jyapayne/Nim
1
0
Fork 0
Code Issues Pull requests Projects Releases Packages Wiki Activity Actions

Generic Complex type (#9590)

Browse source 
* remove `**`
* const `im` can now be used with Complex64
* converters from float|int to Complex are replaced by procs
* converters between various Complex types must stay to allow usage
of `im` with Complex64
* limit types for `+`, `-`, `/`, and `*` between Complex and float
* add `pow` for Complex and a number
* complex type changes
* unpublish approximation function
This commit is contained in:
Arne Döring 2018-11-05 20:27:46 +01:00 • committed by Andreas Rumpf
commit cc5b8c6ad2
2 changed files with 318 additions and 303 deletions
Download patch file Download diff file Expand all files Collapse all files

5
changelog.md
View file

@ -55,7 +55,10 @@ proc enumToString*(enums: openArray[enum]): string =
slightly. The `dumpLisp` macro in this module now outputs an slightly. The `dumpLisp` macro in this module now outputs an
indented proper Lisp, devoid of commas. indented proper Lisp, devoid of commas.
- In `strutils` empty strings now no longer matched as substrings anymore. - In `strutils` empty strings now no longer matched as substrings
anymore.
- Complex type is now generic and not a tuple anymore.
### Language additions ### Language additions

580
lib/pure/complex.nim
View file

@ -9,78 +9,109 @@
## This module implements complex numbers. ## This module implements complex numbers.
## Complex numbers are currently implemented as generic on a 64-bit or 32-bit float.
{.push checks: off, line_dir: off, stack_trace: off, debugger: off.} {.push checks: off, line_dir: off, stack_trace: off, debugger: off.}
# the user does not want to trace a part # the user does not want to trace a part of the standard library!
# of the standard library!
import math
import
math
const
EPS = 1.0e-7 ## Epsilon used for float comparisons.
type type
Complex* = tuple[re, im: float] Complex*[T: SomeFloat] = object
## a complex number, consisting of a real and an imaginary part re, im: T
## A complex number, consisting of a real and an imaginary part.
Complex64* = Complex[float64]
## Alias for a pair of 64-bit floats.
Complex32* = Complex[float32]
## Alias for a pair of 32-bit floats.
const proc complex*[T: SomeFloat](re: T; im: T = 0.0): Complex[T] =
im*: Complex = (re: 0.0, im: 1.0) result.re = re
## The imaginary unit. √-1. result.im = im
proc toComplex*(x: SomeInteger): Complex = proc complex32*(re: float32; im: float32 = 0.0): Complex[float32] =
## Convert some integer ``x`` to a complex number. result.re = re
result.re = x result.im = im
result.im = 0
proc `==` *(x, y: Complex): bool = proc complex64*(re: float64; im: float64 = 0.0): Complex[float64] =
## Compare two complex numbers `x` and `y` for equality. result.re = re
result.im = im
template im*(arg: typedesc[float32]): Complex32 = complex[float32](0, 1)
template im*(arg: typedesc[float64]): Complex64 = complex[float64](0, 1)
template im*(arg : float32): Complex32 = complex[float32](0, arg)
template im*(arg : float64): Complex64 = complex[float64](0, arg)
proc abs*[T](z: Complex[T]): T =
## Return the distance from (0,0) to ``z``.
result = hypot(z.re, z.im)
proc abs2*[T](z: Complex[T]): T =
## Return the squared distance from (0,0) to ``z``.
result = z.re*z.re + z.im*z.im
proc conjugate*[T](z: Complex[T]): Complex[T] =
## Conjugate of complex number ``z``.
result.re = z.re
result.im = -z.im
proc inv*[T](z: Complex[T]): Complex[T] =
## Multiplicative inverse of complex number ``z``.
conjugate(z) / abs2(z)
proc `==` *[T](x, y: Complex[T]): bool =
## Compare two complex numbers ``x`` and ``y`` for equality.
result = x.re == y.re and x.im == y.im result = x.re == y.re and x.im == y.im
proc `=~` *(x, y: Complex): bool = proc `+` *[T](x: T, y: Complex[T]): Complex[T] =
## Compare two complex numbers `x` and `y` approximately. ## Add a real number to a complex number.
result = abs(x.re-y.re)<EPS and abs(x.im-y.im)<EPS result.re = x + y.re
result.im = y.im
proc `+` *(x, y: Complex): Complex = proc `+` *[T](x: Complex[T], y: T): Complex[T] =
## Add a complex number to a real number.
result.re = x.re + y
result.im = x.im
proc `+` *[T](x, y: Complex[T]): Complex[T] =
## Add two complex numbers. ## Add two complex numbers.
result.re = x.re + y.re result.re = x.re + y.re
result.im = x.im + y.im result.im = x.im + y.im
proc `+` *(x: Complex, y: float): Complex = proc `-` *[T](z: Complex[T]): Complex[T] =
## Add complex `x` to float `y`.
result.re = x.re + y
result.im = x.im
proc `+` *(x: float, y: Complex): Complex =
## Add float `x` to complex `y`.
result.re = x + y.re
result.im = y.im
proc `-` *(z: Complex): Complex =
## Unary minus for complex numbers. ## Unary minus for complex numbers.
result.re = -z.re result.re = -z.re
result.im = -z.im result.im = -z.im
proc `-` *(x, y: Complex): Complex = proc `-` *[T](x: T, y: Complex[T]): Complex[T] =
## Subtract a complex number from a real number.
x + (-y)
proc `-` *[T](x: Complex[T], y: T): Complex[T] =
## Subtract a real number from a complex number.
result.re = x.re - y
result.im = x.im
proc `-` *[T](x, y: Complex[T]): Complex[T] =
## Subtract two complex numbers. ## Subtract two complex numbers.
result.re = x.re - y.re result.re = x.re - y.re
result.im = x.im - y.im result.im = x.im - y.im
proc `-` *(x: Complex, y: float): Complex = proc `/` *[T](x: Complex[T], y: T): Complex[T] =
## Subtracts float `y` from complex `x`. ## Divide complex number ``x`` by real number ``y``.
result = x + (-y) result.re = x.re / y
result.im = x.im / y
proc `-` *(x: float, y: Complex): Complex = proc `/` *[T](x: T, y: Complex[T]): Complex[T] =
## Subtracts complex `y` from float `x`. ## Divide real number ``x`` by complex number ``y``.
result = x + (-y) result = x * inv(y)
proc `/` *[T](x, y: Complex[T]): Complex[T] =
proc `/` *(x, y: Complex): Complex = ## Divide ``x`` by ``y``.
## Divide `x` by `y`. var r, den: T
var
r, den: float
if abs(y.re) < abs(y.im): if abs(y.re) < abs(y.im):
r = y.re / y.im r = y.re / y.im
den = y.im + r * y.re den = y.im + r * y.re
@ -92,101 +123,46 @@ proc `/` *(x, y: Complex): Complex =
result.re = (x.re + r * x.im) / den result.re = (x.re + r * x.im) / den
result.im = (x.im - r * x.re) / den result.im = (x.im - r * x.re) / den
proc `/` *(x : Complex, y: float ): Complex = proc `*` *[T](x: T, y: Complex[T]): Complex[T] =
## Divide complex `x` by float `y`. ## Multiply a real number and a complex number.
result.re = x.re/y
result.im = x.im/y
proc `/` *(x : float, y: Complex ): Complex =
## Divide float `x` by complex `y`.
var num : Complex = (x, 0.0)
result = num/y
proc `*` *(x, y: Complex): Complex =
## Multiply `x` with `y`.
result.re = x.re * y.re - x.im * y.im
result.im = x.im * y.re + x.re * y.im
proc `*` *(x: float, y: Complex): Complex =
## Multiply float `x` with complex `y`.
result.re = x * y.re result.re = x * y.re
result.im = x * y.im result.im = x * y.im
proc `*` *(x: Complex, y: float): Complex = proc `*` *[T](x: Complex[T], y: T): Complex[T] =
## Multiply complex `x` with float `y`. ## Multiply a complex number with a real number.
result.re = x.re * y result.re = x.re * y
result.im = x.im * y result.im = x.im * y
proc `*` *[T](x, y: Complex[T]): Complex[T] =
## Multiply ``x`` with ``y``.
result.re = x.re * y.re - x.im * y.im
result.im = x.im * y.re + x.re * y.im
proc `+=` *(x: var Complex, y: Complex) =
## Add `y` to `x`. proc `+=` *[T](x: var Complex[T], y: Complex[T]) =
## Add ``y`` to ``x``.
x.re += y.re x.re += y.re
x.im += y.im x.im += y.im
proc `+=` *(x: var Complex, y: float) = proc `-=` *[T](x: var Complex[T], y: Complex[T]) =
## Add `y` to the complex number `x`. ## Subtract ``y`` from ``x``.
x.re += y
proc `-=` *(x: var Complex, y: Complex) =
## Subtract `y` from `x`.
x.re -= y.re x.re -= y.re
x.im -= y.im x.im -= y.im
proc `-=` *(x: var Complex, y: float) = proc `*=` *[T](x: var Complex[T], y: Complex[T]) =
## Subtract `y` from the complex number `x`. ## Multiply ``y`` to ``x``.
x.re -= y
proc `*=` *(x: var Complex, y: Complex) =
## Multiply `y` to `x`.
let im = x.im * y.re + x.re * y.im let im = x.im * y.re + x.re * y.im
x.re = x.re * y.re - x.im * y.im x.re = x.re * y.re - x.im * y.im
x.im = im x.im = im
proc `*=` *(x: var Complex, y: float) = proc `/=` *[T](x: var Complex[T], y: Complex[T]) =
## Multiply `y` to the complex number `x`. ## Divide ``x`` by ``y`` in place.
x.re *= y
x.im *= y
proc `/=` *(x: var Complex, y: Complex) =
## Divide `x` by `y` in place.
x = x / y x = x / y
proc `/=` *(x : var Complex, y: float) =
## Divide complex `x` by float `y` in place.
x.re /= y
x.im /= y
proc sqrt*[T](z: Complex[T]): Complex[T] =
proc abs*(z: Complex): float = ## Square root for a complex number ``z``.
## Return the distance from (0,0) to `z`. var x, y, w, r: T
# optimized by checking special cases (sqrt is expensive)
var x, y, temp: float
x = abs(z.re)
y = abs(z.im)
if x == 0.0:
result = y
elif y == 0.0:
result = x
elif x > y:
temp = y / x
result = x * sqrt(1.0 + temp * temp)
else:
temp = x / y
result = y * sqrt(1.0 + temp * temp)
proc conjugate*(z: Complex): Complex =
## Conjugate of complex number `z`.
result.re = z.re
result.im = -z.im
proc sqrt*(z: Complex): Complex =
## Square root for a complex number `z`.
var x, y, w, r: float
if z.re == 0.0 and z.im == 0.0: if z.re == 0.0 and z.im == 0.0:
result = z result = z
@ -199,39 +175,37 @@ proc sqrt*(z: Complex): Complex =
else: else:
r = x / y r = x / y
w = sqrt(y) * sqrt(0.5 * (r + sqrt(1.0 + r * r))) w = sqrt(y) * sqrt(0.5 * (r + sqrt(1.0 + r * r)))
if z.re >= 0.0: if z.re >= 0.0:
result.re = w result.re = w
result.im = z.im / (w * 2.0) result.im = z.im / (w * 2.0)
else: else:
if z.im >= 0.0: result.im = w result.im = if z.im >= 0.0: w else: -w
else: result.im = -w
result.re = z.im / (result.im + result.im) result.re = z.im / (result.im + result.im)
proc exp*[T](z: Complex[T]): Complex[T] =
proc exp*(z: Complex): Complex = ## ``e`` raised to the power ``z``.
## e raised to the power `z`. var
var rho = exp(z.re) rho = exp(z.re)
var theta = z.im theta = z.im
result.re = rho * cos(theta) result.re = rho * cos(theta)
result.im = rho * sin(theta) result.im = rho * sin(theta)
proc ln*[T](z: Complex[T]): Complex[T] =
proc ln*(z: Complex): Complex = ## Returns the natural log of ``z``.
## Returns the natural log of `z`.
result.re = ln(abs(z)) result.re = ln(abs(z))
result.im = arctan2(z.im, z.re) result.im = arctan2(z.im, z.re)
proc log10*(z: Complex): Complex = proc log10*[T](z: Complex[T]): Complex[T] =
## Returns the log base 10 of `z`. ## Returns the log base 10 of ``z``.
result = ln(z) / ln(10.0) result = ln(z) / ln(10.0)
proc log2*(z: Complex): Complex = proc log2*[T](z: Complex[T]): Complex[T] =
## Returns the log base 2 of `z`. ## Returns the log base 2 of ``z``.
result = ln(z) / ln(2.0) result = ln(z) / ln(2.0)
proc pow*[T](x, y: Complex[T]): Complex[T] =
proc pow*(x, y: Complex): Complex = ## ``x`` raised to the power ``y``.
## `x` raised to the power `y`.
if x.re == 0.0 and x.im == 0.0: if x.re == 0.0 and x.im == 0.0:
if y.re == 0.0 and y.im == 0.0: if y.re == 0.0 and y.im == 0.0:
result.re = 1.0 result.re = 1.0
@ -242,204 +216,242 @@ proc pow*(x, y: Complex): Complex =
elif y.re == 1.0 and y.im == 0.0: elif y.re == 1.0 and y.im == 0.0:
result = x result = x
elif y.re == -1.0 and y.im == 0.0: elif y.re == -1.0 and y.im == 0.0:
result = 1.0/x result = T(1.0) / x
else: else:
var rho = sqrt(x.re*x.re + x.im*x.im) var
var theta = arctan2(x.im,x.re) rho = abs(x)
var s = pow(rho,y.re) * exp(-y.im*theta) theta = arctan2(x.im, x.re)
var r = y.re*theta + y.im*ln(rho) s = pow(rho, y.re) * exp(-y.im * theta)
r = y.re * theta + y.im * ln(rho)
result.re = s * cos(r) result.re = s * cos(r)
result.im = s * sin(r) result.im = s * sin(r)
proc pow*[T](x: Complex[T], y: T): Complex[T] =
## Complex number ``x`` raised to the power ``y``.
pow(x, complex[T](y))
proc sin*(z: Complex): Complex =
## Returns the sine of `z`. proc sin*[T](z: Complex[T]): Complex[T] =
## Returns the sine of ``z``.
result.re = sin(z.re) * cosh(z.im) result.re = sin(z.re) * cosh(z.im)
result.im = cos(z.re) * sinh(z.im) result.im = cos(z.re) * sinh(z.im)
proc arcsin*(z: Complex): Complex = proc arcsin*[T](z: Complex[T]): Complex[T] =
## Returns the inverse sine of `z`. ## Returns the inverse sine of ``z``.
var i: Complex = (0.0,1.0) result = -im(T) * ln(im(T) * z + sqrt(T(1.0) - z*z))
result = -i*ln(i*z + sqrt(1.0-z*z))
proc cos*(z: Complex): Complex = proc cos*[T](z: Complex[T]): Complex[T] =
## Returns the cosine of `z`. ## Returns the cosine of ``z``.
result.re = cos(z.re) * cosh(z.im) result.re = cos(z.re) * cosh(z.im)
result.im = -sin(z.re) * sinh(z.im) result.im = -sin(z.re) * sinh(z.im)
proc arccos*(z: Complex): Complex = proc arccos*[T](z: Complex[T]): Complex[T] =
## Returns the inverse cosine of `z`. ## Returns the inverse cosine of ``z``.
var i: Complex = (0.0,1.0) result = -im(T) * ln(z + sqrt(z*z - T(1.0)))
result = -i*ln(z + sqrt(z*z-1.0))
proc tan*(z: Complex): Complex = proc tan*[T](z: Complex[T]): Complex[T] =
## Returns the tangent of `z`. ## Returns the tangent of ``z``.
result = sin(z) / cos(z) result = sin(z) / cos(z)
proc arctan*(z: Complex): Complex = proc arctan*[T](z: Complex[T]): Complex[T] =
## Returns the inverse tangent of `z`. ## Returns the inverse tangent of ``z``.
var i: Complex = (0.0,1.0) result = T(0.5)*im(T) * (ln(T(1.0) - im(T)*z) - ln(T(1.0) + im(T)*z))
result = 0.5*i*(ln(1-i*z)-ln(1+i*z))
proc cot*(z: Complex): Complex = proc cot*[T](z: Complex[T]): Complex[T] =
## Returns the cotangent of `z`. ## Returns the cotangent of ``z``.
result = cos(z)/sin(z) result = cos(z)/sin(z)
proc arccot*(z: Complex): Complex = proc arccot*[T](z: Complex[T]): Complex[T] =
## Returns the inverse cotangent of `z`. ## Returns the inverse cotangent of ``z``.
var i: Complex = (0.0,1.0) result = T(0.5)*im(T) * (ln(T(1.0) - im(T)/z) - ln(T(1.0) + im(T)/z))
result = 0.5*i*(ln(1-i/z)-ln(1+i/z))
proc sec*(z: Complex): Complex = proc sec*[T](z: Complex[T]): Complex[T] =
## Returns the secant of `z`. ## Returns the secant of ``z``.
result = 1.0/cos(z) result = T(1.0) / cos(z)
proc arcsec*(z: Complex): Complex = proc arcsec*[T](z: Complex[T]): Complex[T] =
## Returns the inverse secant of `z`. ## Returns the inverse secant of ``z``.
var i: Complex = (0.0,1.0) result = -im(T) * ln(im(T) * sqrt(1.0 - 1.0/(z*z)) + T(1.0)/z)
result = -i*ln(i*sqrt(1-1/(z*z))+1/z)
proc csc*(z: Complex): Complex = proc csc*[T](z: Complex[T]): Complex[T] =
## Returns the cosecant of `z`. ## Returns the cosecant of ``z``.
result = 1.0/sin(z) result = T(1.0) / sin(z)
proc arccsc*(z: Complex): Complex = proc arccsc*[T](z: Complex[T]): Complex[T] =
## Returns the inverse cosecant of `z`. ## Returns the inverse cosecant of ``z``.
var i: Complex = (0.0,1.0) result = -im(T) * ln(sqrt(T(1.0) - T(1.0)/(z*z)) + im(T)/z)
result = -i*ln(sqrt(1-1/(z*z))+i/z)
proc sinh*[T](z: Complex[T]): Complex[T] =
## Returns the hyperbolic sine of ``z``.
result = T(0.5) * (exp(z) - exp(-z))
proc sinh*(z: Complex): Complex = proc arcsinh*[T](z: Complex[T]): Complex[T] =
## Returns the hyperbolic sine of `z`. ## Returns the inverse hyperbolic sine of ``z``.
result = 0.5*(exp(z)-exp(-z)) result = ln(z + sqrt(z*z + 1.0))
proc arcsinh*(z: Complex): Complex = proc cosh*[T](z: Complex[T]): Complex[T] =
## Returns the inverse hyperbolic sine of `z`. ## Returns the hyperbolic cosine of ``z``.
result = ln(z+sqrt(z*z+1)) result = T(0.5) * (exp(z) + exp(-z))
proc cosh*(z: Complex): Complex = proc arccosh*[T](z: Complex[T]): Complex[T] =
## Returns the hyperbolic cosine of `z`. ## Returns the inverse hyperbolic cosine of ``z``.
result = 0.5*(exp(z)+exp(-z)) result = ln(z + sqrt(z*z - T(1.0)))
proc arccosh*(z: Complex): Complex = proc tanh*[T](z: Complex[T]): Complex[T] =
## Returns the inverse hyperbolic cosine of `z`. ## Returns the hyperbolic tangent of ``z``.
result = ln(z+sqrt(z*z-1))
proc tanh*(z: Complex): Complex =
## Returns the hyperbolic tangent of `z`.
result = sinh(z) / cosh(z) result = sinh(z) / cosh(z)
proc arctanh*(z: Complex): Complex = proc arctanh*[T](z: Complex[T]): Complex[T] =
## Returns the inverse hyperbolic tangent of `z`. ## Returns the inverse hyperbolic tangent of ``z``.
result = 0.5*(ln((1+z)/(1-z))) result = T(0.5) * (ln((T(1.0)+z) / (T(1.0)-z)))
proc sech*(z: Complex): Complex = proc sech*[T](z: Complex[T]): Complex[T] =
## Returns the hyperbolic secant of `z`. ## Returns the hyperbolic secant of ``z``.
result = 2/(exp(z)+exp(-z)) result = T(2.0) / (exp(z) + exp(-z))
proc arcsech*(z: Complex): Complex = proc arcsech*[T](z: Complex[T]): Complex[T] =
## Returns the inverse hyperbolic secant of `z`. ## Returns the inverse hyperbolic secant of ``z``.
result = ln(1/z+sqrt(1/z+1)*sqrt(1/z-1)) result = ln(1.0/z + sqrt(T(1.0)/z+T(1.0)) * sqrt(T(1.0)/z-T(1.0)))
proc csch*(z: Complex): Complex = proc csch*[T](z: Complex[T]): Complex[T] =
## Returns the hyperbolic cosecant of `z`. ## Returns the hyperbolic cosecant of ``z``.
result = 2/(exp(z)-exp(-z)) result = T(2.0) / (exp(z) - exp(-z))
proc arccsch*(z: Complex): Complex = proc arccsch*[T](z: Complex[T]): Complex[T] =
## Returns the inverse hyperbolic cosecant of `z`. ## Returns the inverse hyperbolic cosecant of ``z``.
result = ln(1/z+sqrt(1/(z*z)+1)) result = ln(T(1.0)/z + sqrt(T(1.0)/(z*z) + T(1.0)))
proc coth*(z: Complex): Complex = proc coth*[T](z: Complex[T]): Complex[T] =
## Returns the hyperbolic cotangent of `z`. ## Returns the hyperbolic cotangent of ``z``.
result = cosh(z) / sinh(z) result = cosh(z) / sinh(z)
proc arccoth*(z: Complex): Complex = proc arccoth*[T](z: Complex[T]): Complex[T] =
## Returns the inverse hyperbolic cotangent of `z`. ## Returns the inverse hyperbolic cotangent of ``z``.
result = 0.5*(ln(1+1/z)-ln(1-1/z)) result = T(0.5) * (ln(T(1.0) + T(1.0)/z) - ln(T(1.0) - T(1.0)/z))
proc phase*(z: Complex): float = proc phase*[T](z: Complex[T]): T =
## Returns the phase of `z`. ## Returns the phase of ``z``.
arctan2(z.im, z.re) arctan2(z.im, z.re)
proc polar*(z: Complex): tuple[r, phi: float] = proc polar*[T](z: Complex[T]): tuple[r, phi: T] =
## Returns `z` in polar coordinates. ## Returns ``z`` in polar coordinates.
result.r = abs(z) (r: abs(z), phi: phase(z))
result.phi = phase(z)
proc rect*(r: float, phi: float): Complex = proc rect*[T](r, phi: T): Complex[T] =
## Returns the complex number with polar coordinates `r` and `phi`. ## Returns the complex number with polar coordinates ``r`` and ``phi``.
result.re = r * cos(phi) ##
result.im = r * sin(phi) ## | ``result.re = r * cos(phi)``
## | ``result.im = r * sin(phi)``
complex(r * cos(phi), r * sin(phi))
proc `$`*(z: Complex): string = proc `$`*(z: Complex): string =
## Returns `z`'s string representation as ``"(re, im)"``. ## Returns ``z``'s string representation as ``"(re, im)"``.
result = "(" & $z.re & ", " & $z.im & ")" result = "(" & $z.re & ", " & $z.im & ")"
{.pop.} {.pop.}
when isMainModule: when isMainModule:
var z = (0.0, 0.0) proc `=~`[T](x, y: Complex[T]): bool =
var oo = (1.0,1.0) result = abs(x.re-y.re) < 1e-6 and abs(x.im-y.im) < 1e-6
var a = (1.0, 2.0)
var b = (-1.0, -2.0)
var m1 = (-1.0, 0.0)
var i = (0.0,1.0)
var one = (1.0,0.0)
var tt = (10.0, 20.0)
var ipi = (0.0, -PI)
assert( a == a ) proc `=~`[T](x: Complex[T], y: T): bool =
assert( (a-a) == z ) result = abs(x.re-y) < 1e-6 and abs(x.im) < 1e-6
assert( (a+b) == z )
assert( (a/b) == m1 )
assert( (1.0/a) == (0.2, -0.4) )
assert( (a*b) == (3.0, -4.0) )
assert( 10.0*a == tt )
assert( a*10.0 == tt )
assert( tt/10.0 == a )
assert( oo+(-1.0) == i )
assert( (-1.0)+oo == i )
assert( abs(oo) == sqrt(2.0) )
assert( conjugate(a) == (1.0, -2.0) )
assert( sqrt(m1) == i )
assert( exp(ipi) =~ m1 )
assert( pow(a,b) =~ (-3.72999124927876, -1.68815826725068) ) var
assert( pow(z,a) =~ (0.0, 0.0) ) z: Complex64 = complex(0.0, 0.0)
assert( pow(z,z) =~ (1.0, 0.0) ) oo: Complex64 = complex(1.0, 1.0)
assert( pow(a,one) =~ a ) a: Complex64 = complex(1.0, 2.0)
assert( pow(a,m1) =~ (0.2, -0.4) ) b: Complex64 = complex(-1.0, -2.0)
m1: Complex64 = complex(-1.0, 0.0)
i: Complex64 = complex(0.0, 1.0)
one: Complex64 = complex(1.0, 0.0)
tt: Complex64 = complex(10.0, 20.0)
ipi: Complex64 = complex(0.0, -PI)
assert( ln(a) =~ (0.804718956217050, 1.107148717794090) ) doAssert(a/2.0 =~ complex(0.5, 1.0))
assert( log10(a) =~ (0.349485002168009, 0.480828578784234) ) doAssert(a == a)
assert( log2(a) =~ (1.16096404744368, 1.59727796468811) ) doAssert((a-a) == z)
doAssert((a+b) == z)
doAssert((a+b) =~ 0.0)
doAssert((a/b) == m1)
doAssert((1.0/a) == complex(0.2, -0.4))
doAssert((a*b) == complex(3.0, -4.0))
doAssert(10.0*a == tt)
doAssert(a*10.0 == tt)
doAssert(tt/10.0 == a)
doAssert(oo+(-1.0) == i)
doAssert( (-1.0)+oo == i)
doAssert(abs(oo) == sqrt(2.0))
doAssert(conjugate(a) == complex(1.0, -2.0))
doAssert(sqrt(m1) == i)
doAssert(exp(ipi) =~ m1)
assert( sin(a) =~ (3.16577851321617, 1.95960104142161) ) doAssert(pow(a, b) =~ complex(-3.72999124927876, -1.68815826725068))
assert( cos(a) =~ (2.03272300701967, -3.05189779915180) ) doAssert(pow(z, a) =~ complex(0.0, 0.0))
assert( tan(a) =~ (0.0338128260798967, 1.0147936161466335) ) doAssert(pow(z, z) =~ complex(1.0, 0.0))
assert( cot(a) =~ 1.0/tan(a) ) doAssert(pow(a, one) =~ a)
assert( sec(a) =~ 1.0/cos(a) ) doAssert(pow(a, m1) =~ complex(0.2, -0.4))
assert( csc(a) =~ 1.0/sin(a) ) doAssert(pow(a, 2.0) =~ complex(-3.0, 4.0))
assert( arcsin(a) =~ (0.427078586392476, 1.528570919480998) ) doAssert(pow(a, 2) =~ complex(-3.0, 4.0))
assert( arccos(a) =~ (1.14371774040242, -1.52857091948100) ) doAssert(not(pow(a, 2.0) =~ a))
assert( arctan(a) =~ (1.338972522294494, 0.402359478108525) )
assert( cosh(a) =~ (-0.642148124715520, 1.068607421382778) ) doAssert(ln(a) =~ complex(0.804718956217050, 1.107148717794090))
assert( sinh(a) =~ (-0.489056259041294, 1.403119250622040) ) doAssert(log10(a) =~ complex(0.349485002168009, 0.480828578784234))
assert( tanh(a) =~ (1.1667362572409199,-0.243458201185725) ) doAssert(log2(a) =~ complex(1.16096404744368, 1.59727796468811))
assert( sech(a) =~ 1/cosh(a) )
assert( csch(a) =~ 1/sinh(a) )
assert( coth(a) =~ 1/tanh(a) )
assert( arccosh(a) =~ (1.528570919480998, 1.14371774040242) )
assert( arcsinh(a) =~ (1.469351744368185, 1.06344002357775) )
assert( arctanh(a) =~ (0.173286795139986, 1.17809724509617) )
assert( arcsech(a) =~ arccosh(1/a) )
assert( arccsch(a) =~ arcsinh(1/a) )
assert( arccoth(a) =~ arctanh(1/a) )
assert( phase(a) == 1.1071487177940904 ) doAssert(sin(a) =~ complex(3.16577851321617, 1.95960104142161))
doAssert(cos(a) =~ complex(2.03272300701967, -3.05189779915180))
doAssert(tan(a) =~ complex(0.0338128260798967, 1.0147936161466335))
doAssert(cot(a) =~ 1.0 / tan(a))
doAssert(sec(a) =~ 1.0 / cos(a))
doAssert(csc(a) =~ 1.0 / sin(a))
doAssert(arcsin(a) =~ complex(0.427078586392476, 1.528570919480998))
doAssert(arccos(a) =~ complex(1.14371774040242, -1.52857091948100))
doAssert(arctan(a) =~ complex(1.338972522294494, 0.402359478108525))
doAssert(arccot(a) =~ complex(0.2318238045004031, -0.402359478108525))
doAssert(arcsec(a) =~ complex(1.384478272687081, 0.3965682301123288))
doAssert(arccsc(a) =~ complex(0.1863180541078155, -0.3965682301123291))
doAssert(cosh(a) =~ complex(-0.642148124715520, 1.068607421382778))
doAssert(sinh(a) =~ complex(-0.489056259041294, 1.403119250622040))
doAssert(tanh(a) =~ complex(1.1667362572409199, -0.243458201185725))
doAssert(sech(a) =~ 1.0 / cosh(a))
doAssert(csch(a) =~ 1.0 / sinh(a))
doAssert(coth(a) =~ 1.0 / tanh(a))
doAssert(arccosh(a) =~ complex(1.528570919480998, 1.14371774040242))
doAssert(arcsinh(a) =~ complex(1.469351744368185, 1.06344002357775))
doAssert(arctanh(a) =~ complex(0.173286795139986, 1.17809724509617))
doAssert(arcsech(a) =~ arccosh(1.0/a))
doAssert(arccsch(a) =~ arcsinh(1.0/a))
doAssert(arccoth(a) =~ arctanh(1.0/a))
doAssert(phase(a) == 1.1071487177940904)
var t = polar(a) var t = polar(a)
assert( rect(t.r, t.phi) =~ a ) doAssert(rect(t.r, t.phi) =~ a)
assert( rect(1.0, 2.0) =~ (-0.4161468365471424, 0.9092974268256817) ) doAssert(rect(1.0, 2.0) =~ complex(-0.4161468365471424, 0.9092974268256817))
var
i64: Complex32 = complex(0.0f, 1.0f)
a64: Complex32 = 2.0f*i64 + 1.0.float32
b64: Complex32 = complex(-1.0'f32, -2.0'f32)
doAssert(a64 == a64)
doAssert(a64 == -b64)
doAssert(a64 + b64 =~ 0.0'f32)
doAssert(not(pow(a64, b64) =~ a64))
doAssert(pow(a64, 0.5f) =~ sqrt(a64))
doAssert(pow(a64, 2) =~ complex(-3.0'f32, 4.0'f32))
doAssert(sin(arcsin(b64)) =~ b64)
doAssert(cosh(arccosh(a64)) =~ a64)
doAssert(phase(a64) - 1.107149f < 1e-6)
var t64 = polar(a64)
doAssert(rect(t64.r, t64.phi) =~ a64)
doAssert(rect(1.0f, 2.0f) =~ complex(-0.4161468f, 0.90929742f))
doAssert(sizeof(a64) == 8)
doAssert(sizeof(a) == 16)
doAssert 123.0.im + 456.0 == complex64(456, 123)