2 /* @(#)e_log.c 1.3 95/01/18 */
4 * ====================================================
5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
7 * Developed at SunSoft, a Sun Microsystems, Inc. business.
8 * Permission to use, copy, modify, and distribute this
9 * software is freely granted, provided that this notice
11 * ====================================================
15 static char rcsid[] = "$FreeBSD$";
19 * Return the logrithm of x
22 * 1. Argument Reduction: find k and f such that
24 * where sqrt(2)/2 < 1+f < sqrt(2) .
26 * 2. Approximation of log(1+f).
27 * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
28 * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
30 * We use a special Reme algorithm on [0,0.1716] to generate
31 * a polynomial of degree 14 to approximate R The maximum error
32 * of this polynomial approximation is bounded by 2**-58.45. In
35 * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
36 * (the values of Lg1 to Lg7 are listed in the program)
39 * | Lg1*s +...+Lg7*s - R(z) | <= 2
41 * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
42 * In order to guarantee error in log below 1ulp, we compute log
44 * log(1+f) = f - s*(f - R) (if f is not too large)
45 * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
47 * 3. Finally, log(x) = k*ln2 + log(1+f).
48 * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
49 * Here ln2 is split into two floating point number:
51 * where n*ln2_hi is always exact for |n| < 2000.
54 * log(x) is NaN with signal if x < 0 (including -INF) ;
55 * log(+INF) is +INF; log(0) is -INF with signal;
56 * log(NaN) is that NaN with no signal.
59 * according to an error analysis, the error is always less than
60 * 1 ulp (unit in the last place).
63 * The hexadecimal values are the intended ones for the following
64 * constants. The decimal values may be used, provided that the
65 * compiler will convert from decimal to binary accurately enough
66 * to produce the hexadecimal values shown.
70 #include "math_private.h"
73 ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
74 ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
75 two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
76 Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
77 Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
78 Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
79 Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
80 Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
81 Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
82 Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
84 static const double zero = 0.0;
87 __ieee754_log(double x)
89 double hfsq,f,s,z,R,w,t1,t2,dk;
93 EXTRACT_WORDS(hx,lx,x);
96 if (hx < 0x00100000) { /* x < 2**-1022 */
97 if (((hx&0x7fffffff)|lx)==0)
98 return -two54/zero; /* log(+-0)=-inf */
99 if (hx<0) return (x-x)/zero; /* log(-#) = NaN */
100 k -= 54; x *= two54; /* subnormal number, scale up x */
103 if (hx >= 0x7ff00000) return x+x;
106 i = (hx+0x95f64)&0x100000;
107 SET_HIGH_WORD(x,hx|(i^0x3ff00000)); /* normalize x or x/2 */
110 if((0x000fffff&(2+hx))<3) { /* -2**-20 <= f < 2**-20 */
111 if(f==zero) if(k==0) return zero; else {dk=(double)k;
112 return dk*ln2_hi+dk*ln2_lo;}
113 R = f*f*(0.5-0.33333333333333333*f);
114 if(k==0) return f-R; else {dk=(double)k;
115 return dk*ln2_hi-((R-dk*ln2_lo)-f);}
123 t1= w*(Lg2+w*(Lg4+w*Lg6));
124 t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
129 if(k==0) return f-(hfsq-s*(hfsq+R)); else
130 return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
132 if(k==0) return f-s*(f-R); else
133 return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);