// 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
// The original C code, the long comment, and the constants
// below are from FreeBSD's /usr/src/lib/msun/src/s_log1p.c
// and came with this notice. The go code is a simplified
// version of the original C.
//
// ====================================================
// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
//
// Developed at SunPro, a Sun Microsystems, Inc. business.
// Permission to use, copy, modify, and distribute this
// software is freely granted, provided that this notice
// is preserved.
// ====================================================
//
//
// double log1p(double x)
//
// Method :
// 1. Argument Reduction: find k and f such that
// 1+x = 2**k * (1+f),
// where sqrt(2)/2 < 1+f < sqrt(2) .
//
// Note. If k=0, then f=x is exact. However, if k!=0, then f
// may not be representable exactly. In that case, a correction
// term is need. Let u=1+x rounded. Let c = (1+x)-u, then
// log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
// and add back the correction term c/u.
// (Note: when x > 2**53, one can simply return log(x))
//
// 2. Approximation of log1p(f).
// Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
// = 2s + 2/3 s**3 + 2/5 s**5 + .....,
// = 2s + s*R
// We use a special Reme algorithm on [0,0.1716] to generate
// a polynomial of degree 14 to approximate R The maximum error
// of this polynomial approximation is bounded by 2**-58.45. In
// other words,
// 2 4 6 8 10 12 14
// R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s
// (the values of Lp1 to Lp7 are listed in the program)
// and
// | 2 14 | -58.45
// | Lp1*s +...+Lp7*s - R(z) | <= 2
// | |
// Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
// In order to guarantee error in log below 1ulp, we compute log
// by
// log1p(f) = f - (hfsq - s*(hfsq+R)).
//
// 3. Finally, log1p(x) = k*ln2 + log1p(f).
// = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
// Here ln2 is split into two floating point number:
// ln2_hi + ln2_lo,
// where n*ln2_hi is always exact for |n| < 2000.
//
// Special cases:
// log1p(x) is NaN with signal if x < -1 (including -INF) ;
// log1p(+INF) is +INF; log1p(-1) is -INF with signal;
// log1p(NaN) is that NaN with no signal.
//
// Accuracy:
// according to an error analysis, the error is always less than
// 1 ulp (unit in the last place).
//
// 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.
//
// Note: Assuming log() return accurate answer, the following
// algorithm can be used to compute log1p(x) to within a few ULP:
//
// u = 1+x;
// if(u==1.0) return x ; else
// return log(u)*(x/(u-1.0));
//
// See HP-15C Advanced Functions Handbook, p.193.
// Log1p returns the natural logarithm of 1 plus its argument x.
// It is more accurate than Log(1 + x) when x is near zero.
//
// Special cases are:
// Log1p(+Inf) = +Inf
// Log1p(±0) = ±0
// Log1p(-1) = -Inf
// Log1p(x < -1) = NaN
// Log1p(NaN) = NaN
func Log1p(x float64) float64
func log1p(x float64) float64 {
const (
Sqrt2M1 = 4.142135623730950488017e-01 // Sqrt(2)-1 = 0x3fda827999fcef34
Sqrt2HalfM1 = -2.928932188134524755992e-01 // Sqrt(2)/2-1 = 0xbfd2bec333018866
Small = 1.0 / (1 << 29) // 2**-29 = 0x3e20000000000000
Tiny = 1.0 / (1 << 54) // 2**-54
Two53 = 1 << 53 // 2**53
Ln2Hi = 6.93147180369123816490e-01 // 3fe62e42fee00000
Ln2Lo = 1.90821492927058770002e-10 // 3dea39ef35793c76
Lp1 = 6.666666666666735130e-01 // 3FE5555555555593
Lp2 = 3.999999999940941908e-01 // 3FD999999997FA04
Lp3 = 2.857142874366239149e-01 // 3FD2492494229359
Lp4 = 2.222219843214978396e-01 // 3FCC71C51D8E78AF
Lp5 = 1.818357216161805012e-01 // 3FC7466496CB03DE
Lp6 = 1.531383769920937332e-01 // 3FC39A09D078C69F
Lp7 = 1.479819860511658591e-01 // 3FC2F112DF3E5244
)
// special cases
switch {
case x < -1 || IsNaN(x): // includes -Inf
return NaN()
case x == -1:
return Inf(-1)
case IsInf(x, 1):
return Inf(1)
}
absx := x
if absx < 0 {
absx = -absx
}
var f float64
var iu uint64
k := 1
if absx < Sqrt2M1 { // |x| < Sqrt(2)-1
if absx < Small { // |x| < 2**-29
if absx < Tiny { // |x| < 2**-54
return x
}
return x - x*x*0.5
}
if x > Sqrt2HalfM1 { // Sqrt(2)/2-1 < x
// (Sqrt(2)/2-1) < x < (Sqrt(2)-1)
k = 0
f = x
iu = 1
}
}
var c float64
if k != 0 {
var u float64
if absx < Two53 { // 1<<53
u = 1.0 + x
iu = Float64bits(u)
k = int((iu >> 52) - 1023)
if k > 0 {
c = 1.0 - (u - x)
} else {
c = x - (u - 1.0) // correction term
c /= u
}
} else {
u = x
iu = Float64bits(u)
k = int((iu >> 52) - 1023)
c = 0
}
iu &= 0x000fffffffffffff
if iu < 0x0006a09e667f3bcd { // mantissa of Sqrt(2)
u = Float64frombits(iu | 0x3ff0000000000000) // normalize u
} else {
k++
u = Float64frombits(iu | 0x3fe0000000000000) // normalize u/2
iu = (0x0010000000000000 - iu) >> 2
}
f = u - 1.0 // Sqrt(2)/2 < u < Sqrt(2)
}
hfsq := 0.5 * f * f
var s, R, z float64
if iu == 0 { // |f| < 2**-20
if f == 0 {
if k == 0 {
return 0
}
c += float64(k) * Ln2Lo
return float64(k)*Ln2Hi + c
}
R = hfsq * (1.0 - 0.66666666666666666*f) // avoid division
if k == 0 {
return f - R
}
return float64(k)*Ln2Hi - ((R - (float64(k)*Ln2Lo + c)) - f)
}
s = f / (2.0 + f)
z = s * s
R = z * (Lp1 + z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))))
if k == 0 {
return f - (hfsq - s*(hfsq+R))
}
return float64(k)*Ln2Hi - ((hfsq - (s*(hfsq+R) + (float64(k)*Ln2Lo + c))) - f)
}