src/libm/e_log.c
 author Philipp Wiesemann Sun, 28 Dec 2014 22:00:24 +0100 changeset 9301 7377a9a3aed6 parent 6044 35448a5ea044 child 11683 48bcba563d9c permissions -rw-r--r--
```     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 }
```