StdLib/LibC/Math/e_exp.c

   1 /* @(#)e_exp.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 #include  <LibConfig.h>
  13 #include  <sys/EfiCdefs.h>
  14 #if defined(LIBM_SCCS) && !defined(lint)
  15 __RCSID("$NetBSD: e_exp.c,v 1.11 2002/05/26 22:01:49 wiz Exp $");
  16 #endif
  17
  18 #if defined(_MSC_VER)           /* Handle Microsoft VC++ compiler specifics. */
  19   // C4756: overflow in constant arithmetic
  20   #pragma warning ( disable : 4756 )
  21 #endif
  22
  23 /* __ieee754_exp(x)
  24  * Returns the exponential of x.
  25  *
  26  * Method
  27  *   1. Argument reduction:
  28  *      Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
  29  *  Given x, find r and integer k such that
  30  *
  31  *               x = k*ln2 + r,  |r| <= 0.5*ln2.
  32  *
  33  *      Here r will be represented as r = hi-lo for better
  34  *  accuracy.
  35  *
  36  *   2. Approximation of exp(r) by a special rational function on
  37  *  the interval [0,0.34658]:
  38  *  Write
  39  *      R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
  40  *      We use a special Reme algorithm on [0,0.34658] to generate
  41  *  a polynomial of degree 5 to approximate R. The maximum error
  42  *  of this polynomial approximation is bounded by 2**-59. In
  43  *  other words,
  44  *      R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
  45  *    (where z=r*r, and the values of P1 to P5 are listed below)
  46  *  and
  47  *      |                  5          |     -59
  48  *      | 2.0+P1*z+...+P5*z   -  R(z) | <= 2
  49  *      |                             |
  50  *  The computation of exp(r) thus becomes
  51  *                             2*r
  52  *    exp(r) = 1 + -------
  53  *                  R - r
  54  *                                 r*R1(r)
  55  *           = 1 + r + ----------- (for better accuracy)
  56  *                      2 - R1(r)
  57  *  where
  58  *               2       4             10
  59  *    R1(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
  60  *
  61  *   3. Scale back to obtain exp(x):
  62  *  From step 1, we have
  63  *     exp(x) = 2^k * exp(r)
  64  *
  65  * Special cases:
  66  *  exp(INF) is INF, exp(NaN) is NaN;
  67  *  exp(-INF) is 0, and
  68  *  for finite argument, only exp(0)=1 is exact.
  69  *
  70  * Accuracy:
  71  *  according to an error analysis, the error is always less than
  72  *  1 ulp (unit in the last place).
  73  *
  74  * Misc. info.
  75  *  For IEEE double
  76  *      if x >  7.09782712893383973096e+02 then exp(x) overflow
  77  *      if x < -7.45133219101941108420e+02 then exp(x) underflow
  78  *
  79  * Constants:
  80  * The hexadecimal values are the intended ones for the following
  81  * constants. The decimal values may be used, provided that the
  82  * compiler will convert from decimal to binary accurately enough
  83  * to produce the hexadecimal values shown.
  84  */
  85
  86 #include "math.h"
  87 #include "math_private.h"
  88
  89 static const double
  90 one = 1.0,
  91 halF[2] = {0.5,-0.5,},
  92 huge  = 1.0e+300,
  93 twom1000= 9.33263618503218878990e-302,     /* 2**-1000=0x01700000,0*/
  94 o_threshold=  7.09782712893383973096e+02,  /* 0x40862E42, 0xFEFA39EF */
  95 u_threshold= -7.45133219101941108420e+02,  /* 0xc0874910, 0xD52D3051 */
  96 ln2HI[2]   ={ 6.93147180369123816490e-01,  /* 0x3fe62e42, 0xfee00000 */
  97        -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
  98 ln2LO[2]   ={ 1.90821492927058770002e-10,  /* 0x3dea39ef, 0x35793c76 */
  99        -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
 100 invln2 =  1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
 101 P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
 102 P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
 103 P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
 104 P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
 105 P5   =  4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
 106
 107
 108 double
 109 __ieee754_exp(double x) /* default IEEE double exp */
 110 {
 111   double y,hi,lo,c,t;
 112   int32_t k,xsb;
 113   u_int32_t hx;
 114
 115   hi = lo = 0;
 116   k = 0;
 117   GET_HIGH_WORD(hx,x);
 118   xsb = (hx>>31)&1;   /* sign bit of x */
 119   hx &= 0x7fffffff;   /* high word of |x| */
 120
 121     /* filter out non-finite argument */
 122   if(hx >= 0x40862E42) {      /* if |x|>=709.78... */
 123             if(hx>=0x7ff00000) {
 124           u_int32_t lx;
 125     GET_LOW_WORD(lx,x);
 126     if(((hx&0xfffff)|lx)!=0)
 127          return x+x;    /* NaN */
 128     else return (xsb==0)? x:0.0;  /* exp(+-inf)={inf,0} */
 129       }
 130       if(x > o_threshold) return huge*huge; /* overflow */
 131       if(x < u_threshold) return twom1000*twom1000; /* underflow */
 132   }
 133
 134     /* argument reduction */
 135   if(hx > 0x3fd62e42) {   /* if  |x| > 0.5 ln2 */
 136       if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
 137     hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
 138       } else {
 139     k  = (int32_t)(invln2*x+halF[xsb]);
 140     t  = k;
 141     hi = x - t*ln2HI[0];  /* t*ln2HI is exact here */
 142     lo = t*ln2LO[0];
 143       }
 144       x  = hi - lo;
 145   }
 146   else if(hx < 0x3e300000)  { /* when |x|<2**-28 */
 147       if(huge+x>one) return one+x;/* trigger inexact */
 148   }
 149   else k = 0;
 150
 151     /* x is now in primary range */
 152   t  = x*x;
 153   c  = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
 154   if(k==0)  return one-((x*c)/(c-2.0)-x);
 155   else    y = one-((lo-(x*c)/(2.0-c))-hi);
 156   if(k >= -1021) {
 157       u_int32_t hy;
 158       GET_HIGH_WORD(hy,y);
 159       SET_HIGH_WORD(y,hy+(k<<20));  /* add k to y's exponent */
 160       return y;
 161   } else {
 162       u_int32_t hy;
 163       GET_HIGH_WORD(hy,y);
 164       SET_HIGH_WORD(y,hy+((k+1000)<<20)); /* add k to y's exponent */
 165       return y*twom1000;
 166   }
 167 }