1 /** @file
2 Compute the base 10 logrithm of x.
3
4 Copyright (c) 2010 - 2011, Intel Corporation. All rights reserved.<BR>
5 This program and the accompanying materials are licensed and made available under
6 the terms and conditions of the BSD License that accompanies this distribution.
7 The full text of the license may be found at
8 http://opensource.org/licenses/bsd-license.
9
10 THE PROGRAM IS DISTRIBUTED UNDER THE BSD LICENSE ON AN "AS IS" BASIS,
11 WITHOUT WARRANTIES OR REPRESENTATIONS OF ANY KIND, EITHER EXPRESS OR IMPLIED.
12
13 * ====================================================
14 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
15 *
16 * Developed at SunPro, a Sun Microsystems, Inc. business.
17 * Permission to use, copy, modify, and distribute this
18 * software is freely granted, provided that this notice
19 * is preserved.
20 * ====================================================
21
22 e_pow.c 5.1 93/09/24
23 NetBSD: e_pow.c,v 1.13 2004/06/30 18:43:15 drochner Exp
24 **/
25 #include <LibConfig.h>
26 #include <sys/EfiCdefs.h>
27
28 #if defined(_MSC_VER) /* Handle Microsoft VC++ compiler specifics. */
29 // C4723: potential divide by zero.
30 #pragma warning ( disable : 4723 )
31 // C4756: overflow in constant arithmetic
32 #pragma warning ( disable : 4756 )
33 #endif
34
35 /* __ieee754_pow(x,y) return x**y
36 *
37 * n
38 * Method: Let x = 2 * (1+f)
39 * 1. Compute and return log2(x) in two pieces:
40 * log2(x) = w1 + w2,
41 * where w1 has 53-24 = 29 bit trailing zeros.
42 * 2. Perform y*log2(x) = n+y' by simulating multi-precision
43 * arithmetic, where |y'|<=0.5.
44 * 3. Return x**y = 2**n*exp(y'*log2)
45 *
46 * Special cases:
47 * 1. (anything) ** 0 is 1
48 * 2. (anything) ** 1 is itself
49 * 3. (anything) ** NAN is NAN
50 * 4. NAN ** (anything except 0) is NAN
51 * 5. +-(|x| > 1) ** +INF is +INF
52 * 6. +-(|x| > 1) ** -INF is +0
53 * 7. +-(|x| < 1) ** +INF is +0
54 * 8. +-(|x| < 1) ** -INF is +INF
55 * 9. +-1 ** +-INF is NAN
56 * 10. +0 ** (+anything except 0, NAN) is +0
57 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0
58 * 12. +0 ** (-anything except 0, NAN) is +INF
59 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
60 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
61 * 15. +INF ** (+anything except 0,NAN) is +INF
62 * 16. +INF ** (-anything except 0,NAN) is +0
63 * 17. -INF ** (anything) = -0 ** (-anything)
64 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
65 * 19. (-anything except 0 and inf) ** (non-integer) is NAN
66 *
67 * Accuracy:
68 * pow(x,y) returns x**y nearly rounded. In particular
69 * pow(integer,integer)
70 * always returns the correct integer provided it is
71 * representable.
72 *
73 * Constants :
74 * The hexadecimal values are the intended ones for the following
75 * constants. The decimal values may be used, provided that the
76 * compiler will convert from decimal to binary accurately enough
77 * to produce the hexadecimal values shown.
78 */
79
80 #include "math.h"
81 #include "math_private.h"
82 #include <errno.h>
83
84 static const double
85 bp[] = {1.0, 1.5,},
86 dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
87 dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
88 zero = 0.0,
89 one = 1.0,
90 two = 2.0,
91 two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */
92 huge = 1.0e300,
93 tiny = 1.0e-300,
94 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
95 L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
96 L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
97 L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
98 L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
99 L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
100 L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
101 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
102 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
103 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
104 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
105 P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
106 lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
107 lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
108 lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
109 ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
110 cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
111 cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
112 cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
113 ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
114 ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
115 ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
116
117 double
__ieee754_pow(double x,double y)118 __ieee754_pow(double x, double y)
119 {
120 double z,ax,z_h,z_l,p_h,p_l;
121 double y1,t1,t2,r,s,t,u,v,w;
122 int32_t i,j,k,yisint,n;
123 int32_t hx,hy,ix,iy;
124 u_int32_t lx,ly;
125
126 EXTRACT_WORDS(hx,lx,x);
127 EXTRACT_WORDS(hy,ly,y);
128 ix = hx&0x7fffffff; iy = hy&0x7fffffff;
129
130 /* y==zero: x**0 = 1 */
131 if((iy|ly)==0) return one;
132
133 /* +-NaN return x+y */
134 if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
135 iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
136 return x+y;
137
138 /* determine if y is an odd int when x < 0
139 * yisint = 0 ... y is not an integer
140 * yisint = 1 ... y is an odd int
141 * yisint = 2 ... y is an even int
142 */
143 yisint = 0;
144 if(hx<0) {
145 if(iy>=0x43400000) yisint = 2; /* even integer y */
146 else if(iy>=0x3ff00000) {
147 k = (iy>>20)-0x3ff; /* exponent */
148 if(k>20) {
149 j = ly>>(52-k);
150 if((u_int32_t)(j<<(52-k))==ly) yisint = 2-(j&1);
151 } else if(ly==0) {
152 j = iy>>(20-k);
153 if((j<<(20-k))==iy) yisint = 2-(j&1);
154 }
155 }
156 }
157
158 /* special value of y */
159 if(ly==0) {
160 if (iy==0x7ff00000) { /* y is +-inf */
161 if(((ix-0x3ff00000)|lx)==0)
162 return y - y; /* inf**+-1 is NaN */
163 else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
164 return (hy>=0)? y: zero;
165 else /* (|x|<1)**-,+inf = inf,0 */
166 return (hy<0)?-y: zero;
167 }
168 if(iy==0x3ff00000) { /* y is +-1 */
169 if(hy<0) return one/x; else return x;
170 }
171 if(hy==0x40000000) return x*x; /* y is 2 */
172 if(hy==0x3fe00000) { /* y is 0.5 */
173 if(hx>=0) /* x >= +0 */
174 return __ieee754_sqrt(x);
175 }
176 }
177
178 ax = fabs(x);
179 /* special value of x */
180 if(lx==0) {
181 if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
182 z = ax; /*x is +-0,+-inf,+-1*/
183 if(hy<0) z = one/z; /* z = (1/|x|) */
184 if(hx<0) {
185 if(((ix-0x3ff00000)|yisint)==0) {
186 z = (z-z)/(z-z); /* (-1)**non-int is NaN */
187 } else if(yisint==1)
188 z = -z; /* (x<0)**odd = -(|x|**odd) */
189 }
190 return z;
191 }
192 }
193
194 n = (hx>>31)+1;
195
196 /* (x<0)**(non-int) is NaN */
197 if((n|yisint)==0) {
198 errno = EDOM;
199 return (x-x)/(x-x);
200 }
201
202 s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
203 if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */
204
205 /* |y| is huge */
206 if(iy>0x41e00000) { /* if |y| > 2**31 */
207 if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */
208 if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
209 if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
210 }
211 /* over/underflow if x is not close to one */
212 if(ix<0x3fefffff) return (hy<0)? s*huge*huge:s*tiny*tiny;
213 if(ix>0x3ff00000) return (hy>0)? s*huge*huge:s*tiny*tiny;
214 /* now |1-x| is tiny <= 2**-20, suffice to compute
215 log(x) by x-x^2/2+x^3/3-x^4/4 */
216 t = ax-one; /* t has 20 trailing zeros */
217 w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
218 u = ivln2_h*t; /* ivln2_h has 21 sig. bits */
219 v = t*ivln2_l-w*ivln2;
220 t1 = u+v;
221 SET_LOW_WORD(t1,0);
222 t2 = v-(t1-u);
223 } else {
224 double ss,s2,s_h,s_l,t_h,t_l;
225 n = 0;
226 /* take care subnormal number */
227 if(ix<0x00100000)
228 {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); }
229 n += ((ix)>>20)-0x3ff;
230 j = ix&0x000fffff;
231 /* determine interval */
232 ix = j|0x3ff00000; /* normalize ix */
233 if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */
234 else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */
235 else {k=0;n+=1;ix -= 0x00100000;}
236 SET_HIGH_WORD(ax,ix);
237
238 /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
239 u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
240 v = one/(ax+bp[k]);
241 ss = u*v;
242 s_h = ss;
243 SET_LOW_WORD(s_h,0);
244 /* t_h=ax+bp[k] High */
245 t_h = zero;
246 SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18));
247 t_l = ax - (t_h-bp[k]);
248 s_l = v*((u-s_h*t_h)-s_h*t_l);
249 /* compute log(ax) */
250 s2 = ss*ss;
251 r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
252 r += s_l*(s_h+ss);
253 s2 = s_h*s_h;
254 t_h = 3.0+s2+r;
255 SET_LOW_WORD(t_h,0);
256 t_l = r-((t_h-3.0)-s2);
257 /* u+v = ss*(1+...) */
258 u = s_h*t_h;
259 v = s_l*t_h+t_l*ss;
260 /* 2/(3log2)*(ss+...) */
261 p_h = u+v;
262 SET_LOW_WORD(p_h,0);
263 p_l = v-(p_h-u);
264 z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
265 z_l = cp_l*p_h+p_l*cp+dp_l[k];
266 /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
267 t = (double)n;
268 t1 = (((z_h+z_l)+dp_h[k])+t);
269 SET_LOW_WORD(t1,0);
270 t2 = z_l-(((t1-t)-dp_h[k])-z_h);
271 }
272
273 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
274 y1 = y;
275 SET_LOW_WORD(y1,0);
276 p_l = (y-y1)*t1+y*t2;
277 p_h = y1*t1;
278 z = p_l+p_h;
279 EXTRACT_WORDS(j,i,z);
280 if (j>=0x40900000) { /* z >= 1024 */
281 if(((j-0x40900000)|i)!=0) /* if z > 1024 */
282 return s*huge*huge; /* overflow */
283 else {
284 if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */
285 }
286 } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */
287 if(((j-0xc090cc00)|i)!=0) /* z < -1075 */
288 return s*tiny*tiny; /* underflow */
289 else {
290 if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */
291 }
292 }
293 /*
294 * compute 2**(p_h+p_l)
295 */
296 i = j&0x7fffffff;
297 k = (i>>20)-0x3ff;
298 n = 0;
299 if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */
300 n = j+(0x00100000>>(k+1));
301 k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */
302 t = zero;
303 SET_HIGH_WORD(t,n&~(0x000fffff>>k));
304 n = ((n&0x000fffff)|0x00100000)>>(20-k);
305 if(j<0) n = -n;
306 p_h -= t;
307 }
308 t = p_l+p_h;
309 SET_LOW_WORD(t,0);
310 u = t*lg2_h;
311 v = (p_l-(t-p_h))*lg2+t*lg2_l;
312 z = u+v;
313 w = v-(z-u);
314 t = z*z;
315 t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
316 r = (z*t1)/(t1-two)-(w+z*w);
317 z = one-(r-z);
318 GET_HIGH_WORD(j,z);
319 j += (n<<20);
320 if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */
321 else SET_HIGH_WORD(z,j);
322 return s*z;
323 }
324