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+// 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
+
+import "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 -Inf 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
+ )
+
+ // 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;
+ }
+
+ // 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);
+}