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glibc  2.9
e_log2.c
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00001 /* Adapted for log2 by Ulrich Drepper <drepper@cygnus.com>.  */
00002 /*
00003  * ====================================================
00004  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
00005  *
00006  * Developed at SunPro, a Sun Microsystems, Inc. business.
00007  * Permission to use, copy, modify, and distribute this
00008  * software is freely granted, provided that this notice
00009  * is preserved.
00010  * ====================================================
00011  */
00012 
00013 /* __ieee754_log2(x)
00014  * Return the logarithm to base 2 of x
00015  *
00016  * Method :
00017  *   1. Argument Reduction: find k and f such that
00018  *                   x = 2^k * (1+f),
00019  *        where  sqrt(2)/2 < 1+f < sqrt(2) .
00020  *
00021  *   2. Approximation of log(1+f).
00022  *     Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
00023  *             = 2s + 2/3 s**3 + 2/5 s**5 + .....,
00024  *             = 2s + s*R
00025  *      We use a special Reme algorithm on [0,0.1716] to generate
00026  *     a polynomial of degree 14 to approximate R The maximum error
00027  *     of this polynomial approximation is bounded by 2**-58.45. In
00028  *     other words,
00029  *                    2      4      6      8      10      12      14
00030  *         R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
00031  *     (the values of Lg1 to Lg7 are listed in the program)
00032  *     and
00033  *         |      2          14          |     -58.45
00034  *         | Lg1*s +...+Lg7*s    -  R(z) | <= 2
00035  *         |                             |
00036  *     Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
00037  *     In order to guarantee error in log below 1ulp, we compute log
00038  *     by
00039  *            log(1+f) = f - s*(f - R)    (if f is not too large)
00040  *            log(1+f) = f - (hfsq - s*(hfsq+R)).       (better accuracy)
00041  *
00042  *     3. Finally,  log(x) = k + log(1+f).
00043  *                       = k+(f-(hfsq-(s*(hfsq+R))))
00044  *
00045  * Special cases:
00046  *     log2(x) is NaN with signal if x < 0 (including -INF) ;
00047  *     log2(+INF) is +INF; log(0) is -INF with signal;
00048  *     log2(NaN) is that NaN with no signal.
00049  *
00050  * Constants:
00051  * The hexadecimal values are the intended ones for the following
00052  * constants. The decimal values may be used, provided that the
00053  * compiler will convert from decimal to binary accurately enough
00054  * to produce the hexadecimal values shown.
00055  */
00056 
00057 #include "math.h"
00058 #include "math_private.h"
00059 
00060 #ifdef __STDC__
00061 static const double
00062 #else
00063 static double
00064 #endif
00065 ln2 = 0.69314718055994530942,
00066 two54   =  1.80143985094819840000e+16,  /* 43500000 00000000 */
00067 Lg1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
00068 Lg2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
00069 Lg3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
00070 Lg4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
00071 Lg5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
00072 Lg6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
00073 Lg7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
00074 
00075 #ifdef __STDC__
00076 static const double zero   =  0.0;
00077 #else
00078 static double zero   =  0.0;
00079 #endif
00080 
00081 #ifdef __STDC__
00082        double __ieee754_log2(double x)
00083 #else
00084        double __ieee754_log2(x)
00085        double x;
00086 #endif
00087 {
00088        double hfsq,f,s,z,R,w,t1,t2,dk;
00089        int32_t k,hx,i,j;
00090        u_int32_t lx;
00091 
00092        EXTRACT_WORDS(hx,lx,x);
00093 
00094        k=0;
00095        if (hx < 0x00100000) {                    /* x < 2**-1022  */
00096            if (((hx&0x7fffffff)|lx)==0)
00097               return -two54/(x-x);        /* log(+-0)=-inf */
00098            if (hx<0) return (x-x)/(x-x);  /* log(-#) = NaN */
00099            k -= 54; x *= two54; /* subnormal number, scale up x */
00100            GET_HIGH_WORD(hx,x);
00101        }
00102        if (hx >= 0x7ff00000) return x+x;
00103        k += (hx>>20)-1023;
00104        hx &= 0x000fffff;
00105        i = (hx+0x95f64)&0x100000;
00106        SET_HIGH_WORD(x,hx|(i^0x3ff00000));       /* normalize x or x/2 */
00107        k += (i>>20);
00108        dk = (double) k;
00109        f = x-1.0;
00110        if((0x000fffff&(2+hx))<3) { /* |f| < 2**-20 */
00111            if(f==zero) return dk;
00112            R = f*f*(0.5-0.33333333333333333*f);
00113            return dk-(R-f)/ln2;
00114        }
00115        s = f/(2.0+f);
00116        z = s*s;
00117        i = hx-0x6147a;
00118        w = z*z;
00119        j = 0x6b851-hx;
00120        t1= w*(Lg2+w*(Lg4+w*Lg6));
00121        t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
00122        i |= j;
00123        R = t2+t1;
00124        if(i>0) {
00125            hfsq=0.5*f*f;
00126            return dk-((hfsq-(s*(hfsq+R)))-f)/ln2;
00127        } else {
00128            return dk-((s*(f-R))-f)/ln2;
00129        }
00130 }