 */
|
||||
id = 2; x = (x-1.5)/(one+1.5*x);
|
||||
} else { /* 2.4375 <= |x| < 2^66 */
|
||||
id = 3; x = -1.0/x;
|
||||
}
|
||||
}}
|
||||
/* end of argument reduction */
|
||||
z = x*x;
|
||||
w = z*z;
|
||||
/* break sum from i=0 to 10 aT[i]z**(i+1) into odd and even poly */
|
||||
s1 = z*(aT[0]+w*(aT[2]+w*(aT[4]+w*(aT[6]+w*(aT[8]+w*aT[10])))));
|
||||
s2 = w*(aT[1]+w*(aT[3]+w*(aT[5]+w*(aT[7]+w*aT[9]))));
|
||||
if (id<0) return x - x*(s1+s2);
|
||||
else {
|
||||
z = atanhi[id] - ((x*(s1+s2) - atanlo[id]) - x);
|
||||
return (hx<0)? -z:z;
|
||||
}
|
||||
}
|
||||
95
src/math/s_atanf.c
Normal file
95
src/math/s_atanf.c
Normal file
|
|
@ -0,0 +1,95 @@
|
|||
/* s_atanf.c -- float version of s_atan.c.
|
||||
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
||||
*/
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const float atanhi[] = {
|
||||
4.6364760399e-01, /* atan(0.5)hi 0x3eed6338 */
|
||||
7.8539812565e-01, /* atan(1.0)hi 0x3f490fda */
|
||||
9.8279368877e-01, /* atan(1.5)hi 0x3f7b985e */
|
||||
1.5707962513e+00, /* atan(inf)hi 0x3fc90fda */
|
||||
};
|
||||
|
||||
static const float atanlo[] = {
|
||||
5.0121582440e-09, /* atan(0.5)lo 0x31ac3769 */
|
||||
3.7748947079e-08, /* atan(1.0)lo 0x33222168 */
|
||||
3.4473217170e-08, /* atan(1.5)lo 0x33140fb4 */
|
||||
7.5497894159e-08, /* atan(inf)lo 0x33a22168 */
|
||||
};
|
||||
|
||||
static const float aT[] = {
|
||||
3.3333334327e-01, /* 0x3eaaaaaa */
|
||||
-2.0000000298e-01, /* 0xbe4ccccd */
|
||||
1.4285714924e-01, /* 0x3e124925 */
|
||||
-1.1111110449e-01, /* 0xbde38e38 */
|
||||
9.0908870101e-02, /* 0x3dba2e6e */
|
||||
-7.6918758452e-02, /* 0xbd9d8795 */
|
||||
6.6610731184e-02, /* 0x3d886b35 */
|
||||
-5.8335702866e-02, /* 0xbd6ef16b */
|
||||
4.9768779427e-02, /* 0x3d4bda59 */
|
||||
-3.6531571299e-02, /* 0xbd15a221 */
|
||||
1.6285819933e-02, /* 0x3c8569d7 */
|
||||
};
|
||||
|
||||
static const float
|
||||
one = 1.0,
|
||||
huge = 1.0e30;
|
||||
|
||||
float
|
||||
atanf(float x)
|
||||
{
|
||||
float w,s1,s2,z;
|
||||
int32_t ix,hx,id;
|
||||
|
||||
GET_FLOAT_WORD(hx,x);
|
||||
ix = hx&0x7fffffff;
|
||||
if(ix>=0x50800000) { /* if |x| >= 2^34 */
|
||||
if(ix>0x7f800000)
|
||||
return x+x; /* NaN */
|
||||
if(hx>0) return atanhi[3]+atanlo[3];
|
||||
else return -atanhi[3]-atanlo[3];
|
||||
} if (ix < 0x3ee00000) { /* |x| < 0.4375 */
|
||||
if (ix < 0x31000000) { /* |x| < 2^-29 */
|
||||
if(huge+x>one) return x; /* raise inexact */
|
||||
}
|
||||
id = -1;
|
||||
} else {
|
||||
x = fabsf(x);
|
||||
if (ix < 0x3f980000) { /* |x| < 1.1875 */
|
||||
if (ix < 0x3f300000) { /* 7/16 <=|x|<11/16 */
|
||||
id = 0; x = ((float)2.0*x-one)/((float)2.0+x);
|
||||
} else { /* 11/16<=|x|< 19/16 */
|
||||
id = 1; x = (x-one)/(x+one);
|
||||
}
|
||||
} else {
|
||||
if (ix < 0x401c0000) { /* |x| < 2.4375 */
|
||||
id = 2; x = (x-(float)1.5)/(one+(float)1.5*x);
|
||||
} else { /* 2.4375 <= |x| < 2^66 */
|
||||
id = 3; x = -(float)1.0/x;
|
||||
}
|
||||
}}
|
||||
/* end of argument reduction */
|
||||
z = x*x;
|
||||
w = z*z;
|
||||
/* break sum from i=0 to 10 aT[i]z**(i+1) into odd and even poly */
|
||||
s1 = z*(aT[0]+w*(aT[2]+w*(aT[4]+w*(aT[6]+w*(aT[8]+w*aT[10])))));
|
||||
s2 = w*(aT[1]+w*(aT[3]+w*(aT[5]+w*(aT[7]+w*aT[9]))));
|
||||
if (id<0) return x - x*(s1+s2);
|
||||
else {
|
||||
z = atanhi[id] - ((x*(s1+s2) - atanlo[id]) - x);
|
||||
return (hx<0)? -z:z;
|
||||
}
|
||||
}
|
||||
77
src/math/s_cbrt.c
Normal file
77
src/math/s_cbrt.c
Normal file
|
|
@ -0,0 +1,77 @@
|
|||
/* @(#)s_cbrt.c 5.1 93/09/24 */
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
/* cbrt(x)
|
||||
* Return cube root of x
|
||||
*/
|
||||
static const uint32_t
|
||||
B1 = 715094163, /* B1 = (682-0.03306235651)*2**20 */
|
||||
B2 = 696219795; /* B2 = (664-0.03306235651)*2**20 */
|
||||
|
||||
static const double
|
||||
C = 5.42857142857142815906e-01, /* 19/35 = 0x3FE15F15, 0xF15F15F1 */
|
||||
D = -7.05306122448979611050e-01, /* -864/1225 = 0xBFE691DE, 0x2532C834 */
|
||||
E = 1.41428571428571436819e+00, /* 99/70 = 0x3FF6A0EA, 0x0EA0EA0F */
|
||||
F = 1.60714285714285720630e+00, /* 45/28 = 0x3FF9B6DB, 0x6DB6DB6E */
|
||||
G = 3.57142857142857150787e-01; /* 5/14 = 0x3FD6DB6D, 0xB6DB6DB7 */
|
||||
|
||||
double
|
||||
cbrt(double x)
|
||||
{
|
||||
int32_t hx;
|
||||
double r,s,t=0.0,w;
|
||||
uint32_t sign;
|
||||
uint32_t high,low;
|
||||
|
||||
GET_HIGH_WORD(hx,x);
|
||||
sign=hx&0x80000000; /* sign= sign(x) */
|
||||
hx ^=sign;
|
||||
if(hx>=0x7ff00000) return(x+x); /* cbrt(NaN,INF) is itself */
|
||||
GET_LOW_WORD(low,x);
|
||||
if((hx|low)==0)
|
||||
return(x); /* cbrt(0) is itself */
|
||||
|
||||
SET_HIGH_WORD(x,hx); /* x <- |x| */
|
||||
/* rough cbrt to 5 bits */
|
||||
if(hx<0x00100000) /* subnormal number */
|
||||
{SET_HIGH_WORD(t,0x43500000); /* set t= 2**54 */
|
||||
t*=x; GET_HIGH_WORD(high,t); SET_HIGH_WORD(t,high/3+B2);
|
||||
}
|
||||
else
|
||||
SET_HIGH_WORD(t,hx/3+B1);
|
||||
|
||||
|
||||
/* new cbrt to 23 bits, may be implemented in single precision */
|
||||
r=t*t/x;
|
||||
s=C+r*t;
|
||||
t*=G+F/(s+E+D/s);
|
||||
|
||||
/* chopped to 20 bits and make it larger than cbrt(x) */
|
||||
GET_HIGH_WORD(high,t);
|
||||
INSERT_WORDS(t,high+0x00000001,0);
|
||||
|
||||
|
||||
/* one step newton iteration to 53 bits with error less than 0.667 ulps */
|
||||
s=t*t; /* t*t is exact */
|
||||
r=x/s;
|
||||
w=t+t;
|
||||
r=(r-t)/(w+r); /* r-s is exact */
|
||||
t=t+t*r;
|
||||
|
||||
/* retore the sign bit */
|
||||
GET_HIGH_WORD(high,t);
|
||||
SET_HIGH_WORD(t,high|sign);
|
||||
return(t);
|
||||
}
|
||||
67
src/math/s_cbrtf.c
Normal file
67
src/math/s_cbrtf.c
Normal file
|
|
@ -0,0 +1,67 @@
|
|||
/* s_cbrtf.c -- float version of s_cbrt.c.
|
||||
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
||||
*/
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
/* cbrtf(x)
|
||||
* Return cube root of x
|
||||
*/
|
||||
static const unsigned
|
||||
B1 = 709958130, /* B1 = (84+2/3-0.03306235651)*2**23 */
|
||||
B2 = 642849266; /* B2 = (76+2/3-0.03306235651)*2**23 */
|
||||
|
||||
static const float
|
||||
C = 5.4285717010e-01, /* 19/35 = 0x3f0af8b0 */
|
||||
D = -7.0530611277e-01, /* -864/1225 = 0xbf348ef1 */
|
||||
E = 1.4142856598e+00, /* 99/70 = 0x3fb50750 */
|
||||
F = 1.6071428061e+00, /* 45/28 = 0x3fcdb6db */
|
||||
G = 3.5714286566e-01; /* 5/14 = 0x3eb6db6e */
|
||||
|
||||
float
|
||||
cbrtf(float x)
|
||||
{
|
||||
float r,s,t;
|
||||
int32_t hx;
|
||||
uint32_t sign;
|
||||
uint32_t high;
|
||||
|
||||
GET_FLOAT_WORD(hx,x);
|
||||
sign=hx&0x80000000; /* sign= sign(x) */
|
||||
hx ^=sign;
|
||||
if(hx>=0x7f800000) return(x+x); /* cbrt(NaN,INF) is itself */
|
||||
if(hx==0)
|
||||
return(x); /* cbrt(0) is itself */
|
||||
|
||||
SET_FLOAT_WORD(x,hx); /* x <- |x| */
|
||||
/* rough cbrt to 5 bits */
|
||||
if(hx<0x00800000) /* subnormal number */
|
||||
{SET_FLOAT_WORD(t,0x4b800000); /* set t= 2**24 */
|
||||
t*=x; GET_FLOAT_WORD(high,t); SET_FLOAT_WORD(t,high/3+B2);
|
||||
}
|
||||
else
|
||||
SET_FLOAT_WORD(t,hx/3+B1);
|
||||
|
||||
|
||||
/* new cbrt to 23 bits */
|
||||
r=t*t/x;
|
||||
s=C+r*t;
|
||||
t*=G+F/(s+E+D/s);
|
||||
|
||||
/* retore the sign bit */
|
||||
GET_FLOAT_WORD(high,t);
|
||||
SET_FLOAT_WORD(t,high|sign);
|
||||
return(t);
|
||||
}
|
||||
68
src/math/s_ceil.c
Normal file
68
src/math/s_ceil.c
Normal file
|
|
@ -0,0 +1,68 @@
|
|||
/* @(#)s_ceil.c 5.1 93/09/24 */
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/*
|
||||
* ceil(x)
|
||||
* Return x rounded toward -inf to integral value
|
||||
* Method:
|
||||
* Bit twiddling.
|
||||
* Exception:
|
||||
* Inexact flag raised if x not equal to ceil(x).
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const double huge = 1.0e300;
|
||||
|
||||
double
|
||||
ceil(double x)
|
||||
{
|
||||
int32_t i0,i1,j0;
|
||||
uint32_t i,j;
|
||||
EXTRACT_WORDS(i0,i1,x);
|
||||
j0 = ((i0>>20)&0x7ff)-0x3ff;
|
||||
if(j0<20) {
|
||||
if(j0<0) { /* raise inexact if x != 0 */
|
||||
if(huge+x>0.0) {/* return 0*sign(x) if |x|<1 */
|
||||
if(i0<0) {i0=0x80000000;i1=0;}
|
||||
else if((i0|i1)!=0) { i0=0x3ff00000;i1=0;}
|
||||
}
|
||||
} else {
|
||||
i = (0x000fffff)>>j0;
|
||||
if(((i0&i)|i1)==0) return x; /* x is integral */
|
||||
if(huge+x>0.0) { /* raise inexact flag */
|
||||
if(i0>0) i0 += (0x00100000)>>j0;
|
||||
i0 &= (~i); i1=0;
|
||||
}
|
||||
}
|
||||
} else if (j0>51) {
|
||||
if(j0==0x400) return x+x; /* inf or NaN */
|
||||
else return x; /* x is integral */
|
||||
} else {
|
||||
i = ((uint32_t)(0xffffffff))>>(j0-20);
|
||||
if((i1&i)==0) return x; /* x is integral */
|
||||
if(huge+x>0.0) { /* raise inexact flag */
|
||||
if(i0>0) {
|
||||
if(j0==20) i0+=1;
|
||||
else {
|
||||
j = i1 + (1<<(52-j0));
|
||||
if(j<i1) i0+=1; /* got a carry */
|
||||
i1 = j;
|
||||
}
|
||||
}
|
||||
i1 &= (~i);
|
||||
}
|
||||
}
|
||||
INSERT_WORDS(x,i0,i1);
|
||||
return x;
|
||||
}
|
||||
49
src/math/s_ceilf.c
Normal file
49
src/math/s_ceilf.c
Normal file
|
|
@ -0,0 +1,49 @@
|
|||
/* s_ceilf.c -- float version of s_ceil.c.
|
||||
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
||||
*/
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const float huge = 1.0e30;
|
||||
|
||||
float
|
||||
ceilf(float x)
|
||||
{
|
||||
int32_t i0,j0;
|
||||
uint32_t i;
|
||||
|
||||
GET_FLOAT_WORD(i0,x);
|
||||
j0 = ((i0>>23)&0xff)-0x7f;
|
||||
if(j0<23) {
|
||||
if(j0<0) { /* raise inexact if x != 0 */
|
||||
if(huge+x>(float)0.0) {/* return 0*sign(x) if |x|<1 */
|
||||
if(i0<0) {i0=0x80000000;}
|
||||
else if(i0!=0) { i0=0x3f800000;}
|
||||
}
|
||||
} else {
|
||||
i = (0x007fffff)>>j0;
|
||||
if((i0&i)==0) return x; /* x is integral */
|
||||
if(huge+x>(float)0.0) { /* raise inexact flag */
|
||||
if(i0>0) i0 += (0x00800000)>>j0;
|
||||
i0 &= (~i);
|
||||
}
|
||||
}
|
||||
} else {
|
||||
if(j0==0x80) return x+x; /* inf or NaN */
|
||||
else return x; /* x is integral */
|
||||
}
|
||||
SET_FLOAT_WORD(x,i0);
|
||||
return x;
|
||||
}
|
||||
30
src/math/s_copysign.c
Normal file
30
src/math/s_copysign.c
Normal file
|
|
@ -0,0 +1,30 @@
|
|||
/* @(#)s_copysign.c 5.1 93/09/24 */
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/*
|
||||
* copysign(double x, double y)
|
||||
* copysign(x,y) returns a value with the magnitude of x and
|
||||
* with the sign bit of y.
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
double
|
||||
copysign(double x, double y)
|
||||
{
|
||||
uint32_t hx,hy;
|
||||
GET_HIGH_WORD(hx,x);
|
||||
GET_HIGH_WORD(hy,y);
|
||||
SET_HIGH_WORD(x,(hx&0x7fffffff)|(hy&0x80000000));
|
||||
return x;
|
||||
}
|
||||
33
src/math/s_copysignf.c
Normal file
33
src/math/s_copysignf.c
Normal file
|
|
@ -0,0 +1,33 @@
|
|||
/* s_copysignf.c -- float version of s_copysign.c.
|
||||
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
||||
*/
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/*
|
||||
* copysignf(float x, float y)
|
||||
* copysignf(x,y) returns a value with the magnitude of x and
|
||||
* with the sign bit of y.
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
float
|
||||
copysignf(float x, float y)
|
||||
{
|
||||
uint32_t ix,iy;
|
||||
GET_FLOAT_WORD(ix,x);
|
||||
GET_FLOAT_WORD(iy,y);
|
||||
SET_FLOAT_WORD(x,(ix&0x7fffffff)|(iy&0x80000000));
|
||||
return x;
|
||||
}
|
||||
74
src/math/s_cos.c
Normal file
74
src/math/s_cos.c
Normal file
|
|
@ -0,0 +1,74 @@
|
|||
/* @(#)s_cos.c 5.1 93/09/24 */
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/* cos(x)
|
||||
* Return cosine function of x.
|
||||
*
|
||||
* kernel function:
|
||||
* __kernel_sin ... sine function on [-pi/4,pi/4]
|
||||
* __kernel_cos ... cosine function on [-pi/4,pi/4]
|
||||
* __ieee754_rem_pio2 ... argument reduction routine
|
||||
*
|
||||
* Method.
|
||||
* Let S,C and T denote the sin, cos and tan respectively on
|
||||
* [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
|
||||
* in [-pi/4 , +pi/4], and let n = k mod 4.
|
||||
* We have
|
||||
*
|
||||
* n sin(x) cos(x) tan(x)
|
||||
* ----------------------------------------------------------
|
||||
* 0 S C T
|
||||
* 1 C -S -1/T
|
||||
* 2 -S -C T
|
||||
* 3 -C S -1/T
|
||||
* ----------------------------------------------------------
|
||||
*
|
||||
* Special cases:
|
||||
* Let trig be any of sin, cos, or tan.
|
||||
* trig(+-INF) is NaN, with signals;
|
||||
* trig(NaN) is that NaN;
|
||||
*
|
||||
* Accuracy:
|
||||
* TRIG(x) returns trig(x) nearly rounded
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
double
|
||||
cos(double x)
|
||||
{
|
||||
double y[2],z=0.0;
|
||||
int32_t n, ix;
|
||||
|
||||
/* High word of x. */
|
||||
GET_HIGH_WORD(ix,x);
|
||||
|
||||
/* |x| ~< pi/4 */
|
||||
ix &= 0x7fffffff;
|
||||
if(ix <= 0x3fe921fb) return __kernel_cos(x,z);
|
||||
|
||||
/* cos(Inf or NaN) is NaN */
|
||||
else if (ix>=0x7ff00000) return x-x;
|
||||
|
||||
/* argument reduction needed */
|
||||
else {
|
||||
n = __ieee754_rem_pio2(x,y);
|
||||
switch(n&3) {
|
||||
case 0: return __kernel_cos(y[0],y[1]);
|
||||
case 1: return -__kernel_sin(y[0],y[1],1);
|
||||
case 2: return -__kernel_cos(y[0],y[1]);
|
||||
default:
|
||||
return __kernel_sin(y[0],y[1],1);
|
||||
}
|
||||
}
|
||||
}
|
||||
47
src/math/s_cosf.c
Normal file
47
src/math/s_cosf.c
Normal file
|
|
@ -0,0 +1,47 @@
|
|||
/* s_cosf.c -- float version of s_cos.c.
|
||||
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
||||
*/
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const float one=1.0;
|
||||
|
||||
float
|
||||
cosf(float x)
|
||||
{
|
||||
float y[2],z=0.0;
|
||||
int32_t n,ix;
|
||||
|
||||
GET_FLOAT_WORD(ix,x);
|
||||
|
||||
/* |x| ~< pi/4 */
|
||||
ix &= 0x7fffffff;
|
||||
if(ix <= 0x3f490fd8) return __kernel_cosf(x,z);
|
||||
|
||||
/* cos(Inf or NaN) is NaN */
|
||||
else if (ix>=0x7f800000) return x-x;
|
||||
|
||||
/* argument reduction needed */
|
||||
else {
|
||||
n = __ieee754_rem_pio2f(x,y);
|
||||
switch(n&3) {
|
||||
case 0: return __kernel_cosf(y[0],y[1]);
|
||||
case 1: return -__kernel_sinf(y[0],y[1],1);
|
||||
case 2: return -__kernel_cosf(y[0],y[1]);
|
||||
default:
|
||||
return __kernel_sinf(y[0],y[1],1);
|
||||
}
|
||||
}
|
||||
}
|
||||
298
src/math/s_erf.c
Normal file
298
src/math/s_erf.c
Normal file
|
|
@ -0,0 +1,298 @@
|
|||
/* @(#)s_erf.c 5.1 93/09/24 */
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/* double erf(double x)
|
||||
* double erfc(double x)
|
||||
* x
|
||||
* 2 |\
|
||||
* erf(x) = --------- | exp(-t*t)dt
|
||||
* sqrt(pi) \|
|
||||
* 0
|
||||
*
|
||||
* erfc(x) = 1-erf(x)
|
||||
* Note that
|
||||
* erf(-x) = -erf(x)
|
||||
* erfc(-x) = 2 - erfc(x)
|
||||
*
|
||||
* Method:
|
||||
* 1. For |x| in [0, 0.84375]
|
||||
* erf(x) = x + x*R(x^2)
|
||||
* erfc(x) = 1 - erf(x) if x in [-.84375,0.25]
|
||||
* = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375]
|
||||
* where R = P/Q where P is an odd poly of degree 8 and
|
||||
* Q is an odd poly of degree 10.
|
||||
* -57.90
|
||||
* | R - (erf(x)-x)/x | <= 2
|
||||
*
|
||||
*
|
||||
* Remark. The formula is derived by noting
|
||||
* erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
|
||||
* and that
|
||||
* 2/sqrt(pi) = 1.128379167095512573896158903121545171688
|
||||
* is close to one. The interval is chosen because the fix
|
||||
* point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
|
||||
* near 0.6174), and by some experiment, 0.84375 is chosen to
|
||||
* guarantee the error is less than one ulp for erf.
|
||||
*
|
||||
* 2. For |x| in [0.84375,1.25], let s = |x| - 1, and
|
||||
* c = 0.84506291151 rounded to single (24 bits)
|
||||
* erf(x) = sign(x) * (c + P1(s)/Q1(s))
|
||||
* erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0
|
||||
* 1+(c+P1(s)/Q1(s)) if x < 0
|
||||
* |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
|
||||
* Remark: here we use the taylor series expansion at x=1.
|
||||
* erf(1+s) = erf(1) + s*Poly(s)
|
||||
* = 0.845.. + P1(s)/Q1(s)
|
||||
* That is, we use rational approximation to approximate
|
||||
* erf(1+s) - (c = (single)0.84506291151)
|
||||
* Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
|
||||
* where
|
||||
* P1(s) = degree 6 poly in s
|
||||
* Q1(s) = degree 6 poly in s
|
||||
*
|
||||
* 3. For x in [1.25,1/0.35(~2.857143)],
|
||||
* erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
|
||||
* erf(x) = 1 - erfc(x)
|
||||
* where
|
||||
* R1(z) = degree 7 poly in z, (z=1/x^2)
|
||||
* S1(z) = degree 8 poly in z
|
||||
*
|
||||
* 4. For x in [1/0.35,28]
|
||||
* erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
|
||||
* = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
|
||||
* = 2.0 - tiny (if x <= -6)
|
||||
* erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else
|
||||
* erf(x) = sign(x)*(1.0 - tiny)
|
||||
* where
|
||||
* R2(z) = degree 6 poly in z, (z=1/x^2)
|
||||
* S2(z) = degree 7 poly in z
|
||||
*
|
||||
* Note1:
|
||||
* To compute exp(-x*x-0.5625+R/S), let s be a single
|
||||
* precision number and s := x; then
|
||||
* -x*x = -s*s + (s-x)*(s+x)
|
||||
* exp(-x*x-0.5626+R/S) =
|
||||
* exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
|
||||
* Note2:
|
||||
* Here 4 and 5 make use of the asymptotic series
|
||||
* exp(-x*x)
|
||||
* erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
|
||||
* x*sqrt(pi)
|
||||
* We use rational approximation to approximate
|
||||
* g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
|
||||
* Here is the error bound for R1/S1 and R2/S2
|
||||
* |R1/S1 - f(x)| < 2**(-62.57)
|
||||
* |R2/S2 - f(x)| < 2**(-61.52)
|
||||
*
|
||||
* 5. For inf > x >= 28
|
||||
* erf(x) = sign(x) *(1 - tiny) (raise inexact)
|
||||
* erfc(x) = tiny*tiny (raise underflow) if x > 0
|
||||
* = 2 - tiny if x<0
|
||||
*
|
||||
* 7. Special case:
|
||||
* erf(0) = 0, erf(inf) = 1, erf(-inf) = -1,
|
||||
* erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
|
||||
* erfc/erf(NaN) is NaN
|
||||
*/
|
||||
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const double
|
||||
tiny = 1e-300,
|
||||
half= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
|
||||
one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
|
||||
two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
|
||||
/* c = (float)0.84506291151 */
|
||||
erx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
|
||||
/*
|
||||
* Coefficients for approximation to erf on [0,0.84375]
|
||||
*/
|
||||
efx = 1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */
|
||||
efx8= 1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */
|
||||
pp0 = 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
|
||||
pp1 = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
|
||||
pp2 = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
|
||||
pp3 = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
|
||||
pp4 = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */
|
||||
qq1 = 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
|
||||
qq2 = 6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
|
||||
qq3 = 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
|
||||
qq4 = 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
|
||||
qq5 = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */
|
||||
/*
|
||||
* Coefficients for approximation to erf in [0.84375,1.25]
|
||||
*/
|
||||
pa0 = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
|
||||
pa1 = 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
|
||||
pa2 = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
|
||||
pa3 = 3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
|
||||
pa4 = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
|
||||
pa5 = 3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
|
||||
pa6 = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */
|
||||
qa1 = 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
|
||||
qa2 = 5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
|
||||
qa3 = 7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
|
||||
qa4 = 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
|
||||
qa5 = 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
|
||||
qa6 = 1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */
|
||||
/*
|
||||
* Coefficients for approximation to erfc in [1.25,1/0.35]
|
||||
*/
|
||||
ra0 = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
|
||||
ra1 = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
|
||||
ra2 = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
|
||||
ra3 = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
|
||||
ra4 = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
|
||||
ra5 = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
|
||||
ra6 = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
|
||||
ra7 = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */
|
||||
sa1 = 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
|
||||
sa2 = 1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
|
||||
sa3 = 4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
|
||||
sa4 = 6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
|
||||
sa5 = 4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
|
||||
sa6 = 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
|
||||
sa7 = 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
|
||||
sa8 = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */
|
||||
/*
|
||||
* Coefficients for approximation to erfc in [1/.35,28]
|
||||
*/
|
||||
rb0 = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
|
||||
rb1 = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
|
||||
rb2 = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
|
||||
rb3 = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
|
||||
rb4 = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
|
||||
rb5 = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
|
||||
rb6 = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */
|
||||
sb1 = 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
|
||||
sb2 = 3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
|
||||
sb3 = 1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
|
||||
sb4 = 3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
|
||||
sb5 = 2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
|
||||
sb6 = 4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
|
||||
sb7 = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */
|
||||
|
||||
double
|
||||
erf(double x)
|
||||
{
|
||||
int32_t hx,ix,i;
|
||||
double R,S,P,Q,s,y,z,r;
|
||||
GET_HIGH_WORD(hx,x);
|
||||
ix = hx&0x7fffffff;
|
||||
if(ix>=0x7ff00000) { /* erf(nan)=nan */
|
||||
i = ((uint32_t)hx>>31)<<1;
|
||||
return (double)(1-i)+one/x; /* erf(+-inf)=+-1 */
|
||||
}
|
||||
|
||||
if(ix < 0x3feb0000) { /* |x|<0.84375 */
|
||||
if(ix < 0x3e300000) { /* |x|<2**-28 */
|
||||
if (ix < 0x00800000)
|
||||
return 0.125*(8.0*x+efx8*x); /*avoid underflow */
|
||||
return x + efx*x;
|
||||
}
|
||||
z = x*x;
|
||||
r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
|
||||
s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
|
||||
y = r/s;
|
||||
return x + x*y;
|
||||
}
|
||||
if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */
|
||||
s = fabs(x)-one;
|
||||
P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
|
||||
Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
|
||||
if(hx>=0) return erx + P/Q; else return -erx - P/Q;
|
||||
}
|
||||
if (ix >= 0x40180000) { /* inf>|x|>=6 */
|
||||
if(hx>=0) return one-tiny; else return tiny-one;
|
||||
}
|
||||
x = fabs(x);
|
||||
s = one/(x*x);
|
||||
if(ix< 0x4006DB6E) { /* |x| < 1/0.35 */
|
||||
R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
|
||||
ra5+s*(ra6+s*ra7))))));
|
||||
S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
|
||||
sa5+s*(sa6+s*(sa7+s*sa8)))))));
|
||||
} else { /* |x| >= 1/0.35 */
|
||||
R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
|
||||
rb5+s*rb6)))));
|
||||
S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
|
||||
sb5+s*(sb6+s*sb7))))));
|
||||
}
|
||||
z = x;
|
||||
SET_LOW_WORD(z,0);
|
||||
r = exp(-z*z-0.5625)*exp((z-x)*(z+x)+R/S);
|
||||
if(hx>=0) return one-r/x; else return r/x-one;
|
||||
}
|
||||
|
||||
double
|
||||
erfc(double x)
|
||||
{
|
||||
int32_t hx,ix;
|
||||
double R,S,P,Q,s,y,z,r;
|
||||
GET_HIGH_WORD(hx,x);
|
||||
ix = hx&0x7fffffff;
|
||||
if(ix>=0x7ff00000) { /* erfc(nan)=nan */
|
||||
/* erfc(+-inf)=0,2 */
|
||||
return (double)(((uint32_t)hx>>31)<<1)+one/x;
|
||||
}
|
||||
|
||||
if(ix < 0x3feb0000) { /* |x|<0.84375 */
|
||||
if(ix < 0x3c700000) /* |x|<2**-56 */
|
||||
return one-x;
|
||||
z = x*x;
|
||||
r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
|
||||
s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
|
||||
y = r/s;
|
||||
if(hx < 0x3fd00000) { /* x<1/4 */
|
||||
return one-(x+x*y);
|
||||
} else {
|
||||
r = x*y;
|
||||
r += (x-half);
|
||||
return half - r ;
|
||||
}
|
||||
}
|
||||
if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */
|
||||
s = fabs(x)-one;
|
||||
P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
|
||||
Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
|
||||
if(hx>=0) {
|
||||
z = one-erx; return z - P/Q;
|
||||
} else {
|
||||
z = erx+P/Q; return one+z;
|
||||
}
|
||||
}
|
||||
if (ix < 0x403c0000) { /* |x|<28 */
|
||||
x = fabs(x);
|
||||
s = one/(x*x);
|
||||
if(ix< 0x4006DB6D) { /* |x| < 1/.35 ~ 2.857143*/
|
||||
R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
|
||||
ra5+s*(ra6+s*ra7))))));
|
||||
S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
|
||||
sa5+s*(sa6+s*(sa7+s*sa8)))))));
|
||||
} else { /* |x| >= 1/.35 ~ 2.857143 */
|
||||
if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */
|
||||
R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
|
||||
rb5+s*rb6)))));
|
||||
S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
|
||||
sb5+s*(sb6+s*sb7))))));
|
||||
}
|
||||
z = x;
|
||||
SET_LOW_WORD(z,0);
|
||||
r = exp(-z*z-0.5625)*
|
||||
exp((z-x)*(z+x)+R/S);
|
||||
if(hx>0) return r/x; else return two-r/x;
|
||||
} else {
|
||||
if(hx>0) return tiny*tiny; else return two-tiny;
|
||||
}
|
||||
}
|
||||
207
src/math/s_erff.c
Normal file
207
src/math/s_erff.c
Normal file
|
|
@ -0,0 +1,207 @@
|
|||
/* s_erff.c -- float version of s_erf.c.
|
||||
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
||||
*/
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const float
|
||||
tiny = 1e-30,
|
||||
half= 5.0000000000e-01, /* 0x3F000000 */
|
||||
one = 1.0000000000e+00, /* 0x3F800000 */
|
||||
two = 2.0000000000e+00, /* 0x40000000 */
|
||||
/* c = (subfloat)0.84506291151 */
|
||||
erx = 8.4506291151e-01, /* 0x3f58560b */
|
||||
/*
|
||||
* Coefficients for approximation to erf on [0,0.84375]
|
||||
*/
|
||||
efx = 1.2837916613e-01, /* 0x3e0375d4 */
|
||||
efx8= 1.0270333290e+00, /* 0x3f8375d4 */
|
||||
pp0 = 1.2837916613e-01, /* 0x3e0375d4 */
|
||||
pp1 = -3.2504209876e-01, /* 0xbea66beb */
|
||||
pp2 = -2.8481749818e-02, /* 0xbce9528f */
|
||||
pp3 = -5.7702702470e-03, /* 0xbbbd1489 */
|
||||
pp4 = -2.3763017452e-05, /* 0xb7c756b1 */
|
||||
qq1 = 3.9791721106e-01, /* 0x3ecbbbce */
|
||||
qq2 = 6.5022252500e-02, /* 0x3d852a63 */
|
||||
qq3 = 5.0813062117e-03, /* 0x3ba68116 */
|
||||
qq4 = 1.3249473704e-04, /* 0x390aee49 */
|
||||
qq5 = -3.9602282413e-06, /* 0xb684e21a */
|
||||
/*
|
||||
* Coefficients for approximation to erf in [0.84375,1.25]
|
||||
*/
|
||||
pa0 = -2.3621185683e-03, /* 0xbb1acdc6 */
|
||||
pa1 = 4.1485610604e-01, /* 0x3ed46805 */
|
||||
pa2 = -3.7220788002e-01, /* 0xbebe9208 */
|
||||
pa3 = 3.1834661961e-01, /* 0x3ea2fe54 */
|
||||
pa4 = -1.1089469492e-01, /* 0xbde31cc2 */
|
||||
pa5 = 3.5478305072e-02, /* 0x3d1151b3 */
|
||||
pa6 = -2.1663755178e-03, /* 0xbb0df9c0 */
|
||||
qa1 = 1.0642088205e-01, /* 0x3dd9f331 */
|
||||
qa2 = 5.4039794207e-01, /* 0x3f0a5785 */
|
||||
qa3 = 7.1828655899e-02, /* 0x3d931ae7 */
|
||||
qa4 = 1.2617121637e-01, /* 0x3e013307 */
|
||||
qa5 = 1.3637083583e-02, /* 0x3c5f6e13 */
|
||||
qa6 = 1.1984500103e-02, /* 0x3c445aa3 */
|
||||
/*
|
||||
* Coefficients for approximation to erfc in [1.25,1/0.35]
|
||||
*/
|
||||
ra0 = -9.8649440333e-03, /* 0xbc21a093 */
|
||||
ra1 = -6.9385856390e-01, /* 0xbf31a0b7 */
|
||||
ra2 = -1.0558626175e+01, /* 0xc128f022 */
|
||||
ra3 = -6.2375331879e+01, /* 0xc2798057 */
|
||||
ra4 = -1.6239666748e+02, /* 0xc322658c */
|
||||
ra5 = -1.8460508728e+02, /* 0xc3389ae7 */
|
||||
ra6 = -8.1287437439e+01, /* 0xc2a2932b */
|
||||
ra7 = -9.8143291473e+00, /* 0xc11d077e */
|
||||
sa1 = 1.9651271820e+01, /* 0x419d35ce */
|
||||
sa2 = 1.3765776062e+02, /* 0x4309a863 */
|
||||
sa3 = 4.3456588745e+02, /* 0x43d9486f */
|
||||
sa4 = 6.4538726807e+02, /* 0x442158c9 */
|
||||
sa5 = 4.2900814819e+02, /* 0x43d6810b */
|
||||
sa6 = 1.0863500214e+02, /* 0x42d9451f */
|
||||
sa7 = 6.5702495575e+00, /* 0x40d23f7c */
|
||||
sa8 = -6.0424413532e-02, /* 0xbd777f97 */
|
||||
/*
|
||||
* Coefficients for approximation to erfc in [1/.35,28]
|
||||
*/
|
||||
rb0 = -9.8649431020e-03, /* 0xbc21a092 */
|
||||
rb1 = -7.9928326607e-01, /* 0xbf4c9dd4 */
|
||||
rb2 = -1.7757955551e+01, /* 0xc18e104b */
|
||||
rb3 = -1.6063638306e+02, /* 0xc320a2ea */
|
||||
rb4 = -6.3756646729e+02, /* 0xc41f6441 */
|
||||
rb5 = -1.0250950928e+03, /* 0xc480230b */
|
||||
rb6 = -4.8351919556e+02, /* 0xc3f1c275 */
|
||||
sb1 = 3.0338060379e+01, /* 0x41f2b459 */
|
||||
sb2 = 3.2579251099e+02, /* 0x43a2e571 */
|
||||
sb3 = 1.5367296143e+03, /* 0x44c01759 */
|
||||
sb4 = 3.1998581543e+03, /* 0x4547fdbb */
|
||||
sb5 = 2.5530502930e+03, /* 0x451f90ce */
|
||||
sb6 = 4.7452853394e+02, /* 0x43ed43a7 */
|
||||
sb7 = -2.2440952301e+01; /* 0xc1b38712 */
|
||||
|
||||
float
|
||||
erff(float x)
|
||||
{
|
||||
int32_t hx,ix,i;
|
||||
float R,S,P,Q,s,y,z,r;
|
||||
GET_FLOAT_WORD(hx,x);
|
||||
ix = hx&0x7fffffff;
|
||||
if(ix>=0x7f800000) { /* erf(nan)=nan */
|
||||
i = ((uint32_t)hx>>31)<<1;
|
||||
return (float)(1-i)+one/x; /* erf(+-inf)=+-1 */
|
||||
}
|
||||
|
||||
if(ix < 0x3f580000) { /* |x|<0.84375 */
|
||||
if(ix < 0x31800000) { /* |x|<2**-28 */
|
||||
if (ix < 0x04000000)
|
||||
/*avoid underflow */
|
||||
return (float)0.125*((float)8.0*x+efx8*x);
|
||||
return x + efx*x;
|
||||
}
|
||||
z = x*x;
|
||||
r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
|
||||
s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
|
||||
y = r/s;
|
||||
return x + x*y;
|
||||
}
|
||||
if(ix < 0x3fa00000) { /* 0.84375 <= |x| < 1.25 */
|
||||
s = fabsf(x)-one;
|
||||
P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
|
||||
Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
|
||||
if(hx>=0) return erx + P/Q; else return -erx - P/Q;
|
||||
}
|
||||
if (ix >= 0x40c00000) { /* inf>|x|>=6 */
|
||||
if(hx>=0) return one-tiny; else return tiny-one;
|
||||
}
|
||||
x = fabsf(x);
|
||||
s = one/(x*x);
|
||||
if(ix< 0x4036DB6E) { /* |x| < 1/0.35 */
|
||||
R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
|
||||
ra5+s*(ra6+s*ra7))))));
|
||||
S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
|
||||
sa5+s*(sa6+s*(sa7+s*sa8)))))));
|
||||
} else { /* |x| >= 1/0.35 */
|
||||
R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
|
||||
rb5+s*rb6)))));
|
||||
S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
|
||||
sb5+s*(sb6+s*sb7))))));
|
||||
}
|
||||
GET_FLOAT_WORD(ix,x);
|
||||
SET_FLOAT_WORD(z,ix&0xfffff000);
|
||||
r = expf(-z*z-(float)0.5625)*expf((z-x)*(z+x)+R/S);
|
||||
if(hx>=0) return one-r/x; else return r/x-one;
|
||||
}
|
||||
|
||||
float
|
||||
erfcf(float x)
|
||||
{
|
||||
int32_t hx,ix;
|
||||
float R,S,P,Q,s,y,z,r;
|
||||
GET_FLOAT_WORD(hx,x);
|
||||
ix = hx&0x7fffffff;
|
||||
if(ix>=0x7f800000) { /* erfc(nan)=nan */
|
||||
/* erfc(+-inf)=0,2 */
|
||||
return (float)(((uint32_t)hx>>31)<<1)+one/x;
|
||||
}
|
||||
|
||||
if(ix < 0x3f580000) { /* |x|<0.84375 */
|
||||
if(ix < 0x23800000) /* |x|<2**-56 */
|
||||
return one-x;
|
||||
z = x*x;
|
||||
r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
|
||||
s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
|
||||
y = r/s;
|
||||
if(hx < 0x3e800000) { /* x<1/4 */
|
||||
return one-(x+x*y);
|
||||
} else {
|
||||
r = x*y;
|
||||
r += (x-half);
|
||||
return half - r ;
|
||||
}
|
||||
}
|
||||
if(ix < 0x3fa00000) { /* 0.84375 <= |x| < 1.25 */
|
||||
s = fabsf(x)-one;
|
||||
P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
|
||||
Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
|
||||
if(hx>=0) {
|
||||
z = one-erx; return z - P/Q;
|
||||
} else {
|
||||
z = erx+P/Q; return one+z;
|
||||
}
|
||||
}
|
||||
if (ix < 0x41e00000) { /* |x|<28 */
|
||||
x = fabsf(x);
|
||||
s = one/(x*x);
|
||||
if(ix< 0x4036DB6D) { /* |x| < 1/.35 ~ 2.857143*/
|
||||
R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
|
||||
ra5+s*(ra6+s*ra7))))));
|
||||
S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
|
||||
sa5+s*(sa6+s*(sa7+s*sa8)))))));
|
||||
} else { /* |x| >= 1/.35 ~ 2.857143 */
|
||||
if(hx<0&&ix>=0x40c00000) return two-tiny;/* x < -6 */
|
||||
R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
|
||||
rb5+s*rb6)))));
|
||||
S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
|
||||
sb5+s*(sb6+s*sb7))))));
|
||||
}
|
||||
GET_FLOAT_WORD(ix,x);
|
||||
SET_FLOAT_WORD(z,ix&0xfffff000);
|
||||
r = expf(-z*z-(float)0.5625)*
|
||||
expf((z-x)*(z+x)+R/S);
|
||||
if(hx>0) return r/x; else return two-r/x;
|
||||
} else {
|
||||
if(hx>0) return tiny*tiny; else return two-tiny;
|
||||
}
|
||||
}
|
||||
217
src/math/s_expm1.c
Normal file
217
src/math/s_expm1.c
Normal file
|
|
@ -0,0 +1,217 @@
|
|||
/* @(#)s_expm1.c 5.1 93/09/24 */
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/* expm1(x)
|
||||
* Returns exp(x)-1, the exponential of x minus 1.
|
||||
*
|
||||
* Method
|
||||
* 1. Argument reduction:
|
||||
* Given x, find r and integer k such that
|
||||
*
|
||||
* x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658
|
||||
*
|
||||
* Here a correction term c will be computed to compensate
|
||||
* the error in r when rounded to a floating-point number.
|
||||
*
|
||||
* 2. Approximating expm1(r) by a special rational function on
|
||||
* the interval [0,0.34658]:
|
||||
* Since
|
||||
* r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
|
||||
* we define R1(r*r) by
|
||||
* r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
|
||||
* That is,
|
||||
* R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
|
||||
* = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
|
||||
* = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
|
||||
* We use a special Reme algorithm on [0,0.347] to generate
|
||||
* a polynomial of degree 5 in r*r to approximate R1. The
|
||||
* maximum error of this polynomial approximation is bounded
|
||||
* by 2**-61. In other words,
|
||||
* R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
|
||||
* where Q1 = -1.6666666666666567384E-2,
|
||||
* Q2 = 3.9682539681370365873E-4,
|
||||
* Q3 = -9.9206344733435987357E-6,
|
||||
* Q4 = 2.5051361420808517002E-7,
|
||||
* Q5 = -6.2843505682382617102E-9;
|
||||
* (where z=r*r, and the values of Q1 to Q5 are listed below)
|
||||
* with error bounded by
|
||||
* | 5 | -61
|
||||
* | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
|
||||
* | |
|
||||
*
|
||||
* expm1(r) = exp(r)-1 is then computed by the following
|
||||
* specific way which minimize the accumulation rounding error:
|
||||
* 2 3
|
||||
* r r [ 3 - (R1 + R1*r/2) ]
|
||||
* expm1(r) = r + --- + --- * [--------------------]
|
||||
* 2 2 [ 6 - r*(3 - R1*r/2) ]
|
||||
*
|
||||
* To compensate the error in the argument reduction, we use
|
||||
* expm1(r+c) = expm1(r) + c + expm1(r)*c
|
||||
* ~ expm1(r) + c + r*c
|
||||
* Thus c+r*c will be added in as the correction terms for
|
||||
* expm1(r+c). Now rearrange the term to avoid optimization
|
||||
* screw up:
|
||||
* ( 2 2 )
|
||||
* ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )
|
||||
* expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
|
||||
* ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )
|
||||
* ( )
|
||||
*
|
||||
* = r - E
|
||||
* 3. Scale back to obtain expm1(x):
|
||||
* From step 1, we have
|
||||
* expm1(x) = either 2^k*[expm1(r)+1] - 1
|
||||
* = or 2^k*[expm1(r) + (1-2^-k)]
|
||||
* 4. Implementation notes:
|
||||
* (A). To save one multiplication, we scale the coefficient Qi
|
||||
* to Qi*2^i, and replace z by (x^2)/2.
|
||||
* (B). To achieve maximum accuracy, we compute expm1(x) by
|
||||
* (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
|
||||
* (ii) if k=0, return r-E
|
||||
* (iii) if k=-1, return 0.5*(r-E)-0.5
|
||||
* (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)
|
||||
* else return 1.0+2.0*(r-E);
|
||||
* (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
|
||||
* (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else
|
||||
* (vii) return 2^k(1-((E+2^-k)-r))
|
||||
*
|
||||
* Special cases:
|
||||
* expm1(INF) is INF, expm1(NaN) is NaN;
|
||||
* expm1(-INF) is -1, and
|
||||
* for finite argument, only expm1(0)=0 is exact.
|
||||
*
|
||||
* Accuracy:
|
||||
* according to an error analysis, the error is always less than
|
||||
* 1 ulp (unit in the last place).
|
||||
*
|
||||
* Misc. info.
|
||||
* For IEEE double
|
||||
* if x > 7.09782712893383973096e+02 then expm1(x) overflow
|
||||
*
|
||||
* Constants:
|
||||
* The hexadecimal values are the intended ones for the following
|
||||
* constants. The decimal values may be used, provided that the
|
||||
* compiler will convert from decimal to binary accurately enough
|
||||
* to produce the hexadecimal values shown.
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const double
|
||||
one = 1.0,
|
||||
huge = 1.0e+300,
|
||||
tiny = 1.0e-300,
|
||||
o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */
|
||||
ln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */
|
||||
ln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */
|
||||
invln2 = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */
|
||||
/* scaled coefficients related to expm1 */
|
||||
Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */
|
||||
Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
|
||||
Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
|
||||
Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
|
||||
Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */
|
||||
|
||||
double
|
||||
expm1(double x)
|
||||
{
|
||||
double y,hi,lo,c=0.0,t,e,hxs,hfx,r1;
|
||||
int32_t k,xsb;
|
||||
uint32_t hx;
|
||||
|
||||
GET_HIGH_WORD(hx,x);
|
||||
xsb = hx&0x80000000; /* sign bit of x */
|
||||
if(xsb==0) y=x; else y= -x; /* y = |x| */
|
||||
hx &= 0x7fffffff; /* high word of |x| */
|
||||
|
||||
/* filter out huge and non-finite argument */
|
||||
if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */
|
||||
if(hx >= 0x40862E42) { /* if |x|>=709.78... */
|
||||
if(hx>=0x7ff00000) {
|
||||
uint32_t low;
|
||||
GET_LOW_WORD(low,x);
|
||||
if(((hx&0xfffff)|low)!=0)
|
||||
return x+x; /* NaN */
|
||||
else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */
|
||||
}
|
||||
if(x > o_threshold) return huge*huge; /* overflow */
|
||||
}
|
||||
if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */
|
||||
if(x+tiny<0.0) /* raise inexact */
|
||||
return tiny-one; /* return -1 */
|
||||
}
|
||||
}
|
||||
|
||||
/* argument reduction */
|
||||
if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
|
||||
if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
|
||||
if(xsb==0)
|
||||
{hi = x - ln2_hi; lo = ln2_lo; k = 1;}
|
||||
else
|
||||
{hi = x + ln2_hi; lo = -ln2_lo; k = -1;}
|
||||
} else {
|
||||
k = invln2*x+((xsb==0)?0.5:-0.5);
|
||||
t = k;
|
||||
hi = x - t*ln2_hi; /* t*ln2_hi is exact here */
|
||||
lo = t*ln2_lo;
|
||||
}
|
||||
x = hi - lo;
|
||||
c = (hi-x)-lo;
|
||||
}
|
||||
else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */
|
||||
t = huge+x; /* return x with inexact flags when x!=0 */
|
||||
return x - (t-(huge+x));
|
||||
}
|
||||
else k = 0;
|
||||
|
||||
/* x is now in primary range */
|
||||
hfx = 0.5*x;
|
||||
hxs = x*hfx;
|
||||
r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
|
||||
t = 3.0-r1*hfx;
|
||||
e = hxs*((r1-t)/(6.0 - x*t));
|
||||
if(k==0) return x - (x*e-hxs); /* c is 0 */
|
||||
else {
|
||||
e = (x*(e-c)-c);
|
||||
e -= hxs;
|
||||
if(k== -1) return 0.5*(x-e)-0.5;
|
||||
if(k==1) {
|
||||
if(x < -0.25) return -2.0*(e-(x+0.5));
|
||||
else return one+2.0*(x-e);
|
||||
}
|
||||
if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */
|
||||
uint32_t high;
|
||||
y = one-(e-x);
|
||||
GET_HIGH_WORD(high,y);
|
||||
SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */
|
||||
return y-one;
|
||||
}
|
||||
t = one;
|
||||
if(k<20) {
|
||||
uint32_t high;
|
||||
SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k)); /* t=1-2^-k */
|
||||
y = t-(e-x);
|
||||
GET_HIGH_WORD(high,y);
|
||||
SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */
|
||||
} else {
|
||||
uint32_t high;
|
||||
SET_HIGH_WORD(t,((0x3ff-k)<<20)); /* 2^-k */
|
||||
y = x-(e+t);
|
||||
y += one;
|
||||
GET_HIGH_WORD(high,y);
|
||||
SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */
|
||||
}
|
||||
}
|
||||
return y;
|
||||
}
|
||||
122
src/math/s_expm1f.c
Normal file
122
src/math/s_expm1f.c
Normal file
|
|
@ -0,0 +1,122 @@
|
|||
/* s_expm1f.c -- float version of s_expm1.c.
|
||||
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
||||
*/
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const float
|
||||
one = 1.0,
|
||||
huge = 1.0e+30,
|
||||
tiny = 1.0e-30,
|
||||
o_threshold = 8.8721679688e+01,/* 0x42b17180 */
|
||||
ln2_hi = 6.9313812256e-01,/* 0x3f317180 */
|
||||
ln2_lo = 9.0580006145e-06,/* 0x3717f7d1 */
|
||||
invln2 = 1.4426950216e+00,/* 0x3fb8aa3b */
|
||||
/* scaled coefficients related to expm1 */
|
||||
Q1 = -3.3333335072e-02, /* 0xbd088889 */
|
||||
Q2 = 1.5873016091e-03, /* 0x3ad00d01 */
|
||||
Q3 = -7.9365076090e-05, /* 0xb8a670cd */
|
||||
Q4 = 4.0082177293e-06, /* 0x36867e54 */
|
||||
Q5 = -2.0109921195e-07; /* 0xb457edbb */
|
||||
|
||||
float
|
||||
expm1f(float x)
|
||||
{
|
||||
float y,hi,lo,c=0.0,t,e,hxs,hfx,r1;
|
||||
int32_t k,xsb;
|
||||
uint32_t hx;
|
||||
|
||||
GET_FLOAT_WORD(hx,x);
|
||||
xsb = hx&0x80000000; /* sign bit of x */
|
||||
if(xsb==0) y=x; else y= -x; /* y = |x| */
|
||||
hx &= 0x7fffffff; /* high word of |x| */
|
||||
|
||||
/* filter out huge and non-finite argument */
|
||||
if(hx >= 0x4195b844) { /* if |x|>=27*ln2 */
|
||||
if(hx >= 0x42b17218) { /* if |x|>=88.721... */
|
||||
if(hx>0x7f800000)
|
||||
return x+x; /* NaN */
|
||||
if(hx==0x7f800000)
|
||||
return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */
|
||||
if(x > o_threshold) return huge*huge; /* overflow */
|
||||
}
|
||||
if(xsb!=0) { /* x < -27*ln2, return -1.0 with inexact */
|
||||
if(x+tiny<(float)0.0) /* raise inexact */
|
||||
return tiny-one; /* return -1 */
|
||||
}
|
||||
}
|
||||
|
||||
/* argument reduction */
|
||||
if(hx > 0x3eb17218) { /* if |x| > 0.5 ln2 */
|
||||
if(hx < 0x3F851592) { /* and |x| < 1.5 ln2 */
|
||||
if(xsb==0)
|
||||
{hi = x - ln2_hi; lo = ln2_lo; k = 1;}
|
||||
else
|
||||
{hi = x + ln2_hi; lo = -ln2_lo; k = -1;}
|
||||
} else {
|
||||
k = invln2*x+((xsb==0)?(float)0.5:(float)-0.5);
|
||||
t = k;
|
||||
hi = x - t*ln2_hi; /* t*ln2_hi is exact here */
|
||||
lo = t*ln2_lo;
|
||||
}
|
||||
x = hi - lo;
|
||||
c = (hi-x)-lo;
|
||||
}
|
||||
else if(hx < 0x33000000) { /* when |x|<2**-25, return x */
|
||||
t = huge+x; /* return x with inexact flags when x!=0 */
|
||||
return x - (t-(huge+x));
|
||||
}
|
||||
else k = 0;
|
||||
|
||||
/* x is now in primary range */
|
||||
hfx = (float)0.5*x;
|
||||
hxs = x*hfx;
|
||||
r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
|
||||
t = (float)3.0-r1*hfx;
|
||||
e = hxs*((r1-t)/((float)6.0 - x*t));
|
||||
if(k==0) return x - (x*e-hxs); /* c is 0 */
|
||||
else {
|
||||
e = (x*(e-c)-c);
|
||||
e -= hxs;
|
||||
if(k== -1) return (float)0.5*(x-e)-(float)0.5;
|
||||
if(k==1) {
|
||||
if(x < (float)-0.25) return -(float)2.0*(e-(x+(float)0.5));
|
||||
else return one+(float)2.0*(x-e);
|
||||
}
|
||||
if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */
|
||||
int32_t i;
|
||||
y = one-(e-x);
|
||||
GET_FLOAT_WORD(i,y);
|
||||
SET_FLOAT_WORD(y,i+(k<<23)); /* add k to y's exponent */
|
||||
return y-one;
|
||||
}
|
||||
t = one;
|
||||
if(k<23) {
|
||||
int32_t i;
|
||||
SET_FLOAT_WORD(t,0x3f800000 - (0x1000000>>k)); /* t=1-2^-k */
|
||||
y = t-(e-x);
|
||||
GET_FLOAT_WORD(i,y);
|
||||
SET_FLOAT_WORD(y,i+(k<<23)); /* add k to y's exponent */
|
||||
} else {
|
||||
int32_t i;
|
||||
SET_FLOAT_WORD(t,((0x7f-k)<<23)); /* 2^-k */
|
||||
y = x-(e+t);
|
||||
y += one;
|
||||
GET_FLOAT_WORD(i,y);
|
||||
SET_FLOAT_WORD(y,i+(k<<23)); /* add k to y's exponent */
|
||||
}
|
||||
}
|
||||
return y;
|
||||
}
|
||||
27
src/math/s_fabs.c
Normal file
27
src/math/s_fabs.c
Normal file
|
|
@ -0,0 +1,27 @@
|
|||
/* @(#)s_fabs.c 5.1 93/09/24 */
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/*
|
||||
* fabs(x) returns the absolute value of x.
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
double
|
||||
fabs(double x)
|
||||
{
|
||||
uint32_t high;
|
||||
GET_HIGH_WORD(high,x);
|
||||
SET_HIGH_WORD(x,high&0x7fffffff);
|
||||
return x;
|
||||
}
|
||||
30
src/math/s_fabsf.c
Normal file
30
src/math/s_fabsf.c
Normal file
|
|
@ -0,0 +1,30 @@
|
|||
/* s_fabsf.c -- float version of s_fabs.c.
|
||||
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
||||
*/
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/*
|
||||
* fabsf(x) returns the absolute value of x.
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
float
|
||||
fabsf(float x)
|
||||
{
|
||||
uint32_t ix;
|
||||
GET_FLOAT_WORD(ix,x);
|
||||
SET_FLOAT_WORD(x,ix&0x7fffffff);
|
||||
return x;
|
||||
}
|
||||
69
src/math/s_floor.c
Normal file
69
src/math/s_floor.c
Normal file
|
|
@ -0,0 +1,69 @@
|
|||
/* @(#)s_floor.c 5.1 93/09/24 */
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/*
|
||||
* floor(x)
|
||||
* Return x rounded toward -inf to integral value
|
||||
* Method:
|
||||
* Bit twiddling.
|
||||
* Exception:
|
||||
* Inexact flag raised if x not equal to floor(x).
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const double huge = 1.0e300;
|
||||
|
||||
double
|
||||
floor(double x)
|
||||
{
|
||||
int32_t i0,i1,j0;
|
||||
uint32_t i,j;
|
||||
EXTRACT_WORDS(i0,i1,x);
|
||||
j0 = ((i0>>20)&0x7ff)-0x3ff;
|
||||
if(j0<20) {
|
||||
if(j0<0) { /* raise inexact if x != 0 */
|
||||
if(huge+x>0.0) {/* return 0*sign(x) if |x|<1 */
|
||||
if(i0>=0) {i0=i1=0;}
|
||||
else if(((i0&0x7fffffff)|i1)!=0)
|
||||
{ i0=0xbff00000;i1=0;}
|
||||
}
|
||||
} else {
|
||||
i = (0x000fffff)>>j0;
|
||||
if(((i0&i)|i1)==0) return x; /* x is integral */
|
||||
if(huge+x>0.0) { /* raise inexact flag */
|
||||
if(i0<0) i0 += (0x00100000)>>j0;
|
||||
i0 &= (~i); i1=0;
|
||||
}
|
||||
}
|
||||
} else if (j0>51) {
|
||||
if(j0==0x400) return x+x; /* inf or NaN */
|
||||
else return x; /* x is integral */
|
||||
} else {
|
||||
i = ((uint32_t)(0xffffffff))>>(j0-20);
|
||||
if((i1&i)==0) return x; /* x is integral */
|
||||
if(huge+x>0.0) { /* raise inexact flag */
|
||||
if(i0<0) {
|
||||
if(j0==20) i0+=1;
|
||||
else {
|
||||
j = i1+(1<<(52-j0));
|
||||
if(j<i1) i0 +=1 ; /* got a carry */
|
||||
i1=j;
|
||||
}
|
||||
}
|
||||
i1 &= (~i);
|
||||
}
|
||||
}
|
||||
INSERT_WORDS(x,i0,i1);
|
||||
return x;
|
||||
}
|
||||
58
src/math/s_floorf.c
Normal file
58
src/math/s_floorf.c
Normal file
|
|
@ -0,0 +1,58 @@
|
|||
/* s_floorf.c -- float version of s_floor.c.
|
||||
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
||||
*/
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/*
|
||||
* floorf(x)
|
||||
* Return x rounded toward -inf to integral value
|
||||
* Method:
|
||||
* Bit twiddling.
|
||||
* Exception:
|
||||
* Inexact flag raised if x not equal to floorf(x).
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const float huge = 1.0e30;
|
||||
|
||||
float
|
||||
floorf(float x)
|
||||
{
|
||||
int32_t i0,j0;
|
||||
uint32_t i;
|
||||
GET_FLOAT_WORD(i0,x);
|
||||
j0 = ((i0>>23)&0xff)-0x7f;
|
||||
if(j0<23) {
|
||||
if(j0<0) { /* raise inexact if x != 0 */
|
||||
if(huge+x>(float)0.0) {/* return 0*sign(x) if |x|<1 */
|
||||
if(i0>=0) {i0=0;}
|
||||
else if((i0&0x7fffffff)!=0)
|
||||
{ i0=0xbf800000;}
|
||||
}
|
||||
} else {
|
||||
i = (0x007fffff)>>j0;
|
||||
if((i0&i)==0) return x; /* x is integral */
|
||||
if(huge+x>(float)0.0) { /* raise inexact flag */
|
||||
if(i0<0) i0 += (0x00800000)>>j0;
|
||||
i0 &= (~i);
|
||||
}
|
||||
}
|
||||
} else {
|
||||
if(j0==0x80) return x+x; /* inf or NaN */
|
||||
else return x; /* x is integral */
|
||||
}
|
||||
SET_FLOAT_WORD(x,i0);
|
||||
return x;
|
||||
}
|
||||
45
src/math/s_ilogb.c
Normal file
45
src/math/s_ilogb.c
Normal file
|
|
@ -0,0 +1,45 @@
|
|||
/* @(#)s_ilogb.c 5.1 93/09/24 */
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/* ilogb(double x)
|
||||
* return the binary exponent of non-zero x
|
||||
* ilogb(0) = FP_ILOGB0
|
||||
* ilogb(NaN) = FP_ILOGBNAN (no signal is raised)
|
||||
* ilogb(inf) = INT_MAX (no signal is raised)
|
||||
*/
|
||||
|
||||
#include <limits.h>
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
int ilogb(double x)
|
||||
{
|
||||
int32_t hx,lx,ix;
|
||||
|
||||
EXTRACT_WORDS(hx,lx,x);
|
||||
hx &= 0x7fffffff;
|
||||
if(hx<0x00100000) {
|
||||
if((hx|lx)==0)
|
||||
return FP_ILOGB0;
|
||||
else /* subnormal x */
|
||||
if(hx==0) {
|
||||
for (ix = -1043; lx>0; lx<<=1) ix -=1;
|
||||
} else {
|
||||
for (ix = -1022,hx<<=11; hx>0; hx<<=1) ix -=1;
|
||||
}
|
||||
return ix;
|
||||
}
|
||||
else if (hx<0x7ff00000) return (hx>>20)-1023;
|
||||
else if (hx>0x7ff00000 || lx!=0) return FP_ILOGBNAN;
|
||||
else return INT_MAX;
|
||||
}
|
||||
37
src/math/s_ilogbf.c
Normal file
37
src/math/s_ilogbf.c
Normal file
|
|
@ -0,0 +1,37 @@
|
|||
/* s_ilogbf.c -- float version of s_ilogb.c.
|
||||
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
||||
*/
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
#include <limits.h>
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
int ilogbf(float x)
|
||||
{
|
||||
int32_t hx,ix;
|
||||
|
||||
GET_FLOAT_WORD(hx,x);
|
||||
hx &= 0x7fffffff;
|
||||
if(hx<0x00800000) {
|
||||
if(hx==0)
|
||||
return FP_ILOGB0;
|
||||
else /* subnormal x */
|
||||
for (ix = -126,hx<<=8; hx>0; hx<<=1) ix -=1;
|
||||
return ix;
|
||||
}
|
||||
else if (hx<0x7f800000) return (hx>>23)-127;
|
||||
else if (hx>0x7f800000) return FP_ILOGBNAN;
|
||||
else return INT_MAX;
|
||||
}
|
||||
6
src/math/s_ldexp.c
Normal file
6
src/math/s_ldexp.c
Normal file
|
|
@ -0,0 +1,6 @@
|
|||
#include <math.h>
|
||||
|
||||
double ldexp(double x, int n)
|
||||
{
|
||||
return scalbn(x, n);
|
||||
}
|
||||
6
src/math/s_ldexpf.c
Normal file
6
src/math/s_ldexpf.c
Normal file
|
|
@ -0,0 +1,6 @@
|
|||
#include <math.h>
|
||||
|
||||
float ldexpf(float x, int n)
|
||||
{
|
||||
return scalbnf(x, n);
|
||||
}
|
||||
8
src/math/s_llrint.c
Normal file
8
src/math/s_llrint.c
Normal file
|
|
@ -0,0 +1,8 @@
|
|||
#include <math.h>
|
||||
|
||||
// FIXME: incorrect exception behavior
|
||||
|
||||
long long llrint(double x)
|
||||
{
|
||||
return rint(x);
|
||||
}
|
||||
157
src/math/s_log1p.c
Normal file
157
src/math/s_log1p.c
Normal file
|
|
@ -0,0 +1,157 @@
|
|||
/* @(#)s_log1p.c 5.1 93/09/24 */
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/* double log1p(double x)
|
||||
*
|
||||
* Method :
|
||||
* 1. Argument Reduction: find k and f such that
|
||||
* 1+x = 2^k * (1+f),
|
||||
* where sqrt(2)/2 < 1+f < sqrt(2) .
|
||||
*
|
||||
* Note. If k=0, then f=x is exact. However, if k!=0, then f
|
||||
* may not be representable exactly. In that case, a correction
|
||||
* term is need. Let u=1+x rounded. Let c = (1+x)-u, then
|
||||
* log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
|
||||
* and add back the correction term c/u.
|
||||
* (Note: when x > 2**53, one can simply return log(x))
|
||||
*
|
||||
* 2. Approximation of log1p(f).
|
||||
* Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
|
||||
* = 2s + 2/3 s**3 + 2/5 s**5 + .....,
|
||||
* = 2s + s*R
|
||||
* We use a special Reme algorithm on [0,0.1716] to generate
|
||||
* a polynomial of degree 14 to approximate R The maximum error
|
||||
* of this polynomial approximation is bounded by 2**-58.45. In
|
||||
* other words,
|
||||
* 2 4 6 8 10 12 14
|
||||
* R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s
|
||||
* (the values of Lp1 to Lp7 are listed in the program)
|
||||
* and
|
||||
* | 2 14 | -58.45
|
||||
* | Lp1*s +...+Lp7*s - R(z) | <= 2
|
||||
* | |
|
||||
* Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
|
||||
* In order to guarantee error in log below 1ulp, we compute log
|
||||
* by
|
||||
* log1p(f) = f - (hfsq - s*(hfsq+R)).
|
||||
*
|
||||
* 3. Finally, log1p(x) = k*ln2 + log1p(f).
|
||||
* = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
|
||||
* Here ln2 is split into two floating point number:
|
||||
* ln2_hi + ln2_lo,
|
||||
* where n*ln2_hi is always exact for |n| < 2000.
|
||||
*
|
||||
* Special cases:
|
||||
* log1p(x) is NaN with signal if x < -1 (including -INF) ;
|
||||
* log1p(+INF) is +INF; log1p(-1) is -INF with signal;
|
||||
* log1p(NaN) is that NaN with no signal.
|
||||
*
|
||||
* Accuracy:
|
||||
* according to an error analysis, the error is always less than
|
||||
* 1 ulp (unit in the last place).
|
||||
*
|
||||
* Constants:
|
||||
* The hexadecimal values are the intended ones for the following
|
||||
* constants. The decimal values may be used, provided that the
|
||||
* compiler will convert from decimal to binary accurately enough
|
||||
* to produce the hexadecimal values shown.
|
||||
*
|
||||
* Note: Assuming log() return accurate answer, the following
|
||||
* algorithm can be used to compute log1p(x) to within a few ULP:
|
||||
*
|
||||
* u = 1+x;
|
||||
* if(u==1.0) return x ; else
|
||||
* return log(u)*(x/(u-1.0));
|
||||
*
|
||||
* See HP-15C Advanced Functions Handbook, p.193.
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const double
|
||||
ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
|
||||
ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
|
||||
two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
|
||||
Lp1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
|
||||
Lp2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
|
||||
Lp3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
|
||||
Lp4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
|
||||
Lp5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
|
||||
Lp6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
|
||||
Lp7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
|
||||
|
||||
static const double zero = 0.0;
|
||||
|
||||
double
|
||||
log1p(double x)
|
||||
{
|
||||
double hfsq,f=0,c=0,s,z,R,u;
|
||||
int32_t k,hx,hu=0,ax;
|
||||
|
||||
GET_HIGH_WORD(hx,x);
|
||||
ax = hx&0x7fffffff;
|
||||
|
||||
k = 1;
|
||||
if (hx < 0x3FDA827A) { /* x < 0.41422 */
|
||||
if(ax>=0x3ff00000) { /* x <= -1.0 */
|
||||
if(x==-1.0) return -two54/zero; /* log1p(-1)=+inf */
|
||||
else return (x-x)/(x-x); /* log1p(x<-1)=NaN */
|
||||
}
|
||||
if(ax<0x3e200000) { /* |x| < 2**-29 */
|
||||
if(two54+x>zero /* raise inexact */
|
||||
&&ax<0x3c900000) /* |x| < 2**-54 */
|
||||
return x;
|
||||
else
|
||||
return x - x*x*0.5;
|
||||
}
|
||||
if(hx>0||hx<=((int32_t)0xbfd2bec3)) {
|
||||
k=0;f=x;hu=1;} /* -0.2929<x<0.41422 */
|
||||
}
|
||||
if (hx >= 0x7ff00000) return x+x;
|
||||
if(k!=0) {
|
||||
if(hx<0x43400000) {
|
||||
u = 1.0+x;
|
||||
GET_HIGH_WORD(hu,u);
|
||||
k = (hu>>20)-1023;
|
||||
c = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */
|
||||
c /= u;
|
||||
} else {
|
||||
u = x;
|
||||
GET_HIGH_WORD(hu,u);
|
||||
k = (hu>>20)-1023;
|
||||
c = 0;
|
||||
}
|
||||
hu &= 0x000fffff;
|
||||
if(hu<0x6a09e) {
|
||||
SET_HIGH_WORD(u,hu|0x3ff00000); /* normalize u */
|
||||
} else {
|
||||
k += 1;
|
||||
SET_HIGH_WORD(u,hu|0x3fe00000); /* normalize u/2 */
|
||||
hu = (0x00100000-hu)>>2;
|
||||
}
|
||||
f = u-1.0;
|
||||
}
|
||||
hfsq=0.5*f*f;
|
||||
if(hu==0) { /* |f| < 2**-20 */
|
||||
if(f==zero) { if(k==0) return zero;
|
||||
else {c += k*ln2_lo; return k*ln2_hi+c;} }
|
||||
R = hfsq*(1.0-0.66666666666666666*f);
|
||||
if(k==0) return f-R; else
|
||||
return k*ln2_hi-((R-(k*ln2_lo+c))-f);
|
||||
}
|
||||
s = f/(2.0+f);
|
||||
z = s*s;
|
||||
R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))));
|
||||
if(k==0) return f-(hfsq-s*(hfsq+R)); else
|
||||
return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f);
|
||||
}
|
||||
96
src/math/s_log1pf.c
Normal file
96
src/math/s_log1pf.c
Normal file
|
|
@ -0,0 +1,96 @@
|
|||
/* s_log1pf.c -- float version of s_log1p.c.
|
||||
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
||||
*/
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
static const float
|
||||
ln2_hi = 6.9313812256e-01, /* 0x3f317180 */
|
||||
ln2_lo = 9.0580006145e-06, /* 0x3717f7d1 */
|
||||
two25 = 3.355443200e+07, /* 0x4c000000 */
|
||||
Lp1 = 6.6666668653e-01, /* 3F2AAAAB */
|
||||
Lp2 = 4.0000000596e-01, /* 3ECCCCCD */
|
||||
Lp3 = 2.8571429849e-01, /* 3E924925 */
|
||||
Lp4 = 2.2222198546e-01, /* 3E638E29 */
|
||||
Lp5 = 1.8183572590e-01, /* 3E3A3325 */
|
||||
Lp6 = 1.5313838422e-01, /* 3E1CD04F */
|
||||
Lp7 = 1.4798198640e-01; /* 3E178897 */
|
||||
|
||||
static const float zero = 0.0;
|
||||
|
||||
float
|
||||
log1pf(float x)
|
||||
{
|
||||
float hfsq,f=0,c=0,s,z,R,u;
|
||||
int32_t k,hx,hu=0,ax;
|
||||
|
||||
GET_FLOAT_WORD(hx,x);
|
||||
ax = hx&0x7fffffff;
|
||||
|
||||
k = 1;
|
||||
if (hx < 0x3ed413d7) { /* x < 0.41422 */
|
||||
if(ax>=0x3f800000) { /* x <= -1.0 */
|
||||
if(x==(float)-1.0) return -two25/zero; /* log1p(-1)=+inf */
|
||||
else return (x-x)/(x-x); /* log1p(x<-1)=NaN */
|
||||
}
|
||||
if(ax<0x31000000) { /* |x| < 2**-29 */
|
||||
if(two25+x>zero /* raise inexact */
|
||||
&&ax<0x24800000) /* |x| < 2**-54 */
|
||||
return x;
|
||||
else
|
||||
return x - x*x*(float)0.5;
|
||||
}
|
||||
if(hx>0||hx<=((int32_t)0xbe95f61f)) {
|
||||
k=0;f=x;hu=1;} /* -0.2929<x<0.41422 */
|
||||
}
|
||||
if (hx >= 0x7f800000) return x+x;
|
||||
if(k!=0) {
|
||||
if(hx<0x5a000000) {
|
||||
u = (float)1.0+x;
|
||||
GET_FLOAT_WORD(hu,u);
|
||||
k = (hu>>23)-127;
|
||||
/* correction term */
|
||||
c = (k>0)? (float)1.0-(u-x):x-(u-(float)1.0);
|
||||
c /= u;
|
||||
} else {
|
||||
u = x;
|
||||
GET_FLOAT_WORD(hu,u);
|
||||
k = (hu>>23)-127;
|
||||
c = 0;
|
||||
}
|
||||
hu &= 0x007fffff;
|
||||
if(hu<0x3504f7) {
|
||||
SET_FLOAT_WORD(u,hu|0x3f800000);/* normalize u */
|
||||
} else {
|
||||
k += 1;
|
||||
SET_FLOAT_WORD(u,hu|0x3f000000); /* normalize u/2 */
|
||||
hu = (0x00800000-hu)>>2;
|
||||
}
|
||||
f = u-(float)1.0;
|
||||
}
|
||||
hfsq=(float)0.5*f*f;
|
||||
if(hu==0) { /* |f| < 2**-20 */
|
||||
if(f==zero) { if(k==0) return zero;
|
||||
else {c += k*ln2_lo; return k*ln2_hi+c;} }
|
||||
R = hfsq*((float)1.0-(float)0.66666666666666666*f);
|
||||
if(k==0) return f-R; else
|
||||
return k*ln2_hi-((R-(k*ln2_lo+c))-f);
|
||||
}
|
||||
s = f/((float)2.0+f);
|
||||
z = s*s;
|
||||
R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))));
|
||||
if(k==0) return f-(hfsq-s*(hfsq+R)); else
|
||||
return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f);
|
||||
}
|
||||
34
src/math/s_logb.c
Normal file
34
src/math/s_logb.c
Normal file
|
|
@ -0,0 +1,34 @@
|
|||
/* @(#)s_logb.c 5.1 93/09/24 */
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/*
|
||||
* double logb(x)
|
||||
* IEEE 754 logb. Included to pass IEEE test suite. Not recommend.
|
||||
* Use ilogb instead.
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
double
|
||||
logb(double x)
|
||||
{
|
||||
int32_t lx,ix;
|
||||
EXTRACT_WORDS(ix,lx,x);
|
||||
ix &= 0x7fffffff; /* high |x| */
|
||||
if((ix|lx)==0) return -1.0/fabs(x);
|
||||
if(ix>=0x7ff00000) return x*x;
|
||||
if((ix>>=20)==0) /* IEEE 754 logb */
|
||||
return -1022.0;
|
||||
else
|
||||
return (double) (ix-1023);
|
||||
}
|
||||
31
src/math/s_logbf.c
Normal file
31
src/math/s_logbf.c
Normal file
|
|
@ -0,0 +1,31 @@
|
|||
/* s_logbf.c -- float version of s_logb.c.
|
||||
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
||||
*/
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
#include <math.h>
|
||||
#include "math_private.h"
|
||||
|
||||
float
|
||||
logbf(float x)
|
||||
{
|
||||
int32_t ix;
|
||||
GET_FLOAT_WORD(ix,x);
|
||||
ix &= 0x7fffffff; /* high |x| */
|
||||
if(ix==0) return (float)-1.0/fabsf(x);
|
||||
if(ix>=0x7f800000) return x*x;
|
||||
if((ix>>=23)==0) /* IEEE 754 logb */
|
||||
return -126.0;
|
||||
else
|
||||
return (float) (ix-127);
|
||||
}
|
||||
8
src/math/s_lrint.c
Normal file
8
src/math/s_lrint.c
Normal file
|
|
@ -0,0 +1,8 @@
|
|||
#include <math.h>
|
||||
|
||||
// FIXME: incorrect exception behavior
|
||||
|
||||
long lrint(double x)
|
||||
{
|
||||
return rint(x);
|
||||
}
|
||||
Some files were not shown because too many files have changed in this diff Show more
Loading…
Add table
Add a link
Reference in a new issue