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
84static const double
85bp[] = {1.0, 1.5,},
86dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
87dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
88zero    =  0.0,
89one =  1.0,
90two =  2.0,
91two53 =  9007199254740992.0,  /* 0x43400000, 0x00000000 */
92huge  =  1.0e300,
93tiny    =  1.0e-300,
94  /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
95L1  =  5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
96L2  =  4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
97L3  =  3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
98L4  =  2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
99L5  =  2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
100L6  =  2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
101P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
102P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
103P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
104P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
105P5   =  4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
106lg2  =  6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
107lg2_h  =  6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
108lg2_l  = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
109ovt =  8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
110cp    =  9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
111cp_h  =  9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
112cp_l  = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
113ivln2    =  1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
114ivln2_h  =  1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
115ivln2_l  =  1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
116
117double
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