copysign.go\
erf.go\
exp.go\
+ exp_port.go\
exp2.go\
expm1.go\
fabs.go\
}
func TestExp(t *testing.T) {
+ testExp(t, Exp, "Exp")
+ testExp(t, ExpGo, "ExpGo")
+}
+
+func testExp(t *testing.T, Exp func(float64) float64, name string) {
for i := 0; i < len(vf); i++ {
if f := Exp(vf[i]); !close(exp[i], f) {
- t.Errorf("Exp(%g) = %g, want %g", vf[i], f, exp[i])
+ t.Errorf("%s(%g) = %g, want %g", name, vf[i], f, exp[i])
}
}
for i := 0; i < len(vfexpSC); i++ {
if f := Exp(vfexpSC[i]); !alike(expSC[i], f) {
- t.Errorf("Exp(%g) = %g, want %g", vfexpSC[i], f, expSC[i])
+ t.Errorf("%s(%g) = %g, want %g", name, vfexpSC[i], f, expSC[i])
}
}
}
}
func TestExp2(t *testing.T) {
+ testExp2(t, Exp2, "Exp2")
+ testExp2(t, Exp2Go, "Exp2Go")
+}
+
+func testExp2(t *testing.T, Exp2 func(float64) float64, name string) {
for i := 0; i < len(vf); i++ {
if f := Exp2(vf[i]); !close(exp2[i], f) {
- t.Errorf("Exp2(%g) = %g, want %g", vf[i], f, exp2[i])
+ t.Errorf("%s(%g) = %g, want %g", name, vf[i], f, exp2[i])
}
}
for i := 0; i < len(vfexpSC); i++ {
if f := Exp2(vfexpSC[i]); !alike(expSC[i], f) {
- t.Errorf("Exp2(%g) = %g, want %g", vfexpSC[i], f, expSC[i])
+ t.Errorf("%s(%g) = %g, want %g", name, vfexpSC[i], f, expSC[i])
+ }
+ }
+ for n := -1074; n < 1024; n++ {
+ f := Exp2(float64(n))
+ vf := Ldexp(1, n)
+ if f != vf {
+ t.Errorf("%s(%d) = %g, want %g", name, n, f, vf)
}
}
}
}
}
+func BenchmarkExpGo(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ ExpGo(.5)
+ }
+}
+
func BenchmarkExpm1(b *testing.B) {
for i := 0; i < b.N; i++ {
Expm1(.5)
}
}
+func BenchmarkExp2Go(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Exp2Go(.5)
+ }
+}
+
func BenchmarkFabs(b *testing.B) {
for i := 0; i < b.N; i++ {
Fabs(.5)
package math
-
-// The original C code, the long comment, and the constants
-// below are from FreeBSD's /usr/src/lib/msun/src/e_exp.c
-// and came with this notice. The go code is a simplified
-// version of the original C.
-//
-// ====================================================
-// 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.
-
// Exp returns e**x, the base-e exponential of x.
//
// Special cases are:
// Exp(NaN) = NaN
// Very large values overflow to 0 or +Inf.
// Very small values underflow to 1.
-func Exp(x float64) float64 {
- const (
- Ln2Hi = 6.93147180369123816490e-01
- Ln2Lo = 1.90821492927058770002e-10
- Log2e = 1.44269504088896338700e+00
- P1 = 1.66666666666666019037e-01 /* 0x3FC55555; 0x5555553E */
- P2 = -2.77777777770155933842e-03 /* 0xBF66C16C; 0x16BEBD93 */
- 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
- )
-
- // TODO(rsc): Remove manual inlining of IsNaN, IsInf
- // when compiler does it for us
- // special cases
- switch {
- case x != x || x > MaxFloat64: // IsNaN(x) || IsInf(x, 1):
- return x
- case x < -MaxFloat64: // IsInf(x, -1):
- return 0
- case x > Overflow:
- return Inf(1)
- case x < Underflow:
- return 0
- case -NearZero < x && x < NearZero:
- return 1
- }
-
- // reduce; computed as r = hi - lo for extra precision.
- var k int
- switch {
- case x < 0:
- k = int(Log2e*x - 0.5)
- case x > 0:
- 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 Ldexp can handle boundary k
- return Ldexp(y, k)
-}
+func Exp(x float64) float64 { return expGo(x) }
// Exp2 returns 2**x, the base-2 exponential of x.
//
// Special cases are the same as Exp.
-func Exp2(x float64) float64 { return Exp(x * Ln2) }
+func Exp2(x float64) float64 { return exp2Go(x) }
--- /dev/null
+// Copyright 2009 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 math
+
+
+// The original C code, the long comment, and the constants
+// below are from FreeBSD's /usr/src/lib/msun/src/e_exp.c
+// and came with this notice. The go code is a simplified
+// version of the original C.
+//
+// ====================================================
+// 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.
+
+// Exp returns e**x, the base-e exponential of x.
+//
+// Special cases are:
+// Exp(+Inf) = +Inf
+// Exp(NaN) = NaN
+// Very large values overflow to 0 or +Inf.
+// Very small values underflow to 1.
+func expGo(x float64) float64 {
+ const (
+ Ln2Hi = 6.93147180369123816490e-01
+ Ln2Lo = 1.90821492927058770002e-10
+ Log2e = 1.44269504088896338700e+00
+
+ Overflow = 7.09782712893383973096e+02
+ Underflow = -7.45133219101941108420e+02
+ NearZero = 1.0 / (1 << 28) // 2**-28
+ )
+
+ // TODO(rsc): Remove manual inlining of IsNaN, IsInf
+ // when compiler does it for us
+ // special cases
+ switch {
+ case x != x || x > MaxFloat64: // IsNaN(x) || IsInf(x, 1):
+ return x
+ case x < -MaxFloat64: // IsInf(x, -1):
+ return 0
+ case x > Overflow:
+ return Inf(1)
+ case x < Underflow:
+ return 0
+ case -NearZero < x && x < NearZero:
+ return 1 + x
+ }
+
+ // reduce; computed as r = hi - lo for extra precision.
+ var k int
+ switch {
+ case x < 0:
+ k = int(Log2e*x - 0.5)
+ case x > 0:
+ k = int(Log2e*x + 0.5)
+ }
+ hi := x - float64(k)*Ln2Hi
+ lo := float64(k) * Ln2Lo
+
+ // compute
+ return exp(hi, lo, k)
+}
+
+// Exp2 returns 2**x, the base-2 exponential of x.
+//
+// Special cases are the same as Exp.
+func exp2Go(x float64) float64 {
+ const (
+ Ln2Hi = 6.93147180369123816490e-01
+ Ln2Lo = 1.90821492927058770002e-10
+
+ Overflow = 1.0239999999999999e+03
+ Underflow = -1.0740e+03
+ )
+
+ // TODO: remove manual inlining of IsNaN and IsInf
+ // when compiler does it for us
+ // special cases
+ switch {
+ case x != x || x > MaxFloat64: // IsNaN(x) || IsInf(x, 1):
+ return x
+ case x < -MaxFloat64: // IsInf(x, -1):
+ return 0
+ case x > Overflow:
+ return Inf(1)
+ case x < Underflow:
+ return 0
+ }
+
+ // argument reduction; x = r×lg(e) + k with |r| ≤ ln(2)/2.
+ // computed as r = hi - lo for extra precision.
+ var k int
+ switch {
+ case x > 0:
+ k = int(x + 0.5)
+ case x < 0:
+ k = int(x - 0.5)
+ }
+ t := x - float64(k)
+ hi := t * Ln2Hi
+ lo := -t * Ln2Lo
+
+ // compute
+ return exp(hi, lo, k)
+}
+
+// exp returns e**r × 2**k where r = hi - lo and |r| ≤ ln(2)/2.
+func exp(hi, lo float64, k int) float64 {
+ const (
+ P1 = 1.66666666666666019037e-01 /* 0x3FC55555; 0x5555553E */
+ P2 = -2.77777777770155933842e-03 /* 0xBF66C16C; 0x16BEBD93 */
+ P3 = 6.61375632143793436117e-05 /* 0x3F11566A; 0xAF25DE2C */
+ P4 = -1.65339022054652515390e-06 /* 0xBEBBBD41; 0xC5D26BF1 */
+ P5 = 4.13813679705723846039e-08 /* 0x3E663769; 0x72BEA4D0 */
+ )
+
+ r := hi - lo
+ 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 Ldexp can handle boundary k
+ return Ldexp(y, k)
+}
--- /dev/null
+// 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 math
+
+// Make expGo and exp2Go available for testing.
+
+func ExpGo(x float64) float64 { return expGo(x) }
+func Exp2Go(x float64) float64 { return exp2Go(x) }