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more accurate Log, Exp, Pow.

Browse Source 
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 2008-11-20 10:54:02 -08:00
parent c0a01e9665
commit f379ea0b07
6 changed files with 339 additions and 146 deletions
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19
src/lib/math/Makefile
View File

@ -33,6 +33,7 @@ coverage: packages
O1=\ O1=\
atan.$O\ atan.$O\
exp.$O\
fabs.$O\ fabs.$O\
floor.$O\ floor.$O\
fmod.$O\ fmod.$O\
@ -46,32 +47,25 @@ O1=\
O2=\ O2=\
asin.$O\ asin.$O\
atan2.$O\ atan2.$O\
exp.$O\
O3=\
pow.$O\ pow.$O\
sinh.$O\ sinh.$O\
O4=\ O3=\
tanh.$O\ tanh.$O\
math.a: a1 a2 a3 a4 math.a: a1 a2 a3
a1: $(O1) a1: $(O1)
$(AR) grc math.a atan.$O fabs.$O floor.$O fmod.$O hypot.$O log.$O pow10.$O sin.$O sqrt.$O tan.$O $(AR) grc math.a atan.$O exp.$O fabs.$O floor.$O fmod.$O hypot.$O log.$O pow10.$O sin.$O sqrt.$O tan.$O
rm -f $(O1) rm -f $(O1)
a2: $(O2) a2: $(O2)
$(AR) grc math.a asin.$O atan2.$O exp.$O $(AR) grc math.a asin.$O atan2.$O pow.$O sinh.$O
rm -f $(O2) rm -f $(O2)
a3: $(O3) a3: $(O3)
$(AR) grc math.a pow.$O sinh.$O
rm -f $(O3)
a4: $(O4)
$(AR) grc math.a tanh.$O $(AR) grc math.a tanh.$O
rm -f $(O4) rm -f $(O3)
newpkg: clean newpkg: clean
$(AR) grc math.a $(AR) grc math.a
@ -79,7 +73,6 @@ newpkg: clean
$(O1): newpkg $(O1): newpkg
$(O2): a1 $(O2): a1
$(O3): a2 $(O3): a2
$(O4): a3
nuke: clean nuke: clean
rm -f $(GOROOT)/pkg/math.a rm -f $(GOROOT)/pkg/math.a

36
src/lib/math/test.go → src/lib/math/all_test.go
View File

@ -50,7 +50,7 @@ var atan = []float64 {
var exp = []float64 { var exp = []float64 {
1.4533071302642137e+02, 1.4533071302642137e+02,
2.2958822575694450e+03, 2.2958822575694450e+03,
7.5814542574851664e-01, 7.5814542574851666e-01,
6.6668778421791010e-03, 6.6668778421791010e-03,
1.5310493273896035e+04, 1.5310493273896035e+04,
1.8659907517999329e+01, 1.8659907517999329e+01,
@ -156,13 +156,12 @@ var tanh = []float64 {
-9.9999994291374019e-01, -9.9999994291374019e-01,
} }
func Close(a,b float64) bool { func Tolerance(a,b,e float64) bool {
d := a-b; d := a-b;
if d < 0 { if d < 0 {
d = -d; d = -d;
} }
e := float64(1e-14);
if a != 0 { if a != 0 {
e = e*a; e = e*a;
if e < 0 { if e < 0 {
@ -171,10 +170,16 @@ func Close(a,b float64) bool {
} }
return d < e; return d < e;
} }
func Close(a,b float64) bool {
return Tolerance(a, b, 1e-14);
}
func VeryClose(a,b float64) bool {
return Tolerance(a, b, 4e-16);
}
export func TestAsin(t *testing.T) { export func TestAsin(t *testing.T) {
for i := 0; i < len(vf); i++ { for i := 0; i < len(vf); i++ {
if f := math.Asin(vf[i]/10); !Close(asin[i], f) { if f := math.Asin(vf[i]/10); !VeryClose(asin[i], f) {
t.Errorf("math.Asin(%g) = %g, want %g\n", vf[i]/10, f, asin[i]); t.Errorf("math.Asin(%g) = %g, want %g\n", vf[i]/10, f, asin[i]);
} }
} }
@ -182,7 +187,7 @@ export func TestAsin(t *testing.T) {
export func TestAtan(t *testing.T) { export func TestAtan(t *testing.T) {
for i := 0; i < len(vf); i++ { for i := 0; i < len(vf); i++ {
if f := math.Atan(vf[i]); !Close(atan[i], f) { if f := math.Atan(vf[i]); !VeryClose(atan[i], f) {
t.Errorf("math.Atan(%g) = %g, want %g\n", vf[i], f, atan[i]); t.Errorf("math.Atan(%g) = %g, want %g\n", vf[i], f, atan[i]);
} }
} }
@ -190,7 +195,7 @@ export func TestAtan(t *testing.T) {
export func TestExp(t *testing.T) { export func TestExp(t *testing.T) {
for i := 0; i < len(vf); i++ { for i := 0; i < len(vf); i++ {
if f := math.Exp(vf[i]); !Close(exp[i], f) { if f := math.Exp(vf[i]); !VeryClose(exp[i], f) {
t.Errorf("math.Exp(%g) = %g, want %g\n", vf[i], f, exp[i]); t.Errorf("math.Exp(%g) = %g, want %g\n", vf[i], f, exp[i]);
} }
} }
@ -198,7 +203,7 @@ export func TestExp(t *testing.T) {
export func TestFloor(t *testing.T) { export func TestFloor(t *testing.T) {
for i := 0; i < len(vf); i++ { for i := 0; i < len(vf); i++ {
if f := math.Floor(vf[i]); !Close(floor[i], f) { if f := math.Floor(vf[i]); floor[i] != f {
t.Errorf("math.Floor(%g) = %g, want %g\n", vf[i], f, floor[i]); t.Errorf("math.Floor(%g) = %g, want %g\n", vf[i], f, floor[i]);
} }
} }
@ -207,10 +212,14 @@ export func TestFloor(t *testing.T) {
export func TestLog(t *testing.T) { export func TestLog(t *testing.T) {
for i := 0; i < len(vf); i++ { for i := 0; i < len(vf); i++ {
a := math.Fabs(vf[i]); a := math.Fabs(vf[i]);
if f := math.Log(a); !Close(log[i], f) { if f := math.Log(a); log[i] != f {
t.Errorf("math.Log(%g) = %g, want %g\n", a, f, floor[i]); t.Errorf("math.Log(%g) = %g, want %g\n", a, f, log[i]);
} }
} }
const Ln10 = 2.30258509299404568401799145468436421;
if f := math.Log(10); f != Ln10 {
t.Errorf("math.Log(%g) = %g, want %g\n", 10, f, Ln10);
}
} }
export func TestPow(t *testing.T) { export func TestPow(t *testing.T) {
@ -231,7 +240,7 @@ export func TestSin(t *testing.T) {
export func TestSinh(t *testing.T) { export func TestSinh(t *testing.T) {
for i := 0; i < len(vf); i++ { for i := 0; i < len(vf); i++ {
if f := math.Sinh(vf[i]); !Close(sinh[i], f) { if f := math.Sinh(vf[i]); !VeryClose(sinh[i], f) {
t.Errorf("math.Sinh(%g) = %g, want %g\n", vf[i], f, sinh[i]); t.Errorf("math.Sinh(%g) = %g, want %g\n", vf[i], f, sinh[i]);
} }
} }
@ -240,7 +249,7 @@ export func TestSinh(t *testing.T) {
export func TestSqrt(t *testing.T) { export func TestSqrt(t *testing.T) {
for i := 0; i < len(vf); i++ { for i := 0; i < len(vf); i++ {
a := math.Fabs(vf[i]); a := math.Fabs(vf[i]);
if f := math.Sqrt(a); !Close(sqrt[i], f) { if f := math.Sqrt(a); !VeryClose(sqrt[i], f) {
t.Errorf("math.Sqrt(%g) = %g, want %g\n", a, f, floor[i]); t.Errorf("math.Sqrt(%g) = %g, want %g\n", a, f, floor[i]);
} }
} }
@ -256,7 +265,7 @@ export func TestTan(t *testing.T) {
export func TestTanh(t *testing.T) { export func TestTanh(t *testing.T) {
for i := 0; i < len(vf); i++ { for i := 0; i < len(vf); i++ {
if f := math.Tanh(vf[i]); !Close(tanh[i], f) { if f := math.Tanh(vf[i]); !VeryClose(tanh[i], f) {
t.Errorf("math.Tanh(%g) = %g, want %g\n", vf[i], f, tanh[i]); t.Errorf("math.Tanh(%g) = %g, want %g\n", vf[i], f, tanh[i]);
} }
} }
@ -265,9 +274,8 @@ export func TestTanh(t *testing.T) {
export func TestHypot(t *testing.T) { export func TestHypot(t *testing.T) {
for i := 0; i < len(vf); i++ { for i := 0; i < len(vf); i++ {
a := math.Fabs(tanh[i]*math.Sqrt(2)); a := math.Fabs(tanh[i]*math.Sqrt(2));
if f := math.Hypot(tanh[i], tanh[i]); !Close(a, f) { if f := math.Hypot(tanh[i], tanh[i]); !VeryClose(a, f) {
t.Errorf("math.Hypot(%g, %g) = %g, want %g\n", tanh[i], tanh[i], f, a); t.Errorf("math.Hypot(%g, %g) = %g, want %g\n", tanh[i], tanh[i], f, a);
} }
} }
} }

154
src/lib/math/exp.go
View File

@ -6,42 +6,132 @@ package math
import "math" import "math"
/* // The original C code, the long comment, and the constants
* exp returns the exponential func of its // below are from FreeBSD's /usr/src/lib/msun/src/e_exp.c
* floating-point argument. // and came with this notice. The go code is a simplified
* // version of the original C.
* The coefficients are #1069 from Hart and Cheney. (22.35D) //
*/ // ====================================================
// Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
//
// Permission to use, copy, modify, and distribute this
// software is freely granted, provided that this notice
// is preserved.
// ====================================================
//
//
// exp(x)
// Returns the exponential of x.
//
// Method
// 1. Argument reduction:
// Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
// Given x, find r and integer k such that
//
// x = k*ln2 + r, |r| <= 0.5*ln2.
//
// Here r will be represented as r = hi-lo for better
// accuracy.
//
// 2. Approximation of exp(r) by a special rational function on
// the interval [0,0.34658]:
// Write
// R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
// We use a special Remes algorithm on [0,0.34658] to generate
// a polynomial of degree 5 to approximate R. The maximum error
// of this polynomial approximation is bounded by 2**-59. In
// other words,
// R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
// (where z=r*r, and the values of P1 to P5 are listed below)
// and
// | 5 | -59
// | 2.0+P1*z+...+P5*z - R(z) | <= 2
// | |
// The computation of exp(r) thus becomes
// 2*r
// exp(r) = 1 + -------
// R - r
// r*R1(r)
// = 1 + r + ----------- (for better accuracy)
// 2 - R1(r)
// where
// 2 4 10
// R1(r) = r - (P1*r + P2*r + ... + P5*r ).
//
// 3. Scale back to obtain exp(x):
// From step 1, we have
// exp(x) = 2^k * exp(r)
//
// Special cases:
// exp(INF) is INF, exp(NaN) is NaN;
// exp(-INF) is 0, and
// for finite argument, only exp(0)=1 is exact.
//
// Accuracy:
// according to an error analysis, the error is always less than
// 1 ulp (unit in the last place).
//
// Misc. info.
// For IEEE double
// if x > 7.09782712893383973096e+02 then exp(x) overflow
// if x < -7.45133219101941108420e+02 then exp(x) underflow
//
// Constants:
// The hexadecimal values are the intended ones for the following
// constants. The decimal values may be used, provided that the
// compiler will convert from decimal to binary accurately enough
// to produce the hexadecimal values shown.
const export const (
( Ln2 = 0.693147180559945309417232121458176568;
p0 = .2080384346694663001443843411e7; HalfLn2 = 0.346573590279972654708616060729088284;
p1 = .3028697169744036299076048876e5;
p2 = .6061485330061080841615584556e2; Ln2Hi = 6.93147180369123816490e-01;
q0 = .6002720360238832528230907598e7; Ln2Lo = 1.90821492927058770002e-10;
q1 = .3277251518082914423057964422e6; Log2e = 1.44269504088896338700e+00;
q2 = .1749287689093076403844945335e4;
log2e = .14426950408889634073599247e1; P1 = 1.66666666666666019037e-01; /* 0x3FC55555; 0x5555553E */
sqrt2 = .14142135623730950488016887e1; P2 = -2.77777777770155933842e-03; /* 0xBF66C16C; 0x16BEBD93 */
maxf = 10000; P3 = 6.61375632143793436117e-05; /* 0x3F11566A; 0xAF25DE2C */
P4 = -1.65339022054652515390e-06; /* 0xBEBBBD41; 0xC5D26BF1 */
P5 = 4.13813679705723846039e-08; /* 0x3E663769; 0x72BEA4D0 */
Overflow = 7.09782712893383973096e+02;
Underflow = -7.45133219101941108420e+02;
NearZero = 1.0/(1<<28); // 2^-28
) )
export func Exp(arg float64) float64 { export func Exp(x float64) float64 {
if arg == 0. { // special cases
switch {
case sys.isNaN(x) || sys.isInf(x, 1):
return x;
case sys.isInf(x, -1):
return 0;
case x > Overflow:
return sys.Inf(1);
case x < Underflow:
return 0;
case -NearZero < x && x < NearZero:
return 1; return 1;
} }
if arg < -maxf {
return 0;
}
if arg > maxf {
return sys.Inf(1)
}
x := arg*log2e; // reduce; computed as r = hi - lo for extra precision.
ent := int(Floor(x)); var k int;
fract := (x-float64(ent)) - 0.5; switch {
xsq := fract*fract; case x < 0:
temp1 := ((p2*xsq+p1)*xsq+p0)*fract; k = int(Log2e*x - 0.5);
temp2 := ((xsq+q2)*xsq+q1)*xsq + q0; case x > 0:
return sys.ldexp(sqrt2*(temp2+temp1)/(temp2-temp1), ent); k = int(Log2e*x + 0.5);
}
hi := x - float64(k)*Ln2Hi;
lo := float64(k)*Ln2Lo;
r := hi - lo;
// compute
t := r * r;
c := r - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
y := 1 - ((lo - (r*c)/(2-c)) - hi);
// TODO(rsc): make sure sys.ldexp can handle boundary k
return sys.ldexp(y, k);
} }

156
src/lib/math/log.go
View File

@ -4,56 +4,128 @@
package math package math
/* // The original C code, the long comment, and the constants
* Log returns the natural logarithm of its floating // below are from FreeBSD's /usr/src/lib/msun/src/e_log.c
* point argument. // and came with this notice. The go code is a simpler
* // version of the original C.
* The coefficients are #2705 from Hart & Cheney. (19.38D) //
* // ====================================================
* It calls frexp. // Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*/ //
// Developed at SunPro, a Sun Microsystems, Inc. business.
// Permission to use, copy, modify, and distribute this
// software is freely granted, provided that this notice
// is preserved.
// ====================================================
//
// __ieee754_log(x)
// Return the logrithm of x
//
// Method :
// 1. Argument Reduction: find k and f such that
// x = 2^k * (1+f),
// where sqrt(2)/2 < 1+f < sqrt(2) .
//
// 2. Approximation of log(1+f).
// Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
// = 2s + 2/3 s**3 + 2/5 s**5 + .....,
// = 2s + s*R
// We use a special Reme algorithm on [0,0.1716] to generate
// a polynomial of degree 14 to approximate R The maximum error
// of this polynomial approximation is bounded by 2**-58.45. In
// other words,
// 2 4 6 8 10 12 14
// R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
// (the values of Lg1 to Lg7 are listed in the program)
// and
// | 2 14 | -58.45
// | Lg1*s +...+Lg7*s - R(z) | <= 2
// | |
// Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
// In order to guarantee error in log below 1ulp, we compute log
// by
// log(1+f) = f - s*(f - R) (if f is not too large)
// log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
//
// 3. Finally, log(x) = k*ln2 + log(1+f).
// = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
// Here ln2 is split into two floating point number:
// ln2_hi + ln2_lo,
// where n*ln2_hi is always exact for |n| < 2000.
//
// Special cases:
// log(x) is NaN with signal if x < 0 (including -INF) ;
// log(+INF) is +INF; log(0) is -INF with signal;
// log(NaN) is that NaN with no signal.
//
// Accuracy:
// according to an error analysis, the error is always less than
// 1 ulp (unit in the last place).
//
// Constants:
// The hexadecimal values are the intended ones for the following
// constants. The decimal values may be used, provided that the
// compiler will convert from decimal to binary accurately enough
// to produce the hexadecimal values shown.
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 const
( (
log2 = .693147180559945309e0; ln10u1 = .4342944819032518276511;
ln10u1 = .4342944819032518276511;
sqrto2 = .707106781186547524e0;
p0 = -.240139179559210510e2;
p1 = .309572928215376501e2;
p2 = -.963769093377840513e1;
p3 = .421087371217979714e0;
q0 = -.120069589779605255e2;
q1 = .194809660700889731e2;
q2 = -.891110902798312337e1;
) )
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 { export func Log10(arg float64) float64 {
if arg <= 0 { if arg <= 0 {
return sys.NaN(); return sys.NaN();
} }
return Log(arg) * ln10u1; return Log(arg) * ln10u1;
} }

116
src/lib/math/pow.go
View File

@ -6,56 +6,82 @@ package math
import "math" import "math"
/* // x^y: exponentation
arg1 ^ arg2 (exponentiation) export func Pow(x, y float64) float64 {
*/ // TODO: x or y NaN, ±Inf, maybe ±0.
switch {
export func Pow(arg1,arg2 float64) float64 { case y == 0:
if arg2 < 0 { return 1;
return 1/Pow(arg1, -arg2); 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) { absy := y;
if arg2 <= 0 { flip := false;
return sys.NaN(); 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; i >>= 1;
} if i == 0 {
break;
temp := Floor(arg2); }
if temp != arg2 { x1 *= x1;
panic(sys.NaN()); xe <<= 1;
} if x1 < .5 {
x1 += x1;
l := int32(temp); xe--;
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);
} }
return Pow(arg1, temp) * Sqrt(arg1);
} }
return Exp(arg2 * Log(arg1));
} }
l := int32(temp); // ans *= 2^exp
temp = 1; // if flip { ans = 1 / ans }
for { // but in the opposite order
if l&1 != 0 { if flip {
temp = temp*arg1; ans = 1 / ans;
} exp = -exp;
l >>= 1;
if l == 0 {
return temp;
}
arg1 *= arg1;
} }
panic("unreachable") return sys.ldexp(ans, exp);
} }

4
src/lib/math/sin.go
View File

@ -4,6 +4,9 @@
package math package math
/*
Coefficients are #3370 from Hart & Cheney (18.80D).
*/
const const
( (
p0 = .1357884097877375669092680e8; p0 = .1357884097877375669092680e8;
@ -15,6 +18,7 @@ const
q1 = .4081792252343299749395779e6; q1 = .4081792252343299749395779e6;
q2 = .9463096101538208180571257e4; q2 = .9463096101538208180571257e4;
q3 = .1326534908786136358911494e3; q3 = .1326534908786136358911494e3;
piu2 = .6366197723675813430755350e0; // 2/pi piu2 = .6366197723675813430755350e0; // 2/pi
) )