1 /*
2 * ====================================================
3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
4 *
5 * Developed at SunPro, a Sun Microsystems, Inc. business.
6 * Permission to use, copy, modify, and distribute this
7 * software is freely granted, provided that this notice
8 * is preserved.
9 * ====================================================
10 */
11
12 /* __ieee754_log(x)
13 * Return the logrithm of x
14 *
15 * Method :
16 * 1. Argument Reduction: find k and f such that
17 * x = 2^k * (1+f),
18 * where sqrt(2)/2 < 1+f < sqrt(2) .
19 *
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 + .....,
23 * = 2s + s*R
24 * We use a special Reme 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
27 * other words,
28 * 2 4 6 8 10 12 14
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)
31 * and
32 * | 2 14 | -58.45
33 * | Lg1*s +...+Lg7*s - R(z) | <= 2
34 * | |
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
37 * by
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)
40 *
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:
44 * ln2_hi + ln2_lo,
45 * where n*ln2_hi is always exact for |n| < 2000.
46 *
47 * Special cases:
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.
51 *
52 * Accuracy:
53 * according to an error analysis, the error is always less than
54 * 1 ulp (unit in the last place).
55 *
56 * Constants:
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.
61 */
62
63 #include "math.h"
64 #include "math_private.h"
65
66 static const double
67 ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
68 ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
69 two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
70 Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
71 Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
72 Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
73 Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
74 Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
75 Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
76 Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
77
78 static const double zero = 0.0;
79
__ieee754_log(double x)80 double __ieee754_log(double x)
81 {
82 double hfsq,f,s,z,R,w,t1,t2,dk;
83 int32_t k,hx,i,j;
84 u_int32_t lx;
85
86 EXTRACT_WORDS(hx,lx,x);
87
88 k=0;
89 if (hx < 0x00100000) { /* x < 2**-1022 */
90 if (((hx&0x7fffffff)|lx)==0)
91 return -two54/zero; /* log(+-0)=-inf */
92 if (hx<0) return (x-x)/zero; /* log(-#) = NaN */
93 k -= 54; x *= two54; /* subnormal number, scale up x */
94 GET_HIGH_WORD(hx,x);
95 }
96 if (hx >= 0x7ff00000) return x+x;
97 k += (hx>>20)-1023;
98 hx &= 0x000fffff;
99 i = (hx+0x95f64)&0x100000;
100 SET_HIGH_WORD(x,hx|(i^0x3ff00000)); /* normalize x or x/2 */
101 k += (i>>20);
102 f = x-1.0;
103 if((0x000fffff&(2+hx))<3) { /* |f| < 2**-20 */
104 if(f==zero) {if(k==0) return zero; else {dk=(double)k;
105 return dk*ln2_hi+dk*ln2_lo;}
106 }
107 R = f*f*(0.5-0.33333333333333333*f);
108 if(k==0) return f-R; else {dk=(double)k;
109 return dk*ln2_hi-((R-dk*ln2_lo)-f);}
110 }
111 s = f/(2.0+f);
112 dk = (double)k;
113 z = s*s;
114 i = hx-0x6147a;
115 w = z*z;
116 j = 0x6b851-hx;
117 t1= w*(Lg2+w*(Lg4+w*Lg6));
118 t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
119 i |= j;
120 R = t2+t1;
121 if(i>0) {
122 hfsq=0.5*f*f;
123 if(k==0) return f-(hfsq-s*(hfsq+R)); else
124 return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
125 } else {
126 if(k==0) return f-s*(f-R); else
127 return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
128 }
129 }
130
131 /*
132 * wrapper log(x)
133 */
134 #ifndef _IEEE_LIBM
log(double x)135 double log(double x)
136 {
137 double z = __ieee754_log(x);
138 if (_LIB_VERSION == _IEEE_ || isnan(x) || x > 0.0)
139 return z;
140 if (x == 0.0)
141 return __kernel_standard(x, x, 16); /* log(0) */
142 return __kernel_standard(x, x, 17); /* log(x<0) */
143 }
144 #else
145 strong_alias(__ieee754_log, log)
146 #endif
147 libm_hidden_def(log)
148