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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.
 * ====================================================
 */
/* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25,
   for performance improvement on pipelined processors.
*/

#if defined(LIBM_SCCS) && !defined(lint)
static char rcsid[] = "$NetBSD: s_expm1.c,v 1.8 1995/05/10 20:47:09 jtc Exp $";
#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"
#define one Q[0]
#ifdef __STDC__
static const double
#else
static double
#endif
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 */
Q[]  =  {1.0, -3.33333333333331316428e-02, /* BFA11111 111110F4 */
   1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
  -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
   4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
  -2.01099218183624371326e-07}; /* BE8AFDB7 6E09C32D */

#ifdef __STDC__
      double __expm1(double x)
#else
      double __expm1(x)
      double x;
#endif
{
      double y,hi,lo,c,t,e,hxs,hfx,r1,h2,h4,R1,R2,R3;
      int32_t k,xsb;
      u_int32_t hx;

      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  = 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;
#ifdef DO_NOT_USE_THIS
      r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
#else
      R1 = one+hxs*Q[1]; h2 = hxs*hxs;
      R2 = Q[2]+hxs*Q[3]; h4 = h2*h2;
      R3 = Q[4]+hxs*Q[5];
      r1 = R1 + h2*R2 + h4*R3;
#endif
      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;
}
weak_alias (__expm1, expm1)
#ifdef NO_LONG_DOUBLE
strong_alias (__expm1, __expm1l)
weak_alias (__expm1, expm1l)
#endif

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