golang log1p 代码
文件路径：/src/math/log1p.go
/ 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 {
if haveArchLog1p {
return archLog1p(x)
}
return log1p(x)
}
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 := Abs(x)
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)
/ correction term
if k > 0 {
c = 1.0 - (u - x)
} else {
c = x - (u - 1.0)
}
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)
}