mirror of
https://github.com/golang/go
synced 2024-11-23 00:30:07 -07:00
more accurate Log, Exp, Pow.
move test.go to alll_test.go. R=r DELTA=1024 (521 added, 425 deleted, 78 changed) OCL=19687 CL=19695
This commit is contained in:
Russ Cox
parent
c0a01e9665
commit
f379ea0b07
@ -33,6 +33,7 @@ coverage: packages
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O1=\
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atan.$O\
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exp.$O\
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fabs.$O\
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floor.$O\
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fmod.$O\
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@ -46,32 +47,25 @@ O1=\
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O2=\
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asin.$O\
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atan2.$O\
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exp.$O\
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O3=\
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pow.$O\
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sinh.$O\
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O4=\
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O3=\
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tanh.$O\
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math.a: a1 a2 a3 a4
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math.a: a1 a2 a3
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a1: $(O1)
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$(AR) grc math.a atan.$O fabs.$O floor.$O fmod.$O hypot.$O log.$O pow10.$O sin.$O sqrt.$O tan.$O
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$(AR) grc math.a atan.$O exp.$O fabs.$O floor.$O fmod.$O hypot.$O log.$O pow10.$O sin.$O sqrt.$O tan.$O
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rm -f $(O1)
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a2: $(O2)
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$(AR) grc math.a asin.$O atan2.$O exp.$O
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$(AR) grc math.a asin.$O atan2.$O pow.$O sinh.$O
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rm -f $(O2)
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a3: $(O3)
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$(AR) grc math.a pow.$O sinh.$O
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rm -f $(O3)
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a4: $(O4)
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$(AR) grc math.a tanh.$O
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rm -f $(O4)
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rm -f $(O3)
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newpkg: clean
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$(AR) grc math.a
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@ -79,7 +73,6 @@ newpkg: clean
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$(O1): newpkg
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$(O2): a1
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$(O3): a2
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$(O4): a3
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nuke: clean
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rm -f $(GOROOT)/pkg/math.a
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@ -50,7 +50,7 @@ var atan = []float64 {
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var exp = []float64 {
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1.4533071302642137e+02,
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2.2958822575694450e+03,
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7.5814542574851664e-01,
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7.5814542574851666e-01,
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6.6668778421791010e-03,
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1.5310493273896035e+04,
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1.8659907517999329e+01,
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@ -156,13 +156,12 @@ var tanh = []float64 {
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-9.9999994291374019e-01,
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}
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func Close(a,b float64) bool {
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func Tolerance(a,b,e float64) bool {
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d := a-b;
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if d < 0 {
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d = -d;
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}
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e := float64(1e-14);
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if a != 0 {
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e = e*a;
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if e < 0 {
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@ -171,10 +170,16 @@ func Close(a,b float64) bool {
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}
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return d < e;
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}
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func Close(a,b float64) bool {
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return Tolerance(a, b, 1e-14);
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}
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func VeryClose(a,b float64) bool {
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return Tolerance(a, b, 4e-16);
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}
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export func TestAsin(t *testing.T) {
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for i := 0; i < len(vf); i++ {
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if f := math.Asin(vf[i]/10); !Close(asin[i], f) {
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if f := math.Asin(vf[i]/10); !VeryClose(asin[i], f) {
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t.Errorf("math.Asin(%g) = %g, want %g\n", vf[i]/10, f, asin[i]);
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}
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}
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@ -182,7 +187,7 @@ export func TestAsin(t *testing.T) {
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export func TestAtan(t *testing.T) {
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for i := 0; i < len(vf); i++ {
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if f := math.Atan(vf[i]); !Close(atan[i], f) {
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if f := math.Atan(vf[i]); !VeryClose(atan[i], f) {
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t.Errorf("math.Atan(%g) = %g, want %g\n", vf[i], f, atan[i]);
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}
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}
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@ -190,7 +195,7 @@ export func TestAtan(t *testing.T) {
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export func TestExp(t *testing.T) {
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for i := 0; i < len(vf); i++ {
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if f := math.Exp(vf[i]); !Close(exp[i], f) {
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if f := math.Exp(vf[i]); !VeryClose(exp[i], f) {
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t.Errorf("math.Exp(%g) = %g, want %g\n", vf[i], f, exp[i]);
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}
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}
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@ -198,7 +203,7 @@ export func TestExp(t *testing.T) {
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export func TestFloor(t *testing.T) {
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for i := 0; i < len(vf); i++ {
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if f := math.Floor(vf[i]); !Close(floor[i], f) {
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if f := math.Floor(vf[i]); floor[i] != f {
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t.Errorf("math.Floor(%g) = %g, want %g\n", vf[i], f, floor[i]);
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}
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}
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@ -207,10 +212,14 @@ export func TestFloor(t *testing.T) {
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export func TestLog(t *testing.T) {
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for i := 0; i < len(vf); i++ {
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a := math.Fabs(vf[i]);
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if f := math.Log(a); !Close(log[i], f) {
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t.Errorf("math.Log(%g) = %g, want %g\n", a, f, floor[i]);
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if f := math.Log(a); log[i] != f {
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t.Errorf("math.Log(%g) = %g, want %g\n", a, f, log[i]);
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}
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}
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const Ln10 = 2.30258509299404568401799145468436421;
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if f := math.Log(10); f != Ln10 {
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t.Errorf("math.Log(%g) = %g, want %g\n", 10, f, Ln10);
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}
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}
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export func TestPow(t *testing.T) {
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@ -231,7 +240,7 @@ export func TestSin(t *testing.T) {
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export func TestSinh(t *testing.T) {
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for i := 0; i < len(vf); i++ {
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if f := math.Sinh(vf[i]); !Close(sinh[i], f) {
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if f := math.Sinh(vf[i]); !VeryClose(sinh[i], f) {
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t.Errorf("math.Sinh(%g) = %g, want %g\n", vf[i], f, sinh[i]);
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}
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}
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@ -240,7 +249,7 @@ export func TestSinh(t *testing.T) {
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export func TestSqrt(t *testing.T) {
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for i := 0; i < len(vf); i++ {
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a := math.Fabs(vf[i]);
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if f := math.Sqrt(a); !Close(sqrt[i], f) {
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if f := math.Sqrt(a); !VeryClose(sqrt[i], f) {
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t.Errorf("math.Sqrt(%g) = %g, want %g\n", a, f, floor[i]);
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}
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}
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@ -256,7 +265,7 @@ export func TestTan(t *testing.T) {
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export func TestTanh(t *testing.T) {
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for i := 0; i < len(vf); i++ {
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if f := math.Tanh(vf[i]); !Close(tanh[i], f) {
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if f := math.Tanh(vf[i]); !VeryClose(tanh[i], f) {
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t.Errorf("math.Tanh(%g) = %g, want %g\n", vf[i], f, tanh[i]);
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}
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}
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@ -265,9 +274,8 @@ export func TestTanh(t *testing.T) {
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export func TestHypot(t *testing.T) {
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for i := 0; i < len(vf); i++ {
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a := math.Fabs(tanh[i]*math.Sqrt(2));
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if f := math.Hypot(tanh[i], tanh[i]); !Close(a, f) {
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if f := math.Hypot(tanh[i], tanh[i]); !VeryClose(a, f) {
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t.Errorf("math.Hypot(%g, %g) = %g, want %g\n", tanh[i], tanh[i], f, a);
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}
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}
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}
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@ -6,42 +6,132 @@ package math
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import "math"
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/*
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* exp returns the exponential func of its
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* floating-point argument.
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*
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* The coefficients are #1069 from Hart and Cheney. (22.35D)
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*/
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// The original C code, the long comment, and the constants
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// below are from FreeBSD's /usr/src/lib/msun/src/e_exp.c
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// and came with this notice. The go code is a simplified
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// version of the original C.
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//
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// ====================================================
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// Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
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//
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// Permission to use, copy, modify, and distribute this
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// software is freely granted, provided that this notice
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// is preserved.
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// ====================================================
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//
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//
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// exp(x)
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// Returns the exponential of x.
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//
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// Method
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// 1. Argument reduction:
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// Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
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// Given x, find r and integer k such that
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//
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// x = k*ln2 + r, |r| <= 0.5*ln2.
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//
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// Here r will be represented as r = hi-lo for better
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// accuracy.
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//
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// 2. Approximation of exp(r) by a special rational function on
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// the interval [0,0.34658]:
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// Write
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// R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
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// We use a special Remes algorithm on [0,0.34658] to generate
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// a polynomial of degree 5 to approximate R. The maximum error
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// of this polynomial approximation is bounded by 2**-59. In
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// other words,
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// R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
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// (where z=r*r, and the values of P1 to P5 are listed below)
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// and
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// | 5 | -59
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// | 2.0+P1*z+...+P5*z - R(z) | <= 2
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// | |
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// The computation of exp(r) thus becomes
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// 2*r
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// exp(r) = 1 + -------
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// R - r
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// r*R1(r)
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// = 1 + r + ----------- (for better accuracy)
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// 2 - R1(r)
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// where
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// 2 4 10
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// R1(r) = r - (P1*r + P2*r + ... + P5*r ).
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//
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// 3. Scale back to obtain exp(x):
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// From step 1, we have
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// exp(x) = 2^k * exp(r)
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//
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// Special cases:
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// exp(INF) is INF, exp(NaN) is NaN;
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// exp(-INF) is 0, and
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// for finite argument, only exp(0)=1 is exact.
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//
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// Accuracy:
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// according to an error analysis, the error is always less than
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// 1 ulp (unit in the last place).
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//
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// Misc. info.
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// For IEEE double
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// if x > 7.09782712893383973096e+02 then exp(x) overflow
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// if x < -7.45133219101941108420e+02 then exp(x) underflow
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//
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// Constants:
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// The hexadecimal values are the intended ones for the following
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// constants. The decimal values may be used, provided that the
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// compiler will convert from decimal to binary accurately enough
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// to produce the hexadecimal values shown.
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const
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(
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p0 = .2080384346694663001443843411e7;
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p1 = .3028697169744036299076048876e5;
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p2 = .6061485330061080841615584556e2;
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q0 = .6002720360238832528230907598e7;
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q1 = .3277251518082914423057964422e6;
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q2 = .1749287689093076403844945335e4;
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log2e = .14426950408889634073599247e1;
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sqrt2 = .14142135623730950488016887e1;
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maxf = 10000;
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export const (
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Ln2 = 0.693147180559945309417232121458176568;
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HalfLn2 = 0.346573590279972654708616060729088284;
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Ln2Hi = 6.93147180369123816490e-01;
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Ln2Lo = 1.90821492927058770002e-10;
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Log2e = 1.44269504088896338700e+00;
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P1 = 1.66666666666666019037e-01; /* 0x3FC55555; 0x5555553E */
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P2 = -2.77777777770155933842e-03; /* 0xBF66C16C; 0x16BEBD93 */
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P3 = 6.61375632143793436117e-05; /* 0x3F11566A; 0xAF25DE2C */
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P4 = -1.65339022054652515390e-06; /* 0xBEBBBD41; 0xC5D26BF1 */
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P5 = 4.13813679705723846039e-08; /* 0x3E663769; 0x72BEA4D0 */
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Overflow = 7.09782712893383973096e+02;
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Underflow = -7.45133219101941108420e+02;
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NearZero = 1.0/(1<<28); // 2^-28
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)
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export func Exp(arg float64) float64 {
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if arg == 0. {
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export func Exp(x float64) float64 {
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// special cases
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switch {
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case sys.isNaN(x) || sys.isInf(x, 1):
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return x;
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case sys.isInf(x, -1):
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return 0;
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case x > Overflow:
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return sys.Inf(1);
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case x < Underflow:
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return 0;
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case -NearZero < x && x < NearZero:
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return 1;
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}
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if arg < -maxf {
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return 0;
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}
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if arg > maxf {
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return sys.Inf(1)
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}
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x := arg*log2e;
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ent := int(Floor(x));
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fract := (x-float64(ent)) - 0.5;
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xsq := fract*fract;
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temp1 := ((p2*xsq+p1)*xsq+p0)*fract;
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temp2 := ((xsq+q2)*xsq+q1)*xsq + q0;
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return sys.ldexp(sqrt2*(temp2+temp1)/(temp2-temp1), ent);
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// reduce; computed as r = hi - lo for extra precision.
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var k int;
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switch {
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case x < 0:
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k = int(Log2e*x - 0.5);
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case x > 0:
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k = int(Log2e*x + 0.5);
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}
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hi := x - float64(k)*Ln2Hi;
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lo := float64(k)*Ln2Lo;
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r := hi - lo;
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// compute
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t := r * r;
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c := r - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
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y := 1 - ((lo - (r*c)/(2-c)) - hi);
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// TODO(rsc): make sure sys.ldexp can handle boundary k
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return sys.ldexp(y, k);
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}
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@ -4,56 +4,128 @@
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package math
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/*
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* Log returns the natural logarithm of its floating
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* point argument.
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*
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* The coefficients are #2705 from Hart & Cheney. (19.38D)
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*
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* It calls frexp.
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*/
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// The original C code, the long comment, and the constants
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// below are from FreeBSD's /usr/src/lib/msun/src/e_log.c
|
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// and came with this notice. The go code is a simpler
|
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// version of the original C.
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//
|
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// ====================================================
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// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
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//
|
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// Developed at SunPro, a Sun Microsystems, Inc. business.
|
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// Permission to use, copy, modify, and distribute this
|
||||
// software is freely granted, provided that this notice
|
||||
// is preserved.
|
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// ====================================================
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//
|
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// __ieee754_log(x)
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// Return the logrithm of x
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//
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// Method :
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// 1. Argument Reduction: find k and f such that
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// x = 2^k * (1+f),
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// where sqrt(2)/2 < 1+f < sqrt(2) .
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//
|
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// 2. Approximation of log(1+f).
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// Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
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// = 2s + 2/3 s**3 + 2/5 s**5 + .....,
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// = 2s + s*R
|
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// We use a special Reme algorithm on [0,0.1716] to generate
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// a polynomial of degree 14 to approximate R The maximum error
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// of this polynomial approximation is bounded by 2**-58.45. In
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// other words,
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// 2 4 6 8 10 12 14
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// R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
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// (the values of Lg1 to Lg7 are listed in the program)
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// and
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// | 2 14 | -58.45
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// | Lg1*s +...+Lg7*s - R(z) | <= 2
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// | |
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// Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
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// In order to guarantee error in log below 1ulp, we compute log
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// by
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// log(1+f) = f - s*(f - R) (if f is not too large)
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// log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
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//
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// 3. Finally, log(x) = k*ln2 + log(1+f).
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// = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
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// Here ln2 is split into two floating point number:
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// ln2_hi + ln2_lo,
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// where n*ln2_hi is always exact for |n| < 2000.
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//
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// Special cases:
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// log(x) is NaN with signal if x < 0 (including -INF) ;
|
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// log(+INF) is +INF; log(0) is -INF with signal;
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// log(NaN) is that NaN with no signal.
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//
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// Accuracy:
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// 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.
|
||||
|
||||
const (
|
||||
Ln2Hi = 6.93147180369123816490e-01; /* 3fe62e42 fee00000 */
|
||||
Ln2Lo = 1.90821492927058770002e-10; /* 3dea39ef 35793c76 */
|
||||
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 */
|
||||
|
||||
Two54 = 1<<54; // 2^54
|
||||
TwoM20 = 1.0/(1<<20); // 2^-20
|
||||
TwoM1022 = 2.2250738585072014e-308; // 2^-1022
|
||||
Sqrt2 = 1.41421356237309504880168872420969808;
|
||||
)
|
||||
|
||||
export func Log(x float64) float64 {
|
||||
// special cases
|
||||
switch {
|
||||
case sys.isNaN(x) || sys.isInf(x, 1):
|
||||
return x;
|
||||
case x < 0:
|
||||
return sys.NaN();
|
||||
case x == 0:
|
||||
return sys.Inf(-1);
|
||||
}
|
||||
|
||||
// reduce
|
||||
f1, ki := sys.frexp(x);
|
||||
if f1 < Sqrt2/2 {
|
||||
f1 *= 2;
|
||||
ki--;
|
||||
}
|
||||
f := f1 - 1;
|
||||
k := float64(ki);
|
||||
|
||||
// compute
|
||||
s := f/(2+f);
|
||||
s2 := s*s;
|
||||
s4 := s2*s2;
|
||||
t1 := s2*(Lg1 + s4*(Lg3 + s4*(Lg5 + s4*Lg7)));
|
||||
t2 := s4*(Lg2 + s4*(Lg4 + s4*Lg6));
|
||||
R := t1 + t2;
|
||||
hfsq := 0.5*f*f;
|
||||
return k*Ln2Hi - ((hfsq-(s*(hfsq+R)+k*Ln2Lo)) - f);
|
||||
}
|
||||
|
||||
const
|
||||
(
|
||||
log2 = .693147180559945309e0;
|
||||
ln10u1 = .4342944819032518276511;
|
||||
sqrto2 = .707106781186547524e0;
|
||||
p0 = -.240139179559210510e2;
|
||||
p1 = .309572928215376501e2;
|
||||
p2 = -.963769093377840513e1;
|
||||
p3 = .421087371217979714e0;
|
||||
q0 = -.120069589779605255e2;
|
||||
q1 = .194809660700889731e2;
|
||||
q2 = -.891110902798312337e1;
|
||||
ln10u1 = .4342944819032518276511;
|
||||
)
|
||||
|
||||
export func Log(arg float64) float64 {
|
||||
if arg <= 0 {
|
||||
return sys.NaN();
|
||||
}
|
||||
|
||||
x, exp := sys.frexp(arg);
|
||||
for x < 0.5 {
|
||||
x = x*2;
|
||||
exp = exp-1;
|
||||
}
|
||||
if x < sqrto2 {
|
||||
x = x*2;
|
||||
exp = exp-1;
|
||||
}
|
||||
|
||||
z := (x-1) / (x+1);
|
||||
zsq := z*z;
|
||||
|
||||
temp := ((p3*zsq + p2)*zsq + p1)*zsq + p0;
|
||||
temp = temp/(((zsq + q2)*zsq + q1)*zsq + q0);
|
||||
temp = temp*z + float64(exp)*log2;
|
||||
return temp;
|
||||
}
|
||||
|
||||
export func Log10(arg float64) float64 {
|
||||
if arg <= 0 {
|
||||
return sys.NaN();
|
||||
}
|
||||
return Log(arg) * ln10u1;
|
||||
}
|
||||
|
||||
|
||||
|
||||
@ -6,56 +6,82 @@ package math
|
||||
|
||||
import "math"
|
||||
|
||||
/*
|
||||
arg1 ^ arg2 (exponentiation)
|
||||
*/
|
||||
|
||||
export func Pow(arg1,arg2 float64) float64 {
|
||||
if arg2 < 0 {
|
||||
return 1/Pow(arg1, -arg2);
|
||||
// x^y: exponentation
|
||||
export func Pow(x, y float64) float64 {
|
||||
// TODO: x or y NaN, ±Inf, maybe ±0.
|
||||
switch {
|
||||
case y == 0:
|
||||
return 1;
|
||||
case y == 1:
|
||||
return x;
|
||||
case x == 0 && y > 0:
|
||||
return 0;
|
||||
case x == 0 && y < 0:
|
||||
return sys.Inf(1);
|
||||
case y == 0.5:
|
||||
return Sqrt(x);
|
||||
case y == -0.5:
|
||||
return 1 / Sqrt(x);
|
||||
}
|
||||
if arg1 <= 0 {
|
||||
if(arg1 == 0) {
|
||||
if arg2 <= 0 {
|
||||
return sys.NaN();
|
||||
|
||||
absy := y;
|
||||
flip := false;
|
||||
if absy < 0 {
|
||||
absy = -absy;
|
||||
flip = true;
|
||||
}
|
||||
yi, yf := sys.modf(absy);
|
||||
if yf != 0 && x < 0 {
|
||||
return sys.NaN();
|
||||
}
|
||||
if yi >= 1<<63 {
|
||||
return Exp(y * Log(x));
|
||||
}
|
||||
|
||||
ans := float64(1);
|
||||
|
||||
// ans *= x^yf
|
||||
if yf != 0 {
|
||||
if yf > 0.5 {
|
||||
yf--;
|
||||
yi++;
|
||||
}
|
||||
ans = Exp(yf * Log(x));
|
||||
}
|
||||
|
||||
// ans *= x^yi
|
||||
// by multiplying in successive squarings
|
||||
// of x according to bits of yi.
|
||||
// accumulate powers of two into exp.
|
||||
// will still have to do ans *= 2^exp later.
|
||||
x1, xe := sys.frexp(x);
|
||||
exp := 0;
|
||||
if i := int64(yi); i != 0 {
|
||||
for {
|
||||
if i&1 == 1 {
|
||||
ans *= x1;
|
||||
exp += xe;
|
||||
}
|
||||
return 0;
|
||||
}
|
||||
|
||||
temp := Floor(arg2);
|
||||
if temp != arg2 {
|
||||
panic(sys.NaN());
|
||||
}
|
||||
|
||||
l := int32(temp);
|
||||
if l&1 != 0 {
|
||||
return -Pow(-arg1, arg2);
|
||||
}
|
||||
return Pow(-arg1, arg2);
|
||||
}
|
||||
|
||||
temp := Floor(arg2);
|
||||
if temp != arg2 {
|
||||
if arg2-temp == .5 {
|
||||
if temp == 0 {
|
||||
return Sqrt(arg1);
|
||||
i >>= 1;
|
||||
if i == 0 {
|
||||
break;
|
||||
}
|
||||
x1 *= x1;
|
||||
xe <<= 1;
|
||||
if x1 < .5 {
|
||||
x1 += x1;
|
||||
xe--;
|
||||
}
|
||||
return Pow(arg1, temp) * Sqrt(arg1);
|
||||
}
|
||||
return Exp(arg2 * Log(arg1));
|
||||
}
|
||||
|
||||
l := int32(temp);
|
||||
temp = 1;
|
||||
for {
|
||||
if l&1 != 0 {
|
||||
temp = temp*arg1;
|
||||
}
|
||||
l >>= 1;
|
||||
if l == 0 {
|
||||
return temp;
|
||||
}
|
||||
arg1 *= arg1;
|
||||
// ans *= 2^exp
|
||||
// if flip { ans = 1 / ans }
|
||||
// but in the opposite order
|
||||
if flip {
|
||||
ans = 1 / ans;
|
||||
exp = -exp;
|
||||
}
|
||||
panic("unreachable")
|
||||
return sys.ldexp(ans, exp);
|
||||
}
|
||||
|
||||
|
||||
@ -4,6 +4,9 @@
|
||||
|
||||
package math
|
||||
|
||||
/*
|
||||
Coefficients are #3370 from Hart & Cheney (18.80D).
|
||||
*/
|
||||
const
|
||||
(
|
||||
p0 = .1357884097877375669092680e8;
|
||||
@ -15,6 +18,7 @@ const
|
||||
q1 = .4081792252343299749395779e6;
|
||||
q2 = .9463096101538208180571257e4;
|
||||
q3 = .1326534908786136358911494e3;
|
||||
|
||||
piu2 = .6366197723675813430755350e0; // 2/pi
|
||||
)
|
||||
|
||||
|
||||
Loading…
Reference in New Issue
Block a user