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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 | |
6d52618b | 8 | * software is freely granted, provided that this notice |
f7eac6eb RM |
9 | * is preserved. |
10 | * ==================================================== | |
11 | */ | |
923609d1 UD |
12 | /* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25, |
13 | for performance improvement on pipelined processors. | |
14 | */ | |
f7eac6eb RM |
15 | |
16 | #if defined(LIBM_SCCS) && !defined(lint) | |
17 | static char rcsid[] = "$NetBSD: e_log.c,v 1.8 1995/05/10 20:45:49 jtc Exp $"; | |
18 | #endif | |
19 | ||
20 | /* __ieee754_log(x) | |
6d52618b | 21 | * Return the logarithm of x |
f7eac6eb | 22 | * |
6d52618b UD |
23 | * Method : |
24 | * 1. Argument Reduction: find k and f such that | |
25 | * x = 2^k * (1+f), | |
f7eac6eb RM |
26 | * where sqrt(2)/2 < 1+f < sqrt(2) . |
27 | * | |
28 | * 2. Approximation of log(1+f). | |
29 | * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) | |
30 | * = 2s + 2/3 s**3 + 2/5 s**5 + ....., | |
31 | * = 2s + s*R | |
6d52618b UD |
32 | * We use a special Reme algorithm on [0,0.1716] to generate |
33 | * a polynomial of degree 14 to approximate R The maximum error | |
f7eac6eb RM |
34 | * of this polynomial approximation is bounded by 2**-58.45. In |
35 | * other words, | |
36 | * 2 4 6 8 10 12 14 | |
37 | * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s | |
38 | * (the values of Lg1 to Lg7 are listed in the program) | |
39 | * and | |
40 | * | 2 14 | -58.45 | |
6d52618b | 41 | * | Lg1*s +...+Lg7*s - R(z) | <= 2 |
f7eac6eb RM |
42 | * | | |
43 | * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. | |
44 | * In order to guarantee error in log below 1ulp, we compute log | |
45 | * by | |
46 | * log(1+f) = f - s*(f - R) (if f is not too large) | |
47 | * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy) | |
6d52618b UD |
48 | * |
49 | * 3. Finally, log(x) = k*ln2 + log(1+f). | |
f7eac6eb | 50 | * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) |
6d52618b | 51 | * Here ln2 is split into two floating point number: |
f7eac6eb RM |
52 | * ln2_hi + ln2_lo, |
53 | * where n*ln2_hi is always exact for |n| < 2000. | |
54 | * | |
55 | * Special cases: | |
6d52618b | 56 | * log(x) is NaN with signal if x < 0 (including -INF) ; |
f7eac6eb RM |
57 | * log(+INF) is +INF; log(0) is -INF with signal; |
58 | * log(NaN) is that NaN with no signal. | |
59 | * | |
60 | * Accuracy: | |
61 | * according to an error analysis, the error is always less than | |
62 | * 1 ulp (unit in the last place). | |
63 | * | |
64 | * Constants: | |
6d52618b UD |
65 | * The hexadecimal values are the intended ones for the following |
66 | * constants. The decimal values may be used, provided that the | |
67 | * compiler will convert from decimal to binary accurately enough | |
f7eac6eb RM |
68 | * to produce the hexadecimal values shown. |
69 | */ | |
70 | ||
71 | #include "math.h" | |
72 | #include "math_private.h" | |
923609d1 UD |
73 | #define half Lg[8] |
74 | #define two Lg[9] | |
f7eac6eb RM |
75 | #ifdef __STDC__ |
76 | static const double | |
77 | #else | |
78 | static double | |
79 | #endif | |
80 | ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ | |
81 | ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ | |
82 | two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ | |
923609d1 UD |
83 | Lg[] = {0.0, |
84 | 6.666666666666735130e-01, /* 3FE55555 55555593 */ | |
85 | 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ | |
86 | 2.857142874366239149e-01, /* 3FD24924 94229359 */ | |
87 | 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ | |
88 | 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ | |
89 | 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ | |
90 | 1.479819860511658591e-01, /* 3FC2F112 DF3E5244 */ | |
91 | 0.5, | |
92 | 2.0}; | |
f7eac6eb RM |
93 | #ifdef __STDC__ |
94 | static const double zero = 0.0; | |
95 | #else | |
96 | static double zero = 0.0; | |
97 | #endif | |
98 | ||
99 | #ifdef __STDC__ | |
100 | double __ieee754_log(double x) | |
101 | #else | |
102 | double __ieee754_log(x) | |
103 | double x; | |
104 | #endif | |
105 | { | |
48252123 UD |
106 | double hfsq,f,s,z,R,w,dk,t11,t12,t21,t22,w2,zw2; |
107 | #ifdef DO_NOT_USE_THIS | |
108 | double t1,t2; | |
109 | #endif | |
f7eac6eb RM |
110 | int32_t k,hx,i,j; |
111 | u_int32_t lx; | |
112 | ||
113 | EXTRACT_WORDS(hx,lx,x); | |
114 | ||
115 | k=0; | |
116 | if (hx < 0x00100000) { /* x < 2**-1022 */ | |
6d52618b | 117 | if (((hx&0x7fffffff)|lx)==0) |
60c96635 UD |
118 | return -two54/(x-x); /* log(+-0)=-inf */ |
119 | if (hx<0) return (x-x)/(x-x); /* log(-#) = NaN */ | |
f7eac6eb RM |
120 | k -= 54; x *= two54; /* subnormal number, scale up x */ |
121 | GET_HIGH_WORD(hx,x); | |
6d52618b | 122 | } |
f7eac6eb RM |
123 | if (hx >= 0x7ff00000) return x+x; |
124 | k += (hx>>20)-1023; | |
125 | hx &= 0x000fffff; | |
126 | i = (hx+0x95f64)&0x100000; | |
127 | SET_HIGH_WORD(x,hx|(i^0x3ff00000)); /* normalize x or x/2 */ | |
128 | k += (i>>20); | |
129 | f = x-1.0; | |
130 | if((0x000fffff&(2+hx))<3) { /* |f| < 2**-20 */ | |
3bb266e0 UD |
131 | if(f==zero) { |
132 | if(k==0) return zero; else {dk=(double)k; | |
133 | return dk*ln2_hi+dk*ln2_lo;} | |
134 | } | |
923609d1 | 135 | R = f*f*(half-0.33333333333333333*f); |
f7eac6eb RM |
136 | if(k==0) return f-R; else {dk=(double)k; |
137 | return dk*ln2_hi-((R-dk*ln2_lo)-f);} | |
138 | } | |
923609d1 | 139 | s = f/(two+f); |
f7eac6eb RM |
140 | dk = (double)k; |
141 | z = s*s; | |
142 | i = hx-0x6147a; | |
143 | w = z*z; | |
144 | j = 0x6b851-hx; | |
923609d1 | 145 | #ifdef DO_NOT_USE_THIS |
6d52618b UD |
146 | t1= w*(Lg2+w*(Lg4+w*Lg6)); |
147 | t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7))); | |
f7eac6eb | 148 | R = t2+t1; |
923609d1 UD |
149 | #else |
150 | t21 = Lg[5]+w*Lg[7]; w2=w*w; | |
151 | t22 = Lg[1]+w*Lg[3]; zw2=z*w2; | |
152 | t11 = Lg[4]+w*Lg[6]; | |
153 | t12 = w*Lg[2]; | |
154 | R = t12 + w2*t11 + z*t22 + zw2*t21; | |
155 | #endif | |
156 | i |= j; | |
f7eac6eb RM |
157 | if(i>0) { |
158 | hfsq=0.5*f*f; | |
159 | if(k==0) return f-(hfsq-s*(hfsq+R)); else | |
160 | return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f); | |
161 | } else { | |
162 | if(k==0) return f-s*(f-R); else | |
163 | return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f); | |
164 | } | |
165 | } |