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