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 logarithm 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.
67 ln2_hi
= 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
68 ln2_lo
= 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
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 union {double f
; uint64_t i
;} u
= {x
};
80 double_t hfsq
,f
,s
,z
,R
,w
,t1
,t2
,dk
;
86 if (hx
< 0x00100000 || hx
>>31) {
88 return -1/(x
*x
); /* log(+-0)=-inf */
90 return (x
-x
)/0.0; /* log(-#) = NaN */
91 /* subnormal number, scale x up */
96 } else if (hx
>= 0x7ff00000) {
98 } else if (hx
== 0x3ff00000 && u
.i
<<32 == 0)
101 /* reduce x into [sqrt(2)/2, sqrt(2)] */
102 hx
+= 0x3ff00000 - 0x3fe6a09e;
103 k
+= (int)(hx
>>20) - 0x3ff;
104 hx
= (hx
&0x000fffff) + 0x3fe6a09e;
105 u
.i
= (uint64_t)hx
<<32 | (u
.i
&0xffffffff);
113 t1
= w
*(Lg2
+w
*(Lg4
+w
*Lg6
));
114 t2
= z
*(Lg1
+w
*(Lg3
+w
*(Lg5
+w
*Lg7
)));
117 return s
*(hfsq
+R
) + dk
*ln2_lo
- hfsq
+ f
+ dk
*ln2_hi
;