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cmath: new package

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Complex math function package. Still needs more special case checking.

R=rsc
CC=golang-dev
https://golang.org/cl/874041
This commit is contained in:
Charles L. Dorian 2010-04-05 22:10:27 -07:00 committed by Russ Cox
parent d08728f1e1
commit 2e90f66eff
17 changed files with 1715 additions and 0 deletions
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1
src/pkg/Makefile
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@ -26,6 +26,7 @@ DIRS=\
bignum\
bufio\
bytes\
cmath\
compress/flate\
compress/gzip\
compress/zlib\

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src/pkg/cmath/Makefile Normal file
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# Copyright 2010 The Go Authors. All rights reserved.
# Use of this source code is governed by a BSD-style
# license that can be found in the LICENSE file.
include ../../Make.$(GOARCH)
TARG=cmath
GOFILES=\
abs.go\
asin.go\
conj.go\
exp.go\
isinf.go\
isnan.go\
log.go\
phase.go\
polar.go\
pow.go\
rect.go\
sin.go\
sqrt.go\
tan.go\
include ../../Make.pkg

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src/pkg/cmath/abs.go Normal file
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// Copyright 2010 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package cmath
import "math"
// Abs returns the absolute value (also called the modulus) of x.
func Abs(x complex128) float64 { return math.Hypot(real(x), imag(x)) }

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src/pkg/cmath/asin.go Normal file
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// Copyright 2010 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package cmath
import "math"
// The original C code, the long comment, and the constants
// below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c.
// The go code is a simplified version of the original C.
//
// Cephes Math Library Release 2.8: June, 2000
// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
//
// The readme file at http://netlib.sandia.gov/cephes/ says:
// Some software in this archive may be from the book _Methods and
// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
// International, 1989) or from the Cephes Mathematical Library, a
// commercial product. In either event, it is copyrighted by the author.
// What you see here may be used freely but it comes with no support or
// guarantee.
//
// The two known misprints in the book are repaired here in the
// source listings for the gamma function and the incomplete beta
// integral.
//
// Stephen L. Moshier
// moshier@na-net.ornl.gov
// Complex circular arc sine
//
// DESCRIPTION:
//
// Inverse complex sine:
// 2
// w = -i clog( iz + csqrt( 1 - z ) ).
//
// casin(z) = -i casinh(iz)
//
// ACCURACY:
//
// Relative error:
// arithmetic domain # trials peak rms
// DEC -10,+10 10100 2.1e-15 3.4e-16
// IEEE -10,+10 30000 2.2e-14 2.7e-15
// Larger relative error can be observed for z near zero.
// Also tested by csin(casin(z)) = z.
// Asin returns the inverse sine of x.
func Asin(x complex128) complex128 {
if imag(x) == 0 {
if math.Fabs(real(x)) > 1 {
return cmplx(math.Pi/2, 0) // DOMAIN error
}
return cmplx(math.Asin(real(x)), 0)
}
ct := cmplx(-imag(x), real(x)) // i * x
xx := x * x
x1 := cmplx(1-real(xx), -imag(xx)) // 1 - x*x
x2 := Sqrt(x1) // x2 = sqrt(1 - x*x)
w := Log(ct + x2)
return cmplx(imag(w), -real(w)) // -i * w
}
// Asinh returns the inverse hyperbolic sine of x.
func Asinh(x complex128) complex128 {
// TODO check range
if imag(x) == 0 {
if math.Fabs(real(x)) > 1 {
return cmplx(math.Pi/2, 0) // DOMAIN error
}
return cmplx(math.Asinh(real(x)), 0)
}
xx := x * x
x1 := cmplx(1+real(xx), imag(xx)) // 1 + x*x
return Log(x + Sqrt(x1)) // log(x + sqrt(1 + x*x))
}
// Complex circular arc cosine
//
// DESCRIPTION:
//
// w = arccos z = PI/2 - arcsin z.
//
// ACCURACY:
//
// Relative error:
// arithmetic domain # trials peak rms
// DEC -10,+10 5200 1.6e-15 2.8e-16
// IEEE -10,+10 30000 1.8e-14 2.2e-15
// Acos returns the inverse cosine of x.
func Acos(x complex128) complex128 {
w := Asin(x)
return cmplx(math.Pi/2-real(w), -imag(w))
}
// Acosh returns the inverse hyperbolic cosine of x.
func Acosh(x complex128) complex128 {
w := Acos(x)
if imag(w) <= 0 {
return cmplx(-imag(w), real(w)) // i * w
}
return cmplx(imag(w), -real(w)) // -i * w
}
// Complex circular arc tangent
//
// DESCRIPTION:
//
// If
// z = x + iy,
//
// then
// 1 ( 2x )
// Re w = - arctan(-----------) + k PI
// 2 ( 2 2)
// (1 - x - y )
//
// ( 2 2)
// 1 (x + (y+1) )
// Im w = - log(------------)
// 4 ( 2 2)
// (x + (y-1) )
//
// Where k is an arbitrary integer.
//
// catan(z) = -i catanh(iz).
//
// ACCURACY:
//
// Relative error:
// arithmetic domain # trials peak rms
// DEC -10,+10 5900 1.3e-16 7.8e-18
// IEEE -10,+10 30000 2.3e-15 8.5e-17
// The check catan( ctan(z) ) = z, with |x| and |y| < PI/2,
// had peak relative error 1.5e-16, rms relative error
// 2.9e-17. See also clog().
// Atan returns the inverse tangent of x.
func Atan(x complex128) complex128 {
if real(x) == 0 && imag(x) > 1 {
return NaN()
}
x2 := real(x) * real(x)
a := 1 - x2 - imag(x)*imag(x)
if a == 0 {
return NaN()
}
t := 0.5 * math.Atan2(2*real(x), a)
w := reducePi(t)
t = imag(x) - 1
b := x2 + t*t
if b == 0 {
return NaN()
}
t = imag(x) + 1
c := (x2 + t*t) / b
return cmplx(w, 0.25*math.Log(c))
}
// Atanh returns the inverse hyperbolic tangent of x.
func Atanh(x complex128) complex128 {
z := cmplx(-imag(x), real(x)) // z = i * x
z = Atan(z)
return cmplx(imag(z), -real(z)) // z = -i * z
}

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src/pkg/cmath/cmath_test.go Normal file
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// Copyright 2010 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package cmath
import (
"math"
"testing"
)
var vc26 = []complex128{
(4.97901192488367350108546816 + 7.73887247457810456552351752i),
(7.73887247457810456552351752 - 0.27688005719200159404635997i),
(-0.27688005719200159404635997 - 5.01060361827107492160848778i),
(-5.01060361827107492160848778 + 9.63629370719841737980004837i),
(9.63629370719841737980004837 + 2.92637723924396464525443662i),
(2.92637723924396464525443662 + 5.22908343145930665230025625i),
(5.22908343145930665230025625 + 2.72793991043601025126008608i),
(2.72793991043601025126008608 + 1.82530809168085506044576505i),
(1.82530809168085506044576505 - 8.68592476857560136238589621i),
(-8.68592476857560136238589621 + 4.97901192488367350108546816i),
}
var vc = []complex128{
(4.9790119248836735e+00 + 7.7388724745781045e+00i),
(7.7388724745781045e+00 - 2.7688005719200159e-01i),
(-2.7688005719200159e-01 - 5.0106036182710749e+00i),
(-5.0106036182710749e+00 + 9.6362937071984173e+00i),
(9.6362937071984173e+00 + 2.9263772392439646e+00i),
(2.9263772392439646e+00 + 5.2290834314593066e+00i),
(5.2290834314593066e+00 + 2.7279399104360102e+00i),
(2.7279399104360102e+00 + 1.8253080916808550e+00i),
(1.8253080916808550e+00 - 8.6859247685756013e+00i),
(-8.6859247685756013e+00 + 4.9790119248836735e+00i),
}
// The expected results below were computed by the high precision calculators
// at http://keisan.casio.com/. More exact input values (array vc[], above)
// were obtained by printing them with "%.26f". The answers were calculated
// to 26 digits (by using the "Digit number" drop-down control of each
// calculator). Twenty-six digits were chosen so that the answers would be
// accurate even for a float128 type.
var abs = []float64{
9.2022120669932650313380972e+00,
7.7438239742296106616261394e+00,
5.0182478202557746902556648e+00,
1.0861137372799545160704002e+01,
1.0070841084922199607011905e+01,
5.9922447613166942183705192e+00,
5.8978784056736762299945176e+00,
3.2822866700678709020367184e+00,
8.8756430028990417290744307e+00,
1.0011785496777731986390856e+01,
}
var acos = []complex128{
(1.0017679804707456328694569 - 2.9138232718554953784519807i),
(0.03606427612041407369636057 + 2.7358584434576260925091256i),
(1.6249365462333796703711823 + 2.3159537454335901187730929i),
(2.0485650849650740120660391 - 3.0795576791204117911123886i),
(0.29621132089073067282488147 - 3.0007392508200622519398814i),
(1.0664555914934156601503632 - 2.4872865024796011364747111i),
(0.48681307452231387690013905 - 2.463655912283054555225301i),
(0.6116977071277574248407752 - 1.8734458851737055262693056i),
(1.3649311280370181331184214 + 2.8793528632328795424123832i),
(2.6189310485682988308904501 - 2.9956543302898767795858704i),
}
var acosh = []complex128{
(2.9138232718554953784519807 + 1.0017679804707456328694569i),
(2.7358584434576260925091256 - 0.03606427612041407369636057i),
(2.3159537454335901187730929 - 1.6249365462333796703711823i),
(3.0795576791204117911123886 + 2.0485650849650740120660391i),
(3.0007392508200622519398814 + 0.29621132089073067282488147i),
(2.4872865024796011364747111 + 1.0664555914934156601503632i),
(2.463655912283054555225301 + 0.48681307452231387690013905i),
(1.8734458851737055262693056 + 0.6116977071277574248407752i),
(2.8793528632328795424123832 - 1.3649311280370181331184214i),
(2.9956543302898767795858704 + 2.6189310485682988308904501i),
}
var asin = []complex128{
(0.56902834632415098636186476 + 2.9138232718554953784519807i),
(1.5347320506744825455349611 - 2.7358584434576260925091256i),
(-0.054140219438483051139860579 - 2.3159537454335901187730929i),
(-0.47776875817017739283471738 + 3.0795576791204117911123886i),
(1.2745850059041659464064402 + 3.0007392508200622519398814i),
(0.50434073530148095908095852 + 2.4872865024796011364747111i),
(1.0839832522725827423311826 + 2.463655912283054555225301i),
(0.9590986196671391943905465 + 1.8734458851737055262693056i),
(0.20586519875787848611290031 - 2.8793528632328795424123832i),
(-1.0481347217734022116591284 + 2.9956543302898767795858704i),
}
var asinh = []complex128{
(2.9113760469415295679342185 + 0.99639459545704326759805893i),
(2.7441755423994259061579029 - 0.035468308789000500601119392i),
(-2.2962136462520690506126678 - 1.5144663565690151885726707i),
(-3.0771233459295725965402455 + 1.0895577967194013849422294i),
(3.0048366100923647417557027 + 0.29346979169819220036454168i),
(2.4800059370795363157364643 + 1.0545868606049165710424232i),
(2.4718773838309585611141821 + 0.47502344364250803363708842i),
(1.8910743588080159144378396 + 0.56882925572563602341139174i),
(2.8735426423367341878069406 - 1.362376149648891420997548i),
(-2.9981750586172477217567878 + 0.5183571985225367505624207i),
}
var atan = []complex128{
(1.5115747079332741358607654 + 0.091324403603954494382276776i),
(1.4424504323482602560806727 - 0.0045416132642803911503770933i),
(-1.5593488703630532674484026 - 0.20163295409248362456446431i),
(-1.5280619472445889867794105 + 0.081721556230672003746956324i),
(1.4759909163240799678221039 + 0.028602969320691644358773586i),
(1.4877353772046548932715555 + 0.14566877153207281663773599i),
(1.4206983927779191889826 + 0.076830486127880702249439993i),
(1.3162236060498933364869556 + 0.16031313000467530644933363i),
(1.5473450684303703578810093 - 0.11064907507939082484935782i),
(-1.4841462340185253987375812 + 0.049341850305024399493142411i),
}
var atanh = []complex128{
(0.058375027938968509064640438 + 1.4793488495105334458167782i),
(0.12977343497790381229915667 - 1.5661009410463561327262499i),
(-0.010576456067347252072200088 - 1.3743698658402284549750563i),
(-0.042218595678688358882784918 + 1.4891433968166405606692604i),
(0.095218997991316722061828397 + 1.5416884098777110330499698i),
(0.079965459366890323857556487 + 1.4252510353873192700350435i),
(0.15051245471980726221708301 + 1.4907432533016303804884461i),
(0.25082072933993987714470373 + 1.392057665392187516442986i),
(0.022896108815797135846276662 - 1.4609224989282864208963021i),
(-0.08665624101841876130537396 + 1.5207902036935093480142159i),
}
var conj = []complex128{
(4.9790119248836735e+00 - 7.7388724745781045e+00i),
(7.7388724745781045e+00 + 2.7688005719200159e-01i),
(-2.7688005719200159e-01 + 5.0106036182710749e+00i),
(-5.0106036182710749e+00 - 9.6362937071984173e+00i),
(9.6362937071984173e+00 - 2.9263772392439646e+00i),
(2.9263772392439646e+00 - 5.2290834314593066e+00i),
(5.2290834314593066e+00 - 2.7279399104360102e+00i),
(2.7279399104360102e+00 - 1.8253080916808550e+00i),
(1.8253080916808550e+00 + 8.6859247685756013e+00i),
(-8.6859247685756013e+00 - 4.9790119248836735e+00i),
}
var cos = []complex128{
(3.024540920601483938336569e+02 + 1.1073797572517071650045357e+03i),
(1.192858682649064973252758e-01 + 2.7857554122333065540970207e-01i),
(7.2144394304528306603857962e+01 - 2.0500129667076044169954205e+01i),
(2.24921952538403984190541e+03 - 7.317363745602773587049329e+03i),
(-9.148222970032421760015498e+00 + 1.953124661113563541862227e+00i),
(-9.116081175857732248227078e+01 - 1.992669213569952232487371e+01i),
(3.795639179042704640002918e+00 + 6.623513350981458399309662e+00i),
(-2.9144840732498869560679084e+00 - 1.214620271628002917638748e+00i),
(-7.45123482501299743872481e+02 + 2.8641692314488080814066734e+03i),
(-5.371977967039319076416747e+01 + 4.893348341339375830564624e+01i),
}
var cosh = []complex128{
(8.34638383523018249366948e+00 + 7.2181057886425846415112064e+01i),
(1.10421967379919366952251e+03 - 3.1379638689277575379469861e+02i),
(3.051485206773701584738512e-01 - 2.6805384730105297848044485e-01i),
(-7.33294728684187933370938e+01 + 1.574445942284918251038144e+01i),
(-7.478643293945957535757355e+03 + 1.6348382209913353929473321e+03i),
(4.622316522966235701630926e+00 - 8.088695185566375256093098e+00i),
(-8.544333183278877406197712e+01 + 3.7505836120128166455231717e+01i),
(-1.934457815021493925115198e+00 + 7.3725859611767228178358673e+00i),
(-2.352958770061749348353548e+00 - 2.034982010440878358915409e+00i),
(7.79756457532134748165069e+02 + 2.8549350716819176560377717e+03i),
}
var exp = []complex128{
(1.669197736864670815125146e+01 + 1.4436895109507663689174096e+02i),
(2.2084389286252583447276212e+03 - 6.2759289284909211238261917e+02i),
(2.227538273122775173434327e-01 + 7.2468284028334191250470034e-01i),
(-6.5182985958153548997881627e-03 - 1.39965837915193860879044e-03i),
(-1.4957286524084015746110777e+04 + 3.269676455931135688988042e+03i),
(9.218158701983105935659273e+00 - 1.6223985291084956009304582e+01i),
(-1.7088175716853040841444505e+02 + 7.501382609870410713795546e+01i),
(-3.852461315830959613132505e+00 + 1.4808420423156073221970892e+01i),
(-4.586775503301407379786695e+00 - 4.178501081246873415144744e+00i),
(4.451337963005453491095747e-05 - 1.62977574205442915935263e-04i),
}
var log = []complex128{
(2.2194438972179194425697051e+00 + 9.9909115046919291062461269e-01i),
(2.0468956191154167256337289e+00 - 3.5762575021856971295156489e-02i),
(1.6130808329853860438751244e+00 - 1.6259990074019058442232221e+00i),
(2.3851910394823008710032651e+00 + 2.0502936359659111755031062e+00i),
(2.3096442270679923004800651e+00 + 2.9483213155446756211881774e-01i),
(1.7904660933974656106951860e+00 + 1.0605860367252556281902109e+00i),
(1.7745926939841751666177512e+00 + 4.8084556083358307819310911e-01i),
(1.1885403350045342425648780e+00 + 5.8969634164776659423195222e-01i),
(2.1833107837679082586772505e+00 - 1.3636647724582455028314573e+00i),
(2.3037629487273259170991671e+00 + 2.6210913895386013290915234e+00i),
}
var log10 = []complex128{
(9.6389223745559042474184943e-01 + 4.338997735671419492599631e-01i),
(8.8895547241376579493490892e-01 - 1.5531488990643548254864806e-02i),
(7.0055210462945412305244578e-01 - 7.0616239649481243222248404e-01i),
(1.0358753067322445311676952e+00 + 8.9043121238134980156490909e-01i),
(1.003065742975330237172029e+00 + 1.2804396782187887479857811e-01i),
(7.7758954439739162532085157e-01 + 4.6060666333341810869055108e-01i),
(7.7069581462315327037689152e-01 + 2.0882857371769952195512475e-01i),
(5.1617650901191156135137239e-01 + 2.5610186717615977620363299e-01i),
(9.4819982567026639742663212e-01 - 5.9223208584446952284914289e-01i),
(1.0005115362454417135973429e+00 + 1.1383255270407412817250921e+00i),
}
type ff struct {
r, theta float64
}
var polar = []ff{
ff{9.2022120669932650313380972e+00, 9.9909115046919291062461269e-01},
ff{7.7438239742296106616261394e+00, -3.5762575021856971295156489e-02},
ff{5.0182478202557746902556648e+00, -1.6259990074019058442232221e+00},
ff{1.0861137372799545160704002e+01, 2.0502936359659111755031062e+00},
ff{1.0070841084922199607011905e+01, 2.9483213155446756211881774e-01},
ff{5.9922447613166942183705192e+00, 1.0605860367252556281902109e+00},
ff{5.8978784056736762299945176e+00, 4.8084556083358307819310911e-01},
ff{3.2822866700678709020367184e+00, 5.8969634164776659423195222e-01},
ff{8.8756430028990417290744307e+00, -1.3636647724582455028314573e+00},
ff{1.0011785496777731986390856e+01, 2.6210913895386013290915234e+00},
}
var pow = []complex128{
(-2.499956739197529585028819e+00 + 1.759751724335650228957144e+00i),
(7.357094338218116311191939e+04 - 5.089973412479151648145882e+04i),
(1.320777296067768517259592e+01 - 3.165621914333901498921986e+01i),
(-3.123287828297300934072149e-07 - 1.9849567521490553032502223E-7i),
(8.0622651468477229614813e+04 - 7.80028727944573092944363e+04i),
(-1.0268824572103165858577141e+00 - 4.716844738244989776610672e-01i),
(-4.35953819012244175753187e+01 + 2.2036445974645306917648585e+02i),
(8.3556092283250594950239e-01 - 1.2261571947167240272593282e+01i),
(1.582292972120769306069625e+03 + 1.273564263524278244782512e+04i),
(6.592208301642122149025369e-08 + 2.584887236651661903526389e-08i),
}
var sin = []complex128{
(-1.1073801774240233539648544e+03 + 3.024539773002502192425231e+02i),
(1.0317037521400759359744682e+00 - 3.2208979799929570242818e-02i),
(-2.0501952097271429804261058e+01 - 7.2137981348240798841800967e+01i),
(7.3173638080346338642193078e+03 + 2.249219506193664342566248e+03i),
(-1.964375633631808177565226e+00 - 9.0958264713870404464159683e+00i),
(1.992783647158514838337674e+01 - 9.11555769410191350416942e+01i),
(-6.680335650741921444300349e+00 + 3.763353833142432513086117e+00i),
(1.2794028166657459148245993e+00 - 2.7669092099795781155109602e+00i),
(2.8641693949535259594188879e+03 + 7.451234399649871202841615e+02i),
(-4.893811726244659135553033e+01 - 5.371469305562194635957655e+01i),
}
var sinh = []complex128{
(8.34559353341652565758198e+00 + 7.2187893208650790476628899e+01i),
(1.1042192548260646752051112e+03 - 3.1379650595631635858792056e+02i),
(-8.239469336509264113041849e-02 + 9.9273668758439489098514519e-01i),
(7.332295456982297798219401e+01 - 1.574585908122833444899023e+01i),
(-7.4786432301380582103534216e+03 + 1.63483823493980029604071e+03i),
(4.595842179016870234028347e+00 - 8.135290105518580753211484e+00i),
(-8.543842533574163435246793e+01 + 3.750798997857594068272375e+01i),
(-1.918003500809465688017307e+00 + 7.4358344619793504041350251e+00i),
(-2.233816733239658031433147e+00 - 2.143519070805995056229335e+00i),
(-7.797564130187551181105341e+02 - 2.8549352346594918614806877e+03i),
}
var sqrt = []complex128{
(2.6628203086086130543813948e+00 + 1.4531345674282185229796902e+00i),
(2.7823278427251986247149295e+00 - 4.9756907317005224529115567e-02i),
(1.5397025302089642757361015e+00 - 1.6271336573016637535695727e+00i),
(1.7103411581506875260277898e+00 + 2.8170677122737589676157029e+00i),
(3.1390392472953103383607947e+00 + 4.6612625849858653248980849e-01i),
(2.1117080764822417640789287e+00 + 1.2381170223514273234967850e+00i),
(2.3587032281672256703926939e+00 + 5.7827111903257349935720172e-01i),
(1.7335262588873410476661577e+00 + 5.2647258220721269141550382e-01i),
(2.3131094974708716531499282e+00 - 1.8775429304303785570775490e+00i),
(8.1420535745048086240947359e-01 + 3.0575897587277248522656113e+00i),
}
var tan = []complex128{
(-1.928757919086441129134525e-07 + 1.0000003267499169073251826e+00i),
(1.242412685364183792138948e+00 - 3.17149693883133370106696e+00i),
(-4.6745126251587795225571826e-05 - 9.9992439225263959286114298e-01i),
(4.792363401193648192887116e-09 + 1.0000000070589333451557723e+00i),
(2.345740824080089140287315e-03 + 9.947733046570988661022763e-01i),
(-2.396030789494815566088809e-05 + 9.9994781345418591429826779e-01i),
(-7.370204836644931340905303e-03 + 1.0043553413417138987717748e+00i),
(-3.691803847992048527007457e-02 + 9.6475071993469548066328894e-01i),
(-2.781955256713729368401878e-08 - 1.000000049848910609006646e+00i),
(9.4281590064030478879791249e-05 + 9.9999119340863718183758545e-01i),
}
var tanh = []complex128{
(1.0000921981225144748819918e+00 + 2.160986245871518020231507e-05i),
(9.9999967727531993209562591e-01 - 1.9953763222959658873657676e-07i),
(-1.765485739548037260789686e+00 + 1.7024216325552852445168471e+00i),
(-9.999189442732736452807108e-01 + 3.64906070494473701938098e-05i),
(9.9999999224622333738729767e-01 - 3.560088949517914774813046e-09i),
(1.0029324933367326862499343e+00 - 4.948790309797102353137528e-03i),
(9.9996113064788012488693567e-01 - 4.226995742097032481451259e-05i),
(1.0074784189316340029873945e+00 - 4.194050814891697808029407e-03i),
(9.9385534229718327109131502e-01 + 5.144217985914355502713437e-02i),
(-1.0000000491604982429364892e+00 - 2.901873195374433112227349e-08i),
}
// special cases
var vcAbsSC = []complex128{
NaN(),
}
var absSC = []float64{
math.NaN(),
}
var vcAcosSC = []complex128{
NaN(),
}
var acosSC = []complex128{
NaN(),
}
var vcAcoshSC = []complex128{
NaN(),
}
var acoshSC = []complex128{
NaN(),
}
var vcAsinSC = []complex128{
NaN(),
}
var asinSC = []complex128{
NaN(),
}
var vcAsinhSC = []complex128{
NaN(),
}
var asinhSC = []complex128{
NaN(),
}
var vcAtanSC = []complex128{
NaN(),
}
var atanSC = []complex128{
NaN(),
}
var vcAtanhSC = []complex128{
NaN(),
}
var atanhSC = []complex128{
NaN(),
}
var vcConjSC = []complex128{
NaN(),
}
var conjSC = []complex128{
NaN(),
}
var vcCosSC = []complex128{
NaN(),
}
var cosSC = []complex128{
NaN(),
}
var vcCoshSC = []complex128{
NaN(),
}
var coshSC = []complex128{
NaN(),
}
var vcExpSC = []complex128{
NaN(),
}
var expSC = []complex128{
NaN(),
}
var vcLogSC = []complex128{
NaN(),
}
var logSC = []complex128{
NaN(),
}
var vcLog10SC = []complex128{
NaN(),
}
var log10SC = []complex128{
NaN(),
}
var vcPolarSC = []complex128{
NaN(),
}
var polarSC = []ff{
ff{math.NaN(), math.NaN()},
}
var vcPowSC = [][2]complex128{
[2]complex128{NaN(), NaN()},
}
var powSC = []complex128{
NaN(),
}
var vcSinSC = []complex128{
NaN(),
}
var sinSC = []complex128{
NaN(),
}
var vcSinhSC = []complex128{
NaN(),
}
var sinhSC = []complex128{
NaN(),
}
var vcSqrtSC = []complex128{
NaN(),
}
var sqrtSC = []complex128{
NaN(),
}
var vcTanSC = []complex128{
NaN(),
}
var tanSC = []complex128{
NaN(),
}
var vcTanhSC = []complex128{
NaN(),
}
var tanhSC = []complex128{
NaN(),
}
// functions borrowed from pkg/math/all_test.go
func tolerance(a, b, e float64) bool {
d := a - b
if d < 0 {
d = -d
}
if a != 0 {
e = e * a
if e < 0 {
e = -e
}
}
return d < e
}
func soclose(a, b, e float64) bool { return tolerance(a, b, e) }
func veryclose(a, b float64) bool { return tolerance(a, b, 4e-16) }
func alike(a, b float64) bool {
switch {
case a != a && b != b: // math.IsNaN(a) && math.IsNaN(b):
return true
case a == b:
return true
}
return false
}
func cTolerance(a, b complex128, e float64) bool {
d := Abs(a - b)
if a != 0 {
e = e * Abs(a)
if e < 0 {
e = -e
}
}
return d < e
}
func cSoclose(a, b complex128, e float64) bool { return cTolerance(a, b, e) }
func cVeryclose(a, b complex128) bool { return cTolerance(a, b, 4e-16) }
func cAlike(a, b complex128) bool {
switch {
case IsNaN(a) && IsNaN(b):
return true
case a == b:
return true
}
return false
}
func TestAbs(t *testing.T) {
for i := 0; i < len(vc); i++ {
if f := Abs(vc[i]); !veryclose(abs[i], f) {
t.Errorf("Abs(%g) = %g, want %g\n", vc[i], f, abs[i])
}
}
for i := 0; i < len(vcAbsSC); i++ {
if f := Abs(vcAbsSC[i]); !alike(absSC[i], f) {
t.Errorf("Abs(%g) = %g, want %g\n", vcAbsSC[i], f, absSC[i])
}
}
}
func TestAcos(t *testing.T) {
for i := 0; i < len(vc); i++ {
if f := Acos(vc[i]); !cSoclose(acos[i], f, 1e-14) {
t.Errorf("Acos(%g) = %g, want %g\n", vc[i], f, acos[i])
}
}
for i := 0; i < len(vcAcosSC); i++ {
if f := Acos(vcAcosSC[i]); !cAlike(acosSC[i], f) {
t.Errorf("Acos(%g) = %g, want %g\n", vcAcosSC[i], f, acosSC[i])
}
}
}
func TestAcosh(t *testing.T) {
for i := 0; i < len(vc); i++ {
if f := Acosh(vc[i]); !cSoclose(acosh[i], f, 1e-14) {
t.Errorf("Acosh(%g) = %g, want %g\n", vc[i], f, acosh[i])
}
}
for i := 0; i < len(vcAcoshSC); i++ {
if f := Acosh(vcAcoshSC[i]); !cAlike(acoshSC[i], f) {
t.Errorf("Acosh(%g) = %g, want %g\n", vcAcoshSC[i], f, acoshSC[i])
}
}
}
func TestAsin(t *testing.T) {
for i := 0; i < len(vc); i++ {
if f := Asin(vc[i]); !cSoclose(asin[i], f, 1e-14) {
t.Errorf("Asin(%g) = %g, want %g\n", vc[i], f, asin[i])
}
}
for i := 0; i < len(vcAsinSC); i++ {
if f := Asin(vcAsinSC[i]); !cAlike(asinSC[i], f) {
t.Errorf("Asin(%g) = %g, want %g\n", vcAsinSC[i], f, asinSC[i])
}
}
}
func TestAsinh(t *testing.T) {
for i := 0; i < len(vc); i++ {
if f := Asinh(vc[i]); !cSoclose(asinh[i], f, 4e-15) {
t.Errorf("Asinh(%g) = %g, want %g\n", vc[i], f, asinh[i])
}
}
for i := 0; i < len(vcAsinhSC); i++ {
if f := Asinh(vcAsinhSC[i]); !cAlike(asinhSC[i], f) {
t.Errorf("Asinh(%g) = %g, want %g\n", vcAsinhSC[i], f, asinhSC[i])
}
}
}
func TestAtan(t *testing.T) {
for i := 0; i < len(vc); i++ {
if f := Atan(vc[i]); !cVeryclose(atan[i], f) {
t.Errorf("Atan(%g) = %g, want %g\n", vc[i], f, atan[i])
}
}
for i := 0; i < len(vcAtanSC); i++ {
if f := Atan(vcAtanSC[i]); !cAlike(atanSC[i], f) {
t.Errorf("Atan(%g) = %g, want %g\n", vcAtanSC[i], f, atanSC[i])
}
}
}
func TestAtanh(t *testing.T) {
for i := 0; i < len(vc); i++ {
if f := Atanh(vc[i]); !cVeryclose(atanh[i], f) {
t.Errorf("Atanh(%g) = %g, want %g\n", vc[i], f, atanh[i])
}
}
for i := 0; i < len(vcAtanhSC); i++ {
if f := Atanh(vcAtanhSC[i]); !cAlike(atanhSC[i], f) {
t.Errorf("Atanh(%g) = %g, want %g\n", vcAtanhSC[i], f, atanhSC[i])
}
}
}
func TestConj(t *testing.T) {
for i := 0; i < len(vc); i++ {
if f := Conj(vc[i]); !cVeryclose(conj[i], f) {
t.Errorf("Conj(%g) = %g, want %g\n", vc[i], f, conj[i])
}
}
for i := 0; i < len(vcConjSC); i++ {
if f := Conj(vcConjSC[i]); !cAlike(conjSC[i], f) {
t.Errorf("Conj(%g) = %g, want %g\n", vcConjSC[i], f, conjSC[i])
}
}
}
func TestCos(t *testing.T) {
for i := 0; i < len(vc); i++ {
if f := Cos(vc[i]); !cSoclose(cos[i], f, 3e-15) {
t.Errorf("Cos(%g) = %g, want %g\n", vc[i], f, cos[i])
}
}
for i := 0; i < len(vcCosSC); i++ {
if f := Cos(vcCosSC[i]); !cAlike(cosSC[i], f) {
t.Errorf("Cos(%g) = %g, want %g\n", vcCosSC[i], f, cosSC[i])
}
}
}
func TestCosh(t *testing.T) {
for i := 0; i < len(vc); i++ {
if f := Cosh(vc[i]); !cSoclose(cosh[i], f, 2e-15) {
t.Errorf("Cosh(%g) = %g, want %g\n", vc[i], f, cosh[i])
}
}
for i := 0; i < len(vcCoshSC); i++ {
if f := Cosh(vcCoshSC[i]); !cAlike(coshSC[i], f) {
t.Errorf("Cosh(%g) = %g, want %g\n", vcCoshSC[i], f, coshSC[i])
}
}
}
func TestExp(t *testing.T) {
for i := 0; i < len(vc); i++ {
if f := Exp(vc[i]); !cSoclose(exp[i], f, 1e-15) {
t.Errorf("Exp(%g) = %g, want %g\n", vc[i], f, exp[i])
}
}
for i := 0; i < len(vcExpSC); i++ {
if f := Exp(vcExpSC[i]); !cAlike(expSC[i], f) {
t.Errorf("Exp(%g) = %g, want %g\n", vcExpSC[i], f, expSC[i])
}
}
}
func TestLog(t *testing.T) {
for i := 0; i < len(vc); i++ {
if f := Log(vc[i]); !cVeryclose(log[i], f) {
t.Errorf("Log(%g) = %g, want %g\n", vc[i], f, log[i])
}
}
for i := 0; i < len(vcLogSC); i++ {
if f := Log(vcLogSC[i]); !cAlike(logSC[i], f) {
t.Errorf("Log(%g) = %g, want %g\n", vcLogSC[i], f, logSC[i])
}
}
}
func TestLog10(t *testing.T) {
for i := 0; i < len(vc); i++ {
if f := Log10(vc[i]); !cVeryclose(log10[i], f) {
t.Errorf("Log10(%g) = %g, want %g\n", vc[i], f, log10[i])
}
}
for i := 0; i < len(vcLog10SC); i++ {
if f := Log10(vcLog10SC[i]); !cAlike(log10SC[i], f) {
t.Errorf("Log10(%g) = %g, want %g\n", vcLog10SC[i], f, log10SC[i])
}
}
}
func TestPolar(t *testing.T) {
for i := 0; i < len(vc); i++ {
if r, theta := Polar(vc[i]); !veryclose(polar[i].r, r) && !veryclose(polar[i].theta, theta) {
t.Errorf("Polar(%g) = %g, %g want %g, %g\n", vc[i], r, theta, polar[i].r, polar[i].theta)
}
}
for i := 0; i < len(vcPolarSC); i++ {
if r, theta := Polar(vcPolarSC[i]); !alike(polarSC[i].r, r) && !alike(polarSC[i].theta, theta) {
t.Errorf("Polar(%g) = %g, %g, want %g, %g\n", vcPolarSC[i], r, theta, polarSC[i].r, polarSC[i].theta)
}
}
}
func TestPow(t *testing.T) {
var a = cmplx(float64(3), float64(3))
for i := 0; i < len(vc); i++ {
if f := Pow(a, vc[i]); !cSoclose(pow[i], f, 4e-15) {
t.Errorf("Pow(%g, %g) = %g, want %g\n", a, vc[i], f, pow[i])
}
}
for i := 0; i < len(vcPowSC); i++ {
if f := Pow(vcPowSC[i][0], vcPowSC[i][0]); !cAlike(powSC[i], f) {
t.Errorf("Pow(%g, %g) = %g, want %g\n", vcPowSC[i][0], vcPowSC[i][0], f, powSC[i])
}
}
}
func TestRect(t *testing.T) {
for i := 0; i < len(vc); i++ {
if f := Rect(polar[i].r, polar[i].theta); !cVeryclose(vc[i], f) {
t.Errorf("Rect(%g, %g) = %g want %g\n", polar[i].r, polar[i].theta, f, vc[i])
}
}
for i := 0; i < len(vcPolarSC); i++ {
if f := Rect(polarSC[i].r, polarSC[i].theta); !cAlike(vcPolarSC[i], f) {
t.Errorf("Rect(%g, %g) = %g, want %g\n", polarSC[i].r, polarSC[i].theta, f, vcPolarSC[i])
}
}
}
func TestSin(t *testing.T) {
for i := 0; i < len(vc); i++ {
if f := Sin(vc[i]); !cSoclose(sin[i], f, 2e-15) {
t.Errorf("Sin(%g) = %g, want %g\n", vc[i], f, sin[i])
}
}
for i := 0; i < len(vcSinSC); i++ {
if f := Sin(vcSinSC[i]); !cAlike(sinSC[i], f) {
t.Errorf("Sin(%g) = %g, want %g\n", vcSinSC[i], f, sinSC[i])
}
}
}
func TestSinh(t *testing.T) {
for i := 0; i < len(vc); i++ {
if f := Sinh(vc[i]); !cSoclose(sinh[i], f, 2e-15) {
t.Errorf("Sinh(%g) = %g, want %g\n", vc[i], f, sinh[i])
}
}
for i := 0; i < len(vcSinhSC); i++ {
if f := Sinh(vcSinhSC[i]); !cAlike(sinhSC[i], f) {
t.Errorf("Sinh(%g) = %g, want %g\n", vcSinhSC[i], f, sinhSC[i])
}
}
}
func TestSqrt(t *testing.T) {
for i := 0; i < len(vc); i++ {
if f := Sqrt(vc[i]); !cVeryclose(sqrt[i], f) {
t.Errorf("Sqrt(%g) = %g, want %g\n", vc[i], f, sqrt[i])
}
}
for i := 0; i < len(vcSqrtSC); i++ {
if f := Sqrt(vcSqrtSC[i]); !cAlike(sqrtSC[i], f) {
t.Errorf("Sqrt(%g) = %g, want %g\n", vcSqrtSC[i], f, sqrtSC[i])
}
}
}
func TestTan(t *testing.T) {
for i := 0; i < len(vc); i++ {
if f := Tan(vc[i]); !cSoclose(tan[i], f, 3e-15) {
t.Errorf("Tan(%g) = %g, want %g\n", vc[i], f, tan[i])
}
}
for i := 0; i < len(vcTanSC); i++ {
if f := Tan(vcTanSC[i]); !cAlike(tanSC[i], f) {
t.Errorf("Tan(%g) = %g, want %g\n", vcTanSC[i], f, tanSC[i])
}
}
}
func TestTanh(t *testing.T) {
for i := 0; i < len(vc); i++ {
if f := Tanh(vc[i]); !cSoclose(tanh[i], f, 2e-15) {
t.Errorf("Tanh(%g) = %g, want %g\n", vc[i], f, tanh[i])
}
}
for i := 0; i < len(vcTanhSC); i++ {
if f := Tanh(vcTanhSC[i]); !cAlike(tanhSC[i], f) {
t.Errorf("Tanh(%g) = %g, want %g\n", vcTanhSC[i], f, tanhSC[i])
}
}
}
func BenchmarkAbs(b *testing.B) {
for i := 0; i < b.N; i++ {
Abs(cmplx(2.5, 3.5))
}
}
func BenchmarkAcos(b *testing.B) {
for i := 0; i < b.N; i++ {
Acos(cmplx(2.5, 3.5))
}
}
func BenchmarkAcosh(b *testing.B) {
for i := 0; i < b.N; i++ {
Acosh(cmplx(2.5, 3.5))
}
}
func BenchmarkAsin(b *testing.B) {
for i := 0; i < b.N; i++ {
Asin(cmplx(2.5, 3.5))
}
}
func BenchmarkAsinh(b *testing.B) {
for i := 0; i < b.N; i++ {
Asinh(cmplx(2.5, 3.5))
}
}
func BenchmarkAtan(b *testing.B) {
for i := 0; i < b.N; i++ {
Atan(cmplx(2.5, 3.5))
}
}
func BenchmarkAtanh(b *testing.B) {
for i := 0; i < b.N; i++ {
Atanh(cmplx(2.5, 3.5))
}
}
func BenchmarkConj(b *testing.B) {
for i := 0; i < b.N; i++ {
Conj(cmplx(2.5, 3.5))
}
}
func BenchmarkCos(b *testing.B) {
for i := 0; i < b.N; i++ {
Cos(cmplx(2.5, 3.5))
}
}
func BenchmarkCosh(b *testing.B) {
for i := 0; i < b.N; i++ {
Cosh(cmplx(2.5, 3.5))
}
}
func BenchmarkExp(b *testing.B) {
for i := 0; i < b.N; i++ {
Exp(cmplx(2.5, 3.5))
}
}
func BenchmarkLog(b *testing.B) {
for i := 0; i < b.N; i++ {
Log(cmplx(2.5, 3.5))
}
}
func BenchmarkLog10(b *testing.B) {
for i := 0; i < b.N; i++ {
Log10(cmplx(2.5, 3.5))
}
}
func BenchmarkPhase(b *testing.B) {
for i := 0; i < b.N; i++ {
Phase(cmplx(2.5, 3.5))
}
}
func BenchmarkPolar(b *testing.B) {
for i := 0; i < b.N; i++ {
Polar(cmplx(2.5, 3.5))
}
}
func BenchmarkPow(b *testing.B) {
for i := 0; i < b.N; i++ {
Pow(cmplx(2.5, 3.5), cmplx(2.5, 3.5))
}
}
func BenchmarkRect(b *testing.B) {
for i := 0; i < b.N; i++ {
Rect(2.5, 1.5)
}
}
func BenchmarkSin(b *testing.B) {
for i := 0; i < b.N; i++ {
Sin(cmplx(2.5, 3.5))
}
}
func BenchmarkSinh(b *testing.B) {
for i := 0; i < b.N; i++ {
Sinh(cmplx(2.5, 3.5))
}
}
func BenchmarkSqrt(b *testing.B) {
for i := 0; i < b.N; i++ {
Sqrt(cmplx(2.5, 3.5))
}
}
func BenchmarkTan(b *testing.B) {
for i := 0; i < b.N; i++ {
Tan(cmplx(2.5, 3.5))
}
}
func BenchmarkTanh(b *testing.B) {
for i := 0; i < b.N; i++ {
Tanh(cmplx(2.5, 3.5))
}
}

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// Copyright 2010 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package cmath
// Conj returns the complex conjugate of x.
func Conj(x complex128) complex128 { return cmplx(real(x), -imag(x)) }

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// Copyright 2010 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package cmath
import "math"
// The original C code, the long comment, and the constants
// below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c.
// The go code is a simplified version of the original C.
//
// Cephes Math Library Release 2.8: June, 2000
// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
//
// The readme file at http://netlib.sandia.gov/cephes/ says:
// Some software in this archive may be from the book _Methods and
// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
// International, 1989) or from the Cephes Mathematical Library, a
// commercial product. In either event, it is copyrighted by the author.
// What you see here may be used freely but it comes with no support or
// guarantee.
//
// The two known misprints in the book are repaired here in the
// source listings for the gamma function and the incomplete beta
// integral.
//
// Stephen L. Moshier
// moshier@na-net.ornl.gov
// Complex exponential function
//
// DESCRIPTION:
//
// Returns the complex exponential of the complex argument z.
//
// If
// z = x + iy,
// r = exp(x),
// then
// w = r cos y + i r sin y.
//
// ACCURACY:
//
// Relative error:
// arithmetic domain # trials peak rms
// DEC -10,+10 8700 3.7e-17 1.1e-17
// IEEE -10,+10 30000 3.0e-16 8.7e-17
// Exp returns e^x, the base-e exponential of x.
func Exp(x complex128) complex128 {
r := math.Exp(real(x))
s, c := math.Sincos(imag(x))
return cmplx(r*c, r*s)
}

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// Copyright 2010 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package cmath
import "math"
// IsInf returns true if either real(x) or imag(x) is an infinity.
func IsInf(x complex128) bool {
if math.IsInf(real(x), 0) || math.IsInf(imag(x), 0) {
return true
}
return false
}
// Inf returns a complex infinity, cmplx(+Inf, +Inf).
func Inf() complex128 {
inf := math.Inf(1)
return cmplx(inf, inf)
}

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// Copyright 2010 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package cmath
import "math"
// IsNaN returns true if either real(x) or imag(x) is NaN.
func IsNaN(x complex128) bool {
if math.IsNaN(real(x)) || math.IsNaN(imag(x)) {
return true
}
return false
}
// NaN returns a complex ``not-a-number'' value.
func NaN() complex128 {
nan := math.NaN()
return cmplx(nan, nan)
}

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// Copyright 2010 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package cmath
import "math"
// The original C code, the long comment, and the constants
// below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c.
// The go code is a simplified version of the original C.
//
// Cephes Math Library Release 2.8: June, 2000
// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
//
// The readme file at http://netlib.sandia.gov/cephes/ says:
// Some software in this archive may be from the book _Methods and
// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
// International, 1989) or from the Cephes Mathematical Library, a
// commercial product. In either event, it is copyrighted by the author.
// What you see here may be used freely but it comes with no support or
// guarantee.
//
// The two known misprints in the book are repaired here in the
// source listings for the gamma function and the incomplete beta
// integral.
//
// Stephen L. Moshier
// moshier@na-net.ornl.gov
// Complex natural logarithm
//
// DESCRIPTION:
//
// Returns complex logarithm to the base e (2.718...) of
// the complex argument z.
//
// If
// z = x + iy, r = sqrt( x**2 + y**2 ),
// then
// w = log(r) + i arctan(y/x).
//
// The arctangent ranges from -PI to +PI.
//
// ACCURACY:
//
// Relative error:
// arithmetic domain # trials peak rms
// DEC -10,+10 7000 8.5e-17 1.9e-17
// IEEE -10,+10 30000 5.0e-15 1.1e-16
//
// Larger relative error can be observed for z near 1 +i0.
// In IEEE arithmetic the peak absolute error is 5.2e-16, rms
// absolute error 1.0e-16.
// Log returns the natural logarithm of x.
func Log(x complex128) complex128 {
return cmplx(math.Log(Abs(x)), Phase(x))
}
// Log10 returns the decimal logarithm of x.
func Log10(x complex128) complex128 {
return math.Log10E * Log(x)
}

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// Copyright 2010 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package cmath
import "math"
// Phase returns the phase (also called the argument) of x.
// The returned value is in the range (-Pi, Pi].
func Phase(x complex128) float64 { return math.Atan2(imag(x), real(x)) }

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// Copyright 2010 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package cmath
// Polar returns the absolute value r and phase θ of x,
// such that x = r * e^θi.
// The phase is in the range (-Pi, Pi].
func Polar(x complex128) (r, θ float64) {
return Abs(x), Phase(x)
}

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// Copyright 2010 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package cmath
import "math"
// The original C code, the long comment, and the constants
// below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c.
// The go code is a simplified version of the original C.
//
// Cephes Math Library Release 2.8: June, 2000
// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
//
// The readme file at http://netlib.sandia.gov/cephes/ says:
// Some software in this archive may be from the book _Methods and
// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
// International, 1989) or from the Cephes Mathematical Library, a
// commercial product. In either event, it is copyrighted by the author.
// What you see here may be used freely but it comes with no support or
// guarantee.
//
// The two known misprints in the book are repaired here in the
// source listings for the gamma function and the incomplete beta
// integral.
//
// Stephen L. Moshier
// moshier@na-net.ornl.gov
// Complex power function
//
// DESCRIPTION:
//
// Raises complex A to the complex Zth power.
// Definition is per AMS55 # 4.2.8,
// analytically equivalent to cpow(a,z) = cexp(z clog(a)).
//
// ACCURACY:
//
// Relative error:
// arithmetic domain # trials peak rms
// IEEE -10,+10 30000 9.4e-15 1.5e-15
// Pow returns x^y, the base-x exponential of y.
func Pow(x, y complex128) complex128 {
modulus := Abs(x)
if modulus == 0 {
return cmplx(0, 0)
}
r := math.Pow(modulus, real(y))
arg := Phase(x)
theta := real(y) * arg
if imag(y) != 0 {
r *= math.Exp(-imag(y) * arg)
theta += imag(y) * math.Log(modulus)
}
s, c := math.Sincos(theta)
return cmplx(r*c, r*s)
}

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// Copyright 2010 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package cmath
import "math"
// Rect returns the complex number x with polar coordinates r, θ.
func Rect(r, θ float64) complex128 {
s, c := math.Sincos(θ)
return cmplx(r*c, r*s)
}

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// Copyright 2010 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package cmath
import "math"
// The original C code, the long comment, and the constants
// below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c.
// The go code is a simplified version of the original C.
//
// Cephes Math Library Release 2.8: June, 2000
// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
//
// The readme file at http://netlib.sandia.gov/cephes/ says:
// Some software in this archive may be from the book _Methods and
// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
// International, 1989) or from the Cephes Mathematical Library, a
// commercial product. In either event, it is copyrighted by the author.
// What you see here may be used freely but it comes with no support or
// guarantee.
//
// The two known misprints in the book are repaired here in the
// source listings for the gamma function and the incomplete beta
// integral.
//
// Stephen L. Moshier
// moshier@na-net.ornl.gov
// Complex circular sine
//
// DESCRIPTION:
//
// If
// z = x + iy,
//
// then
//
// w = sin x cosh y + i cos x sinh y.
//
// csin(z) = -i csinh(iz).
//
// ACCURACY:
//
// Relative error:
// arithmetic domain # trials peak rms
// DEC -10,+10 8400 5.3e-17 1.3e-17
// IEEE -10,+10 30000 3.8e-16 1.0e-16
// Also tested by csin(casin(z)) = z.
// Sin returns the sine of x.
func Sin(x complex128) complex128 {
s, c := math.Sincos(real(x))
sh, ch := sinhcosh(imag(x))
return cmplx(s*ch, c*sh)
}
// Complex hyperbolic sine
//
// DESCRIPTION:
//
// csinh z = (cexp(z) - cexp(-z))/2
// = sinh x * cos y + i cosh x * sin y .
//
// ACCURACY:
//
// Relative error:
// arithmetic domain # trials peak rms
// IEEE -10,+10 30000 3.1e-16 8.2e-17
// Sinh returns the hyperbolic sine of x.
func Sinh(x complex128) complex128 {
s, c := math.Sincos(imag(x))
sh, ch := sinhcosh(real(x))
return cmplx(c*sh, s*ch)
}
// Complex circular cosine
//
// DESCRIPTION:
//
// If
// z = x + iy,
//
// then
//
// w = cos x cosh y - i sin x sinh y.
//
// ACCURACY:
//
// Relative error:
// arithmetic domain # trials peak rms
// DEC -10,+10 8400 4.5e-17 1.3e-17
// IEEE -10,+10 30000 3.8e-16 1.0e-16
// Cos returns the cosine of x.
func Cos(x complex128) complex128 {
s, c := math.Sincos(real(x))
sh, ch := sinhcosh(imag(x))
return cmplx(c*ch, -s*sh)
}
// Complex hyperbolic cosine
//
// DESCRIPTION:
//
// ccosh(z) = cosh x cos y + i sinh x sin y .
//
// ACCURACY:
//
// Relative error:
// arithmetic domain # trials peak rms
// IEEE -10,+10 30000 2.9e-16 8.1e-17
// Cosh returns the hyperbolic cosine of x.
func Cosh(x complex128) complex128 {
s, c := math.Sincos(imag(x))
sh, ch := sinhcosh(real(x))
return cmplx(c*ch, s*sh)
}
// calculate sinh and cosh
func sinhcosh(x float64) (sh, ch float64) {
if math.Fabs(x) <= 0.5 {
return math.Sinh(x), math.Cosh(x)
}
e := math.Exp(x)
ei := 0.5 / e
e *= 0.5
return e - ei, e + ei
}

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// Copyright 2010 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package cmath
import "math"
// The original C code, the long comment, and the constants
// below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c.
// The go code is a simplified version of the original C.
//
// Cephes Math Library Release 2.8: June, 2000
// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
//
// The readme file at http://netlib.sandia.gov/cephes/ says:
// Some software in this archive may be from the book _Methods and
// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
// International, 1989) or from the Cephes Mathematical Library, a
// commercial product. In either event, it is copyrighted by the author.
// What you see here may be used freely but it comes with no support or
// guarantee.
//
// The two known misprints in the book are repaired here in the
// source listings for the gamma function and the incomplete beta
// integral.
//
// Stephen L. Moshier
// moshier@na-net.ornl.gov
// Complex square root
//
// DESCRIPTION:
//
// If z = x + iy, r = |z|, then
//
// 1/2
// Re w = [ (r + x)/2 ] ,
//
// 1/2
// Im w = [ (r - x)/2 ] .
//
// Cancellation error in r-x or r+x is avoided by using the
// identity 2 Re w Im w = y.
//
// Note that -w is also a square root of z. The root chosen
// is always in the right half plane and Im w has the same sign as y.
//
// ACCURACY:
//
// Relative error:
// arithmetic domain # trials peak rms
// DEC -10,+10 25000 3.2e-17 9.6e-18
// IEEE -10,+10 1,000,000 2.9e-16 6.1e-17
// Sqrt returns the square root of x.
func Sqrt(x complex128) complex128 {
if imag(x) == 0 {
if real(x) == 0 {
return cmplx(0, 0)
}
if real(x) < 0 {
return cmplx(0, math.Sqrt(-real(x)))
}
return cmplx(math.Sqrt(real(x)), 0)
}
if real(x) == 0 {
if imag(x) < 0 {
r := math.Sqrt(-0.5 * imag(x))
return cmplx(r, -r)
}
r := math.Sqrt(0.5 * imag(x))
return cmplx(r, r)
}
a := real(x)
b := imag(x)
var scale float64
// Rescale to avoid internal overflow or underflow.
if math.Fabs(a) > 4 || math.Fabs(b) > 4 {
a *= 0.25
b *= 0.25
scale = 2
} else {
a *= 1.8014398509481984e16 // 2^54
b *= 1.8014398509481984e16
scale = 7.450580596923828125e-9 // 2^-27
}
r := math.Hypot(a, b)
var t float64
if a > 0 {
t = math.Sqrt(0.5*r + 0.5*a)
r = scale * math.Fabs((0.5*b)/t)
t *= scale
} else {
r = math.Sqrt(0.5*r - 0.5*a)
t = scale * math.Fabs((0.5*b)/r)
r *= scale
}
if b < 0 {
return cmplx(t, -r)
}
return cmplx(t, r)
}

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// Copyright 2010 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package cmath
import "math"
// The original C code, the long comment, and the constants
// below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c.
// The go code is a simplified version of the original C.
//
// Cephes Math Library Release 2.8: June, 2000
// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
//
// The readme file at http://netlib.sandia.gov/cephes/ says:
// Some software in this archive may be from the book _Methods and
// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
// International, 1989) or from the Cephes Mathematical Library, a
// commercial product. In either event, it is copyrighted by the author.
// What you see here may be used freely but it comes with no support or
// guarantee.
//
// The two known misprints in the book are repaired here in the
// source listings for the gamma function and the incomplete beta
// integral.
//
// Stephen L. Moshier
// moshier@na-net.ornl.gov
// Complex circular tangent
//
// DESCRIPTION:
//
// If
// z = x + iy,
//
// then
//
// sin 2x + i sinh 2y
// w = --------------------.
// cos 2x + cosh 2y
//
// On the real axis the denominator is zero at odd multiples
// of PI/2. The denominator is evaluated by its Taylor
// series near these points.
//
// ctan(z) = -i ctanh(iz).
//
// ACCURACY:
//
// Relative error:
// arithmetic domain # trials peak rms
// DEC -10,+10 5200 7.1e-17 1.6e-17
// IEEE -10,+10 30000 7.2e-16 1.2e-16
// Also tested by ctan * ccot = 1 and catan(ctan(z)) = z.
// Tan returns the tangent of x.
func Tan(x complex128) complex128 {
d := math.Cos(2*real(x)) + math.Cosh(2*imag(x))
if math.Fabs(d) < 0.25 {
d = tanSeries(x)
}
if d == 0 {
return Inf()
}
return cmplx(math.Sin(2*real(x))/d, math.Sinh(2*imag(x))/d)
}
// Complex hyperbolic tangent
//
// DESCRIPTION:
//
// tanh z = (sinh 2x + i sin 2y) / (cosh 2x + cos 2y) .
//
// ACCURACY:
//
// Relative error:
// arithmetic domain # trials peak rms
// IEEE -10,+10 30000 1.7e-14 2.4e-16
// Tanh returns the hyperbolic tangent of x.
func Tanh(x complex128) complex128 {
d := math.Cosh(2*real(x)) + math.Cos(2*imag(x))
if d == 0 {
return Inf()
}
return cmplx(math.Sinh(2*real(x))/d, math.Sin(2*imag(x))/d)
}
// Program to subtract nearest integer multiple of PI
func reducePi(x float64) float64 {
const (
// extended precision value of PI:
DP1 = 3.14159265160560607910E0 // ?? 0x400921fb54000000
DP2 = 1.98418714791870343106E-9 // ?? 0x3e210b4610000000
DP3 = 1.14423774522196636802E-17 // ?? 0x3c6a62633145c06e
)
t := x / math.Pi
if t >= 0 {
t += 0.5
} else {
t -= 0.5
}
t = float64(int64(t)) // int64(t) = the multiple
return ((x - t*DP1) - t*DP2) - t*DP3
}
// Taylor series expansion for cosh(2y) - cos(2x)
func tanSeries(z complex128) float64 {
const MACHEP = 1.0 / (1 << 53)
x := math.Fabs(2 * real(z))
y := math.Fabs(2 * imag(z))
x = reducePi(x)
x = x * x
y = y * y
x2 := float64(1)
y2 := float64(1)
f := float64(1)
rn := float64(0)
d := float64(0)
for {
rn += 1
f *= rn
rn += 1
f *= rn
x2 *= x
y2 *= y
t := y2 + x2
t /= f
d += t
rn += 1
f *= rn
rn += 1
f *= rn
x2 *= x
y2 *= y
t = y2 - x2
t /= f
d += t
if math.Fabs(t/d) <= MACHEP {
break
}
}
return d
}
// Complex circular cotangent
//
// DESCRIPTION:
//
// If
// z = x + iy,
//
// then
//
// sin 2x - i sinh 2y
// w = --------------------.
// cosh 2y - cos 2x
//
// On the real axis, the denominator has zeros at even
// multiples of PI/2. Near these points it is evaluated
// by a Taylor series.
//
// ACCURACY:
//
// Relative error:
// arithmetic domain # trials peak rms
// DEC -10,+10 3000 6.5e-17 1.6e-17
// IEEE -10,+10 30000 9.2e-16 1.2e-16
// Also tested by ctan * ccot = 1 + i0.
// Cot returns the cotangent of x.
func Cot(x complex128) complex128 {
d := math.Cosh(2*imag(x)) - math.Cos(2*real(x))
if math.Fabs(d) < 0.25 {
d = tanSeries(x)
}
if d == 0 {
return Inf()
}
return cmplx(math.Sin(2*real(x))/d, -math.Sinh(2*imag(x))/d)
}