src/libm/e_log.c
 changeset 2756 a98604b691c8 child 3162 dc1eb82ffdaa
equal inserted replaced
2755:2a3ec308d995 2756:a98604b691c8
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`     1 /* @(#)e_log.c 5.1 93/09/24 */`
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`     2 /*`
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`     3  * ====================================================`
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`     4  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.`
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`     5  *`
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`     6  * Developed at SunPro, a Sun Microsystems, Inc. business.`
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`     7  * Permission to use, copy, modify, and distribute this`
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`     8  * software is freely granted, provided that this notice`
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`     9  * is preserved.`
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`    10  * ====================================================`
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`    11  */`
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`    12 `
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`    13 #if defined(LIBM_SCCS) && !defined(lint)`
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`    14 static char rcsid[] = "\$NetBSD: e_log.c,v 1.8 1995/05/10 20:45:49 jtc Exp \$";`
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`    15 #endif`
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`    16 `
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`    17 /* __ieee754_log(x)`
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`    18  * Return the logrithm of x`
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`    19  *`
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`    20  * Method :`
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`    21  *   1. Argument Reduction: find k and f such that`
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`    22  *			x = 2^k * (1+f),`
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`    23  *	   where  sqrt(2)/2 < 1+f < sqrt(2) .`
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`    24  *`
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`    25  *   2. Approximation of log(1+f).`
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`    26  *	Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)`
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`    27  *		 = 2s + 2/3 s**3 + 2/5 s**5 + .....,`
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`    28  *	     	 = 2s + s*R`
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`    29  *      We use a special Reme algorithm on [0,0.1716] to generate`
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`    30  * 	a polynomial of degree 14 to approximate R The maximum error`
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`    31  *	of this polynomial approximation is bounded by 2**-58.45. In`
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`    32  *	other words,`
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`    33  *		        2      4      6      8      10      12      14`
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`    34  *	    R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s`
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`    35  *  	(the values of Lg1 to Lg7 are listed in the program)`
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`    36  *	and`
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`    37  *	    |      2          14          |     -58.45`
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`    38  *	    | Lg1*s +...+Lg7*s    -  R(z) | <= 2`
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`    39  *	    |                             |`
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`    40  *	Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.`
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`    41  *	In order to guarantee error in log below 1ulp, we compute log`
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`    42  *	by`
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`    43  *		log(1+f) = f - s*(f - R)	(if f is not too large)`
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`    44  *		log(1+f) = f - (hfsq - s*(hfsq+R)).	(better accuracy)`
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`    45  *`
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`    46  *	3. Finally,  log(x) = k*ln2 + log(1+f).`
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`    47  *			    = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))`
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`    48  *	   Here ln2 is split into two floating point number:`
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`    49  *			ln2_hi + ln2_lo,`
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`    50  *	   where n*ln2_hi is always exact for |n| < 2000.`
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`    51  *`
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`    52  * Special cases:`
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`    53  *	log(x) is NaN with signal if x < 0 (including -INF) ;`
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`    54  *	log(+INF) is +INF; log(0) is -INF with signal;`
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`    55  *	log(NaN) is that NaN with no signal.`
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`    56  *`
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`    57  * Accuracy:`
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`    58  *	according to an error analysis, the error is always less than`
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`    59  *	1 ulp (unit in the last place).`
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`    60  *`
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`    61  * Constants:`
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`    62  * The hexadecimal values are the intended ones for the following`
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`    63  * constants. The decimal values may be used, provided that the`
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`    64  * compiler will convert from decimal to binary accurately enough`
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`    65  * to produce the hexadecimal values shown.`
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`    66  */`
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`    67 `
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`    68 #include "math.h"`
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`    69 #include "math_private.h"`
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`    70 `
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`    71 #ifdef __STDC__`
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`    72 static const double`
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`    73 #else`
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`    74 static double`
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`    75 #endif`
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`    76   ln2_hi = 6.93147180369123816490e-01,  /* 3fe62e42 fee00000 */`
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`    77     ln2_lo = 1.90821492927058770002e-10,        /* 3dea39ef 35793c76 */`
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`    78     two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */`
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`    79     Lg1 = 6.666666666666735130e-01,     /* 3FE55555 55555593 */`
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`    80     Lg2 = 3.999999999940941908e-01,     /* 3FD99999 9997FA04 */`
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`    81     Lg3 = 2.857142874366239149e-01,     /* 3FD24924 94229359 */`
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`    82     Lg4 = 2.222219843214978396e-01,     /* 3FCC71C5 1D8E78AF */`
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`    83     Lg5 = 1.818357216161805012e-01,     /* 3FC74664 96CB03DE */`
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`    84     Lg6 = 1.531383769920937332e-01,     /* 3FC39A09 D078C69F */`
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`    85     Lg7 = 1.479819860511658591e-01;     /* 3FC2F112 DF3E5244 */`
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`    86 `
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`    87 #ifdef __STDC__`
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`    88 static const double zero = 0.0;`
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`    89 #else`
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`    90 static double zero = 0.0;`
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`    91 #endif`
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`    92 `
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`    93 #ifdef __STDC__`
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`    94 double attribute_hidden`
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`    95 __ieee754_log(double x)`
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`    96 #else`
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`    97 double attribute_hidden`
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`    98 __ieee754_log(x)`
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`    99      double x;`
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`   100 #endif`
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`   101 {`
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`   102     double hfsq, f, s, z, R, w, t1, t2, dk;`
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`   103     int32_t k, hx, i, j;`
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`   104     u_int32_t lx;`
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`   105 `
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`   106     EXTRACT_WORDS(hx, lx, x);`
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`   107 `
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`   108     k = 0;`
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`   109     if (hx < 0x00100000) {      /* x < 2**-1022  */`
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`   110         if (((hx & 0x7fffffff) | lx) == 0)`
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`   111             return -two54 / zero;       /* log(+-0)=-inf */`
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`   112         if (hx < 0)`
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`   113             return (x - x) / zero;      /* log(-#) = NaN */`
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`   114         k -= 54;`
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`   115         x *= two54;             /* subnormal number, scale up x */`
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`   116         GET_HIGH_WORD(hx, x);`
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`   117     }`
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`   118     if (hx >= 0x7ff00000)`
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`   119         return x + x;`
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`   120     k += (hx >> 20) - 1023;`
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`   121     hx &= 0x000fffff;`
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`   122     i = (hx + 0x95f64) & 0x100000;`
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`   123     SET_HIGH_WORD(x, hx | (i ^ 0x3ff00000));    /* normalize x or x/2 */`
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`   124     k += (i >> 20);`
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`   125     f = x - 1.0;`
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`   126     if ((0x000fffff & (2 + hx)) < 3) {  /* |f| < 2**-20 */`
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`   127         if (f == zero) {`
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`   128             if (k == 0)`
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`   129                 return zero;`
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`   130             else {`
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`   131                 dk = (double) k;`
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`   132                 return dk * ln2_hi + dk * ln2_lo;`
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`   133             }`
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`   134         }`
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`   135         R = f * f * (0.5 - 0.33333333333333333 * f);`
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`   136         if (k == 0)`
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`   137             return f - R;`
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`   138         else {`
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`   139             dk = (double) k;`
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`   140             return dk * ln2_hi - ((R - dk * ln2_lo) - f);`
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`   141         }`
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`   142     }`
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`   143     s = f / (2.0 + f);`
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`   144     dk = (double) k;`
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`   145     z = s * s;`
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`   146     i = hx - 0x6147a;`
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`   147     w = z * z;`
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`   148     j = 0x6b851 - hx;`
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`   149     t1 = w * (Lg2 + w * (Lg4 + w * Lg6));`
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`   150     t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7)));`
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`   151     i |= j;`
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`   152     R = t2 + t1;`
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`   153     if (i > 0) {`
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`   154         hfsq = 0.5 * f * f;`
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`   155         if (k == 0)`
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`   156             return f - (hfsq - s * (hfsq + R));`
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`   157         else`
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`   158             return dk * ln2_hi - ((hfsq - (s * (hfsq + R) + dk * ln2_lo)) -`
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`   159                                   f);`
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`   160     } else {`
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`   161         if (k == 0)`
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`   162             return f - s * (f - R);`
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`   163         else`
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`   164             return dk * ln2_hi - ((s * (f - R) - dk * ln2_lo) - f);`
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`   165     }`
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`   166 }`