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 | |
83 | static const double |
84 | one = 1.0, |
85 | halF[2] = {0.5,-0.5,}, |
86 | huge = 1.0e+300, |
87 | twom1000= 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/ |
88 | o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */ |
89 | u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */ |
90 | ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */ |
91 | -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */ |
92 | ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */ |
93 | -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */ |
94 | invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */ |
95 | P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ |
96 | P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ |
97 | P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ |
98 | P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ |
99 | P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */ |
100 | |
101 | union { |
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 | |
108 | double __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 |
182 | double 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 |
199 | strong_alias(__ieee754_exp, exp) |
200 | #endif |
201 | libm_hidden_def(exp) |
202 | |