lib/libm/src/s_expm1.c

   1 /* @(#)s_expm1.c 5.1 93/09/24 */
   2 /*
   3  * ====================================================
   4  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   5  *
   6  * Developed at SunPro, a Sun Microsystems, Inc. business.
   7  * Permission to use, copy, modify, and distribute this
   8  * software is freely granted, provided that this notice
   9  * is preserved.
  10  * ====================================================
  11  *
  12  * $NetBSD: s_expm1.c,v 1.12 2002/05/26 22:01:55 wiz Exp $
  13  * $DragonFly: src/lib/libm/src/s_expm1.c,v 1.1 2005/07/26 21:15:20 joerg Exp $
  14  */
  15
  16 /* expm1(x)
  17  * Returns exp(x)-1, the exponential of x minus 1.
  18  *
  19  * Method
  20  *   1. Argument reduction:
  21  *      Given x, find r and integer k such that
  22  *
  23  *               x = k*ln2 + r,  |r| <= 0.5*ln2 ~ 0.34658
  24  *
  25  *      Here a correction term c will be computed to compensate
  26  *      the error in r when rounded to a floating-point number.
  27  *
  28  *   2. Approximating expm1(r) by a special rational function on
  29  *      the interval [0,0.34658]:
  30  *      Since
  31  *          r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
  32  *      we define R1(r*r) by
  33  *          r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
  34  *      That is,
  35  *          R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
  36  *                   = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
  37  *                   = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
  38  *      We use a special Reme algorithm on [0,0.347] to generate
  39  *      a polynomial of degree 5 in r*r to approximate R1. The
  40  *      maximum error of this polynomial approximation is bounded
  41  *      by 2**-61. In other words,
  42  *          R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
  43  *      where   Q1  =  -1.6666666666666567384E-2,
  44  *              Q2  =   3.9682539681370365873E-4,
  45  *              Q3  =  -9.9206344733435987357E-6,
  46  *              Q4  =   2.5051361420808517002E-7,
  47  *              Q5  =  -6.2843505682382617102E-9;
  48  *      (where z=r*r, and the values of Q1 to Q5 are listed below)
  49  *      with error bounded by
  50  *          |                  5           |     -61
  51  *          | 1.0+Q1*z+...+Q5*z   -  R1(z) | <= 2
  52  *          |                              |
  53  *
  54  *      expm1(r) = exp(r)-1 is then computed by the following
  55  *      specific way which minimize the accumulation rounding error:
  56  *                             2     3
  57  *                            r     r    [ 3 - (R1 + R1*r/2)  ]
  58  *            expm1(r) = r + --- + --- * [--------------------]
  59  *                            2     2    [ 6 - r*(3 - R1*r/2) ]
  60  *
  61  *      To compensate the error in the argument reduction, we use
  62  *              expm1(r+c) = expm1(r) + c + expm1(r)*c
  63  *                         ~ expm1(r) + c + r*c
  64  *      Thus c+r*c will be added in as the correction terms for
  65  *      expm1(r+c). Now rearrange the term to avoid optimization
  66  *      screw up:
  67  *                      (      2                                    2 )
  68  *                      ({  ( r    [ R1 -  (3 - R1*r/2) ]  )  }    r  )
  69  *       expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
  70  *                      ({  ( 2    [ 6 - r*(3 - R1*r/2) ]  )  }    2  )
  71  *                      (                                             )
  72  *
  73  *                 = r - E
  74  *   3. Scale back to obtain expm1(x):
  75  *      From step 1, we have
  76  *         expm1(x) = either 2^k*[expm1(r)+1] - 1
  77  *                  = or     2^k*[expm1(r) + (1-2^-k)]
  78  *   4. Implementation notes:
  79  *      (A). To save one multiplication, we scale the coefficient Qi
  80  *           to Qi*2^i, and replace z by (x^2)/2.
  81  *      (B). To achieve maximum accuracy, we compute expm1(x) by
  82  *        (i)   if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
  83  *        (ii)  if k=0, return r-E
  84  *        (iii) if k=-1, return 0.5*(r-E)-0.5
  85  *        (iv)  if k=1 if r < -0.25, return 2*((r+0.5)- E)
  86  *                     else          return  1.0+2.0*(r-E);
  87  *        (v)   if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
  88  *        (vi)  if k <= 20, return 2^k((1-2^-k)-(E-r)), else
  89  *        (vii) return 2^k(1-((E+2^-k)-r))
  90  *
  91  * Special cases:
  92  *      expm1(INF) is INF, expm1(NaN) is NaN;
  93  *      expm1(-INF) is -1, and
  94  *      for finite argument, only expm1(0)=0 is exact.
  95  *
  96  * Accuracy:
  97  *      according to an error analysis, the error is always less than
  98  *      1 ulp (unit in the last place).
  99  *
 100  * Misc. info.
 101  *      For IEEE double
 102  *          if x >  7.09782712893383973096e+02 then expm1(x) overflow
 103  *
 104  * Constants:
 105  * The hexadecimal values are the intended ones for the following
 106  * constants. The decimal values may be used, provided that the
 107  * compiler will convert from decimal to binary accurately enough
 108  * to produce the hexadecimal values shown.
 109  */
 110
 111 #include <math.h>
 112 #include "math_private.h"
 113
 114 static const double
 115 one             = 1.0,
 116 huge            = 1.0e+300,
 117 tiny            = 1.0e-300,
 118 o_threshold     = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */
 119 ln2_hi          = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */
 120 ln2_lo          = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */
 121 invln2          = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */
 122         /* scaled coefficients related to expm1 */
 123 Q1  =  -3.33333333333331316428e-02, /* BFA11111 111110F4 */
 124 Q2  =   1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
 125 Q3  =  -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
 126 Q4  =   4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
 127 Q5  =  -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */
 128
 129 double
 130 expm1(double x)
 131 {
 132         double y,hi,lo,c,t,e,hxs,hfx,r1;
 133         int32_t k,xsb;
 134         u_int32_t hx;
 135
 136         c = 0;
 137         GET_HIGH_WORD(hx,x);
 138         xsb = hx&0x80000000;            /* sign bit of x */
 139         hx &= 0x7fffffff;               /* high word of |x| */
 140
 141     /* filter out huge and non-finite argument */
 142         if(hx >= 0x4043687A) {                  /* if |x|>=56*ln2 */
 143             if(hx >= 0x40862E42) {              /* if |x|>=709.78... */
 144                 if(hx>=0x7ff00000) {
 145                     u_int32_t low;
 146                     GET_LOW_WORD(low,x);
 147                     if(((hx&0xfffff)|low)!=0)
 148                          return x+x;     /* NaN */
 149                     else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */
 150                 }
 151                 if(x > o_threshold) return huge*huge; /* overflow */
 152             }
 153             if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */
 154                 if(x+tiny<0.0)          /* raise inexact */
 155                 return tiny-one;        /* return -1 */
 156             }
 157         }
 158
 159     /* argument reduction */
 160         if(hx > 0x3fd62e42) {           /* if  |x| > 0.5 ln2 */
 161             if(hx < 0x3FF0A2B2) {       /* and |x| < 1.5 ln2 */
 162                 if(xsb==0)
 163                     {hi = x - ln2_hi; lo =  ln2_lo;  k =  1;}
 164                 else
 165                     {hi = x + ln2_hi; lo = -ln2_lo;  k = -1;}
 166             } else {
 167                 k  = invln2*x+((xsb==0)?0.5:-0.5);
 168                 t  = k;
 169                 hi = x - t*ln2_hi;      /* t*ln2_hi is exact here */
 170                 lo = t*ln2_lo;
 171             }
 172             x  = hi - lo;
 173             c  = (hi-x)-lo;
 174         }
 175         else if(hx < 0x3c900000) {      /* when |x|<2**-54, return x */
 176             t = huge+x; /* return x with inexact flags when x!=0 */
 177             return x - (t-(huge+x));
 178         }
 179         else k = 0;
 180
 181     /* x is now in primary range */
 182         hfx = 0.5*x;
 183         hxs = x*hfx;
 184         r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
 185         t  = 3.0-r1*hfx;
 186         e  = hxs*((r1-t)/(6.0 - x*t));
 187         if(k==0) return x - (x*e-hxs);          /* c is 0 */
 188         else {
 189             e  = (x*(e-c)-c);
 190             e -= hxs;
 191             if(k== -1) return 0.5*(x-e)-0.5;
 192             if(k==1)  {
 193                 if(x < -0.25) return -2.0*(e-(x+0.5));
 194                 else          return  one+2.0*(x-e);
 195             }
 196             if (k <= -2 || k>56) {   /* suffice to return exp(x)-1 */
 197                 u_int32_t high;
 198                 y = one-(e-x);
 199                 GET_HIGH_WORD(high,y);
 200                 SET_HIGH_WORD(y,high+(k<<20));  /* add k to y's exponent */
 201                 return y-one;
 202             }
 203             t = one;
 204             if(k<20) {
 205                 u_int32_t high;
 206                 SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k));  /* t=1-2^-k */
 207                 y = t-(e-x);
 208                 GET_HIGH_WORD(high,y);
 209                 SET_HIGH_WORD(y,high+(k<<20));  /* add k to y's exponent */
 210            } else {
 211                 u_int32_t high;
 212                 SET_HIGH_WORD(t,((0x3ff-k)<<20));       /* 2^-k */
 213                 y = x-(e+t);
 214                 y += one;
 215                 GET_HIGH_WORD(high,y);
 216                 SET_HIGH_WORD(y,high+(k<<20));  /* add k to y's exponent */
 217             }
 218         }
 219         return y;
 220 }