xref: /freebsd-11-stable/lib/msun/src/e_logf.c

/* e_logf.c -- float version of e_log.c.
 * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
 */

/*
 * ====================================================
 * 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.
 * ====================================================
 */

#include <sys/cdefs.h>
__FBSDID("$FreeBSD: head/lib/msun/src/e_logf.c 241755 2012-10-19 22:46:48Z imp $");

#include "math.h"
#include "math_private.h"

/* __ieee754_log(x)
 * Return the logrithm of x
 *
 * 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(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 8 to approximate R The maximum error
 *      of this polynomial approximation is bounded by 2**-34.24. In
 *      other words,
 *                      2      4      6      8
 *          R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s
 *      (the values of Lg1 to Lg7 are listed in the program)
 *      and
 *          |      2          8           |     -34.24
 *          | Lg1*s +...+Lg4*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 float
ln2_hi =   6.9313812256e-01,	/* 0x3f317180 */
ln2_lo =   9.0580006145e-06,	/* 0x3717f7d1 */
two25 =    3.355443200e+07,	/* 0x4c000000 */
/* |(log(1+s)-log(1-s))/s - Lg(s)| < 2**-34.24 (~[-4.95e-11, 4.97e-11]). */
Lg1 =      0xaaaaaa.0p-24,	/* 0.66666662693 */
Lg2 =      0xccce13.0p-25,	/* 0.40000972152 */
Lg3 =      0x91e9ee.0p-25,	/* 0.28498786688 */
Lg4 =      0xf89e26.0p-26;	/* 0.24279078841 */

static const float zero   =  0.0;

float
__ieee754_logf(float x)
{
	float hfsq,f,s,z,R,w,t1,t2,dk;
	int32_t k,ix,i,j;

	GET_FLOAT_WORD(ix,x);

	k=0;
	if (ix < 0x00800000) {			/* x < 2**-126  */
	    if ((ix&0x7fffffff)==0)
		return -two25/zero;		/* log(+-0)=-inf */
	    if (ix<0) return (x-x)/zero;	/* log(-#) = NaN */
	    k -= 25; x *= two25; /* subnormal number, scale up x */
	    GET_FLOAT_WORD(ix,x);
	}
	if (ix >= 0x7f800000) return x+x;
	k += (ix>>23)-127;
	ix &= 0x007fffff;
	i = (ix+(0x95f64<<3))&0x800000;
	SET_FLOAT_WORD(x,ix|(i^0x3f800000));	/* normalize x or x/2 */
	k += (i>>23);
	f = x-(float)1.0;
	if((0x007fffff&(0x8000+ix))<0xc000) {	/* -2**-9 <= f < 2**-9 */
	    if(f==zero) {
		if(k==0) {
		    return zero;
		} else {
		    dk=(float)k;
		    return dk*ln2_hi+dk*ln2_lo;
		}
	    }
	    R = f*f*((float)0.5-(float)0.33333333333333333*f);
	    if(k==0) return f-R; else {dk=(float)k;
	    	     return dk*ln2_hi-((R-dk*ln2_lo)-f);}
	}
 	s = f/((float)2.0+f);
	dk = (float)k;
	z = s*s;
	i = ix-(0x6147a<<3);
	w = z*z;
	j = (0x6b851<<3)-ix;
	t1= w*(Lg2+w*Lg4);
	t2= z*(Lg1+w*Lg3);
	i |= j;
	R = t2+t1;
	if(i>0) {
	    hfsq=(float)0.5*f*f;
	    if(k==0) return f-(hfsq-s*(hfsq+R)); else
		     return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
	} else {
	    if(k==0) return f-s*(f-R); else
		     return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
	}
}