musl/src/math/exp.c

/* origin: FreeBSD /usr/src/lib/msun/src/e_exp.c */
/*
 * ====================================================
 * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
 *
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice
 * is preserved.
 * ====================================================
 */
/* exp(x)
 * Returns the exponential of x.
 *
 * Method
 *   1. Argument reduction:
 *      Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
 *      Given x, find r and integer k such that
 *
 *               x = k*ln2 + r,  |r| <= 0.5*ln2.
 *
 *      Here r will be represented as r = hi-lo for better
 *      accuracy.
 *
 *   2. Approximation of exp(r) by a special rational function on
 *      the interval [0,0.34658]:
 *      Write
 *          R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
 *      We use a special Remes algorithm on [0,0.34658] to generate
 *      a polynomial of degree 5 to approximate R. The maximum error
 *      of this polynomial approximation is bounded by 2**-59. In
 *      other words,
 *          R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
 *      (where z=r*r, and the values of P1 to P5 are listed below)
 *      and
 *          |                  5          |     -59
 *          | 2.0+P1*z+...+P5*z   -  R(z) | <= 2
 *          |                             |
 *      The computation of exp(r) thus becomes
 *                             2*r
 *              exp(r) = 1 + -------
 *                            R - r
 *                                 r*R1(r)
 *                     = 1 + r + ----------- (for better accuracy)
 *                                2 - R1(r)
 *      where
 *                               2       4             10
 *              R1(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
 *
 *   3. Scale back to obtain exp(x):
 *      From step 1, we have
 *         exp(x) = 2^k * exp(r)
 *
 * Special cases:
 *      exp(INF) is INF, exp(NaN) is NaN;
 *      exp(-INF) is 0, and
 *      for finite argument, only exp(0)=1 is exact.
 *
 * Accuracy:
 *      according to an error analysis, the error is always less than
 *      1 ulp (unit in the last place).
 *
 * Misc. info.
 *      For IEEE double
 *          if x >  7.09782712893383973096e+02 then exp(x) overflow
 *          if x < -7.45133219101941108420e+02 then exp(x) underflow
 *
 * 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.
 */

#include "libm.h"

static const double
halF[2] = {0.5,-0.5,},
huge    = 1.0e+300,
o_threshold =  7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
u_threshold = -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */
ln2HI[2]   = { 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
              -6.93147180369123816490e-01},/* 0xbfe62e42, 0xfee00000 */
ln2LO[2]   = { 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
              -1.90821492927058770002e-10},/* 0xbdea39ef, 0x35793c76 */
invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
P5   =  4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */

static const volatile double
twom1000 = 9.33263618503218878990e-302; /* 2**-1000=0x01700000,0 */

double exp(double x)
{
	double y,hi=0.0,lo=0.0,c,t,twopk;
	int32_t k=0,xsb;
	uint32_t hx;

	GET_HIGH_WORD(hx, x);
	xsb = (hx>>31)&1;  /* sign bit of x */
	hx &= 0x7fffffff;  /* high word of |x| */

	/* filter out non-finite argument */
	if (hx >= 0x40862E42) {  /* if |x| >= 709.78... */
		if (hx >= 0x7ff00000) {
			uint32_t lx;
	
			GET_LOW_WORD(lx,x);
			if (((hx&0xfffff)|lx) != 0)  /* NaN */
				 return x+x;
			return xsb==0 ? x : 0.0;  /* exp(+-inf)={inf,0} */
		}
		if (x > o_threshold)
			return huge*huge; /* overflow */
		if (x < u_threshold)
			return twom1000*twom1000; /* underflow */
	}

	/* argument reduction */
	if (hx > 0x3fd62e42) {  /* if  |x| > 0.5 ln2 */
		if (hx < 0x3FF0A2B2) {  /* and |x| < 1.5 ln2 */
			hi = x-ln2HI[xsb];
			lo = ln2LO[xsb];
			k = 1 - xsb - xsb;
		} else {
			k  = (int)(invln2*x+halF[xsb]);
			t  = k;
			hi = x - t*ln2HI[0];  /* t*ln2HI is exact here */
			lo = t*ln2LO[0];
		}
		STRICT_ASSIGN(double, x, hi - lo);
	} else if(hx < 0x3e300000)  {  /* |x| < 2**-28 */
		/* raise inexact */
		if (huge+x > 1.0)
			return 1.0+x;
	} else
		k = 0;

	/* x is now in primary range */
	t  = x*x;
	if (k >= -1021)
		INSERT_WORDS(twopk, 0x3ff00000+(k<<20), 0);
	else
		INSERT_WORDS(twopk, 0x3ff00000+((k+1000)<<20), 0);
	c  = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
	if (k == 0)
		return 1.0 - ((x*c)/(c-2.0) - x);
	y = 1.0-((lo-(x*c)/(2.0-c))-hi);
	if (k < -1021)
		return y*twopk*twom1000;
	if (k == 1024)
		return y*2.0*0x1p1023;
	return y*twopk;
}