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/* origin: FreeBSD /usr/src/lib/msun/src/k_log.h */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunSoft, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/*
* __log1p(f):
* Return log(1+f)  f for 1+f in ~[sqrt(2)/2, sqrt(2)].
*
* The following describes the overall strategy for computing
* logarithms in base e. The argument reduction and adding the final
* term of the polynomial are done by the caller for increased accuracy
* when different bases are used.
*
* Method :
* 1. Argument Reduction: find k and f such that
* x = 2^k * (1+f),
* where sqrt(2)/2 < 1+f < sqrt(2) .
*
* 2. Approximation of log(1+f).
* Let s = f/(2+f) ; based on log(1+f) = log(1+s)  log(1s)
* = 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) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
* (the values of Lg1 to Lg7 are listed in the program)
* and
*  2 14  58.45
*  Lg1*s +...+Lg7*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
* log(1+f) = f  s*(f  R) (if f is not too large)
* log(1+f) = f  (hfsq  s*(hfsq+R)). (better accuracy)
*
* 3. Finally, log(x) = k*ln2 + log(1+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:
* log(x) is NaN with signal if x < 0 (including INF) ;
* log(+INF) is +INF; log(0) is INF with signal;
* log(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.
*/
static const double
Lg1 = 6.666666666666735130e01, /* 3FE55555 55555593 */
Lg2 = 3.999999999940941908e01, /* 3FD99999 9997FA04 */
Lg3 = 2.857142874366239149e01, /* 3FD24924 94229359 */
Lg4 = 2.222219843214978396e01, /* 3FCC71C5 1D8E78AF */
Lg5 = 1.818357216161805012e01, /* 3FC74664 96CB03DE */
Lg6 = 1.531383769920937332e01, /* 3FC39A09 D078C69F */
Lg7 = 1.479819860511658591e01; /* 3FC2F112 DF3E5244 */
/*
* We always inline __log1p(), since doing so produces a
* substantial performance improvement (~40% on amd64).
*/
static inline double __log1p(double f)
{
double_t hfsq,s,z,R,w,t1,t2;
s = f/(2.0+f);
z = s*s;
w = z*z;
t1= w*(Lg2+w*(Lg4+w*Lg6));
t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
R = t2+t1;
hfsq = 0.5*f*f;
return s*(hfsq+R);
}
