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/* __ieee754_exp(x)
14 * Returns the exponential of x.
15 *
16 * Method
17 * 1. Argument reduction:
18 * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
19 * Given x, find r and integer k such that
20 *
21 * x = k*ln2 + r, |r| <= 0.5*ln2.
22 *
23 * Here r will be represented as r = hi-lo for better
24 * accuracy.
25 *
26 * 2. Approximation of exp(r) by a special rational function on
27 * the interval [0,0.34658]:
28 * Write
29 * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
30 * We use a special Reme algorithm on [0,0.34658] to generate
31 * a polynomial of degree 5 to approximate R. The maximum error
32 * of this polynomial approximation is bounded by 2**-59. In
33 * other words,
34 * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
35 * (where z=r*r, and the values of P1 to P5 are listed below)
36 * and
37 * | 5 | -59
38 * | 2.0+P1*z+...+P5*z - R(z) | <= 2
39 * | |
40 * The computation of exp(r) thus becomes
41 * 2*r
42 * exp(r) = 1 + -------
43 * R - r
44 * r*R1(r)
45 * = 1 + r + ----------- (for better accuracy)
46 * 2 - R1(r)
47 * where
48 * 2 4 10
49 * R1(r) = r - (P1*r + P2*r + ... + P5*r ).
50 *
51 * 3. Scale back to obtain exp(x):
52 * From step 1, we have
53 * exp(x) = 2^k * exp(r)
54 *
55 * Special cases:
56 * exp(INF) is INF, exp(NaN) is NaN;
57 * exp(-INF) is 0, and
58 * for finite argument, only exp(0)=1 is exact.
59 *
60 * Accuracy:
61 * according to an error analysis, the error is always less than
62 * 1 ulp (unit in the last place).
63 *
64 * Misc. info.
65 * For IEEE double
66 * if x > 7.09782712893383973096e+02 then exp(x) overflow
67 * if x < -7.45133219101941108420e+02 then exp(x) underflow
68 *
69 * Constants:
70 * The hexadecimal values are the intended ones for the following
71 * constants. The decimal values may be used, provided that the
72 * compiler will convert from decimal to binary accurately enough
73 * to produce the hexadecimal values shown.
74 */
75
76#include "math_libm.h"
77#include "math_private.h"
78
79#ifdef __WATCOMC__ /* Watcom defines huge=__huge */
80#undef huge
81#endif
82
83static const double
84one = 1.0,
85halF[2] = {0.5,-0.5,},
86huge = 1.0e+300,
87twom1000= 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/
88o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
89u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */
90ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
91 -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
92ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
93 -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
94invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
95P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
96P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
97P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
98P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
99P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
100
101union {
102 Uint64 u64;
103 double d;
104} inf_union = {
105 SDL_UINT64_C(0x7ff0000000000000) /* Binary representation of a 64-bit infinite double (sign=0, exponent=2047, mantissa=0) */
106};
107
108double __ieee754_exp(double x) /* default IEEE double exp */
109{
110 double y;
111 double hi = 0.0;
112 double lo = 0.0;
113 double c;
114 double t;
115 int32_t k=0;
116 int32_t xsb;
117 u_int32_t hx;
118
119 GET_HIGH_WORD(hx,x);
120 xsb = (hx>>31)&1; /* sign bit of x */
121 hx &= 0x7fffffff; /* high word of |x| */
122
123 /* filter out non-finite argument */
124 if(hx >= 0x40862E42) { /* if |x|>=709.78... */
125 if(hx>=0x7ff00000) {
126 u_int32_t lx;
127 GET_LOW_WORD(lx,x);
128 if(((hx&0xfffff)|lx)!=0)
129 return x+x; /* NaN */
130 else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */
131 }
132 #if 1
133 if(x > o_threshold) return inf_union.d; /* overflow */
134 #elif 1
135 if(x > o_threshold) return huge*huge; /* overflow */
136 #else /* !!! FIXME: check this: "huge * huge" is a compiler warning, maybe they wanted +Inf? */
137 if(x > o_threshold) return INFINITY; /* overflow */
138 #endif
139
140 if(x < u_threshold) return twom1000*twom1000; /* underflow */
141 }
142
143 /* argument reduction */
144 if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
145 if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
146 hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
147 } else {
148 k = (int32_t) (invln2*x+halF[xsb]);
149 t = k;
150 hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */
151 lo = t*ln2LO[0];
152 }
153 x = hi - lo;
154 }
155 else if(hx < 0x3e300000) { /* when |x|<2**-28 */
156 if(huge+x>one) return one+x;/* trigger inexact */
157 }
158 else k = 0;
159
160 /* x is now in primary range */
161 t = x*x;
162 c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
163 if(k==0) return one-((x*c)/(c-2.0)-x);
164 else y = one-((lo-(x*c)/(2.0-c))-hi);
165 if(k >= -1021) {
166 u_int32_t hy;
167 GET_HIGH_WORD(hy,y);
168 SET_HIGH_WORD(y,hy+(k<<20)); /* add k to y's exponent */
169 return y;
170 } else {
171 u_int32_t hy;
172 GET_HIGH_WORD(hy,y);
173 SET_HIGH_WORD(y,hy+((k+1000)<<20)); /* add k to y's exponent */
174 return y*twom1000;
175 }
176}
177
178/*
179 * wrapper exp(x)
180 */
181#ifndef _IEEE_LIBM
182double exp(double x)
183{
184 static const double o_threshold = 7.09782712893383973096e+02; /* 0x40862E42, 0xFEFA39EF */
185 static const double u_threshold = -7.45133219101941108420e+02; /* 0xc0874910, 0xD52D3051 */
186
187 double z = __ieee754_exp(x);
188 if (_LIB_VERSION == _IEEE_)
189 return z;
190 if (isfinite(x)) {
191 if (x > o_threshold)
192 return __kernel_standard(x, x, 6); /* exp overflow */
193 if (x < u_threshold)
194 return __kernel_standard(x, x, 7); /* exp underflow */
195 }
196 return z;
197}
198#else
199strong_alias(__ieee754_exp, exp)
200#endif
201libm_hidden_def(exp)
202