toolchain/gcc/libjava/java/lang/e_log.c

   1
   2 /* @(#)e_log.c 5.1 93/09/24 */
   3 /*
   4  * ====================================================
   5  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   6  *
   7  * Developed at SunPro, a Sun Microsystems, Inc. business.
   8  * Permission to use, copy, modify, and distribute this
   9  * software is freely granted, provided that this notice
  10  * is preserved.
  11  * ====================================================
  12  */
  13
  14 /* __ieee754_log(x)
  15  * Return the logrithm of x
  16  *
  17  * Method :
  18  *   1. Argument Reduction: find k and f such that
  19  *                      x = 2^k * (1+f),
  20  *         where  sqrt(2)/2 < 1+f < sqrt(2) .
  21  *
  22  *   2. Approximation of log(1+f).
  23  *      Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
  24  *               = 2s + 2/3 s**3 + 2/5 s**5 + .....,
  25  *               = 2s + s*R
  26  *      We use a special Reme algorithm on [0,0.1716] to generate
  27  *      a polynomial of degree 14 to approximate R The maximum error
  28  *      of this polynomial approximation is bounded by 2**-58.45. In
  29  *      other words,
  30  *                      2      4      6      8      10      12      14
  31  *          R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
  32  *      (the values of Lg1 to Lg7 are listed in the program)
  33  *      and
  34  *          |      2          14          |     -58.45
  35  *          | Lg1*s +...+Lg7*s    -  R(z) | <= 2
  36  *          |                             |
  37  *      Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
  38  *      In order to guarantee error in log below 1ulp, we compute log
  39  *      by
  40  *              log(1+f) = f - s*(f - R)        (if f is not too large)
  41  *              log(1+f) = f - (hfsq - s*(hfsq+R)).     (better accuracy)
  42  *
  43  *      3. Finally,  log(x) = k*ln2 + log(1+f).
  44  *                          = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
  45  *         Here ln2 is split into two floating point number:
  46  *                      ln2_hi + ln2_lo,
  47  *         where n*ln2_hi is always exact for |n| < 2000.
  48  *
  49  * Special cases:
  50  *      log(x) is NaN with signal if x < 0 (including -INF) ;
  51  *      log(+INF) is +INF; log(0) is -INF with signal;
  52  *      log(NaN) is that NaN with no signal.
  53  *
  54  * Accuracy:
  55  *      according to an error analysis, the error is always less than
  56  *      1 ulp (unit in the last place).
  57  *
  58  * Constants:
  59  * The hexadecimal values are the intended ones for the following
  60  * constants. The decimal values may be used, provided that the
  61  * compiler will convert from decimal to binary accurately enough
  62  * to produce the hexadecimal values shown.
  63  */
  64
  65 #include "fdlibm.h"
  66
  67 #ifndef _DOUBLE_IS_32BITS
  68
  69 #ifdef __STDC__
  70 static const double
  71 #else
  72 static double
  73 #endif
  74 ln2_hi  =  6.93147180369123816490e-01,  /* 3fe62e42 fee00000 */
  75 ln2_lo  =  1.90821492927058770002e-10,  /* 3dea39ef 35793c76 */
  76 two54   =  1.80143985094819840000e+16,  /* 43500000 00000000 */
  77 Lg1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
  78 Lg2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
  79 Lg3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
  80 Lg4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
  81 Lg5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
  82 Lg6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
  83 Lg7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
  84
  85 #ifdef __STDC__
  86 static const double zero   =  0.0;
  87 #else
  88 static double zero   =  0.0;
  89 #endif
  90
  91 #ifdef __STDC__
  92         double __ieee754_log(double x)
  93 #else
  94         double __ieee754_log(x)
  95         double x;
  96 #endif
  97 {
  98         double hfsq,f,s,z,R,w,t1,t2,dk;
  99         int32_t k,hx,i,j;
 100         uint32_t lx;
 101
 102         EXTRACT_WORDS(hx,lx,x);
 103
 104         k=0;
 105         if (hx < 0x00100000) {                  /* x < 2**-1022  */
 106             if (((hx&0x7fffffff)|lx)==0)
 107                 return -two54/zero;             /* log(+-0)=-inf */
 108             if (hx<0) return (x-x)/zero;        /* log(-#) = NaN */
 109             k -= 54; x *= two54; /* subnormal number, scale up x */
 110             GET_HIGH_WORD(hx,x);
 111         }
 112         if (hx >= 0x7ff00000) return x+x;
 113         k += (hx>>20)-1023;
 114         hx &= 0x000fffff;
 115         i = (hx+0x95f64)&0x100000;
 116         SET_HIGH_WORD(x,hx|(i^0x3ff00000));     /* normalize x or x/2 */
 117         k += (i>>20);
 118         f = x-1.0;
 119         if((0x000fffff&(2+hx))<3) {     /* |f| < 2**-20 */
 120             if(f==zero) {
 121               if(k==0)
 122                 return zero;
 123               else {
 124                 dk=(double)k;
 125                 return dk*ln2_hi+dk*ln2_lo;
 126               }
 127             }
 128             R = f*f*(0.5-0.33333333333333333*f);
 129             if(k==0) return f-R; else {dk=(double)k;
 130                      return dk*ln2_hi-((R-dk*ln2_lo)-f);}
 131         }
 132         s = f/(2.0+f);
 133         dk = (double)k;
 134         z = s*s;
 135         i = hx-0x6147a;
 136         w = z*z;
 137         j = 0x6b851-hx;
 138         t1= w*(Lg2+w*(Lg4+w*Lg6));
 139         t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
 140         i |= j;
 141         R = t2+t1;
 142         if(i>0) {
 143             hfsq=0.5*f*f;
 144             if(k==0) return f-(hfsq-s*(hfsq+R)); else
 145                      return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
 146         } else {
 147             if(k==0) return f-s*(f-R); else
 148                      return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
 149         }
 150 }
 151
 152 #endif /* defined(_DOUBLE_IS_32BITS) */