libm/e_exp.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 /* __ieee754_exp(x)
  13  * Returns the exponential of x.
  14  *
  15  * Method
  16  *   1. Argument reduction:
  17  *      Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
  18  *      Given x, find r and integer k such that
  19  *
  20  *               x = k*ln2 + r,  |r| <= 0.5*ln2.
  21  *
  22  *      Here r will be represented as r = hi-lo for better
  23  *      accuracy.
  24  *
  25  *   2. Approximation of exp(r) by a special rational function on
  26  *      the interval [0,0.34658]:
  27  *      Write
  28  *          R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
  29  *      We use a special Reme algorithm on [0,0.34658] to generate
  30  *      a polynomial of degree 5 to approximate R. The maximum error
  31  *      of this polynomial approximation is bounded by 2**-59. In
  32  *      other words,
  33  *          R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
  34  *      (where z=r*r, and the values of P1 to P5 are listed below)
  35  *      and
  36  *          |                  5          |     -59
  37  *          | 2.0+P1*z+...+P5*z   -  R(z) | <= 2
  38  *          |                             |
  39  *      The computation of exp(r) thus becomes
  40  *                             2*r
  41  *              exp(r) = 1 + -------
  42  *                            R - r
  43  *                                 r*R1(r)
  44  *                     = 1 + r + ----------- (for better accuracy)
  45  *                                2 - R1(r)
  46  *      where
  47  *                               2       4             10
  48  *              R1(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
  49  *
  50  *   3. Scale back to obtain exp(x):
  51  *      From step 1, we have
  52  *         exp(x) = 2^k * exp(r)
  53  *
  54  * Special cases:
  55  *      exp(INF) is INF, exp(NaN) is NaN;
  56  *      exp(-INF) is 0, and
  57  *      for finite argument, only exp(0)=1 is exact.
  58  *
  59  * Accuracy:
  60  *      according to an error analysis, the error is always less than
  61  *      1 ulp (unit in the last place).
  62  *
  63  * Misc. info.
  64  *      For IEEE double
  65  *          if x >  7.09782712893383973096e+02 then exp(x) overflow
  66  *          if x < -7.45133219101941108420e+02 then exp(x) underflow
  67  *
  68  * Constants:
  69  * The hexadecimal values are the intended ones for the following
  70  * constants. The decimal values may be used, provided that the
  71  * compiler will convert from decimal to binary accurately enough
  72  * to produce the hexadecimal values shown.
  73  */
  74
  75 #include "math.h"
  76 #include "math_private.h"
  77
  78 static const double
  79 one     = 1.0,
  80 halF[2] = {0.5,-0.5,},
  81 huge    = 1.0e+300,
  82 twom1000= 9.33263618503218878990e-302,     /* 2**-1000=0x01700000,0*/
  83 o_threshold=  7.09782712893383973096e+02,  /* 0x40862E42, 0xFEFA39EF */
  84 u_threshold= -7.45133219101941108420e+02,  /* 0xc0874910, 0xD52D3051 */
  85 ln2HI[2]   ={ 6.93147180369123816490e-01,  /* 0x3fe62e42, 0xfee00000 */
  86              -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
  87 ln2LO[2]   ={ 1.90821492927058770002e-10,  /* 0x3dea39ef, 0x35793c76 */
  88              -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
  89 invln2 =  1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
  90 P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
  91 P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
  92 P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
  93 P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
  94 P5   =  4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
  95
  96 double __ieee754_exp(double x)  /* default IEEE double exp */
  97 {
  98         double y;
  99         double hi = 0.0;
 100         double lo = 0.0;
 101         double c;
 102         double t;
 103         int32_t k=0;
 104         int32_t xsb;
 105         u_int32_t hx;
 106
 107         GET_HIGH_WORD(hx,x);
 108         xsb = (hx>>31)&1;               /* sign bit of x */
 109         hx &= 0x7fffffff;               /* high word of |x| */
 110
 111     /* filter out non-finite argument */
 112         if(hx >= 0x40862E42) {                  /* if |x|>=709.78... */
 113             if(hx>=0x7ff00000) {
 114                 u_int32_t lx;
 115                 GET_LOW_WORD(lx,x);
 116                 if(((hx&0xfffff)|lx)!=0)
 117                      return x+x;                /* NaN */
 118                 else return (xsb==0)? x:0.0;    /* exp(+-inf)={inf,0} */
 119             }
 120             if(x > o_threshold) return huge*huge; /* overflow */
 121             if(x < u_threshold) return twom1000*twom1000; /* underflow */
 122         }
 123
 124     /* argument reduction */
 125         if(hx > 0x3fd62e42) {           /* if  |x| > 0.5 ln2 */
 126             if(hx < 0x3FF0A2B2) {       /* and |x| < 1.5 ln2 */
 127                 hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
 128             } else {
 129                 k  = (int32_t)(invln2*x+halF[xsb]);
 130                 t  = k;
 131                 hi = x - t*ln2HI[0];    /* t*ln2HI is exact here */
 132                 lo = t*ln2LO[0];
 133             }
 134             x  = hi - lo;
 135         }
 136         else if(hx < 0x3e300000)  {     /* when |x|<2**-28 */
 137             if(huge+x>one) return one+x;/* trigger inexact */
 138         }
 139         else k = 0;
 140
 141     /* x is now in primary range */
 142         t  = x*x;
 143         c  = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
 144         if(k==0)        return one-((x*c)/(c-2.0)-x);
 145         else            y = one-((lo-(x*c)/(2.0-c))-hi);
 146         if(k >= -1021) {
 147             u_int32_t hy;
 148             GET_HIGH_WORD(hy,y);
 149             SET_HIGH_WORD(y,hy+(k<<20));        /* add k to y's exponent */
 150             return y;
 151         } else {
 152             u_int32_t hy;
 153             GET_HIGH_WORD(hy,y);
 154             SET_HIGH_WORD(y,hy+((k+1000)<<20)); /* add k to y's exponent */
 155             return y*twom1000;
 156         }
 157 }