newlib/libm/common/s_expm1.c

   1
   2 /* @(#)s_expm1.c 5.1 93/09/24 */
   3 /*
   4  * ====================================================
   5  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   6  *
   7  * Developed at SunPro, a Sun Microsystems, Inc. business.
   8  * Permission to use, copy, modify, and distribute this
   9  * software is freely granted, provided that this notice
  10  * is preserved.
  11  * ====================================================
  12  */
  13
  14 /*
  15 FUNCTION
  16         <<expm1>>, <<expm1f>>---exponential minus 1
  17 INDEX
  18         expm1
  19 INDEX
  20         expm1f
  21
  22 SYNOPSIS
  23         #include <math.h>
  24         double expm1(double <[x]>);
  25         float expm1f(float <[x]>);
  26
  27 DESCRIPTION
  28         <<expm1>> and <<expm1f>> calculate the exponential of <[x]>
  29         and subtract 1, that is,
  30         @ifnottex
  31         e raised to the power <[x]> minus 1 (where e
  32         @end ifnottex
  33         @tex
  34         $e^x - 1$ (where $e$
  35         @end tex
  36         is the base of the natural system of logarithms, approximately
  37         2.71828).  The result is accurate even for small values of
  38         <[x]>, where using <<exp(<[x]>)-1>> would lose many
  39         significant digits.
  40
  41 RETURNS
  42         e raised to the power <[x]>, minus 1.
  43
  44 PORTABILITY
  45         Neither <<expm1>> nor <<expm1f>> is required by ANSI C or by
  46         the System V Interface Definition (Issue 2).
  47 */
  48
  49 /* expm1(x)
  50  * Returns exp(x)-1, the exponential of x minus 1.
  51  *
  52  * Method
  53  *   1. Argument reduction:
  54  *      Given x, find r and integer k such that
  55  *
  56  *               x = k*ln2 + r,  |r| <= 0.5*ln2 ~ 0.34658
  57  *
  58  *      Here a correction term c will be computed to compensate
  59  *      the error in r when rounded to a floating-point number.
  60  *
  61  *   2. Approximating expm1(r) by a special rational function on
  62  *      the interval [0,0.34658]:
  63  *      Since
  64  *          r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
  65  *      we define R1(r*r) by
  66  *          r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
  67  *      That is,
  68  *          R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
  69  *                   = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
  70  *                   = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
  71  *      We use a special Remez algorithm on [0,0.347] to generate
  72  *      a polynomial of degree 5 in r*r to approximate R1. The
  73  *      maximum error of this polynomial approximation is bounded
  74  *      by 2**-61. In other words,
  75  *          R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
  76  *      where   Q1  =  -1.6666666666666567384E-2,
  77  *              Q2  =   3.9682539681370365873E-4,
  78  *              Q3  =  -9.9206344733435987357E-6,
  79  *              Q4  =   2.5051361420808517002E-7,
  80  *              Q5  =  -6.2843505682382617102E-9;
  81  *      (where z=r*r, and the values of Q1 to Q5 are listed below)
  82  *      with error bounded by
  83  *          |                  5           |     -61
  84  *          | 1.0+Q1*z+...+Q5*z   -  R1(z) | <= 2
  85  *          |                              |
  86  *
  87  *      expm1(r) = exp(r)-1 is then computed by the following
  88  *      specific way which minimize the accumulation rounding error:
  89  *                             2     3
  90  *                            r     r    [ 3 - (R1 + R1*r/2)  ]
  91  *            expm1(r) = r + --- + --- * [--------------------]
  92  *                            2     2    [ 6 - r*(3 - R1*r/2) ]
  93  *
  94  *      To compensate the error in the argument reduction, we use
  95  *              expm1(r+c) = expm1(r) + c + expm1(r)*c
  96  *                         ~ expm1(r) + c + r*c
  97  *      Thus c+r*c will be added in as the correction terms for
  98  *      expm1(r+c). Now rearrange the term to avoid optimization
  99  *      screw up:
 100  *                      (      2                                    2 )
 101  *                      ({  ( r    [ R1 -  (3 - R1*r/2) ]  )  }    r  )
 102  *       expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
 103  *                      ({  ( 2    [ 6 - r*(3 - R1*r/2) ]  )  }    2  )
 104  *                      (                                             )
 105  *
 106  *                 = r - E
 107  *   3. Scale back to obtain expm1(x):
 108  *      From step 1, we have
 109  *         expm1(x) = either 2^k*[expm1(r)+1] - 1
 110  *                  = or     2^k*[expm1(r) + (1-2^-k)]
 111  *   4. Implementation notes:
 112  *      (A). To save one multiplication, we scale the coefficient Qi
 113  *           to Qi*2^i, and replace z by (x^2)/2.
 114  *      (B). To achieve maximum accuracy, we compute expm1(x) by
 115  *        (i)   if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
 116  *        (ii)  if k=0, return r-E
 117  *        (iii) if k=-1, return 0.5*(r-E)-0.5
 118  *        (iv)  if k=1 if r < -0.25, return 2*((r+0.5)- E)
 119  *                     else          return  1.0+2.0*(r-E);
 120  *        (v)   if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
 121  *        (vi)  if k <= 20, return 2^k((1-2^-k)-(E-r)), else
 122  *        (vii) return 2^k(1-((E+2^-k)-r))
 123  *
 124  * Special cases:
 125  *      expm1(INF) is INF, expm1(NaN) is NaN;
 126  *      expm1(-INF) is -1, and
 127  *      for finite argument, only expm1(0)=0 is exact.
 128  *
 129  * Accuracy:
 130  *      according to an error analysis, the error is always less than
 131  *      1 ulp (unit in the last place).
 132  *
 133  * Misc. info.
 134  *      For IEEE double
 135  *          if x >  7.09782712893383973096e+02 then expm1(x) overflow
 136  *
 137  * Constants:
 138  * The hexadecimal values are the intended ones for the following
 139  * constants. The decimal values may be used, provided that the
 140  * compiler will convert from decimal to binary accurately enough
 141  * to produce the hexadecimal values shown.
 142  */
 143
 144 #include "fdlibm.h"
 145 #include "math_config.h"
 146
 147 #ifndef _DOUBLE_IS_32BITS
 148
 149 #ifdef __STDC__
 150 static const double
 151 #else
 152 static double
 153 #endif
 154 one             = 1.0,
 155 huge            = 1.0e+300,
 156 tiny            = 1.0e-300,
 157 o_threshold     = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */
 158 ln2_hi          = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */
 159 ln2_lo          = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */
 160 invln2          = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */
 161         /* scaled coefficients related to expm1 */
 162 Q1  =  -3.33333333333331316428e-02, /* BFA11111 111110F4 */
 163 Q2  =   1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
 164 Q3  =  -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
 165 Q4  =   4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
 166 Q5  =  -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */
 167
 168 #ifdef __STDC__
 169         double expm1(double x)
 170 #else
 171         double expm1(x)
 172         double x;
 173 #endif
 174 {
 175         double y,hi,lo,c,t,e,hxs,hfx,r1;
 176         __int32_t k,xsb;
 177         __uint32_t hx;
 178
 179         GET_HIGH_WORD(hx,x);
 180         xsb = hx&0x80000000;            /* sign bit of x */
 181         if(xsb==0) y=x; else y= -x;     /* y = |x| */
 182         hx &= 0x7fffffff;               /* high word of |x| */
 183
 184     /* filter out huge and non-finite argument */
 185         if(hx >= 0x4043687A) {                  /* if |x|>=56*ln2 */
 186             if(hx >= 0x40862E42) {              /* if |x|>=709.78... */
 187                 if(hx>=0x7ff00000) {
 188                     __uint32_t low;
 189                     GET_LOW_WORD(low,x);
 190                     if(((hx&0xfffff)|low)!=0)
 191                          return x+x;     /* NaN */
 192                     else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */
 193                 }
 194                 if(x > o_threshold) return __math_oflow (0); /* overflow */
 195             }
 196             if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */
 197                 if(x+tiny<0.0)          /* raise inexact */
 198                 return tiny-one;        /* return -1 */
 199             }
 200         }
 201
 202     /* argument reduction */
 203         if(hx > 0x3fd62e42) {           /* if  |x| > 0.5 ln2 */
 204             if(hx < 0x3FF0A2B2) {       /* and |x| < 1.5 ln2 */
 205                 if(xsb==0)
 206                     {hi = x - ln2_hi; lo =  ln2_lo;  k =  1;}
 207                 else
 208                     {hi = x + ln2_hi; lo = -ln2_lo;  k = -1;}
 209             } else {
 210                 k  = invln2*x+((xsb==0)?0.5:-0.5);
 211                 t  = k;
 212                 hi = x - t*ln2_hi;      /* t*ln2_hi is exact here */
 213                 lo = t*ln2_lo;
 214             }
 215             x  = hi - lo;
 216             c  = (hi-x)-lo;
 217         }
 218         else if(hx < 0x3c900000) {      /* when |x|<2**-54, return x */
 219             t = huge+x; /* return x with inexact flags when x!=0 */
 220             return x - (t-(huge+x));
 221         }
 222         else k = 0;
 223
 224     /* x is now in primary range */
 225         hfx = 0.5*x;
 226         hxs = x*hfx;
 227         r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
 228         t  = 3.0-r1*hfx;
 229         e  = hxs*((r1-t)/(6.0 - x*t));
 230         if(k==0) return x - (x*e-hxs);          /* c is 0 */
 231         else {
 232             e  = (x*(e-c)-c);
 233             e -= hxs;
 234             if(k== -1) return 0.5*(x-e)-0.5;
 235           if(k==1) {
 236                 if(x < -0.25) return -2.0*(e-(x+0.5));
 237                 else          return  one+2.0*(x-e);
 238           }
 239             if (k <= -2 || k>56) {   /* suffice to return exp(x)-1 */
 240                 __uint32_t high;
 241                 y = one-(e-x);
 242                 GET_HIGH_WORD(high,y);
 243                 SET_HIGH_WORD(y,high+(k<<20));  /* add k to y's exponent */
 244                 return y-one;
 245             }
 246             t = one;
 247             if(k<20) {
 248                 __uint32_t high;
 249                 SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k));  /* t=1-2^-k */
 250                 y = t-(e-x);
 251                 GET_HIGH_WORD(high,y);
 252                 SET_HIGH_WORD(y,high+(k<<20));  /* add k to y's exponent */
 253            } else {
 254                 __uint32_t high;
 255                 SET_HIGH_WORD(t,((0x3ff-k)<<20));       /* 2^-k */
 256                 y = x-(e+t);
 257                 y += one;
 258                 GET_HIGH_WORD(high,y);
 259                 SET_HIGH_WORD(y,high+(k<<20));  /* add k to y's exponent */
 260             }
 261         }
 262         return y;
 263 }
 264
 265 #endif /* _DOUBLE_IS_32BITS */