1 /* origin: FreeBSD /usr/src/lib/msun/src/e_log.c */
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6 * Developed at SunSoft, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
10 * ====================================================
13 * Return the logrithm of x
16 * 1. Argument Reduction: find k and f such that
18 * where sqrt(2)/2 < 1+f < sqrt(2) .
20 * 2. Approximation of log(1+f).
21 * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
22 * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
24 * We use a special Remez algorithm on [0,0.1716] to generate
25 * a polynomial of degree 14 to approximate R The maximum error
26 * of this polynomial approximation is bounded by 2**-58.45. In
29 * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
30 * (the values of Lg1 to Lg7 are listed in the program)
33 * | Lg1*s +...+Lg7*s - R(z) | <= 2
35 * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
36 * In order to guarantee error in log below 1ulp, we compute log
38 * log(1+f) = f - s*(f - R) (if f is not too large)
39 * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
41 * 3. Finally, log(x) = k*ln2 + log(1+f).
42 * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
43 * Here ln2 is split into two floating point number:
45 * where n*ln2_hi is always exact for |n| < 2000.
48 * log(x) is NaN with signal if x < 0 (including -INF) ;
49 * log(+INF) is +INF; log(0) is -INF with signal;
50 * log(NaN) is that NaN with no signal.
53 * according to an error analysis, the error is always less than
54 * 1 ulp (unit in the last place).
57 * The hexadecimal values are the intended ones for the following
58 * constants. The decimal values may be used, provided that the
59 * compiler will convert from decimal to binary accurately enough
60 * to produce the hexadecimal values shown.
66 ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
67 ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
68 two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
69 Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
70 Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
71 Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
72 Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
73 Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
74 Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
75 Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
79 double hfsq,f,s,z,R,w,t1,t2,dk;
83 EXTRACT_WORDS(hx, lx, x);
86 if (hx < 0x00100000) { /* x < 2**-1022 */
87 if (((hx&0x7fffffff)|lx) == 0)
88 return -two54/0.0; /* log(+-0)=-inf */
90 return (x-x)/0.0; /* log(-#) = NaN */
91 /* subnormal number, scale up x */
100 i = (hx+0x95f64)&0x100000;
101 SET_HIGH_WORD(x, hx|(i^0x3ff00000)); /* normalize x or x/2 */
104 if ((0x000fffff&(2+hx)) < 3) { /* -2**-20 <= f < 2**-20 */
110 return dk*ln2_hi + dk*ln2_lo;
112 R = f*f*(0.5-0.33333333333333333*f);
116 return dk*ln2_hi - ((R-dk*ln2_lo)-f);
124 t1 = w*(Lg2+w*(Lg4+w*Lg6));
125 t2 = z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
131 return f - (hfsq-s*(hfsq+R));
132 return dk*ln2_hi - ((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
136 return dk*ln2_hi - ((s*(f-R)-dk*ln2_lo)-f);