1*f504f610SAugustin Cavalier /* origin: FreeBSD /usr/src/lib/msun/src/e_log10.c */
2*f504f610SAugustin Cavalier /*
3*f504f610SAugustin Cavalier * ====================================================
4*f504f610SAugustin Cavalier * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5*f504f610SAugustin Cavalier *
6*f504f610SAugustin Cavalier * Developed at SunSoft, a Sun Microsystems, Inc. business.
7*f504f610SAugustin Cavalier * Permission to use, copy, modify, and distribute this
8*f504f610SAugustin Cavalier * software is freely granted, provided that this notice
9*f504f610SAugustin Cavalier * is preserved.
10*f504f610SAugustin Cavalier * ====================================================
11*f504f610SAugustin Cavalier */
12*f504f610SAugustin Cavalier /*
13*f504f610SAugustin Cavalier * Return the base 10 logarithm of x. See log.c for most comments.
14*f504f610SAugustin Cavalier *
15*f504f610SAugustin Cavalier * Reduce x to 2^k (1+f) and calculate r = log(1+f) - f + f*f/2
16*f504f610SAugustin Cavalier * as in log.c, then combine and scale in extra precision:
17*f504f610SAugustin Cavalier * log10(x) = (f - f*f/2 + r)/log(10) + k*log10(2)
18*f504f610SAugustin Cavalier */
19*f504f610SAugustin Cavalier
20*f504f610SAugustin Cavalier #include <math.h>
21*f504f610SAugustin Cavalier #include <stdint.h>
22*f504f610SAugustin Cavalier
23*f504f610SAugustin Cavalier static const double
24*f504f610SAugustin Cavalier ivln10hi = 4.34294481878168880939e-01, /* 0x3fdbcb7b, 0x15200000 */
25*f504f610SAugustin Cavalier ivln10lo = 2.50829467116452752298e-11, /* 0x3dbb9438, 0xca9aadd5 */
26*f504f610SAugustin Cavalier log10_2hi = 3.01029995663611771306e-01, /* 0x3FD34413, 0x509F6000 */
27*f504f610SAugustin Cavalier log10_2lo = 3.69423907715893078616e-13, /* 0x3D59FEF3, 0x11F12B36 */
28*f504f610SAugustin Cavalier Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
29*f504f610SAugustin Cavalier Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
30*f504f610SAugustin Cavalier Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
31*f504f610SAugustin Cavalier Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
32*f504f610SAugustin Cavalier Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
33*f504f610SAugustin Cavalier Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
34*f504f610SAugustin Cavalier Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
35*f504f610SAugustin Cavalier
log10(double x)36*f504f610SAugustin Cavalier double log10(double x)
37*f504f610SAugustin Cavalier {
38*f504f610SAugustin Cavalier union {double f; uint64_t i;} u = {x};
39*f504f610SAugustin Cavalier double_t hfsq,f,s,z,R,w,t1,t2,dk,y,hi,lo,val_hi,val_lo;
40*f504f610SAugustin Cavalier uint32_t hx;
41*f504f610SAugustin Cavalier int k;
42*f504f610SAugustin Cavalier
43*f504f610SAugustin Cavalier hx = u.i>>32;
44*f504f610SAugustin Cavalier k = 0;
45*f504f610SAugustin Cavalier if (hx < 0x00100000 || hx>>31) {
46*f504f610SAugustin Cavalier if (u.i<<1 == 0)
47*f504f610SAugustin Cavalier return -1/(x*x); /* log(+-0)=-inf */
48*f504f610SAugustin Cavalier if (hx>>31)
49*f504f610SAugustin Cavalier return (x-x)/0.0; /* log(-#) = NaN */
50*f504f610SAugustin Cavalier /* subnormal number, scale x up */
51*f504f610SAugustin Cavalier k -= 54;
52*f504f610SAugustin Cavalier x *= 0x1p54;
53*f504f610SAugustin Cavalier u.f = x;
54*f504f610SAugustin Cavalier hx = u.i>>32;
55*f504f610SAugustin Cavalier } else if (hx >= 0x7ff00000) {
56*f504f610SAugustin Cavalier return x;
57*f504f610SAugustin Cavalier } else if (hx == 0x3ff00000 && u.i<<32 == 0)
58*f504f610SAugustin Cavalier return 0;
59*f504f610SAugustin Cavalier
60*f504f610SAugustin Cavalier /* reduce x into [sqrt(2)/2, sqrt(2)] */
61*f504f610SAugustin Cavalier hx += 0x3ff00000 - 0x3fe6a09e;
62*f504f610SAugustin Cavalier k += (int)(hx>>20) - 0x3ff;
63*f504f610SAugustin Cavalier hx = (hx&0x000fffff) + 0x3fe6a09e;
64*f504f610SAugustin Cavalier u.i = (uint64_t)hx<<32 | (u.i&0xffffffff);
65*f504f610SAugustin Cavalier x = u.f;
66*f504f610SAugustin Cavalier
67*f504f610SAugustin Cavalier f = x - 1.0;
68*f504f610SAugustin Cavalier hfsq = 0.5*f*f;
69*f504f610SAugustin Cavalier s = f/(2.0+f);
70*f504f610SAugustin Cavalier z = s*s;
71*f504f610SAugustin Cavalier w = z*z;
72*f504f610SAugustin Cavalier t1 = w*(Lg2+w*(Lg4+w*Lg6));
73*f504f610SAugustin Cavalier t2 = z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
74*f504f610SAugustin Cavalier R = t2 + t1;
75*f504f610SAugustin Cavalier
76*f504f610SAugustin Cavalier /* See log2.c for details. */
77*f504f610SAugustin Cavalier /* hi+lo = f - hfsq + s*(hfsq+R) ~ log(1+f) */
78*f504f610SAugustin Cavalier hi = f - hfsq;
79*f504f610SAugustin Cavalier u.f = hi;
80*f504f610SAugustin Cavalier u.i &= (uint64_t)-1<<32;
81*f504f610SAugustin Cavalier hi = u.f;
82*f504f610SAugustin Cavalier lo = f - hi - hfsq + s*(hfsq+R);
83*f504f610SAugustin Cavalier
84*f504f610SAugustin Cavalier /* val_hi+val_lo ~ log10(1+f) + k*log10(2) */
85*f504f610SAugustin Cavalier val_hi = hi*ivln10hi;
86*f504f610SAugustin Cavalier dk = k;
87*f504f610SAugustin Cavalier y = dk*log10_2hi;
88*f504f610SAugustin Cavalier val_lo = dk*log10_2lo + (lo+hi)*ivln10lo + lo*ivln10hi;
89*f504f610SAugustin Cavalier
90*f504f610SAugustin Cavalier /*
91*f504f610SAugustin Cavalier * Extra precision in for adding y is not strictly needed
92*f504f610SAugustin Cavalier * since there is no very large cancellation near x = sqrt(2) or
93*f504f610SAugustin Cavalier * x = 1/sqrt(2), but we do it anyway since it costs little on CPUs
94*f504f610SAugustin Cavalier * with some parallelism and it reduces the error for many args.
95*f504f610SAugustin Cavalier */
96*f504f610SAugustin Cavalier w = y + val_hi;
97*f504f610SAugustin Cavalier val_lo += (y - w) + val_hi;
98*f504f610SAugustin Cavalier val_hi = w;
99*f504f610SAugustin Cavalier
100*f504f610SAugustin Cavalier return val_lo + val_hi;
101*f504f610SAugustin Cavalier }
102