+++ /dev/null
-/* @(#)s_log1p.c 5.1 93/09/24 */
-/*
- * ====================================================
- * 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.
- */
-
-#include <math.h>
-#include "math_private.h"
-
-static const double
-ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
-ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
-two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
-Lp1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
-Lp2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
-Lp3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
-Lp4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
-Lp5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
-Lp6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
-Lp7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
-
-static const double zero = 0.0;
-
-double
-log1p(double x)
-{
- double hfsq,f=0,c=0,s,z,R,u;
- int32_t k,hx,hu=0,ax;
-
- GET_HIGH_WORD(hx,x);
- ax = hx&0x7fffffff;
-
- k = 1;
- if (hx < 0x3FDA827A) { /* x < 0.41422 */
- if(ax>=0x3ff00000) { /* x <= -1.0 */
- if(x==-1.0) return -two54/zero; /* log1p(-1)=+inf */
- else return (x-x)/(x-x); /* log1p(x<-1)=NaN */
- }
- if(ax<0x3e200000) { /* |x| < 2**-29 */
- if(two54+x>zero /* raise inexact */
- &&ax<0x3c900000) /* |x| < 2**-54 */
- return x;
- else
- return x - x*x*0.5;
- }
- if(hx>0||hx<=((int32_t)0xbfd2bec3)) {
- k=0;f=x;hu=1;} /* -0.2929<x<0.41422 */
- }
- if (hx >= 0x7ff00000) return x+x;
- if(k!=0) {
- if(hx<0x43400000) {
- u = 1.0+x;
- GET_HIGH_WORD(hu,u);
- k = (hu>>20)-1023;
- c = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */
- c /= u;
- } else {
- u = x;
- GET_HIGH_WORD(hu,u);
- k = (hu>>20)-1023;
- c = 0;
- }
- hu &= 0x000fffff;
- if(hu<0x6a09e) {
- SET_HIGH_WORD(u,hu|0x3ff00000); /* normalize u */
- } else {
- k += 1;
- SET_HIGH_WORD(u,hu|0x3fe00000); /* normalize u/2 */
- hu = (0x00100000-hu)>>2;
- }
- f = u-1.0;
- }
- hfsq=0.5*f*f;
- if(hu==0) { /* |f| < 2**-20 */
- if(f==zero) { if(k==0) return zero;
- else {c += k*ln2_lo; return k*ln2_hi+c;} }
- R = hfsq*(1.0-0.66666666666666666*f);
- if(k==0) return f-R; else
- return k*ln2_hi-((R-(k*ln2_lo+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)); else
- return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f);
-}
projects / musl / blobdiff