src/math/log1p.c

/* origin: FreeBSD /usr/src/lib/msun/src/s_log1p.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.
 */

#include "libm.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 */

double log1p(double x)
{
	double hfsq,f,c,s,z,R,u;
	int32_t k,hx,hu,ax;

	GET_HIGH_WORD(hx, x);
	ax = hx & 0x7fffffff;

	k = 1;
	if (hx < 0x3FDA827A) {  /* 1+x < sqrt(2)+ */
		if (ax >= 0x3ff00000) {  /* x <= -1.0 */
			if (x == -1.0)
				return -two54/0.0; /* log1p(-1)=+inf */
			return (x-x)/(x-x);         /* log1p(x<-1)=NaN */
		}
		if (ax < 0x3e200000) {   /* |x| < 2**-29 */
			/* if 0x1p-1022 <= |x| < 0x1p-54, avoid raising underflow */
			if (ax < 0x3c900000 && ax >= 0x00100000)
				return x;
#if FLT_EVAL_METHOD != 0
			FORCE_EVAL((float)x);
#endif
			return x - x*x*0.5;
		}
		if (hx > 0 || hx <= (int32_t)0xbfd2bec4) {  /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
			k = 0;
			f = x;
			hu = 1;
		}
	}
	if (hx >= 0x7ff00000)
		return x+x;
	if (k != 0) {
		if (hx < 0x43400000) {
			STRICT_ASSIGN(double, 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;
		/*
		 * The approximation to sqrt(2) used in thresholds is not
		 * critical.  However, the ones used above must give less
		 * strict bounds than the one here so that the k==0 case is
		 * never reached from here, since here we have committed to
		 * using the correction term but don't use it if k==0.
		 */
		if (hu < 0x6a09e) {  /* u ~< sqrt(2) */
			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 == 0.0) {
			if(k == 0)
				return 0.0;
			c += k*ln2_lo;
			return k*ln2_hi + c;
		}
		R = hfsq*(1.0 - 0.66666666666666666*f);
		if (k == 0)
			return f - R;
		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));
	return k*ln2_hi - ((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f);
}