1df930be7Sderaadt /* @(#)e_pow.c 5.1 93/09/24 */
2df930be7Sderaadt /*
3df930be7Sderaadt * ====================================================
4df930be7Sderaadt * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5df930be7Sderaadt *
6df930be7Sderaadt * Developed at SunPro, a Sun Microsystems, Inc. business.
7df930be7Sderaadt * Permission to use, copy, modify, and distribute this
8df930be7Sderaadt * software is freely granted, provided that this notice
9df930be7Sderaadt * is preserved.
10df930be7Sderaadt * ====================================================
11df930be7Sderaadt */
12df930be7Sderaadt
137b36286aSmartynas /* pow(x,y) return x**y
14df930be7Sderaadt *
15df930be7Sderaadt * n
16df930be7Sderaadt * Method: Let x = 2 * (1+f)
17df930be7Sderaadt * 1. Compute and return log2(x) in two pieces:
18df930be7Sderaadt * log2(x) = w1 + w2,
19df930be7Sderaadt * where w1 has 53-24 = 29 bit trailing zeros.
2017d217faSmartynas * 2. Perform y*log2(x) = n+y' by simulating multi-precision
21df930be7Sderaadt * arithmetic, where |y'|<=0.5.
22df930be7Sderaadt * 3. Return x**y = 2**n*exp(y'*log2)
23df930be7Sderaadt *
24df930be7Sderaadt * Special cases:
25df930be7Sderaadt * 1. (anything) ** 0 is 1
26df930be7Sderaadt * 2. (anything) ** 1 is itself
27d205ab02Skettenis * 3. (anything except 1) ** NAN is NAN
28df930be7Sderaadt * 4. NAN ** (anything except 0) is NAN
29df930be7Sderaadt * 5. +-(|x| > 1) ** +INF is +INF
30df930be7Sderaadt * 6. +-(|x| > 1) ** -INF is +0
31df930be7Sderaadt * 7. +-(|x| < 1) ** +INF is +0
32df930be7Sderaadt * 8. +-(|x| < 1) ** -INF is +INF
33d205ab02Skettenis * 9. +-1 ** +-INF is 1
34df930be7Sderaadt * 10. +0 ** (+anything except 0, NAN) is +0
35df930be7Sderaadt * 11. -0 ** (+anything except 0, NAN, odd integer) is +0
36df930be7Sderaadt * 12. +0 ** (-anything except 0, NAN) is +INF
37df930be7Sderaadt * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
38df930be7Sderaadt * 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
39df930be7Sderaadt * 15. +INF ** (+anything except 0,NAN) is +INF
40df930be7Sderaadt * 16. +INF ** (-anything except 0,NAN) is +0
41df930be7Sderaadt * 17. -INF ** (anything) = -0 ** (-anything)
42df930be7Sderaadt * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
43df930be7Sderaadt * 19. (-anything except 0 and inf) ** (non-integer) is NAN
44df930be7Sderaadt *
45df930be7Sderaadt * Accuracy:
46df930be7Sderaadt * pow(x,y) returns x**y nearly rounded. In particular
47df930be7Sderaadt * pow(integer,integer)
48df930be7Sderaadt * always returns the correct integer provided it is
49df930be7Sderaadt * representable.
50df930be7Sderaadt *
51df930be7Sderaadt * Constants :
52df930be7Sderaadt * The hexadecimal values are the intended ones for the following
53df930be7Sderaadt * constants. The decimal values may be used, provided that the
54df930be7Sderaadt * compiler will convert from decimal to binary accurately enough
55df930be7Sderaadt * to produce the hexadecimal values shown.
56df930be7Sderaadt */
57df930be7Sderaadt
5849393c00Smartynas #include <float.h>
5949393c00Smartynas #include <math.h>
6049393c00Smartynas
61df930be7Sderaadt #include "math_private.h"
62df930be7Sderaadt
63df930be7Sderaadt static const double
64df930be7Sderaadt bp[] = {1.0, 1.5,},
65df930be7Sderaadt dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
66df930be7Sderaadt dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
67df930be7Sderaadt zero = 0.0,
68df930be7Sderaadt one = 1.0,
69df930be7Sderaadt two = 2.0,
70df930be7Sderaadt two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */
71df930be7Sderaadt huge = 1.0e300,
72df930be7Sderaadt tiny = 1.0e-300,
73df930be7Sderaadt /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
74df930be7Sderaadt L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
75df930be7Sderaadt L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
76df930be7Sderaadt L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
77df930be7Sderaadt L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
78df930be7Sderaadt L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
79df930be7Sderaadt L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
80df930be7Sderaadt P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
81df930be7Sderaadt P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
82df930be7Sderaadt P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
83df930be7Sderaadt P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
84df930be7Sderaadt P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
85df930be7Sderaadt lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
86df930be7Sderaadt lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
87df930be7Sderaadt lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
88df930be7Sderaadt ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
89df930be7Sderaadt cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
90df930be7Sderaadt cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
91df930be7Sderaadt cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
92df930be7Sderaadt ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
93df930be7Sderaadt ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
94df930be7Sderaadt ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
95df930be7Sderaadt
96e7beb4a7Smillert double
pow(double x,double y)977b36286aSmartynas pow(double x, double y)
98df930be7Sderaadt {
99df930be7Sderaadt double z,ax,z_h,z_l,p_h,p_l;
10055fa043fSotto double yy1,t1,t2,r,s,t,u,v,w;
101df930be7Sderaadt int32_t i,j,k,yisint,n;
102df930be7Sderaadt int32_t hx,hy,ix,iy;
103df930be7Sderaadt u_int32_t lx,ly;
104df930be7Sderaadt
105df930be7Sderaadt EXTRACT_WORDS(hx,lx,x);
106df930be7Sderaadt EXTRACT_WORDS(hy,ly,y);
107df930be7Sderaadt ix = hx&0x7fffffff; iy = hy&0x7fffffff;
108df930be7Sderaadt
109df930be7Sderaadt /* y==zero: x**0 = 1 */
110df930be7Sderaadt if((iy|ly)==0) return one;
111df930be7Sderaadt
112d205ab02Skettenis /* x==1: 1**y = 1, even if y is NaN */
113d205ab02Skettenis if (hx==0x3ff00000 && lx == 0) return one;
114d205ab02Skettenis
115df930be7Sderaadt /* +-NaN return x+y */
116df930be7Sderaadt if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
117df930be7Sderaadt iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
118df930be7Sderaadt return x+y;
119df930be7Sderaadt
120df930be7Sderaadt /* determine if y is an odd int when x < 0
121df930be7Sderaadt * yisint = 0 ... y is not an integer
122df930be7Sderaadt * yisint = 1 ... y is an odd int
123df930be7Sderaadt * yisint = 2 ... y is an even int
124df930be7Sderaadt */
125df930be7Sderaadt yisint = 0;
126df930be7Sderaadt if(hx<0) {
127df930be7Sderaadt if(iy>=0x43400000) yisint = 2; /* even integer y */
128df930be7Sderaadt else if(iy>=0x3ff00000) {
129df930be7Sderaadt k = (iy>>20)-0x3ff; /* exponent */
130df930be7Sderaadt if(k>20) {
131df930be7Sderaadt j = ly>>(52-k);
132df930be7Sderaadt if((j<<(52-k))==ly) yisint = 2-(j&1);
133df930be7Sderaadt } else if(ly==0) {
134df930be7Sderaadt j = iy>>(20-k);
135df930be7Sderaadt if((j<<(20-k))==iy) yisint = 2-(j&1);
136df930be7Sderaadt }
137df930be7Sderaadt }
138df930be7Sderaadt }
139df930be7Sderaadt
140df930be7Sderaadt /* special value of y */
141df930be7Sderaadt if(ly==0) {
142df930be7Sderaadt if (iy==0x7ff00000) { /* y is +-inf */
143df930be7Sderaadt if(((ix-0x3ff00000)|lx)==0)
144d205ab02Skettenis return one; /* (-1)**+-inf is 1 */
145df930be7Sderaadt else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
146df930be7Sderaadt return (hy>=0)? y: zero;
147df930be7Sderaadt else /* (|x|<1)**-,+inf = inf,0 */
148df930be7Sderaadt return (hy<0)?-y: zero;
149df930be7Sderaadt }
150df930be7Sderaadt if(iy==0x3ff00000) { /* y is +-1 */
151df930be7Sderaadt if(hy<0) return one/x; else return x;
152df930be7Sderaadt }
153df930be7Sderaadt if(hy==0x40000000) return x*x; /* y is 2 */
154df930be7Sderaadt if(hy==0x3fe00000) { /* y is 0.5 */
155df930be7Sderaadt if(hx>=0) /* x >= +0 */
1567b36286aSmartynas return sqrt(x);
157df930be7Sderaadt }
158df930be7Sderaadt }
159df930be7Sderaadt
160df930be7Sderaadt ax = fabs(x);
161df930be7Sderaadt /* special value of x */
162df930be7Sderaadt if(lx==0) {
163df930be7Sderaadt if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
164df930be7Sderaadt z = ax; /*x is +-0,+-inf,+-1*/
165df930be7Sderaadt if(hy<0) z = one/z; /* z = (1/|x|) */
166df930be7Sderaadt if(hx<0) {
167df930be7Sderaadt if(((ix-0x3ff00000)|yisint)==0) {
168df930be7Sderaadt z = (z-z)/(z-z); /* (-1)**non-int is NaN */
169df930be7Sderaadt } else if(yisint==1)
170df930be7Sderaadt z = -z; /* (x<0)**odd = -(|x|**odd) */
171df930be7Sderaadt }
172df930be7Sderaadt return z;
173df930be7Sderaadt }
174df930be7Sderaadt }
175df930be7Sderaadt
17617d217faSmartynas n = (hx>>31)+1;
17717d217faSmartynas
178df930be7Sderaadt /* (x<0)**(non-int) is NaN */
17917d217faSmartynas if((n|yisint)==0) return (x-x)/(x-x);
18017d217faSmartynas
18117d217faSmartynas s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
18217d217faSmartynas if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */
183df930be7Sderaadt
184df930be7Sderaadt /* |y| is huge */
185df930be7Sderaadt if(iy>0x41e00000) { /* if |y| > 2**31 */
186df930be7Sderaadt if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */
187df930be7Sderaadt if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
188df930be7Sderaadt if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
189df930be7Sderaadt }
190df930be7Sderaadt /* over/underflow if x is not close to one */
19117d217faSmartynas if(ix<0x3fefffff) return (hy<0)? s*huge*huge:s*tiny*tiny;
19217d217faSmartynas if(ix>0x3ff00000) return (hy>0)? s*huge*huge:s*tiny*tiny;
193df930be7Sderaadt /* now |1-x| is tiny <= 2**-20, suffice to compute
194df930be7Sderaadt log(x) by x-x^2/2+x^3/3-x^4/4 */
19517d217faSmartynas t = ax-one; /* t has 20 trailing zeros */
196df930be7Sderaadt w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
197df930be7Sderaadt u = ivln2_h*t; /* ivln2_h has 21 sig. bits */
198df930be7Sderaadt v = t*ivln2_l-w*ivln2;
199df930be7Sderaadt t1 = u+v;
200df930be7Sderaadt SET_LOW_WORD(t1,0);
201df930be7Sderaadt t2 = v-(t1-u);
202df930be7Sderaadt } else {
20317d217faSmartynas double ss,s2,s_h,s_l,t_h,t_l;
204df930be7Sderaadt n = 0;
205df930be7Sderaadt /* take care subnormal number */
206df930be7Sderaadt if(ix<0x00100000)
207df930be7Sderaadt {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); }
208df930be7Sderaadt n += ((ix)>>20)-0x3ff;
209df930be7Sderaadt j = ix&0x000fffff;
210df930be7Sderaadt /* determine interval */
211df930be7Sderaadt ix = j|0x3ff00000; /* normalize ix */
212df930be7Sderaadt if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */
213df930be7Sderaadt else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */
214df930be7Sderaadt else {k=0;n+=1;ix -= 0x00100000;}
215df930be7Sderaadt SET_HIGH_WORD(ax,ix);
216df930be7Sderaadt
21717d217faSmartynas /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
218df930be7Sderaadt u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
219df930be7Sderaadt v = one/(ax+bp[k]);
22017d217faSmartynas ss = u*v;
22117d217faSmartynas s_h = ss;
222df930be7Sderaadt SET_LOW_WORD(s_h,0);
223df930be7Sderaadt /* t_h=ax+bp[k] High */
224df930be7Sderaadt t_h = zero;
225df930be7Sderaadt SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18));
226df930be7Sderaadt t_l = ax - (t_h-bp[k]);
227df930be7Sderaadt s_l = v*((u-s_h*t_h)-s_h*t_l);
228df930be7Sderaadt /* compute log(ax) */
22917d217faSmartynas s2 = ss*ss;
230df930be7Sderaadt r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
23117d217faSmartynas r += s_l*(s_h+ss);
232df930be7Sderaadt s2 = s_h*s_h;
233df930be7Sderaadt t_h = 3.0+s2+r;
234df930be7Sderaadt SET_LOW_WORD(t_h,0);
235df930be7Sderaadt t_l = r-((t_h-3.0)-s2);
23617d217faSmartynas /* u+v = ss*(1+...) */
237df930be7Sderaadt u = s_h*t_h;
23817d217faSmartynas v = s_l*t_h+t_l*ss;
23917d217faSmartynas /* 2/(3log2)*(ss+...) */
240df930be7Sderaadt p_h = u+v;
241df930be7Sderaadt SET_LOW_WORD(p_h,0);
242df930be7Sderaadt p_l = v-(p_h-u);
243df930be7Sderaadt z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
244df930be7Sderaadt z_l = cp_l*p_h+p_l*cp+dp_l[k];
24517d217faSmartynas /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
246df930be7Sderaadt t = (double)n;
247df930be7Sderaadt t1 = (((z_h+z_l)+dp_h[k])+t);
248df930be7Sderaadt SET_LOW_WORD(t1,0);
249df930be7Sderaadt t2 = z_l-(((t1-t)-dp_h[k])-z_h);
250df930be7Sderaadt }
251df930be7Sderaadt
25255fa043fSotto /* split up y into yy1+y2 and compute (yy1+y2)*(t1+t2) */
25355fa043fSotto yy1 = y;
25455fa043fSotto SET_LOW_WORD(yy1,0);
25555fa043fSotto p_l = (y-yy1)*t1+y*t2;
25655fa043fSotto p_h = yy1*t1;
257df930be7Sderaadt z = p_l+p_h;
258df930be7Sderaadt EXTRACT_WORDS(j,i,z);
259df930be7Sderaadt if (j>=0x40900000) { /* z >= 1024 */
260df930be7Sderaadt if(((j-0x40900000)|i)!=0) /* if z > 1024 */
261df930be7Sderaadt return s*huge*huge; /* overflow */
262df930be7Sderaadt else {
263df930be7Sderaadt if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */
264df930be7Sderaadt }
265df930be7Sderaadt } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */
266df930be7Sderaadt if(((j-0xc090cc00)|i)!=0) /* z < -1075 */
267df930be7Sderaadt return s*tiny*tiny; /* underflow */
268df930be7Sderaadt else {
269df930be7Sderaadt if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */
270df930be7Sderaadt }
271df930be7Sderaadt }
272df930be7Sderaadt /*
273df930be7Sderaadt * compute 2**(p_h+p_l)
274df930be7Sderaadt */
275df930be7Sderaadt i = j&0x7fffffff;
276df930be7Sderaadt k = (i>>20)-0x3ff;
277df930be7Sderaadt n = 0;
278df930be7Sderaadt if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */
279df930be7Sderaadt n = j+(0x00100000>>(k+1));
280df930be7Sderaadt k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */
281df930be7Sderaadt t = zero;
282df930be7Sderaadt SET_HIGH_WORD(t,n&~(0x000fffff>>k));
283df930be7Sderaadt n = ((n&0x000fffff)|0x00100000)>>(20-k);
284df930be7Sderaadt if(j<0) n = -n;
285df930be7Sderaadt p_h -= t;
286df930be7Sderaadt }
287df930be7Sderaadt t = p_l+p_h;
288df930be7Sderaadt SET_LOW_WORD(t,0);
289df930be7Sderaadt u = t*lg2_h;
290df930be7Sderaadt v = (p_l-(t-p_h))*lg2+t*lg2_l;
291df930be7Sderaadt z = u+v;
292df930be7Sderaadt w = v-(z-u);
293df930be7Sderaadt t = z*z;
294df930be7Sderaadt t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
295df930be7Sderaadt r = (z*t1)/(t1-two)-(w+z*w);
296df930be7Sderaadt z = one-(r-z);
297df930be7Sderaadt GET_HIGH_WORD(j,z);
298df930be7Sderaadt j += (n<<20);
299df930be7Sderaadt if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */
300df930be7Sderaadt else SET_HIGH_WORD(z,j);
301df930be7Sderaadt return s*z;
302df930be7Sderaadt }
303*2f2c0062Sguenther DEF_STD(pow);
304*2f2c0062Sguenther LDBL_MAYBE_CLONE(pow);
305