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Rev	Author	Line No.	Line
2	pj	1	/* @(#)s_log1p.c 5.1 93/09/24 */
		2	/*
		3	* ====================================================
		4	* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
		5	*
		6	* Developed at SunPro, a Sun Microsystems, Inc. business.
		7	* Permission to use, copy, modify, and distribute this
		8	* software is freely granted, provided that this notice
		9	* is preserved.
		10	* ====================================================
		11	*/
		12
		13	#ifndef lint
		14	static char rcsid[] = "$\Id: s_log1p.c,v 1.2 1995/05/30 05:49:57 rgrimes Exp $";
		15	#endif
		16
		17	/* double log1p(double x)
		18	*
		19	* Method :
		20	* 1. Argument Reduction: find k and f such that
		21	* 1+x = 2^k * (1+f),
		22	* where sqrt(2)/2 < 1+f < sqrt(2) .
		23	*
		24	* Note. If k=0, then f=x is exact. However, if k!=0, then f
		25	* may not be representable exactly. In that case, a correction
		26	* term is need. Let u=1+x rounded. Let c = (1+x)-u, then
		27	* log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
		28	* and add back the correction term c/u.
		29	* (Note: when x > 2**53, one can simply return log(x))
		30	*
		31	* 2. Approximation of log1p(f).
		32	* Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
		33	* = 2s + 2/3 s3 + 2/5 s5 + .....,
		34	* = 2s + s*R
		35	* We use a special Reme algorithm on [0,0.1716] to generate
		36	* a polynomial of degree 14 to approximate R The maximum error
		37	* of this polynomial approximation is bounded by 2**-58.45. In
		38	* other words,
		39	* 2 4 6 8 10 12 14
		40	* R(z) ~ Lp1s +Lp2s +Lp3s +Lp4s +Lp5s +Lp6s +Lp7*s
		41	* (the values of Lp1 to Lp7 are listed in the program)
		42	* and
		43	* \| 2 14 \| -58.45
		44	* \| Lp1s +...+Lp7s - R(z) \| <= 2
		45	* \| \|
		46	* Note that 2s = f - sf = f - hfsq + shfsq, where hfsq = f*f/2.
		47	* In order to guarantee error in log below 1ulp, we compute log
		48	* by
		49	* log1p(f) = f - (hfsq - s*(hfsq+R)).
		50	*
		51	* 3. Finally, log1p(x) = k*ln2 + log1p(f).
		52	* = kln2_hi+(f-(hfsq-(s(hfsq+R)+k*ln2_lo)))
		53	* Here ln2 is split into two floating point number:
		54	* ln2_hi + ln2_lo,
		55	* where n*ln2_hi is always exact for \|n\| < 2000.
		56	*
		57	* Special cases:
		58	* log1p(x) is NaN with signal if x < -1 (including -INF) ;
		59	* log1p(+INF) is +INF; log1p(-1) is -INF with signal;
		60	* log1p(NaN) is that NaN with no signal.
		61	*
		62	* Accuracy:
		63	* according to an error analysis, the error is always less than
		64	* 1 ulp (unit in the last place).
		65	*
		66	* Constants:
		67	* The hexadecimal values are the intended ones for the following
		68	* constants. The decimal values may be used, provided that the
		69	* compiler will convert from decimal to binary accurately enough
		70	* to produce the hexadecimal values shown.
		71	*
		72	* Note: Assuming log() return accurate answer, the following
		73	* algorithm can be used to compute log1p(x) to within a few ULP:
		74	*
		75	* u = 1+x;
		76	* if(u==1.0) return x ; else
		77	* return log(u)*(x/(u-1.0));
		78	*
		79	* See HP-15C Advanced Functions Handbook, p.193.
		80	*/
		81
		82	#include "math.h"
		83	#include "math_private.h"
		84
		85	#ifdef __STDC__
		86	static const double
		87	#else
		88	static double
		89	#endif
		90	ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
		91	ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
		92	two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
		93	Lp1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
		94	Lp2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
		95	Lp3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
		96	Lp4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
		97	Lp5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
		98	Lp6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
		99	Lp7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
		100
		101	#ifdef __STDC__
		102	static const double zero = 0.0;
		103	#else
		104	static double zero = 0.0;
		105	#endif
		106
		107	#ifdef __STDC__
		108	double log1p(double x)
		109	#else
		110	double log1p(x)
		111	double x;
		112	#endif
		113	{
		114	double hfsq,f,c,s,z,R,u;
		115	int32_t k,hx,hu,ax;
		116
		117	GET_HIGH_WORD(hx,x);
		118	ax = hx&0x7fffffff;
		119
		120	k = 1;
		121	if (hx < 0x3FDA827A) { /* x < 0.41422 */
		122	if(ax>=0x3ff00000) { /* x <= -1.0 */
		123	if(x==-1.0) return -two54/zero; /* log1p(-1)=+inf */
		124	else return (x-x)/(x-x); /* log1p(x<-1)=NaN */
		125	}
		126	if(ax<0x3e200000) { /* \|x\| < 2*-29 /
		127	if(two54+x>zero /* raise inexact */
		128	&&ax<0x3c900000) /* \|x\| < 2*-54 /
		129	return x;
		130	else
		131	return x - xx0.5;
		132	}
		133	if(hx>0\|\|hx<=((int32_t)0xbfd2bec3)) {
		134	k=0;f=x;hu=1;} /* -0.2929<x<0.41422 */
		135	}
		136	if (hx >= 0x7ff00000) return x+x;
		137	if(k!=0) {
		138	if(hx<0x43400000) {
		139	u = 1.0+x;
		140	GET_HIGH_WORD(hu,u);
		141	k = (hu>>20)-1023;
		142	c = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */
		143	c /= u;
		144	} else {
		145	u = x;
		146	GET_HIGH_WORD(hu,u);
		147	k = (hu>>20)-1023;
		148	c = 0;
		149	}
		150	hu &= 0x000fffff;
		151	if(hu<0x6a09e) {
		152	SET_HIGH_WORD(u,hu\|0x3ff00000); /* normalize u */
		153	} else {
		154	k += 1;
		155	SET_HIGH_WORD(u,hu\|0x3fe00000); /* normalize u/2 */
		156	hu = (0x00100000-hu)>>2;
		157	}
		158	f = u-1.0;
		159	}
		160	hfsq=0.5ff;
		161	if(hu==0) { /* \|f\| < 2*-20 /
		162	if(f==zero) if(k==0) return zero;
		163	else {c += kln2_lo; return kln2_hi+c;}
		164	R = hfsq(1.0-0.66666666666666666f);
		165	if(k==0) return f-R; else
		166	return kln2_hi-((R-(kln2_lo+c))-f);
		167	}
		168	s = f/(2.0+f);
		169	z = s*s;
		170	R = z(Lp1+z(Lp2+z(Lp3+z(Lp4+z(Lp5+z(Lp6+z*Lp7))))));
		171	if(k==0) return f-(hfsq-s*(hfsq+R)); else
		172	return kln2_hi-((hfsq-(s(hfsq+R)+(k*ln2_lo+c)))-f);
		173	}

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