1#include "SDL_internal.h"
2/*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunPro, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
9 * is preserved.
10 * ====================================================
11 */
12
13#if defined(_MSC_VER) /* Handle Microsoft VC++ compiler specifics. */
14/* C4723: potential divide by zero. */
15#pragma warning ( disable : 4723 )
16#endif
17
18/* __ieee754_log(x)
19 * Return the logrithm of x
20 *
21 * Method :
22 * 1. Argument Reduction: find k and f such that
23 * x = 2^k * (1+f),
24 * where sqrt(2)/2 < 1+f < sqrt(2) .
25 *
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 + .....,
29 * = 2s + s*R
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
33 * other words,
34 * 2 4 6 8 10 12 14
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)
37 * and
38 * | 2 14 | -58.45
39 * | Lg1*s +...+Lg7*s - R(z) | <= 2
40 * | |
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
43 * by
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)
46 *
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:
50 * ln2_hi + ln2_lo,
51 * where n*ln2_hi is always exact for |n| < 2000.
52 *
53 * Special cases:
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.
57 *
58 * Accuracy:
59 * according to an error analysis, the error is always less than
60 * 1 ulp (unit in the last place).
61 *
62 * Constants:
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.
67 */
68
69#include "math_libm.h"
70#include "math_private.h"
71
72static const double
73ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
74ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
75two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
76Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
77Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
78Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
79Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
80Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
81Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
82Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
83
84static const double zero = 0.0;
85
86double attribute_hidden __ieee754_log(double x)
87{
88 double hfsq,f,s,z,R,w,t1,t2,dk;
89 int32_t k,hx,i,j;
90 u_int32_t lx;
91
92 EXTRACT_WORDS(hx,lx,x);
93
94 k=0;
95 if (hx < 0x00100000) { /* x < 2**-1022 */
96 if (((hx&0x7fffffff)|lx)==0)
97 return -two54/zero; /* log(+-0)=-inf */
98 if (hx<0) return (x-x)/zero; /* log(-#) = NaN */
99 k -= 54; x *= two54; /* subnormal number, scale up x */
100 GET_HIGH_WORD(hx,x);
101 }
102 if (hx >= 0x7ff00000) return x+x;
103 k += (hx>>20)-1023;
104 hx &= 0x000fffff;
105 i = (hx+0x95f64)&0x100000;
106 SET_HIGH_WORD(x,hx|(i^0x3ff00000)); /* normalize x or x/2 */
107 k += (i>>20);
108 f = x-1.0;
109 if((0x000fffff&(2+hx))<3) { /* |f| < 2**-20 */
110 if(f==zero) {if(k==0) return zero; else {dk=(double)k;
111 return dk*ln2_hi+dk*ln2_lo;}
112 }
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);}
116 }
117 s = f/(2.0+f);
118 dk = (double)k;
119 z = s*s;
120 i = hx-0x6147a;
121 w = z*z;
122 j = 0x6b851-hx;
123 t1= w*(Lg2+w*(Lg4+w*Lg6));
124 t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
125 i |= j;
126 R = t2+t1;
127 if(i>0) {
128 hfsq=0.5*f*f;
129 if(k==0) return f-(hfsq-s*(hfsq+R)); else
130 return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
131 } else {
132 if(k==0) return f-s*(f-R); else
133 return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
134 }
135}
136
137/*
138 * wrapper log(x)
139 */
140#ifndef _IEEE_LIBM
141double log(double x)
142{
143 double z = __ieee754_log(x);
144 if (_LIB_VERSION == _IEEE_ || isnan(x) || x > 0.0)
145 return z;
146 if (x == 0.0)
147 return __kernel_standard(x, x, 16); /* log(0) */
148 return __kernel_standard(x, x, 17); /* log(x<0) */
149}
150#else
151strong_alias(__ieee754_log, log)
152#endif
153libm_hidden_def(log)
154