libm/e_log2.c

   1 /* Adapted for log2 by Ulrich Drepper <drepper@cygnus.com>.  */
   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 /* __ieee754_log2(x)
  14  * Return the logarithm to base 2 of x
  15  *
  16  * Method :
  17  *   1. Argument Reduction: find k and f such that
  18  *                      x = 2^k * (1+f),
  19  *         where  sqrt(2)/2 < 1+f < sqrt(2) .
  20  *
  21  *   2. Approximation of log(1+f).
  22  *      Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
  23  *               = 2s + 2/3 s**3 + 2/5 s**5 + .....,
  24  *               = 2s + s*R
  25  *      We use a special Reme algorithm on [0,0.1716] to generate
  26  *      a polynomial of degree 14 to approximate R The maximum error
  27  *      of this polynomial approximation is bounded by 2**-58.45. In
  28  *      other words,
  29  *                      2      4      6      8      10      12      14
  30  *          R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
  31  *      (the values of Lg1 to Lg7 are listed in the program)
  32  *      and
  33  *          |      2          14          |     -58.45
  34  *          | Lg1*s +...+Lg7*s    -  R(z) | <= 2
  35  *          |                             |
  36  *      Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
  37  *      In order to guarantee error in log below 1ulp, we compute log
  38  *      by
  39  *              log(1+f) = f - s*(f - R)        (if f is not too large)
  40  *              log(1+f) = f - (hfsq - s*(hfsq+R)).     (better accuracy)
  41  *
  42  *      3. Finally,  log(x) = k + log(1+f).
  43  *                          = k+(f-(hfsq-(s*(hfsq+R))))
  44  *
  45  * Special cases:
  46  *      log2(x) is NaN with signal if x < 0 (including -INF) ;
  47  *      log2(+INF) is +INF; log(0) is -INF with signal;
  48  *      log2(NaN) is that NaN with no signal.
  49  *
  50  * Constants:
  51  * The hexadecimal values are the intended ones for the following
  52  * constants. The decimal values may be used, provided that the
  53  * compiler will convert from decimal to binary accurately enough
  54  * to produce the hexadecimal values shown.
  55  */
  56
  57 #include "math.h"
  58 #include "math_private.h"
  59
  60 static const double
  61 ln2 = 0.69314718055994530942,
  62 two54   =  1.80143985094819840000e+16,  /* 43500000 00000000 */
  63 Lg1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
  64 Lg2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
  65 Lg3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
  66 Lg4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
  67 Lg5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
  68 Lg6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
  69 Lg7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
  70
  71 static const double zero   =  0.0;
  72
  73 double __ieee754_log2(double x)
  74 {
  75         double hfsq,f,s,z,R,w,t1,t2,dk;
  76         int32_t k,hx,i,j;
  77         u_int32_t lx;
  78
  79         EXTRACT_WORDS(hx,lx,x);
  80
  81         k=0;
  82         if (hx < 0x00100000) {                  /* x < 2**-1022  */
  83             if (((hx&0x7fffffff)|lx)==0)
  84                 return -two54/(x-x);            /* log(+-0)=-inf */
  85             if (hx<0) return (x-x)/(x-x);       /* log(-#) = NaN */
  86             k -= 54; x *= two54; /* subnormal number, scale up x */
  87             GET_HIGH_WORD(hx,x);
  88         }
  89         if (hx >= 0x7ff00000) return x+x;
  90         k += (hx>>20)-1023;
  91         hx &= 0x000fffff;
  92         i = (hx+0x95f64)&0x100000;
  93         SET_HIGH_WORD(x,hx|(i^0x3ff00000));     /* normalize x or x/2 */
  94         k += (i>>20);
  95         dk = (double) k;
  96         f = x-1.0;
  97         if((0x000fffff&(2+hx))<3) {     /* |f| < 2**-20 */
  98             if(f==zero) return dk;
  99             R = f*f*(0.5-0.33333333333333333*f);
 100             return dk-(R-f)/ln2;
 101         }
 102         s = f/(2.0+f);
 103         z = s*s;
 104         i = hx-0x6147a;
 105         w = z*z;
 106         j = 0x6b851-hx;
 107         t1= w*(Lg2+w*(Lg4+w*Lg6));
 108         t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
 109         i |= j;
 110         R = t2+t1;
 111         if(i>0) {
 112             hfsq=0.5*f*f;
 113             return dk-((hfsq-(s*(hfsq+R)))-f)/ln2;
 114         } else {
 115             return dk-((s*(f-R))-f)/ln2;
 116         }
 117 }