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| -rw-r--r-- | src/math/exp.go | 191 |
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diff --git a/src/math/exp.go b/src/math/exp.go new file mode 100644 index 0000000..bd4c5c9 --- /dev/null +++ b/src/math/exp.go @@ -0,0 +1,191 @@ +// 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 + +// 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 Exp(x float64) float64 + +// 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 Remez 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. + +func exp(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 + ) + + // special cases + switch { + case IsNaN(x) || IsInf(x, 1): + return x + case 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 expmulti(hi, lo, k) +} + +// Exp2 returns 2**x, the base-2 exponential of x. +// +// Special cases are the same as Exp. +func Exp2(x float64) float64 + +func exp2(x float64) float64 { + const ( + Ln2Hi = 6.93147180369123816490e-01 + Ln2Lo = 1.90821492927058770002e-10 + + Overflow = 1.0239999999999999e+03 + Underflow = -1.0740e+03 + ) + + // special cases + switch { + case IsNaN(x) || IsInf(x, 1): + return x + case 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 expmulti(hi, lo, k) +} + +// exp1 returns e**r × 2**k where r = hi - lo and |r| ≤ ln(2)/2. +func expmulti(hi, lo float64, k int) float64 { + const ( + P1 = 1.66666666666666657415e-01 /* 0x3FC55555; 0x55555555 */ + 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) +} |