LibC/Math/s_expm1.c

/* @(#)s_expm1.c 5.1 93/09/24 */
/*
 * ====================================================
 * 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  <LibConfig.h>
#include  <sys/EfiCdefs.h>
#if defined(LIBM_SCCS) && !defined(lint)
__RCSID("$NetBSD: s_expm1.c,v 1.12 2002/05/26 22:01:55 wiz Exp $");
#endif

#if defined(_MSC_VER)           /* Handle Microsoft VC++ compiler specifics. */
  // C4756: overflow in constant arithmetic
  #pragma warning ( disable : 4756 )
#endif

/* 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 Reme 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;
 *    (where z=r*r, and the values of Q1 to Q5 are listed below)
 *  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 "math.h"
#include "math_private.h"

static const double
one   = 1.0,
huge    = 1.0e+300,
tiny    = 1.0e-300,
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 coefficients related to expm1 */
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 y,hi,lo,c,t,e,hxs,hfx,r1;
  int32_t k,xsb;
  u_int32_t hx;

  c = 0;
  GET_HIGH_WORD(hx,x);
  xsb = hx&0x80000000;    /* sign bit of x */
  if(xsb==0) y=x; else y= -x; /* y = |x| */
  hx &= 0x7fffffff;   /* high word of |x| */

    /* filter out huge and non-finite argument */
  if(hx >= 0x4043687A) {      /* if |x|>=56*ln2 */
      if(hx >= 0x40862E42) {    /* if |x|>=709.78... */
                if(hx>=0x7ff00000) {
        u_int32_t low;
        GET_LOW_WORD(low,x);
        if(((hx&0xfffff)|low)!=0)
             return x+x;   /* NaN */
        else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */
          }
          if(x > o_threshold) return huge*huge; /* overflow */
      }
      if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */
    if(x+tiny<0.0)    /* raise inexact */
    return tiny-one;  /* return -1 */
      }
  }

    /* argument reduction */
  if(hx > 0x3fd62e42) {   /* if  |x| > 0.5 ln2 */
      if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
    if(xsb==0)
        {hi = x - ln2_hi; lo =  ln2_lo;  k =  1;}
    else
        {hi = x + ln2_hi; lo = -ln2_lo;  k = -1;}
      } else {
    k  = (int32_t)(invln2*x+((xsb==0)?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) {    /* when |x|<2**-54, return x */
      t = huge+x; /* return x with inexact flags when x!=0 */
      return x - (t-(huge+x));
  }
  else k = 0;

    /* x is now in primary range */
  hfx = 0.5*x;
  hxs = x*hfx;
  r1 = one+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) return x - (x*e-hxs);    /* c is 0 */
  else {
      e  = (x*(e-c)-c);
      e -= hxs;
      if(k== -1) return 0.5*(x-e)-0.5;
      if(k==1)  {
          if(x < -0.25) return -2.0*(e-(x+0.5));
          else        return  one+2.0*(x-e);
      }
      if (k <= -2 || k>56) {   /* suffice to return exp(x)-1 */
          u_int32_t high;
          y = one-(e-x);
    GET_HIGH_WORD(high,y);
    SET_HIGH_WORD(y,high+(k<<20));  /* add k to y's exponent */
          return y-one;
      }
      t = one;
      if(k<20) {
          u_int32_t high;
          SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k));  /* t=1-2^-k */
          y = t-(e-x);
    GET_HIGH_WORD(high,y);
    SET_HIGH_WORD(y,high+(k<<20));  /* add k to y's exponent */
     } else {
          u_int32_t high;
    SET_HIGH_WORD(t,((0x3ff-k)<<20)); /* 2^-k */
          y = x-(e+t);
          y += one;
    GET_HIGH_WORD(high,y);
    SET_HIGH_WORD(y,high+(k<<20));  /* add k to y's exponent */
      }
  }
  return y;
}