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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
+
+// 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 {
+ if haveArchExp {
+ return archExp(x)
+ }
+ return exp(x)
+}
+
+// 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 {
+ if haveArchExp2 {
+ return archExp2(x)
+ }
+ return exp2(x)
+}
+
+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)
+}