gonzui

    1: /* @(#)e_log.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: /* log(x)
   16:  * Return the logarithm of x
   17:  *
   18:  * Method :                  
   19:  *   1. Argument Reduction: find k and f such that 
   20:  *                      x = 2^k * (1+f), 
   21:  *         where  sqrt(2)/2 < 1+f < sqrt(2) .
   22:  *
   23:  *   2. Approximation of log(1+f).
   24:  *      Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
   25:  *               = 2s + 2/3 s**3 + 2/5 s**5 + .....,
   26:  *                    = 2s + s*R
   27:  *      We use a special Remes algorithm on [0,0.1716] to generate 
   28:  *      a polynomial of degree 14 to approximate R The maximum error 
   29:  *      of this polynomial approximation is bounded by 2**-58.45. In
   30:  *      other words,
   31:  *                      2      4      6      8      10      12      14
   32:  *          R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
   33:  *      (the values of Lg1 to Lg7 are listed in the program)
   34:  *      and
   35:  *          |      2          14          |     -58.45
   36:  *          | Lg1*s +...+Lg7*s    -  R(z) | <= 2 
   37:  *          |                             |
   38:  *      Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
   39:  *      In order to guarantee error in log below 1ulp, we compute log
   40:  *      by
   41:  *              log(1+f) = f - s*(f - R)    (if f is not too large)
   42:  *              log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
   43:  *      
   44:  *      3. Finally,  log(x) = k*ln2 + log(1+f).  
   45:  *                          = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
   46:  *         Here ln2 is split into two floating point number: 
   47:  *                      ln2_hi + ln2_lo,
   48:  *         where n*ln2_hi is always exact for |n| < 2000.
   49:  *
   50:  * Special cases:
   51:  *      log(x) is NaN with signal if x < 0 (including -INF) ; 
   52:  *      log(+INF) is +INF; log(0) is -INF with signal;
   53:  *      log(NaN) is that NaN with no signal.
   54:  *
   55:  * Accuracy:
   56:  *      according to an error analysis, the error is always less than
   57:  *      1 ulp (unit in the last place).
   58:  *
   59:  * Constants:
   60:  * The hexadecimal values are the intended ones for the following 
   61:  * constants. The decimal values may be used, provided that the 
   62:  * compiler will convert from decimal to binary accurately enough 
   63:  * to produce the hexadecimal values shown.
   64:  */
   65: 
   66: #include <sys/cdefs.h>
   67: #include <float.h>
   68: #include <math.h>
   69: 
   70: #include "math_private.h"
   71: 
   72: static const double
   73: ln2_hi  =  6.93147180369123816490e-01,  /* 3fe62e42 fee00000 */
   74: ln2_lo  =  1.90821492927058770002e-10,  /* 3dea39ef 35793c76 */
   75: two54   =  1.80143985094819840000e+16,  /* 43500000 00000000 */
   76: Lg1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
   77: Lg2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
   78: Lg3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
   79: Lg4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
   80: Lg5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
   81: Lg6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
   82: Lg7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
   83: 
   84: static const double zero   =  0.0;
   85: 
   86: double
   87: log(double x)
   88: {
   89:         double hfsq,f,s,z,R,w,t1,t2,dk;
   90:         int32_t k,hx,i,j;
   91:         u_int32_t lx;
   92: 
   93:         EXTRACT_WORDS(hx,lx,x);
   94: 
   95:         k=0;
   96:         if (hx < 0x00100000) {                 /* x < 2**-1022  */
   97:             if (((hx&0x7fffffff)|lx)==0) 
   98:                 return -two54/zero;           /* log(+-0)=-inf */
   99:             if (hx<0) return (x-x)/zero;       /* log(-#) = NaN */
  100:             k -= 54; x *= two54; /* subnormal number, scale up x */
  101:             GET_HIGH_WORD(hx,x);
  102:         } 
  103:         if (hx >= 0x7ff00000) return x+x;
  104:         k += (hx>>20)-1023;
  105:         hx &= 0x000fffff;
  106:         i = (hx+0x95f64)&0x100000;
  107:         SET_HIGH_WORD(x,hx|(i^0x3ff00000));    /* normalize x or x/2 */
  108:         k += (i>>20);
  109:         f = x-1.0;
  110:         if((0x000fffff&(2+hx))<3) {    /* |f| < 2**-20 */
  111:             if(f==zero) if(k==0) return zero;  else {dk=(double)k;
  112:                                  return dk*ln2_hi+dk*ln2_lo;}
  113:             R = f*f*(0.5-0.33333333333333333*f);
  114:             if(k==0) return f-R; else {dk=(double)k;
  115:                     return dk*ln2_hi-((R-dk*ln2_lo)-f);}
  116:         }
  117:         s = f/(2.0+f); 
  118:         dk = (double)k;
  119:         z = s*s;
  120:         i = hx-0x6147a;
  121:         w = z*z;
  122:         j = 0x6b851-hx;
  123:         t1= w*(Lg2+w*(Lg4+w*Lg6)); 
  124:         t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7))); 
  125:         i |= j;
  126:         R = t2+t1;
  127:         if(i>0) {
  128:             hfsq=0.5*f*f;
  129:             if(k==0) return f-(hfsq-s*(hfsq+R)); else
  130:                      return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
  131:         } else {
  132:             if(k==0) return f-s*(f-R); else
  133:                      return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
  134:         }
  135: }
  136: 
  137: #if     LDBL_MANT_DIG == 53
  138: #ifdef  lint
  139: /* PROTOLIB1 */
  140: long double logl(long double);
  141: #else   /* lint */
  142: __weak_alias(logl, log);
  143: #endif  /* lint */
  144: #endif  /* LDBL_MANT_DIG == 53 */