1
2 /* @(#)e_log.c 5.1 93/09/24 */
3 /*
4 * ====================================================
5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6 *
7 * Developed at SunPro, a Sun Microsystems, Inc. business.
8 * Permission to use, copy, modify, and distribute this
9 * software is freely granted, provided that this notice
10 * is preserved.
11 * ====================================================
12 */
13
14 /* log(x)
15 * Return the logrithm of x
16 *
17 * Method :
18 * 1. Argument Reduction: find k and f such that
19 * x = 2^k * (1+f),
20 * where sqrt(2)/2 < 1+f < sqrt(2) .
21 *
22 * 2. Approximation of log(1+f).
23 * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
24 * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
25 * = 2s + s*R
26 * We use a special Reme algorithm on [0,0.1716] to generate
27 * a polynomial of degree 14 to approximate R The maximum error
28 * of this polynomial approximation is bounded by 2**-58.45. In
29 * other words,
30 * 2 4 6 8 10 12 14
31 * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
32 * (the values of Lg1 to Lg7 are listed in the program)
33 * and
34 * | 2 14 | -58.45
35 * | Lg1*s +...+Lg7*s - R(z) | <= 2
36 * | |
37 * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
38 * In order to guarantee error in log below 1ulp, we compute log
39 * by
40 * log(1+f) = f - s*(f - R) (if f is not too large)
41 * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
42 *
43 * 3. Finally, log(x) = k*ln2 + log(1+f).
44 * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
45 * Here ln2 is split into two floating point number:
46 * ln2_hi + ln2_lo,
47 * where n*ln2_hi is always exact for |n| < 2000.
48 *
49 * Special cases:
50 * log(x) is NaN with signal if x < 0 (including -INF) ;
51 * log(+INF) is +INF; log(0) is -INF with signal;
52 * log(NaN) is that NaN with no signal.
53 *
54 * Accuracy:
55 * according to an error analysis, the error is always less than
56 * 1 ulp (unit in the last place).
57 *
58 * Constants:
59 * The hexadecimal values are the intended ones for the following
60 * constants. The decimal values may be used, provided that the
61 * compiler will convert from decimal to binary accurately enough
62 * to produce the hexadecimal values shown.
63 */
64
65 #include "fdlibm.h"
66 #if __OBSOLETE_MATH_DOUBLE
67
68 #ifdef _NEED_FLOAT64
69
70 static const __float64 ln2_hi = _F_64(6.93147180369123816490e-01), /* 3fe62e42 fee00000 */
71 ln2_lo = _F_64(1.90821492927058770002e-10), /* 3dea39ef 35793c76 */
72 two54 = _F_64(1.80143985094819840000e+16), /* 43500000 00000000 */
73 Lg1 = _F_64(6.666666666666735130e-01), /* 3FE55555 55555593 */
74 Lg2 = _F_64(3.999999999940941908e-01), /* 3FD99999 9997FA04 */
75 Lg3 = _F_64(2.857142874366239149e-01), /* 3FD24924 94229359 */
76 Lg4 = _F_64(2.222219843214978396e-01), /* 3FCC71C5 1D8E78AF */
77 Lg5 = _F_64(1.818357216161805012e-01), /* 3FC74664 96CB03DE */
78 Lg6 = _F_64(1.531383769920937332e-01), /* 3FC39A09 D078C69F */
79 Lg7 = _F_64(1.479819860511658591e-01); /* 3FC2F112 DF3E5244 */
80
81 static const __float64 zero = _F_64(0.0);
82
83 __float64
log64(__float64 x)84 log64(__float64 x)
85 {
86 __float64 hfsq, f, s, z, R, w, t1, t2, dk;
87 __int32_t k, hx, i, j;
88 __uint32_t lx;
89
90 EXTRACT_WORDS(hx, lx, x);
91
92 k = 0;
93 if (hx < 0x00100000) { /* x < 2**-1022 */
94 if (((hx & 0x7fffffff) | lx) == 0)
95 return __math_divzero(1); /* log(+-0)=-inf */
96 if (hx < 0)
97 return __math_invalid(x); /* log(-#) = NaN */
98 k -= 54;
99 x *= two54; /* subnormal number, scale up x */
100 GET_HIGH_WORD(hx, x);
101 }
102 if (hx >= 0x7ff00000)
103 return x + x;
104 k += (hx >> 20) - 1023;
105 hx &= 0x000fffff;
106 i = (hx + 0x95f64) & 0x100000;
107 SET_HIGH_WORD(x, hx | (i ^ 0x3ff00000)); /* normalize x or x/2 */
108 k += (i >> 20);
109 f = x - _F_64(1.0);
110 if ((0x000fffff & (2 + hx)) < 3) { /* |f| < 2**-20 */
111 if (f == zero) {
112 if (k == 0)
113 return zero;
114 else {
115 dk = (__float64)k;
116 return dk * ln2_hi + dk * ln2_lo;
117 }
118 }
119 R = f * f * (_F_64(0.5) - _F_64(0.33333333333333333) * f);
120 if (k == 0)
121 return f - R;
122 else {
123 dk = (__float64)k;
124 return dk * ln2_hi - ((R - dk * ln2_lo) - f);
125 }
126 }
127 s = f / (_F_64(2.0) + f);
128 dk = (__float64)k;
129 z = s * s;
130 i = hx - 0x6147a;
131 w = z * z;
132 j = 0x6b851 - hx;
133 t1 = w * (Lg2 + w * (Lg4 + w * Lg6));
134 t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7)));
135 i |= j;
136 R = t2 + t1;
137 if (i > 0) {
138 hfsq = _F_64(0.5) * f * f;
139 if (k == 0)
140 return f - (hfsq - s * (hfsq + R));
141 else
142 return dk * ln2_hi - ((hfsq - (s * (hfsq + R) + dk * ln2_lo)) - f);
143 } else {
144 if (k == 0)
145 return f - s * (f - R);
146 else
147 return dk * ln2_hi - ((s * (f - R) - dk * ln2_lo) - f);
148 }
149 }
150
151 _MATH_ALIAS_d_d(log)
152
153 #endif /* _NEED_FLOAT64 */
154 #else
155 #include "../common/log.c"
156 #endif /*__OBSOLETE_MATH_DOUBLE */
157
