libm/s_expm1.c

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