* simplify sin_pi(x) (don't care about inexact here, the result is
inexact anyway, and x is not so small to underflow)
* in lgammal add the previously removed special case for x==1 and
x==2 (to fix the sign of zero in downward rounding mode)
* only define lgammal on supported long double platforms
* change tgamma so the generated code is a bit smaller
The log, log2 and log10 functions share a lot of code and to a lesser
extent log1p too. A small part of the code was kept separately in
__log1p.h, but since it did not capture much of the common code and
it was inlined anyway, it did not solve the issue properly. Now the
log functions have significant code duplication, which may be resolved
later, until then they need to be modified together.
logl, log10l, log2l, log1pl:
* Fix the sign when the return value should be -inf.
* Remove the volatile hack from log10l (seems unnecessary)
log1p, log1pf:
* Change the handling of small inputs: only |x|<2^-53 is special
(then it is enough to return x with the usual subnormal handling)
this fixes the sign of log1p(0) in downward rounding.
* Do not handle the k==0 case specially (other than skipping the
elaborate argument reduction)
* Do not handle 1+x close to power-of-two specially (this code was
used rarely, did not give much speed up and the precision wasn't
better than the general)
* Fix the correction term formula (c=1-(u-x) was used incorrectly
when x<1 but (double)(x+1)==2, this was not a critical issue)
* Use the exact same method for calculating log(1+f) as in log
(except in log1p the c correction term is added to the result).
log, logf, log10, log10f, log2, log2f:
* Use double_t and float_t consistently.
* Now the first part of log10 and log2 is identical to log (until the
return statement, hopefully this makes maintainence easier).
* Most special case formulas were removed (close to power-of-two and
k==0 cases), they increase the code size without providing precision
or performance benefits (and obfuscate the code).
Only x==1 is handled specially so in downward rounding mode the
sign of zero is correct (the general formula happens to give -0).
* For x==0 instead of -1/0.0 or -two54/0.0, return -1/(x*x) to force
raising the exception at runtime.
* Arg reduction code is changed (slightly simplified)
* The thresholds for arg reduction to [sqrt(2)/2,sqrt(2)] are now
consistently the [0x3fe6a09e00000000,0x3ff6a09dffffffff] and the
[0x3f3504f3,0x3fb504f2] intervals for double and float reductions
respectively (the exact threshold values are not critical)
* Remove the obsolete comment for the FLT_EVAL_METHOD!=0 case in log2f
(The same code is used for all eval methods now, on i386 slightly
simpler code could be used, but we have asm there anyway)
all:
* Fix signed int arithmetics (using unsigned for bitmanipulation)
* Fix various comments
the underlying problem was not incorrect sign extension (fixed in the
previous commit to this file by nsz) but that code that treats "long"
as 32-bit was copied blindly from i386 to x86_64.
now lrintl is identical to llrintl on x86_64, as it should be.
gcc did not always drop excess precision according to c99 at assignments
before version 4.5 even if -std=c99 was requested which caused badly
broken mathematical functions on i386 when FLT_EVAL_METHOD!=0
but STRICT_ASSIGN was not used consistently and it is worked around for
old compilers with -ffloat-store so it is no longer needed
the new convention is to get the compiler respect c99 semantics and when
excess precision is not harmful use float_t or double_t or to specialize
code using FLT_EVAL_METHOD
apparently gnulib requires invalid long double representations
to be handled correctly in printf so we classify them according
to how the fpu treats them: bad inf is nan, bad nan is nan,
bad normal is nan and bad subnormal/zero is minimal normal
in atanh exception handling was left to the called log functions,
but the argument to those functions could underflow or overflow.
use double_t and float_t to avoid some useless stores on x86
acosh(x) is invalid for x<1, acoshf tried to be clever using
signed comparisions to handle all x<2 the same way, but the
formula was wrong on large negative values.
there were two problems:
* omitted underflow on subnormal results: exp2l(-16383.5) was calculated
as sqrt(2)*2^-16384, the last bits of sqrt(2) are zero so the down scaling
does not underflow eventhough the result is in subnormal range
* spurious underflow for subnormal inputs: exp2l(0x1p-16400) was evaluated
as f2xm1(x)+1 and f2xm1 raised underflow (because inexact subnormal result)
the first issue is fixed by raising underflow manually if x is in
(-32768,-16382] and not integer (x-0x1p63+0x1p63 != x)
the second issue is fixed by treating x in (-0x1p64,0x1p64) specially
for these fixes the special case handling was completely rewritten
* use float_t and double_t
* cleanup subnormal handling
* bithacks according to the new convention (ldshape for long double
and explicit unions for float and double)
* don't care about inexact flag
* use double_t and float_t (faster, smaller, more precise on x86)
* exp: underflow when result is zero or subnormal and not -inf
* exp2: underflow when result is zero or subnormal and not exact
* expm1: underflow when result is zero or subnormal
* expl: don't underflow on -inf
* exp2: fix incorrect comment
* expm1: simplify special case handling and overflow properly
* expm1: cleanup final scaling and fix negative left shift ub (twopk)
ld128 support was added to internal kernel functions (__cosl, __sinl,
__tanl, __rem_pio2l) from freebsd (not tested, but should be a good
start for when ld128 arch arrives)
__rem_pio2l had some code cleanup, the freebsd ld128 code seems to
gather the results of a large reduction with precision loss (fixed
the bug but a todo comment was added for later investigation)
the old copyright was removed from the non-kernel wrapper functions
(cosl, sinl, sincosl, tanl) since these are trivial and the interesting
parts and comments had been already rewritten.
method: if there is a large difference between the scale of x and y
then the larger magnitude dominates, otherwise reduce x,y so the
argument of sqrt (x*x+y*y) does not overflow or underflow and calculate
the argument precisely using exact multiplication. If the argument
has less error than 1/sqrt(2) ~ 0.7 ulp, then the result has less error
than 1 ulp in nearest rounding mode.
the original fdlibm method was the same, except it used bit hacks
instead of dekker-veltkamp algorithm, which is problematic for long
double where different representations are supported. (the new hypot
and hypotl code should be smaller and faster on 32bit cpu archs with
fast fpu), the new code behaves differently in non-nearest rounding,
but the error should be still less than 2ulps.
ld80 and ld128 are supported
* results are exact
* modfl follows truncl (raises inexact flag spuriously now)
* modf and modff only had cosmetic cleanup
* remainder is just a wrapper around remquo now
* using iterative shift+subtract for remquo and fmod
* ld80 and ld128 are supported as well
* faster, smaller, cleaner implementation than the bit hacks of fdlibm
* use arithmetics like y=(double)(x+0x1p52)-0x1p52, which is an integer
neighbor of x in all rounding modes (0<=x<0x1p52) and only use bithacks
when that's faster and smaller (for float it usually is)
* the code assumes standard excess precision handling for casts
* long double code supports both ld80 and ld128
* nearbyint is not changed (it is a wrapper around rint)
* consistent code style
* explicit union instead of typedef for double and float bit access
* turn FENV_ACCESS ON to make 0/0.0f raise invalid flag
* (untested) ld128 version of ilogbl (used by logbl which has ld128 support)
new ldshape union, ld128 support is kept, code that used the old
ldshape union was rewritten (IEEEl2bits union of freebsd libm is
not touched yet)
ld80 __fpclassifyl no longer tries to handle invalid representation
if FLT_EVAL_METHOD!=0 check if (double)(1/x) is subnormal and not a
power of 2 (if 1/x is power of 2 then either it is exact or the
long double to double rounding already raised inexact and underflow)
* remove volatile hacks
* don't care about inexact flag for now (removed all the +-tiny)
* fix atanl to raise underflow properly
* remove signed int arithmetics
* use pi/2 instead of pi_o_2 (gcc generates the same code, which is not
correct, but it does not matter: we mainly care about nearest rounding)
underflow is raised by an inexact subnormal float store,
since subnormal operations are slow, check the underflow
flag and skip the store if it's already raised
for these functions f(x)=x for small inputs, because f(0)=0 and
f'(0)=1, but for subnormal values they should raise the underflow
flag (required by annex F), if they are approximated by a polynomial
around 0 then spurious underflow should be avoided (not required by
annex F)
all these functions should raise inexact flag for small x if x!=0,
but it's not required by the standard and it does not seem a worthy
goal, so support for it is removed in some cases.
raising underflow:
- x*x may not raise underflow for subnormal x if FLT_EVAL_METHOD!=0
- x*x may raise spurious underflow for normal x if FLT_EVAL_METHOD==0
- in case of double subnormal x, store x as float
- in case of float subnormal x, store x*x as float
The underflow exception is not raised correctly in some
cornercases (see previous fma commit), added comments
with examples for fmaf, fmal and non-x86 fma.
In fmaf store the result before returning so it has the
correct precision when FLT_EVAL_METHOD!=0
1) in downward rounding fma(1,1,-1) should be -0 but it was 0 with
gcc, the code was correct but gcc does not support FENV_ACCESS ON
so it used common subexpression elimination where it shouldn't have.
now volatile memory access is used as a barrier after fesetround.
2) in directed rounding modes there is no double rounding issue
so the complicated adjustments done for nearest rounding mode are
not needed. the only exception to this rule is raising the underflow
flag: assume "small" is an exactly representible subnormal value in
double precision and "verysmall" is a much smaller value so that
(long double)(small plus verysmall) == small
then
(double)(small plus verysmall)
raises underflow because the result is an inexact subnormal, but
(double)(long double)(small plus verysmall)
does not because small is not a subnormal in long double precision
and it is exact in double precision.
now this problem is fixed by checking inexact using fenv when the
result is subnormal
* use unsigned arithmetics
* use unsigned to store arg reduction quotient (so n&3 is understood)
* remove z=0.0 variables, use literal 0
* raise underflow and inexact exceptions properly when x is small
* fix spurious underflow in tanl
* use unsigned arithmetics on the representation
* store arg reduction quotient in unsigned (so n%2 would work like n&1)
* use different convention to pass the arg reduction bit to __tan
(this argument used to be 1 for even and -1 for odd reduction
which meant obscure bithacks, the new n&1 is cleaner)
* raise inexact and underflow flags correctly for small x
(tanl(x) may still raise spurious underflow for small but normal x)
(this exception raising code increases codesize a bit, similar fixes
are needed in many other places, it may worth investigating at some
point if the inexact and underflow flags are worth raising correctly
as this is not strictly required by the standard)
* tanf manual reduction optimization is kept for now
* tanl code path is cleaned up to follow similar logic to tan and tanf
When FLT_EVAL_METHOD!=0 (only i386 with x87 fp) the excess
precision of an expression must be removed in an assignment.
(gcc needs -fexcess-precision=standard or -std=c99 for this)
This is done by extra load/store instructions which adds code
bloat when lot of temporaries are used and it makes the result
less precise in many cases.
Using double_t and float_t avoids these issues on i386 and
it makes no difference on other archs.
For now only a few functions are modified where the excess
precision is clearly beneficial (mostly polynomial evaluations
with temporaries).
object size differences on i386, gcc-4.8:
old new
__cosdf.o 123 95
__cos.o 199 169
__sindf.o 131 95
__sin.o 225 203
__tandf.o 207 151
__tan.o 605 499
erff.o 1470 1416
erf.o 1703 1649
j0f.o 1779 1745
j0.o 2308 2274
j1f.o 1602 1568
j1.o 2286 2252
tgamma.o 1431 1424
math/*.o 64164 63635
common part of erf and erfc was put in a separate function which
saved some space and the new code is using unsigned arithmetics
erfcf had a bug: for some inputs in [7.95,8] the result had
more than 60ulp error: in expf(-z*z - 0.5625f) the argument
must be exact but not enough lowbits of z were zeroed,
-SET_FLOAT_WORD(z, ix&0xfffff000);
+SET_FLOAT_WORD(z, ix&0xffffe000);
fixed the issue
