gonzui


Format: Advanced Search

t2ex/bsd_source/lib/libc/src_bsd/math/e_exp.cbare sourcepermlink (0.01 seconds)

Search this content:

    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: 
   13: /* LINTLIBRARY */
   14: 
   15: /* exp(x)
   16:  * Returns the exponential of x.
   17:  *
   18:  * Method
   19:  *   1. Argument reduction:
   20:  *      Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
   21:  *      Given x, find r and integer k such that
   22:  *
   23:  *               x = k*ln2 + r,  |r| <= 0.5*ln2.  
   24:  *
   25:  *      Here r will be represented as r = hi-lo for better 
   26:  *      accuracy.
   27:  *
   28:  *   2. Approximation of exp(r) by a special rational function on
   29:  *      the interval [0,0.34658]:
   30:  *      Write
   31:  *          R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
   32:  *      We use a special Remes algorithm on [0,0.34658] to generate 
   33:  *      a polynomial of degree 5 to approximate R. The maximum error 
   34:  *      of this polynomial approximation is bounded by 2**-59. In
   35:  *      other words,
   36:  *          R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
   37:  *      (where z=r*r, and the values of P1 to P5 are listed below)
   38:  *      and
   39:  *          |                  5          |     -59
   40:  *          | 2.0+P1*z+...+P5*z   -  R(z) | <= 2 
   41:  *          |                             |
   42:  *      The computation of exp(r) thus becomes
   43:  *                             2*r
   44:  *              exp(r) = 1 + -------
   45:  *                            R - r
   46:  *                                 r*R1(r)      
   47:  *                     = 1 + r + ----------- (for better accuracy)
   48:  *                                2 - R1(r)
   49:  *      where
   50:  *                               2       4             10
   51:  *              R1(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
   52:  *      
   53:  *   3. Scale back to obtain exp(x):
   54:  *      From step 1, we have
   55:  *         exp(x) = 2^k * exp(r)
   56:  *
   57:  * Special cases:
   58:  *      exp(INF) is INF, exp(NaN) is NaN;
   59:  *      exp(-INF) is 0, and
   60:  *      for finite argument, only exp(0)=1 is exact.
   61:  *
   62:  * Accuracy:
   63:  *      according to an error analysis, the error is always less than
   64:  *      1 ulp (unit in the last place).
   65:  *
   66:  * Misc. info.
   67:  *      For IEEE double 
   68:  *          if x >  7.09782712893383973096e+02 then exp(x) overflow
   69:  *          if x < -7.45133219101941108420e+02 then exp(x) underflow
   70:  *
   71:  * Constants:
   72:  * The hexadecimal values are the intended ones for the following 
   73:  * constants. The decimal values may be used, provided that the 
   74:  * compiler will convert from decimal to binary accurately enough
   75:  * to produce the hexadecimal values shown.
   76:  */
   77: 
   78: #include <sys/cdefs.h>
   79: #include <float.h>
   80: #include <math.h>
   81: 
   82: #include "math_private.h"
   83: 
   84: static const double
   85: one     = 1.0,
   86: halF[2] = {0.5,-0.5,},
   87: huge    = 1.0e+300,
   88: twom1000= 9.33263618503218878990e-302,     /* 2**-1000=0x01700000,0*/
   89: o_threshold=  7.09782712893383973096e+02,  /* 0x40862E42, 0xFEFA39EF */
   90: u_threshold= -7.45133219101941108420e+02,  /* 0xc0874910, 0xD52D3051 */
   91: ln2HI[2]   ={ 6.93147180369123816490e-01,  /* 0x3fe62e42, 0xfee00000 */
   92:              -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
   93: ln2LO[2]   ={ 1.90821492927058770002e-10,  /* 0x3dea39ef, 0x35793c76 */
   94:              -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
   95: invln2 =  1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
   96: P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
   97: P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
   98: P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
   99: P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
  100: P5   =  4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
  101: 
  102: 
  103: double
  104: exp(double x)   /* default IEEE double exp */
  105: {
  106:         double y,hi,lo,c,t;
  107:         int32_t k,xsb;
  108:         u_int32_t hx;
  109: 
  110:         GET_HIGH_WORD(hx,x);
  111:         xsb = (hx>>31)&1;              /* sign bit of x */
  112:         hx &= 0x7fffffff;              /* high word of |x| */
  113: 
  114:     /* filter out non-finite argument */
  115:         if(hx >= 0x40862E42) {                 /* if |x|>=709.78... */
  116:             if(hx>=0x7ff00000) {
  117:                 u_int32_t lx;
  118:                 GET_LOW_WORD(lx,x);
  119:                 if(((hx&0xfffff)|lx)!=0) 
  120:                      return x+x;              /* NaN */
  121:                 else return (xsb==0)? x:0.0;  /* exp(+-inf)={inf,0} */
  122:             }
  123:             if(x > o_threshold) return huge*huge; /* overflow */
  124:             if(x < u_threshold) return twom1000*twom1000; /* underflow */
  125:         }
  126: 
  127:     /* argument reduction */
  128:         if(hx > 0x3fd62e42) {          /* if  |x| > 0.5 ln2 */ 
  129:             if(hx < 0x3FF0A2B2) {      /* and |x| < 1.5 ln2 */
  130:                 hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
  131:             } else {
  132:                 k  = invln2*x+halF[xsb];
  133:                 t  = k;
  134:                 hi = x - t*ln2HI[0];  /* t*ln2HI is exact here */
  135:                 lo = t*ln2LO[0];
  136:             }
  137:             x  = hi - lo;
  138:         } 
  139:         else if(hx < 0x3e300000)  {    /* when |x|<2**-28 */
  140:             if(huge+x>one) return one+x;/* trigger inexact */
  141:         }
  142:         else k = 0;
  143: 
  144:     /* x is now in primary range */
  145:         t  = x*x;
  146:         c  = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
  147:         if(k==0)       return one-((x*c)/(c-2.0)-x); 
  148:         else           y = one-((lo-(x*c)/(2.0-c))-hi);
  149:         if(k >= -1021) {
  150:             u_int32_t hy;
  151:             GET_HIGH_WORD(hy,y);
  152:             SET_HIGH_WORD(y,hy+(k<<20));       /* add k to y's exponent */
  153:             return y;
  154:         } else {
  155:             u_int32_t hy;
  156:             GET_HIGH_WORD(hy,y);
  157:             SET_HIGH_WORD(y,hy+((k+1000)<<20));        /* add k to y's exponent */
  158:             return y*twom1000;
  159:         }
  160: }
  161: 
  162: #if     LDBL_MANT_DIG == 53
  163: #ifdef  lint
  164: /* PROTOLIB1 */
  165: long double expl(long double);
  166: #else   /* lint */
  167: __weak_alias(expl, exp);
  168: #endif  /* lint */
  169: #endif  /* LDBL_MANT_DIG == 53 */