105a0b428SJohn Marino /* e_powf.c -- float version of e_pow.c.
205a0b428SJohn Marino * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
305a0b428SJohn Marino */
405a0b428SJohn Marino
505a0b428SJohn Marino /*
605a0b428SJohn Marino * ====================================================
705a0b428SJohn Marino * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
805a0b428SJohn Marino *
905a0b428SJohn Marino * Developed at SunPro, a Sun Microsystems, Inc. business.
1005a0b428SJohn Marino * Permission to use, copy, modify, and distribute this
1105a0b428SJohn Marino * software is freely granted, provided that this notice
1205a0b428SJohn Marino * is preserved.
1305a0b428SJohn Marino * ====================================================
1405a0b428SJohn Marino */
1505a0b428SJohn Marino
1605a0b428SJohn Marino #include "math.h"
1705a0b428SJohn Marino #include "math_private.h"
1805a0b428SJohn Marino
1905a0b428SJohn Marino static const volatile float huge = 1.0e+30, tiny = 1.0e-30;
2005a0b428SJohn Marino
2105a0b428SJohn Marino static const float
2205a0b428SJohn Marino bp[] = {1.0, 1.5,},
2305a0b428SJohn Marino dp_h[] = { 0.0, 5.84960938e-01,}, /* 0x3f15c000 */
2405a0b428SJohn Marino dp_l[] = { 0.0, 1.56322085e-06,}, /* 0x35d1cfdc */
2505a0b428SJohn Marino zero = 0.0,
2605a0b428SJohn Marino one = 1.0,
2705a0b428SJohn Marino two = 2.0,
2805a0b428SJohn Marino two24 = 16777216.0, /* 0x4b800000 */
2905a0b428SJohn Marino /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
3005a0b428SJohn Marino L1 = 6.0000002384e-01, /* 0x3f19999a */
3105a0b428SJohn Marino L2 = 4.2857143283e-01, /* 0x3edb6db7 */
3205a0b428SJohn Marino L3 = 3.3333334327e-01, /* 0x3eaaaaab */
3305a0b428SJohn Marino L4 = 2.7272811532e-01, /* 0x3e8ba305 */
3405a0b428SJohn Marino L5 = 2.3066075146e-01, /* 0x3e6c3255 */
3505a0b428SJohn Marino L6 = 2.0697501302e-01, /* 0x3e53f142 */
3605a0b428SJohn Marino P1 = 1.6666667163e-01, /* 0x3e2aaaab */
3705a0b428SJohn Marino P2 = -2.7777778450e-03, /* 0xbb360b61 */
3805a0b428SJohn Marino P3 = 6.6137559770e-05, /* 0x388ab355 */
3905a0b428SJohn Marino P4 = -1.6533901999e-06, /* 0xb5ddea0e */
4005a0b428SJohn Marino P5 = 4.1381369442e-08, /* 0x3331bb4c */
4105a0b428SJohn Marino lg2 = 6.9314718246e-01, /* 0x3f317218 */
4205a0b428SJohn Marino lg2_h = 6.93145752e-01, /* 0x3f317200 */
4305a0b428SJohn Marino lg2_l = 1.42860654e-06, /* 0x35bfbe8c */
4405a0b428SJohn Marino ovt = 4.2995665694e-08, /* -(128-log2(ovfl+.5ulp)) */
4505a0b428SJohn Marino cp = 9.6179670095e-01, /* 0x3f76384f =2/(3ln2) */
4605a0b428SJohn Marino cp_h = 9.6179199219e-01, /* 0x3f763800 =head of cp */
4705a0b428SJohn Marino cp_l = 4.7017383622e-06, /* 0x369dc3a0 =tail of cp_h */
4805a0b428SJohn Marino ivln2 = 1.4426950216e+00, /* 0x3fb8aa3b =1/ln2 */
4905a0b428SJohn Marino ivln2_h = 1.4426879883e+00, /* 0x3fb8aa00 =16b 1/ln2*/
5005a0b428SJohn Marino ivln2_l = 7.0526075433e-06; /* 0x36eca570 =1/ln2 tail*/
5105a0b428SJohn Marino
5205a0b428SJohn Marino float
powf(float x,float y)5305a0b428SJohn Marino powf(float x, float y)
5405a0b428SJohn Marino {
5505a0b428SJohn Marino float z,ax,z_h,z_l,p_h,p_l;
5605a0b428SJohn Marino float yy1,t1,t2,r,s,t,u,v,w;
5705a0b428SJohn Marino int32_t i,j,k,yisint,n;
5805a0b428SJohn Marino int32_t hx,hy,ix,iy,is;
5905a0b428SJohn Marino
6005a0b428SJohn Marino GET_FLOAT_WORD(hx,x);
6105a0b428SJohn Marino GET_FLOAT_WORD(hy,y);
6205a0b428SJohn Marino ix = hx&0x7fffffff; iy = hy&0x7fffffff;
6305a0b428SJohn Marino
6405a0b428SJohn Marino /* y==zero: x**0 = 1 */
6505a0b428SJohn Marino if(iy==0) return one;
6605a0b428SJohn Marino
6705a0b428SJohn Marino /* x==1: 1**y = 1, even if y is NaN */
6805a0b428SJohn Marino if (hx==0x3f800000) return one;
6905a0b428SJohn Marino
7005a0b428SJohn Marino /* +-NaN return x+y */
7105a0b428SJohn Marino if(ix > 0x7f800000 ||
7205a0b428SJohn Marino iy > 0x7f800000)
7305a0b428SJohn Marino return x+y;
7405a0b428SJohn Marino
7505a0b428SJohn Marino /* determine if y is an odd int when x < 0
7605a0b428SJohn Marino * yisint = 0 ... y is not an integer
7705a0b428SJohn Marino * yisint = 1 ... y is an odd int
7805a0b428SJohn Marino * yisint = 2 ... y is an even int
7905a0b428SJohn Marino */
8005a0b428SJohn Marino yisint = 0;
8105a0b428SJohn Marino if(hx<0) {
8205a0b428SJohn Marino if(iy>=0x4b800000) yisint = 2; /* even integer y */
8305a0b428SJohn Marino else if(iy>=0x3f800000) {
8405a0b428SJohn Marino k = (iy>>23)-0x7f; /* exponent */
8505a0b428SJohn Marino j = iy>>(23-k);
8605a0b428SJohn Marino if((j<<(23-k))==iy) yisint = 2-(j&1);
8705a0b428SJohn Marino }
8805a0b428SJohn Marino }
8905a0b428SJohn Marino
9005a0b428SJohn Marino /* special value of y */
9105a0b428SJohn Marino if (iy==0x7f800000) { /* y is +-inf */
9205a0b428SJohn Marino if (ix==0x3f800000)
9305a0b428SJohn Marino return one; /* (-1)**+-inf is NaN */
9405a0b428SJohn Marino else if (ix > 0x3f800000)/* (|x|>1)**+-inf = inf,0 */
9505a0b428SJohn Marino return (hy>=0)? y: zero;
9605a0b428SJohn Marino else /* (|x|<1)**-,+inf = inf,0 */
9705a0b428SJohn Marino return (hy<0)?-y: zero;
9805a0b428SJohn Marino }
9905a0b428SJohn Marino if(iy==0x3f800000) { /* y is +-1 */
10005a0b428SJohn Marino if(hy<0) return one/x; else return x;
10105a0b428SJohn Marino }
10205a0b428SJohn Marino if(hy==0x40000000) return x*x; /* y is 2 */
10305a0b428SJohn Marino if(hy==0x3f000000) { /* y is 0.5 */
10405a0b428SJohn Marino if(hx>=0) /* x >= +0 */
10505a0b428SJohn Marino return sqrtf(x);
10605a0b428SJohn Marino }
10705a0b428SJohn Marino
10805a0b428SJohn Marino ax = fabsf(x);
10905a0b428SJohn Marino /* special value of x */
11005a0b428SJohn Marino if(ix==0x7f800000||ix==0||ix==0x3f800000){
11105a0b428SJohn Marino z = ax; /*x is +-0,+-inf,+-1*/
11205a0b428SJohn Marino if(hy<0) z = one/z; /* z = (1/|x|) */
11305a0b428SJohn Marino if(hx<0) {
11405a0b428SJohn Marino if(((ix-0x3f800000)|yisint)==0) {
11505a0b428SJohn Marino z = (z-z)/(z-z); /* (-1)**non-int is NaN */
11605a0b428SJohn Marino } else if(yisint==1)
11705a0b428SJohn Marino z = -z; /* (x<0)**odd = -(|x|**odd) */
11805a0b428SJohn Marino }
11905a0b428SJohn Marino return z;
12005a0b428SJohn Marino }
12105a0b428SJohn Marino
12205a0b428SJohn Marino /* (x<0)**(non-int) is NaN */
12305a0b428SJohn Marino if(((((u_int32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x);
12405a0b428SJohn Marino
12505a0b428SJohn Marino /* |y| is huge */
12605a0b428SJohn Marino if(iy>0x4d000000) { /* if |y| > 2**27 */
12705a0b428SJohn Marino /* over/underflow if x is not close to one */
12805a0b428SJohn Marino if(ix<0x3f7ffff8) return (hy<0)? huge*huge:tiny*tiny;
12905a0b428SJohn Marino if(ix>0x3f800007) return (hy>0)? huge*huge:tiny*tiny;
13005a0b428SJohn Marino /* now |1-x| is tiny <= 2**-20, suffice to compute
13105a0b428SJohn Marino log(x) by x-x^2/2+x^3/3-x^4/4 */
13205a0b428SJohn Marino t = ax-one; /* t has 20 trailing zeros */
13305a0b428SJohn Marino w = (t*t)*((float)0.5-t*((float)0.333333333333-t*(float)0.25));
13405a0b428SJohn Marino u = ivln2_h*t; /* ivln2_h has 16 sig. bits */
13505a0b428SJohn Marino v = t*ivln2_l-w*ivln2;
13605a0b428SJohn Marino t1 = u+v;
13705a0b428SJohn Marino GET_FLOAT_WORD(is,t1);
13805a0b428SJohn Marino SET_FLOAT_WORD(t1,is&0xfffff000);
13905a0b428SJohn Marino t2 = v-(t1-u);
14005a0b428SJohn Marino } else {
14105a0b428SJohn Marino float s2,s_h,s_l,t_h,t_l;
14205a0b428SJohn Marino n = 0;
14305a0b428SJohn Marino /* take care subnormal number */
14405a0b428SJohn Marino if(ix<0x00800000)
14505a0b428SJohn Marino {ax *= two24; n -= 24; GET_FLOAT_WORD(ix,ax); }
14605a0b428SJohn Marino n += ((ix)>>23)-0x7f;
14705a0b428SJohn Marino j = ix&0x007fffff;
14805a0b428SJohn Marino /* determine interval */
14905a0b428SJohn Marino ix = j|0x3f800000; /* normalize ix */
15005a0b428SJohn Marino if(j<=0x1cc471) k=0; /* |x|<sqrt(3/2) */
15105a0b428SJohn Marino else if(j<0x5db3d7) k=1; /* |x|<sqrt(3) */
15205a0b428SJohn Marino else {k=0;n+=1;ix -= 0x00800000;}
15305a0b428SJohn Marino SET_FLOAT_WORD(ax,ix);
15405a0b428SJohn Marino
15505a0b428SJohn Marino /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
15605a0b428SJohn Marino u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
15705a0b428SJohn Marino v = one/(ax+bp[k]);
15805a0b428SJohn Marino s = u*v;
15905a0b428SJohn Marino s_h = s;
16005a0b428SJohn Marino GET_FLOAT_WORD(is,s_h);
16105a0b428SJohn Marino SET_FLOAT_WORD(s_h,is&0xfffff000);
16205a0b428SJohn Marino /* t_h=ax+bp[k] High */
16305a0b428SJohn Marino SET_FLOAT_WORD(t_h,((ix>>1)|0x20000000)+0x0040000+(k<<21));
16405a0b428SJohn Marino t_l = ax - (t_h-bp[k]);
16505a0b428SJohn Marino s_l = v*((u-s_h*t_h)-s_h*t_l);
16605a0b428SJohn Marino /* compute log(ax) */
16705a0b428SJohn Marino s2 = s*s;
16805a0b428SJohn Marino r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
16905a0b428SJohn Marino r += s_l*(s_h+s);
17005a0b428SJohn Marino s2 = s_h*s_h;
17105a0b428SJohn Marino t_h = (float)3.0+s2+r;
17205a0b428SJohn Marino GET_FLOAT_WORD(is,t_h);
17305a0b428SJohn Marino SET_FLOAT_WORD(t_h,is&0xfffff000);
17405a0b428SJohn Marino t_l = r-((t_h-(float)3.0)-s2);
17505a0b428SJohn Marino /* u+v = s*(1+...) */
17605a0b428SJohn Marino u = s_h*t_h;
17705a0b428SJohn Marino v = s_l*t_h+t_l*s;
17805a0b428SJohn Marino /* 2/(3log2)*(s+...) */
17905a0b428SJohn Marino p_h = u+v;
18005a0b428SJohn Marino GET_FLOAT_WORD(is,p_h);
18105a0b428SJohn Marino SET_FLOAT_WORD(p_h,is&0xfffff000);
18205a0b428SJohn Marino p_l = v-(p_h-u);
18305a0b428SJohn Marino z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
18405a0b428SJohn Marino z_l = cp_l*p_h+p_l*cp+dp_l[k];
18505a0b428SJohn Marino /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
18605a0b428SJohn Marino t = (float)n;
18705a0b428SJohn Marino t1 = (((z_h+z_l)+dp_h[k])+t);
18805a0b428SJohn Marino GET_FLOAT_WORD(is,t1);
18905a0b428SJohn Marino SET_FLOAT_WORD(t1,is&0xfffff000);
19005a0b428SJohn Marino t2 = z_l-(((t1-t)-dp_h[k])-z_h);
19105a0b428SJohn Marino }
19205a0b428SJohn Marino
19305a0b428SJohn Marino s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
19405a0b428SJohn Marino if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0)
19505a0b428SJohn Marino s = -one; /* (-ve)**(odd int) */
19605a0b428SJohn Marino
19705a0b428SJohn Marino /* split up y into yy1+y2 and compute (yy1+y2)*(t1+t2) */
19805a0b428SJohn Marino GET_FLOAT_WORD(is,y);
19905a0b428SJohn Marino SET_FLOAT_WORD(yy1,is&0xfffff000);
20005a0b428SJohn Marino p_l = (y-yy1)*t1+y*t2;
20105a0b428SJohn Marino p_h = yy1*t1;
20205a0b428SJohn Marino z = p_l+p_h;
20305a0b428SJohn Marino GET_FLOAT_WORD(j,z);
20405a0b428SJohn Marino if (j>0x43000000) /* if z > 128 */
20505a0b428SJohn Marino return s*huge*huge; /* overflow */
20605a0b428SJohn Marino else if (j==0x43000000) { /* if z == 128 */
20705a0b428SJohn Marino if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */
20805a0b428SJohn Marino }
20905a0b428SJohn Marino else if ((j&0x7fffffff)>0x43160000) /* z <= -150 */
21005a0b428SJohn Marino return s*tiny*tiny; /* underflow */
211*74b7c7a8SJohn Marino else if ((u_int32_t)j==0xc3160000){ /* z == -150 */
21205a0b428SJohn Marino if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */
21305a0b428SJohn Marino }
21405a0b428SJohn Marino /*
21505a0b428SJohn Marino * compute 2**(p_h+p_l)
21605a0b428SJohn Marino */
21705a0b428SJohn Marino i = j&0x7fffffff;
21805a0b428SJohn Marino k = (i>>23)-0x7f;
21905a0b428SJohn Marino n = 0;
22005a0b428SJohn Marino if(i>0x3f000000) { /* if |z| > 0.5, set n = [z+0.5] */
22105a0b428SJohn Marino n = j+(0x00800000>>(k+1));
22205a0b428SJohn Marino k = ((n&0x7fffffff)>>23)-0x7f; /* new k for n */
22305a0b428SJohn Marino SET_FLOAT_WORD(t,n&~(0x007fffff>>k));
22405a0b428SJohn Marino n = ((n&0x007fffff)|0x00800000)>>(23-k);
22505a0b428SJohn Marino if(j<0) n = -n;
22605a0b428SJohn Marino p_h -= t;
22705a0b428SJohn Marino }
22805a0b428SJohn Marino t = p_l+p_h;
22905a0b428SJohn Marino GET_FLOAT_WORD(is,t);
23005a0b428SJohn Marino SET_FLOAT_WORD(t,is&0xfffff000);
23105a0b428SJohn Marino u = t*lg2_h;
23205a0b428SJohn Marino v = (p_l-(t-p_h))*lg2+t*lg2_l;
23305a0b428SJohn Marino z = u+v;
23405a0b428SJohn Marino w = v-(z-u);
23505a0b428SJohn Marino t = z*z;
23605a0b428SJohn Marino t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
23705a0b428SJohn Marino r = (z*t1)/(t1-two)-(w+z*w);
23805a0b428SJohn Marino z = one-(r-z);
23905a0b428SJohn Marino GET_FLOAT_WORD(j,z);
24005a0b428SJohn Marino j += (n<<23);
24105a0b428SJohn Marino if((j>>23)<=0) z = scalbnf(z,n); /* subnormal output */
24205a0b428SJohn Marino else SET_FLOAT_WORD(z,j);
24305a0b428SJohn Marino return s*z;
24405a0b428SJohn Marino }
245