1/*
2 * Copyright (c) 1998, 2001, Oracle and/or its affiliates. All rights reserved.
3 * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
4 *
5 * This code is free software; you can redistribute it and/or modify it
6 * under the terms of the GNU General Public License version 2 only, as
7 * published by the Free Software Foundation. Oracle designates this
8 * particular file as subject to the "Classpath" exception as provided
9 * by Oracle in the LICENSE file that accompanied this code.
10 *
11 * This code is distributed in the hope that it will be useful, but WITHOUT
12 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
13 * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
14 * version 2 for more details (a copy is included in the LICENSE file that
15 * accompanied this code).
16 *
17 * You should have received a copy of the GNU General Public License version
18 * 2 along with this work; if not, write to the Free Software Foundation,
19 * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
20 *
21 * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
22 * or visit www.oracle.com if you need additional information or have any
23 * questions.
24 */
25
26/* __ieee754_log(x)
27 * Return the logrithm of x
28 *
29 * Method :
30 * 1. Argument Reduction: find k and f such that
31 * x = 2^k * (1+f),
32 * where sqrt(2)/2 < 1+f < sqrt(2) .
33 *
34 * 2. Approximation of log(1+f).
35 * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
36 * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
37 * = 2s + s*R
38 * We use a special Reme algorithm on [0,0.1716] to generate
39 * a polynomial of degree 14 to approximate R The maximum error
40 * of this polynomial approximation is bounded by 2**-58.45. In
41 * other words,
42 * 2 4 6 8 10 12 14
43 * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
44 * (the values of Lg1 to Lg7 are listed in the program)
45 * and
46 * | 2 14 | -58.45
47 * | Lg1*s +...+Lg7*s - R(z) | <= 2
48 * | |
49 * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
50 * In order to guarantee error in log below 1ulp, we compute log
51 * by
52 * log(1+f) = f - s*(f - R) (if f is not too large)
53 * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
54 *
55 * 3. Finally, log(x) = k*ln2 + log(1+f).
56 * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
57 * Here ln2 is split into two floating point number:
58 * ln2_hi + ln2_lo,
59 * where n*ln2_hi is always exact for |n| < 2000.
60 *
61 * Special cases:
62 * log(x) is NaN with signal if x < 0 (including -INF) ;
63 * log(+INF) is +INF; log(0) is -INF with signal;
64 * log(NaN) is that NaN with no signal.
65 *
66 * Accuracy:
67 * according to an error analysis, the error is always less than
68 * 1 ulp (unit in the last place).
69 *
70 * Constants:
71 * The hexadecimal values are the intended ones for the following
72 * constants. The decimal values may be used, provided that the
73 * compiler will convert from decimal to binary accurately enough
74 * to produce the hexadecimal values shown.
75 */
76
77#include "fdlibm.h"
78
79#ifdef __STDC__
80static const double
81#else
82static double
83#endif
84ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
85ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
86two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
87Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
88Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
89Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
90Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
91Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
92Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
93Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
94
95static double zero = 0.0;
96
97#ifdef __STDC__
98 double __ieee754_log(double x)
99#else
100 double __ieee754_log(x)
101 double x;
102#endif
103{
104 double hfsq,f,s,z,R,w,t1,t2,dk;
105 int k,hx,i,j;
106 unsigned lx;
107
108 hx = __HI(x); /* high word of x */
109 lx = __LO(x); /* low word of x */
110
111 k=0;
112 if (hx < 0x00100000) { /* x < 2**-1022 */
113 if (((hx&0x7fffffff)|lx)==0)
114 return -two54/zero; /* log(+-0)=-inf */
115 if (hx<0) return (x-x)/zero; /* log(-#) = NaN */
116 k -= 54; x *= two54; /* subnormal number, scale up x */
117 hx = __HI(x); /* high word of x */
118 }
119 if (hx >= 0x7ff00000) return x+x;
120 k += (hx>>20)-1023;
121 hx &= 0x000fffff;
122 i = (hx+0x95f64)&0x100000;
123 __HI(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) return zero;
129 else {dk=(double)k; return dk*ln2_hi+dk*ln2_lo;}
130 }
131 R = f*f*(0.5-0.33333333333333333*f);
132 if(k==0) return f-R; else {dk=(double)k;
133 return dk*ln2_hi-((R-dk*ln2_lo)-f);}
134 }
135 s = f/(2.0+f);
136 dk = (double)k;
137 z = s*s;
138 i = hx-0x6147a;
139 w = z*z;
140 j = 0x6b851-hx;
141 t1= w*(Lg2+w*(Lg4+w*Lg6));
142 t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
143 i |= j;
144 R = t2+t1;
145 if(i>0) {
146 hfsq=0.5*f*f;
147 if(k==0) return f-(hfsq-s*(hfsq+R)); else
148 return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
149 } else {
150 if(k==0) return f-s*(f-R); else
151 return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
152 }
153}
154