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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
// 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)
}
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