src/libm/e_log.c
 author Ryan C. Gordon 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  * ====================================================
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  */
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
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  */
69 #include "math_libm.h"
70 #include "math_private.h"
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 */
88 #ifdef __STDC__
89 static const double zero = 0.0;
90 #else
91 static double zero = 0.0;
92 #endif
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;
107     EXTRACT_WORDS(hx, lx, x);
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 }