src/libm/e_log.c
changeset 2756 a98604b691c8
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2755:2a3ec308d995 2756:a98604b691c8
       
     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 char rcsid[] = "$NetBSD: e_log.c,v 1.8 1995/05/10 20:45:49 jtc Exp $";
       
    15 #endif
       
    16 
       
    17 /* __ieee754_log(x)
       
    18  * Return the logrithm of x
       
    19  *
       
    20  * Method :
       
    21  *   1. Argument Reduction: find k and f such that
       
    22  *			x = 2^k * (1+f),
       
    23  *	   where  sqrt(2)/2 < 1+f < sqrt(2) .
       
    24  *
       
    25  *   2. Approximation of log(1+f).
       
    26  *	Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
       
    27  *		 = 2s + 2/3 s**3 + 2/5 s**5 + .....,
       
    28  *	     	 = 2s + s*R
       
    29  *      We use a special Reme algorithm on [0,0.1716] to generate
       
    30  * 	a polynomial of degree 14 to approximate R The maximum error
       
    31  *	of this polynomial approximation is bounded by 2**-58.45. In
       
    32  *	other words,
       
    33  *		        2      4      6      8      10      12      14
       
    34  *	    R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
       
    35  *  	(the values of Lg1 to Lg7 are listed in the program)
       
    36  *	and
       
    37  *	    |      2          14          |     -58.45
       
    38  *	    | Lg1*s +...+Lg7*s    -  R(z) | <= 2
       
    39  *	    |                             |
       
    40  *	Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
       
    41  *	In order to guarantee error in log below 1ulp, we compute log
       
    42  *	by
       
    43  *		log(1+f) = f - s*(f - R)	(if f is not too large)
       
    44  *		log(1+f) = f - (hfsq - s*(hfsq+R)).	(better accuracy)
       
    45  *
       
    46  *	3. Finally,  log(x) = k*ln2 + log(1+f).
       
    47  *			    = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
       
    48  *	   Here ln2 is split into two floating point number:
       
    49  *			ln2_hi + ln2_lo,
       
    50  *	   where n*ln2_hi is always exact for |n| < 2000.
       
    51  *
       
    52  * Special cases:
       
    53  *	log(x) is NaN with signal if x < 0 (including -INF) ;
       
    54  *	log(+INF) is +INF; log(0) is -INF with signal;
       
    55  *	log(NaN) is that NaN with no signal.
       
    56  *
       
    57  * Accuracy:
       
    58  *	according to an error analysis, the error is always less than
       
    59  *	1 ulp (unit in the last place).
       
    60  *
       
    61  * Constants:
       
    62  * The hexadecimal values are the intended ones for the following
       
    63  * constants. The decimal values may be used, provided that the
       
    64  * compiler will convert from decimal to binary accurately enough
       
    65  * to produce the hexadecimal values shown.
       
    66  */
       
    67 
       
    68 #include "math.h"
       
    69 #include "math_private.h"
       
    70 
       
    71 #ifdef __STDC__
       
    72 static const double
       
    73 #else
       
    74 static double
       
    75 #endif
       
    76   ln2_hi = 6.93147180369123816490e-01,  /* 3fe62e42 fee00000 */
       
    77     ln2_lo = 1.90821492927058770002e-10,        /* 3dea39ef 35793c76 */
       
    78     two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
       
    79     Lg1 = 6.666666666666735130e-01,     /* 3FE55555 55555593 */
       
    80     Lg2 = 3.999999999940941908e-01,     /* 3FD99999 9997FA04 */
       
    81     Lg3 = 2.857142874366239149e-01,     /* 3FD24924 94229359 */
       
    82     Lg4 = 2.222219843214978396e-01,     /* 3FCC71C5 1D8E78AF */
       
    83     Lg5 = 1.818357216161805012e-01,     /* 3FC74664 96CB03DE */
       
    84     Lg6 = 1.531383769920937332e-01,     /* 3FC39A09 D078C69F */
       
    85     Lg7 = 1.479819860511658591e-01;     /* 3FC2F112 DF3E5244 */
       
    86 
       
    87 #ifdef __STDC__
       
    88 static const double zero = 0.0;
       
    89 #else
       
    90 static double zero = 0.0;
       
    91 #endif
       
    92 
       
    93 #ifdef __STDC__
       
    94 double attribute_hidden
       
    95 __ieee754_log(double x)
       
    96 #else
       
    97 double attribute_hidden
       
    98 __ieee754_log(x)
       
    99      double x;
       
   100 #endif
       
   101 {
       
   102     double hfsq, f, s, z, R, w, t1, t2, dk;
       
   103     int32_t k, hx, i, j;
       
   104     u_int32_t lx;
       
   105 
       
   106     EXTRACT_WORDS(hx, lx, x);
       
   107 
       
   108     k = 0;
       
   109     if (hx < 0x00100000) {      /* x < 2**-1022  */
       
   110         if (((hx & 0x7fffffff) | lx) == 0)
       
   111             return -two54 / zero;       /* log(+-0)=-inf */
       
   112         if (hx < 0)
       
   113             return (x - x) / zero;      /* log(-#) = NaN */
       
   114         k -= 54;
       
   115         x *= two54;             /* subnormal number, scale up x */
       
   116         GET_HIGH_WORD(hx, x);
       
   117     }
       
   118     if (hx >= 0x7ff00000)
       
   119         return x + x;
       
   120     k += (hx >> 20) - 1023;
       
   121     hx &= 0x000fffff;
       
   122     i = (hx + 0x95f64) & 0x100000;
       
   123     SET_HIGH_WORD(x, hx | (i ^ 0x3ff00000));    /* normalize x or x/2 */
       
   124     k += (i >> 20);
       
   125     f = x - 1.0;
       
   126     if ((0x000fffff & (2 + hx)) < 3) {  /* |f| < 2**-20 */
       
   127         if (f == zero) {
       
   128             if (k == 0)
       
   129                 return zero;
       
   130             else {
       
   131                 dk = (double) k;
       
   132                 return dk * ln2_hi + dk * ln2_lo;
       
   133             }
       
   134         }
       
   135         R = f * f * (0.5 - 0.33333333333333333 * f);
       
   136         if (k == 0)
       
   137             return f - R;
       
   138         else {
       
   139             dk = (double) k;
       
   140             return dk * ln2_hi - ((R - dk * ln2_lo) - f);
       
   141         }
       
   142     }
       
   143     s = f / (2.0 + f);
       
   144     dk = (double) k;
       
   145     z = s * s;
       
   146     i = hx - 0x6147a;
       
   147     w = z * z;
       
   148     j = 0x6b851 - hx;
       
   149     t1 = w * (Lg2 + w * (Lg4 + w * Lg6));
       
   150     t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7)));
       
   151     i |= j;
       
   152     R = t2 + t1;
       
   153     if (i > 0) {
       
   154         hfsq = 0.5 * f * f;
       
   155         if (k == 0)
       
   156             return f - (hfsq - s * (hfsq + R));
       
   157         else
       
   158             return dk * ln2_hi - ((hfsq - (s * (hfsq + R) + dk * ln2_lo)) -
       
   159                                   f);
       
   160     } else {
       
   161         if (k == 0)
       
   162             return f - s * (f - R);
       
   163         else
       
   164             return dk * ln2_hi - ((s * (f - R) - dk * ln2_lo) - f);
       
   165     }
       
   166 }