src/libm/e_log.c
author Ryan C. Gordon <icculus@icculus.org>
Tue, 26 May 2015 21:19:23 -0400
changeset 9649 d7762e30ba24
parent 6044 35448a5ea044
child 11683 48bcba563d9c
permissions -rw-r--r--
Stack hint should look for 0, not -1, and not care about environment variables.
     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 #if defined(LIBM_SCCS) && !defined(lint)
    14 static const char rcsid[] =
    15     "$NetBSD: e_log.c,v 1.8 1995/05/10 20:45:49 jtc Exp $";
    16 #endif
    17 
    18 /* __ieee754_log(x)
    19  * Return the logrithm of x
    20  *
    21  * Method :
    22  *   1. Argument Reduction: find k and f such that
    23  *			x = 2^k * (1+f),
    24  *	   where  sqrt(2)/2 < 1+f < sqrt(2) .
    25  *
    26  *   2. Approximation of log(1+f).
    27  *	Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
    28  *		 = 2s + 2/3 s**3 + 2/5 s**5 + .....,
    29  *	     	 = 2s + s*R
    30  *      We use a special Reme algorithm on [0,0.1716] to generate
    31  * 	a polynomial of degree 14 to approximate R The maximum error
    32  *	of this polynomial approximation is bounded by 2**-58.45. In
    33  *	other words,
    34  *		        2      4      6      8      10      12      14
    35  *	    R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
    36  *  	(the values of Lg1 to Lg7 are listed in the program)
    37  *	and
    38  *	    |      2          14          |     -58.45
    39  *	    | Lg1*s +...+Lg7*s    -  R(z) | <= 2
    40  *	    |                             |
    41  *	Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
    42  *	In order to guarantee error in log below 1ulp, we compute log
    43  *	by
    44  *		log(1+f) = f - s*(f - R)	(if f is not too large)
    45  *		log(1+f) = f - (hfsq - s*(hfsq+R)).	(better accuracy)
    46  *
    47  *	3. Finally,  log(x) = k*ln2 + log(1+f).
    48  *			    = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
    49  *	   Here ln2 is split into two floating point number:
    50  *			ln2_hi + ln2_lo,
    51  *	   where n*ln2_hi is always exact for |n| < 2000.
    52  *
    53  * Special cases:
    54  *	log(x) is NaN with signal if x < 0 (including -INF) ;
    55  *	log(+INF) is +INF; log(0) is -INF with signal;
    56  *	log(NaN) is that NaN with no signal.
    57  *
    58  * Accuracy:
    59  *	according to an error analysis, the error is always less than
    60  *	1 ulp (unit in the last place).
    61  *
    62  * Constants:
    63  * The hexadecimal values are the intended ones for the following
    64  * constants. The decimal values may be used, provided that the
    65  * compiler will convert from decimal to binary accurately enough
    66  * to produce the hexadecimal values shown.
    67  */
    68 
    69 #include "math_libm.h"
    70 #include "math_private.h"
    71 
    72 #ifdef __STDC__
    73 static const double
    74 #else
    75 static double
    76 #endif
    77   ln2_hi = 6.93147180369123816490e-01,  /* 3fe62e42 fee00000 */
    78     ln2_lo = 1.90821492927058770002e-10,        /* 3dea39ef 35793c76 */
    79     two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
    80     Lg1 = 6.666666666666735130e-01,     /* 3FE55555 55555593 */
    81     Lg2 = 3.999999999940941908e-01,     /* 3FD99999 9997FA04 */
    82     Lg3 = 2.857142874366239149e-01,     /* 3FD24924 94229359 */
    83     Lg4 = 2.222219843214978396e-01,     /* 3FCC71C5 1D8E78AF */
    84     Lg5 = 1.818357216161805012e-01,     /* 3FC74664 96CB03DE */
    85     Lg6 = 1.531383769920937332e-01,     /* 3FC39A09 D078C69F */
    86     Lg7 = 1.479819860511658591e-01;     /* 3FC2F112 DF3E5244 */
    87 
    88 #ifdef __STDC__
    89 static const double zero = 0.0;
    90 #else
    91 static double zero = 0.0;
    92 #endif
    93 
    94 #ifdef __STDC__
    95 double attribute_hidden
    96 __ieee754_log(double x)
    97 #else
    98 double attribute_hidden
    99 __ieee754_log(x)
   100      double x;
   101 #endif
   102 {
   103     double hfsq, f, s, z, R, w, t1, t2, dk;
   104     int32_t k, hx, i, j;
   105     u_int32_t lx;
   106 
   107     EXTRACT_WORDS(hx, lx, x);
   108 
   109     k = 0;
   110     if (hx < 0x00100000) {      /* x < 2**-1022  */
   111         if (((hx & 0x7fffffff) | lx) == 0)
   112             return -two54 / zero;       /* log(+-0)=-inf */
   113         if (hx < 0)
   114             return (x - x) / zero;      /* log(-#) = NaN */
   115         k -= 54;
   116         x *= two54;             /* subnormal number, scale up x */
   117         GET_HIGH_WORD(hx, x);
   118     }
   119     if (hx >= 0x7ff00000)
   120         return x + x;
   121     k += (hx >> 20) - 1023;
   122     hx &= 0x000fffff;
   123     i = (hx + 0x95f64) & 0x100000;
   124     SET_HIGH_WORD(x, hx | (i ^ 0x3ff00000));    /* normalize x or x/2 */
   125     k += (i >> 20);
   126     f = x - 1.0;
   127     if ((0x000fffff & (2 + hx)) < 3) {  /* |f| < 2**-20 */
   128         if (f == zero) {
   129             if (k == 0)
   130                 return zero;
   131             else {
   132                 dk = (double) k;
   133                 return dk * ln2_hi + dk * ln2_lo;
   134             }
   135         }
   136         R = f * f * (0.5 - 0.33333333333333333 * f);
   137         if (k == 0)
   138             return f - R;
   139         else {
   140             dk = (double) k;
   141             return dk * ln2_hi - ((R - dk * ln2_lo) - f);
   142         }
   143     }
   144     s = f / (2.0 + f);
   145     dk = (double) k;
   146     z = s * s;
   147     i = hx - 0x6147a;
   148     w = z * z;
   149     j = 0x6b851 - hx;
   150     t1 = w * (Lg2 + w * (Lg4 + w * Lg6));
   151     t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7)));
   152     i |= j;
   153     R = t2 + t1;
   154     if (i > 0) {
   155         hfsq = 0.5 * f * f;
   156         if (k == 0)
   157             return f - (hfsq - s * (hfsq + R));
   158         else
   159             return dk * ln2_hi - ((hfsq - (s * (hfsq + R) + dk * ln2_lo)) -
   160                                   f);
   161     } else {
   162         if (k == 0)
   163             return f - s * (f - R);
   164         else
   165             return dk * ln2_hi - ((s * (f - R) - dk * ln2_lo) - f);
   166     }
   167 }