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_pow(x,y) return x**y
14 *
15 * n
16 * Method: Let x = 2 * (1+f)
17 * 1. Compute and return log2(x) in two pieces:
18 * log2(x) = w1 + w2,
19 * where w1 has 53-24 = 29 bit trailing zeros.
20 * 2. Perform y*log2(x) = n+y' by simulating muti-precision
21 * arithmetic, where |y'|<=0.5.
22 * 3. Return x**y = 2**n*exp(y'*log2)
23 *
24 * Special cases:
25 * 1. +-1 ** anything is 1.0
26 * 2. +-1 ** +-INF is 1.0
27 * 3. (anything) ** 0 is 1
28 * 4. (anything) ** 1 is itself
29 * 5. (anything) ** NAN is NAN
30 * 6. NAN ** (anything except 0) is NAN
31 * 7. +-(|x| > 1) ** +INF is +INF
32 * 8. +-(|x| > 1) ** -INF is +0
33 * 9. +-(|x| < 1) ** +INF is +0
34 * 10 +-(|x| < 1) ** -INF is +INF
35 * 11. +0 ** (+anything except 0, NAN) is +0
36 * 12. -0 ** (+anything except 0, NAN, odd integer) is +0
37 * 13. +0 ** (-anything except 0, NAN) is +INF
38 * 14. -0 ** (-anything except 0, NAN, odd integer) is +INF
39 * 15. -0 ** (odd integer) = -( +0 ** (odd integer) )
40 * 16. +INF ** (+anything except 0,NAN) is +INF
41 * 17. +INF ** (-anything except 0,NAN) is +0
42 * 18. -INF ** (anything) = -0 ** (-anything)
43 * 19. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
44 * 20. (-anything except 0 and inf) ** (non-integer) is NAN
45 *
46 * Accuracy:
47 * pow(x,y) returns x**y nearly rounded. In particular
48 * pow(integer,integer)
49 * always returns the correct integer provided it is
50 * representable.
51 *
52 * Constants :
53 * The hexadecimal values are the intended ones for the following
54 * constants. The decimal values may be used, provided that the
55 * compiler will convert from decimal to binary accurately enough
56 * to produce the hexadecimal values shown.
57 */
58
59#include "math_libm.h"
60#include "math_private.h"
61
62#if defined(_MSC_VER) /* Handle Microsoft VC++ compiler specifics. */
63/* C4756: overflow in constant arithmetic */
64#pragma warning ( disable : 4756 )
65#endif
66
67#ifdef __WATCOMC__ /* Watcom defines huge=__huge */
68#undef huge
69#endif
70
71static const double
72bp[] = {1.0, 1.5,},
73dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
74dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
75zero = 0.0,
76one = 1.0,
77two = 2.0,
78two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */
79huge = 1.0e300,
80tiny = 1.0e-300,
81 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
82L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
83L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
84L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
85L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
86L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
87L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
88P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
89P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
90P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
91P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
92P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
93lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
94lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
95lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
96ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
97cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
98cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
99cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
100ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
101ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
102ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
103
104double attribute_hidden __ieee754_pow(double x, double y)
105{
106 double z,ax,z_h,z_l,p_h,p_l;
107 double y1,t1,t2,r,s,t,u,v,w;
108 int32_t i,j,k,yisint,n;
109 int32_t hx,hy,ix,iy;
110 u_int32_t lx,ly;
111
112 EXTRACT_WORDS(hx,lx,x);
113 /* x==1: 1**y = 1 (even if y is NaN) */
114 if (hx==0x3ff00000 && lx==0) {
115 return x;
116 }
117 ix = hx&0x7fffffff;
118
119 EXTRACT_WORDS(hy,ly,y);
120 iy = hy&0x7fffffff;
121
122 /* y==zero: x**0 = 1 */
123 if((iy|ly)==0) return one;
124
125 /* +-NaN return x+y */
126 if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
127 iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
128 return x+y;
129
130 /* determine if y is an odd int when x < 0
131 * yisint = 0 ... y is not an integer
132 * yisint = 1 ... y is an odd int
133 * yisint = 2 ... y is an even int
134 */
135 yisint = 0;
136 if(hx<0) {
137 if(iy>=0x43400000) yisint = 2; /* even integer y */
138 else if(iy>=0x3ff00000) {
139 k = (iy>>20)-0x3ff; /* exponent */
140 if(k>20) {
141 j = ly>>(52-k);
142 if(((u_int32_t)j<<(52-k))==ly) yisint = 2-(j&1);
143 } else if(ly==0) {
144 j = iy>>(20-k);
145 if((j<<(20-k))==iy) yisint = 2-(j&1);
146 }
147 }
148 }
149
150 /* special value of y */
151 if(ly==0) {
152 if (iy==0x7ff00000) { /* y is +-inf */
153 if (((ix-0x3ff00000)|lx)==0)
154 return one; /* +-1**+-inf is 1 (yes, weird rule) */
155 if (ix >= 0x3ff00000) /* (|x|>1)**+-inf = inf,0 */
156 return (hy>=0) ? y : zero;
157 /* (|x|<1)**-,+inf = inf,0 */
158 return (hy<0) ? -y : zero;
159 }
160 if(iy==0x3ff00000) { /* y is +-1 */
161 if(hy<0) return one/x; else return x;
162 }
163 if(hy==0x40000000) return x*x; /* y is 2 */
164 if(hy==0x3fe00000) { /* y is 0.5 */
165 if(hx>=0) /* x >= +0 */
166 return __ieee754_sqrt(x);
167 }
168 }
169
170 ax = fabs(x);
171 /* special value of x */
172 if(lx==0) {
173 if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
174 z = ax; /*x is +-0,+-inf,+-1*/
175 if(hy<0) z = one/z; /* z = (1/|x|) */
176 if(hx<0) {
177 if(((ix-0x3ff00000)|yisint)==0) {
178 z = (z-z)/(z-z); /* (-1)**non-int is NaN */
179 } else if(yisint==1)
180 z = -z; /* (x<0)**odd = -(|x|**odd) */
181 }
182 return z;
183 }
184 }
185
186 /* (x<0)**(non-int) is NaN */
187 if(((((u_int32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x);
188
189 /* |y| is huge */
190 if(iy>0x41e00000) { /* if |y| > 2**31 */
191 if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */
192 if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
193 if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
194 }
195 /* over/underflow if x is not close to one */
196 if(ix<0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
197 if(ix>0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
198 /* now |1-x| is tiny <= 2**-20, suffice to compute
199 log(x) by x-x^2/2+x^3/3-x^4/4 */
200 t = x-1; /* t has 20 trailing zeros */
201 w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
202 u = ivln2_h*t; /* ivln2_h has 21 sig. bits */
203 v = t*ivln2_l-w*ivln2;
204 t1 = u+v;
205 SET_LOW_WORD(t1,0);
206 t2 = v-(t1-u);
207 } else {
208 double s2,s_h,s_l,t_h,t_l;
209 n = 0;
210 /* take care subnormal number */
211 if(ix<0x00100000)
212 {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); }
213 n += ((ix)>>20)-0x3ff;
214 j = ix&0x000fffff;
215 /* determine interval */
216 ix = j|0x3ff00000; /* normalize ix */
217 if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */
218 else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */
219 else {k=0;n+=1;ix -= 0x00100000;}
220 SET_HIGH_WORD(ax,ix);
221
222 /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
223 u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
224 v = one/(ax+bp[k]);
225 s = u*v;
226 s_h = s;
227 SET_LOW_WORD(s_h,0);
228 /* t_h=ax+bp[k] High */
229 t_h = zero;
230 SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18));
231 t_l = ax - (t_h-bp[k]);
232 s_l = v*((u-s_h*t_h)-s_h*t_l);
233 /* compute log(ax) */
234 s2 = s*s;
235 r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
236 r += s_l*(s_h+s);
237 s2 = s_h*s_h;
238 t_h = 3.0+s2+r;
239 SET_LOW_WORD(t_h,0);
240 t_l = r-((t_h-3.0)-s2);
241 /* u+v = s*(1+...) */
242 u = s_h*t_h;
243 v = s_l*t_h+t_l*s;
244 /* 2/(3log2)*(s+...) */
245 p_h = u+v;
246 SET_LOW_WORD(p_h,0);
247 p_l = v-(p_h-u);
248 z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
249 z_l = cp_l*p_h+p_l*cp+dp_l[k];
250 /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
251 t = (double)n;
252 t1 = (((z_h+z_l)+dp_h[k])+t);
253 SET_LOW_WORD(t1,0);
254 t2 = z_l-(((t1-t)-dp_h[k])-z_h);
255 }
256
257 s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
258 if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0)
259 s = -one;/* (-ve)**(odd int) */
260
261 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
262 y1 = y;
263 SET_LOW_WORD(y1,0);
264 p_l = (y-y1)*t1+y*t2;
265 p_h = y1*t1;
266 z = p_l+p_h;
267 EXTRACT_WORDS(j,i,z);
268 if (j>=0x40900000) { /* z >= 1024 */
269 if(((j-0x40900000)|i)!=0) /* if z > 1024 */
270 return s*huge*huge; /* overflow */
271 else {
272 if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */
273 }
274 } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */
275 if(((j-0xc090cc00)|i)!=0) /* z < -1075 */
276 return s*tiny*tiny; /* underflow */
277 else {
278 if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */
279 }
280 }
281 /*
282 * compute 2**(p_h+p_l)
283 */
284 i = j&0x7fffffff;
285 k = (i>>20)-0x3ff;
286 n = 0;
287 if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */
288 n = j+(0x00100000>>(k+1));
289 k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */
290 t = zero;
291 SET_HIGH_WORD(t,n&~(0x000fffff>>k));
292 n = ((n&0x000fffff)|0x00100000)>>(20-k);
293 if(j<0) n = -n;
294 p_h -= t;
295 }
296 t = p_l+p_h;
297 SET_LOW_WORD(t,0);
298 u = t*lg2_h;
299 v = (p_l-(t-p_h))*lg2+t*lg2_l;
300 z = u+v;
301 w = v-(z-u);
302 t = z*z;
303 t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
304 r = (z*t1)/(t1-two)-(w+z*w);
305 z = one-(r-z);
306 GET_HIGH_WORD(j,z);
307 j += (n<<20);
308 if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */
309 else SET_HIGH_WORD(z,j);
310 return s*z;
311}
312
313/*
314 * wrapper pow(x,y) return x**y
315 */
316#ifndef _IEEE_LIBM
317double pow(double x, double y)
318{
319 double z = __ieee754_pow(x, y);
320 if (_LIB_VERSION == _IEEE_|| isnan(y))
321 return z;
322 if (isnan(x)) {
323 if (y == 0.0)
324 return __kernel_standard(x, y, 42); /* pow(NaN,0.0) */
325 return z;
326 }
327 if (x == 0.0) {
328 if (y == 0.0)
329 return __kernel_standard(x, y, 20); /* pow(0.0,0.0) */
330 if (isfinite(y) && y < 0.0)
331 return __kernel_standard(x,y,23); /* pow(0.0,negative) */
332 return z;
333 }
334 if (!isfinite(z)) {
335 if (isfinite(x) && isfinite(y)) {
336 if (isnan(z))
337 return __kernel_standard(x, y, 24); /* pow neg**non-int */
338 return __kernel_standard(x, y, 21); /* pow overflow */
339 }
340 }
341 if (z == 0.0 && isfinite(x) && isfinite(y))
342 return __kernel_standard(x, y, 22); /* pow underflow */
343 return z;
344}
345#else
346strong_alias(__ieee754_pow, pow)
347#endif
348libm_hidden_def(pow)
349