expm1.c

/* origin: FreeBSD /usr/src/lib/msun/src/s_expm1.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.
 * ====================================================
 */
/* expm1(x)
 * Returns exp(x)-1, the exponential of x minus 1.
 *
 * Method
 *   1. Argument reduction:
 *      Given x, find r and integer k such that
 *
 *               x = k*ln2 + r,  |r| <= 0.5*ln2 ~ 0.34658
 *
 *      Here a correction term c will be computed to compensate
 *      the error in r when rounded to a floating-point number.
 *
 *   2. Approximating expm1(r) by a special rational function on
 *      the interval [0,0.34658]:
 *      Since
 *          r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
 *      we define R1(r*r) by
 *          r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
 *      That is,
 *          R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
 *                   = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
 *                   = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
 *      We use a special Remez algorithm on [0,0.347] to generate
 *      a polynomial of degree 5 in r*r to approximate R1. The
 *      maximum error of this polynomial approximation is bounded
 *      by 2**-61. In other words,
 *          R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
 *      where   Q1  =  -1.6666666666666567384E-2,
 *              Q2  =   3.9682539681370365873E-4,
 *              Q3  =  -9.9206344733435987357E-6,
 *              Q4  =   2.5051361420808517002E-7,
 *              Q5  =  -6.2843505682382617102E-9;
 *              z   =  r*r,
 *      with error bounded by
 *          |                  5           |     -61
 *          | 1.0+Q1*z+...+Q5*z   -  R1(z) | <= 2
 *          |                              |
 *
 *      expm1(r) = exp(r)-1 is then computed by the following
 *      specific way which minimize the accumulation rounding error:
 *                             2     3
 *                            r     r    [ 3 - (R1 + R1*r/2)  ]
 *            expm1(r) = r + --- + --- * [--------------------]
 *                            2     2    [ 6 - r*(3 - R1*r/2) ]
 *
 *      To compensate the error in the argument reduction, we use
 *              expm1(r+c) = expm1(r) + c + expm1(r)*c
 *                         ~ expm1(r) + c + r*c
 *      Thus c+r*c will be added in as the correction terms for
 *      expm1(r+c). Now rearrange the term to avoid optimization
 *      screw up:
 *                      (      2                                    2 )
 *                      ({  ( r    [ R1 -  (3 - R1*r/2) ]  )  }    r  )
 *       expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
 *                      ({  ( 2    [ 6 - r*(3 - R1*r/2) ]  )  }    2  )
 *                      (                                             )
 *
 *                 = r - E
 *   3. Scale back to obtain expm1(x):
 *      From step 1, we have
 *         expm1(x) = either 2^k*[expm1(r)+1] - 1
 *                  = or     2^k*[expm1(r) + (1-2^-k)]
 *   4. Implementation notes:
 *      (A). To save one multiplication, we scale the coefficient Qi
 *           to Qi*2^i, and replace z by (x^2)/2.
 *      (B). To achieve maximum accuracy, we compute expm1(x) by
 *        (i)   if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
 *        (ii)  if k=0, return r-E
 *        (iii) if k=-1, return 0.5*(r-E)-0.5
 *        (iv)  if k=1 if r < -0.25, return 2*((r+0.5)- E)
 *                     else          return  1.0+2.0*(r-E);
 *        (v)   if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
 *        (vi)  if k <= 20, return 2^k((1-2^-k)-(E-r)), else
 *        (vii) return 2^k(1-((E+2^-k)-r))
 *
 * Special cases:
 *      expm1(INF) is INF, expm1(NaN) is NaN;
 *      expm1(-INF) is -1, and
 *      for finite argument, only expm1(0)=0 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 expm1(x) overflow
 *
 * 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
o_threshold = 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
ln2_hi      = 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
ln2_lo      = 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
invln2      = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
/* Scaled Q's: Qn_here = 2**n * Qn_above, for R(2*z) where z = hxs = x*x/2: */
Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */
Q2 =  1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
Q4 =  4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */

double expm1(double x)
{
	double_t y,hi,lo,c,t,e,hxs,hfx,r1,twopk;
	union {double f; uint64_t i;} u = {x};
	uint32_t hx = u.i>>32 & 0x7fffffff;
	int k, sign = u.i>>63;

	/* filter out huge and non-finite argument */
	if (hx >= 0x4043687A) {  /* if |x|>=56*ln2 */
		if (isnan(x))
			return x;
		if (sign)
			return -1;
		if (x > o_threshold) {
			x *= 0x1p1023;
			return x;
		}
	}

	/* argument reduction */
	if (hx > 0x3fd62e42) {  /* if  |x| > 0.5 ln2 */
		if (hx < 0x3FF0A2B2) {  /* and |x| < 1.5 ln2 */
			if (!sign) {
				hi = x - ln2_hi;
				lo = ln2_lo;
				k =  1;
			} else {
				hi = x + ln2_hi;
				lo = -ln2_lo;
				k = -1;
			}
		} else {
			k  = invln2*x + (sign ? -0.5 : 0.5);
			t  = k;
			hi = x - t*ln2_hi;  /* t*ln2_hi is exact here */
			lo = t*ln2_lo;
		}
		x = hi-lo;
		c = (hi-x)-lo;
	} else if (hx < 0x3c900000) {  /* |x| < 2**-54, return x */
		if (hx < 0x00100000)
			FORCE_EVAL((float)x);
		return x;
	} else
		k = 0;

	/* x is now in primary range */
	hfx = 0.5*x;
	hxs = x*hfx;
	r1 = 1.0+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
	t  = 3.0-r1*hfx;
	e  = hxs*((r1-t)/(6.0 - x*t));
	if (k == 0)   /* c is 0 */
		return x - (x*e-hxs);
	e  = x*(e-c) - c;
	e -= hxs;
	/* exp(x) ~ 2^k (x_reduced - e + 1) */
	if (k == -1)
		return 0.5*(x-e) - 0.5;
	if (k == 1) {
		if (x < -0.25)
			return -2.0*(e-(x+0.5));
		return 1.0+2.0*(x-e);
	}
	u.i = (uint64_t)(0x3ff + k)<<52;  /* 2^k */
	twopk = u.f;
	if (k < 0 || k > 56) {  /* suffice to return exp(x)-1 */
		y = x - e + 1.0;
		if (k == 1024)
			y = y*2.0*0x1p1023;
		else
			y = y*twopk;
		return y - 1.0;
	}
	u.i = (uint64_t)(0x3ff - k)<<52;  /* 2^-k */
	if (k < 20)
		y = (x-e+(1-u.f))*twopk;
	else
		y = (x-(e+u.f)+1)*twopk;
	return y;
}