js/src/fdlibm/e_log.c

   1 /* -*- Mode: C; tab-width: 8; indent-tabs-mode: nil; c-basic-offset: 4 -*-
   2  *
   3  * ***** BEGIN LICENSE BLOCK *****
   4  * Version: MPL 1.1/GPL 2.0/LGPL 2.1
   5  *
   6  * The contents of this file are subject to the Mozilla Public License Version
   7  * 1.1 (the "License"); you may not use this file except in compliance with
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   9  * http://www.mozilla.org/MPL/
  10  *
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  12  * WITHOUT WARRANTY OF ANY KIND, either express or implied. See the License
  13  * for the specific language governing rights and limitations under the
  14  * License.
  15  *
  16  * The Original Code is Mozilla Communicator client code, released
  17  * March 31, 1998.
  18  *
  19  * The Initial Developer of the Original Code is
  20  * Sun Microsystems, Inc.
  21  * Portions created by the Initial Developer are Copyright (C) 1998
  22  * the Initial Developer. All Rights Reserved.
  23  *
  24  * Contributor(s):
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  26  * Alternatively, the contents of this file may be used under the terms of
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  38  * ***** END LICENSE BLOCK ***** */
  39
  40 /* @(#)e_log.c 1.3 95/01/18 */
  41 /*
  42  * ====================================================
  43  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
  44  *
  45  * Developed at SunSoft, a Sun Microsystems, Inc. business.
  46  * Permission to use, copy, modify, and distribute this
  47  * software is freely granted, provided that this notice
  48  * is preserved.
  49  * ====================================================
  50  */
  51
  52 /* __ieee754_log(x)
  53  * Return the logrithm of x
  54  *
  55  * Method :
  56  *   1. Argument Reduction: find k and f such that
  57  *                      x = 2^k * (1+f),
  58  *         where  sqrt(2)/2 < 1+f < sqrt(2) .
  59  *
  60  *   2. Approximation of log(1+f).
  61  *      Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
  62  *               = 2s + 2/3 s**3 + 2/5 s**5 + .....,
  63  *               = 2s + s*R
  64  *      We use a special Reme algorithm on [0,0.1716] to generate
  65  *      a polynomial of degree 14 to approximate R The maximum error
  66  *      of this polynomial approximation is bounded by 2**-58.45. In
  67  *      other words,
  68  *                      2      4      6      8      10      12      14
  69  *          R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
  70  *      (the values of Lg1 to Lg7 are listed in the program)
  71  *      and
  72  *          |      2          14          |     -58.45
  73  *          | Lg1*s +...+Lg7*s    -  R(z) | <= 2
  74  *          |                             |
  75  *      Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
  76  *      In order to guarantee error in log below 1ulp, we compute log
  77  *      by
  78  *              log(1+f) = f - s*(f - R)        (if f is not too large)
  79  *              log(1+f) = f - (hfsq - s*(hfsq+R)).     (better accuracy)
  80  *
  81  *      3. Finally,  log(x) = k*ln2 + log(1+f).
  82  *                          = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
  83  *         Here ln2 is split into two floating point number:
  84  *                      ln2_hi + ln2_lo,
  85  *         where n*ln2_hi is always exact for |n| < 2000.
  86  *
  87  * Special cases:
  88  *      log(x) is NaN with signal if x < 0 (including -INF) ;
  89  *      log(+INF) is +INF; log(0) is -INF with signal;
  90  *      log(NaN) is that NaN with no signal.
  91  *
  92  * Accuracy:
  93  *      according to an error analysis, the error is always less than
  94  *      1 ulp (unit in the last place).
  95  *
  96  * Constants:
  97  * The hexadecimal values are the intended ones for the following
  98  * constants. The decimal values may be used, provided that the
  99  * compiler will convert from decimal to binary accurately enough
 100  * to produce the hexadecimal values shown.
 101  */
 102
 103 #include "fdlibm.h"
 104
 105 #ifdef __STDC__
 106 static const double
 107 #else
 108 static double
 109 #endif
 110 ln2_hi  =  6.93147180369123816490e-01,  /* 3fe62e42 fee00000 */
 111 ln2_lo  =  1.90821492927058770002e-10,  /* 3dea39ef 35793c76 */
 112 two54   =  1.80143985094819840000e+16,  /* 43500000 00000000 */
 113 Lg1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
 114 Lg2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
 115 Lg3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
 116 Lg4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
 117 Lg5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
 118 Lg6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
 119 Lg7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
 120
 121 static double zero   =  0.0;
 122
 123 #ifdef __STDC__
 124         double __ieee754_log(double x)
 125 #else
 126         double __ieee754_log(x)
 127         double x;
 128 #endif
 129 {
 130         fd_twoints u;
 131         double hfsq,f,s,z,R,w,t1,t2,dk;
 132         int k,hx,i,j;
 133         unsigned lx;
 134
 135         u.d = x;
 136         hx = __HI(u);           /* high word of x */
 137         lx = __LO(u);           /* low  word of x */
 138
 139         k=0;
 140         if (hx < 0x00100000) {                  /* x < 2**-1022  */
 141             if (((hx&0x7fffffff)|lx)==0)
 142                 return -two54/zero;             /* log(+-0)=-inf */
 143             if (hx<0) return (x-x)/zero;        /* log(-#) = NaN */
 144             k -= 54; x *= two54; /* subnormal number, scale up x */
 145             u.d = x;
 146             hx = __HI(u);               /* high word of x */
 147         }
 148         if (hx >= 0x7ff00000) return x+x;
 149         k += (hx>>20)-1023;
 150         hx &= 0x000fffff;
 151         i = (hx+0x95f64)&0x100000;
 152         u.d = x;
 153         __HI(u) = hx|(i^0x3ff00000);    /* normalize x or x/2 */
 154         x = u.d;
 155         k += (i>>20);
 156         f = x-1.0;
 157         if((0x000fffff&(2+hx))<3) {     /* |f| < 2**-20 */
 158             if(f==zero) {
 159                 if(k==0) return zero; else {dk=(double)k;
 160                                             return dk*ln2_hi+dk*ln2_lo;}
 161             }
 162             R = f*f*(0.5-0.33333333333333333*f);
 163             if(k==0) return f-R; else {dk=(double)k;
 164                      return dk*ln2_hi-((R-dk*ln2_lo)-f);}
 165         }
 166         s = f/(2.0+f);
 167         dk = (double)k;
 168         z = s*s;
 169         i = hx-0x6147a;
 170         w = z*z;
 171         j = 0x6b851-hx;
 172         t1= w*(Lg2+w*(Lg4+w*Lg6));
 173         t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
 174         i |= j;
 175         R = t2+t1;
 176         if(i>0) {
 177             hfsq=0.5*f*f;
 178             if(k==0) return f-(hfsq-s*(hfsq+R)); else
 179                      return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
 180         } else {
 181             if(k==0) return f-s*(f-R); else
 182                      return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
 183         }
 184 }