1 /* e_powf.c -- float version of e_pow.c. 2 * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. 3 */ 4 5 /* 6 * ==================================================== 7 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 8 * 9 * Developed at SunPro, a Sun Microsystems, Inc. business. 10 * Permission to use, copy, modify, and distribute this 11 * software is freely granted, provided that this notice 12 * is preserved. 13 * ==================================================== 14 */ 15 16 #include <sys/cdefs.h> 17 #if defined(LIBM_SCCS) && !defined(lint) 18 __RCSID("$NetBSD: e_powf.c,v 1.10 2002/05/26 22:01:51 wiz Exp $"); 19 #endif 20 21 #include "math.h" 22 #include "math_private.h" 23 24 static const float huge = 1.0e+30, tiny = 1.0e-30; 25 26 static const float 27 bp[] = {1.0, 1.5,}, 28 dp_h[] = { 0.0, 5.84960938e-01,}, /* 0x3f15c000 */ 29 dp_l[] = { 0.0, 1.56322085e-06,}, /* 0x35d1cfdc */ 30 zero = 0.0, 31 one = 1.0, 32 two = 2.0, 33 two24 = 16777216.0, /* 0x4b800000 */ 34 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */ 35 L1 = 6.0000002384e-01, /* 0x3f19999a */ 36 L2 = 4.2857143283e-01, /* 0x3edb6db7 */ 37 L3 = 3.3333334327e-01, /* 0x3eaaaaab */ 38 L4 = 2.7272811532e-01, /* 0x3e8ba305 */ 39 L5 = 2.3066075146e-01, /* 0x3e6c3255 */ 40 L6 = 2.0697501302e-01, /* 0x3e53f142 */ 41 P1 = 1.6666667163e-01, /* 0x3e2aaaab */ 42 P2 = -2.7777778450e-03, /* 0xbb360b61 */ 43 P3 = 6.6137559770e-05, /* 0x388ab355 */ 44 P4 = -1.6533901999e-06, /* 0xb5ddea0e */ 45 P5 = 4.1381369442e-08, /* 0x3331bb4c */ 46 lg2 = 6.9314718246e-01, /* 0x3f317218 */ 47 lg2_h = 6.93145752e-01, /* 0x3f317200 */ 48 lg2_l = 1.42860654e-06, /* 0x35bfbe8c */ 49 ovt = 4.2995665694e-08, /* -(128-log2(ovfl+.5ulp)) */ 50 cp = 9.6179670095e-01, /* 0x3f76384f =2/(3ln2) */ 51 cp_h = 9.6179199219e-01, /* 0x3f763800 =head of cp */ 52 cp_l = 4.7017383622e-06, /* 0x369dc3a0 =tail of cp_h */ 53 ivln2 = 1.4426950216e+00, /* 0x3fb8aa3b =1/ln2 */ 54 ivln2_h = 1.4426879883e+00, /* 0x3fb8aa00 =16b 1/ln2*/ 55 ivln2_l = 7.0526075433e-06; /* 0x36eca570 =1/ln2 tail*/ 56 57 float 58 __ieee754_powf(float x, float y) 59 { 60 float z,ax,z_h,z_l,p_h,p_l; 61 float y1,t1,t2,r,s,t,u,v,w; 62 int32_t i,j,k,yisint,n; 63 int32_t hx,hy,ix,iy,is; 64 65 GET_FLOAT_WORD(hx,x); 66 GET_FLOAT_WORD(hy,y); 67 ix = hx&0x7fffffff; iy = hy&0x7fffffff; 68 69 /* y==zero: x**0 = 1 */ 70 if(iy==0) return one; 71 72 /* +-NaN return x+y */ 73 if(ix > 0x7f800000 || 74 iy > 0x7f800000) 75 return x+y; 76 77 /* determine if y is an odd int when x < 0 78 * yisint = 0 ... y is not an integer 79 * yisint = 1 ... y is an odd int 80 * yisint = 2 ... y is an even int 81 */ 82 yisint = 0; 83 if(hx<0) { 84 if(iy>=0x4b800000) yisint = 2; /* even integer y */ 85 else if(iy>=0x3f800000) { 86 k = (iy>>23)-0x7f; /* exponent */ 87 j = iy>>(23-k); 88 if((j<<(23-k))==iy) yisint = 2-(j&1); 89 } 90 } 91 92 /* special value of y */ 93 if (iy==0x7f800000) { /* y is +-inf */ 94 if (ix==0x3f800000) 95 return y - y; /* inf**+-1 is NaN */ 96 else if (ix > 0x3f800000)/* (|x|>1)**+-inf = inf,0 */ 97 return (hy>=0)? y: zero; 98 else /* (|x|<1)**-,+inf = inf,0 */ 99 return (hy<0)?-y: zero; 100 } 101 if(iy==0x3f800000) { /* y is +-1 */ 102 if(hy<0) return one/x; else return x; 103 } 104 if(hy==0x40000000) return x*x; /* y is 2 */ 105 if(hy==0x3f000000) { /* y is 0.5 */ 106 if(hx>=0) /* x >= +0 */ 107 return __ieee754_sqrtf(x); 108 } 109 110 ax = fabsf(x); 111 /* special value of x */ 112 if(ix==0x7f800000||ix==0||ix==0x3f800000){ 113 z = ax; /*x is +-0,+-inf,+-1*/ 114 if(hy<0) z = one/z; /* z = (1/|x|) */ 115 if(hx<0) { 116 if(((ix-0x3f800000)|yisint)==0) { 117 z = (z-z)/(z-z); /* (-1)**non-int is NaN */ 118 } else if(yisint==1) 119 z = -z; /* (x<0)**odd = -(|x|**odd) */ 120 } 121 return z; 122 } 123 124 /* (x<0)**(non-int) is NaN */ 125 if(((((u_int32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x); 126 127 /* |y| is huge */ 128 if(iy>0x4d000000) { /* if |y| > 2**27 */ 129 /* over/underflow if x is not close to one */ 130 if(ix<0x3f7ffff8) return (hy<0)? huge*huge:tiny*tiny; 131 if(ix>0x3f800007) return (hy>0)? huge*huge:tiny*tiny; 132 /* now |1-x| is tiny <= 2**-20, suffice to compute 133 log(x) by x-x^2/2+x^3/3-x^4/4 */ 134 t = x-1; /* t has 20 trailing zeros */ 135 w = (t*t)*((float)0.5-t*((float)0.333333333333-t*(float)0.25)); 136 u = ivln2_h*t; /* ivln2_h has 16 sig. bits */ 137 v = t*ivln2_l-w*ivln2; 138 t1 = u+v; 139 GET_FLOAT_WORD(is,t1); 140 SET_FLOAT_WORD(t1,is&0xfffff000); 141 t2 = v-(t1-u); 142 } else { 143 float s2,s_h,s_l,t_h,t_l; 144 n = 0; 145 /* take care subnormal number */ 146 if(ix<0x00800000) 147 {ax *= two24; n -= 24; GET_FLOAT_WORD(ix,ax); } 148 n += ((ix)>>23)-0x7f; 149 j = ix&0x007fffff; 150 /* determine interval */ 151 ix = j|0x3f800000; /* normalize ix */ 152 if(j<=0x1cc471) k=0; /* |x|<sqrt(3/2) */ 153 else if(j<0x5db3d7) k=1; /* |x|<sqrt(3) */ 154 else {k=0;n+=1;ix -= 0x00800000;} 155 SET_FLOAT_WORD(ax,ix); 156 157 /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ 158 u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ 159 v = one/(ax+bp[k]); 160 s = u*v; 161 s_h = s; 162 GET_FLOAT_WORD(is,s_h); 163 SET_FLOAT_WORD(s_h,is&0xfffff000); 164 /* t_h=ax+bp[k] High */ 165 SET_FLOAT_WORD(t_h,((ix>>1)|0x20000000)+0x0040000+(k<<21)); 166 t_l = ax - (t_h-bp[k]); 167 s_l = v*((u-s_h*t_h)-s_h*t_l); 168 /* compute log(ax) */ 169 s2 = s*s; 170 r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6))))); 171 r += s_l*(s_h+s); 172 s2 = s_h*s_h; 173 t_h = (float)3.0+s2+r; 174 GET_FLOAT_WORD(is,t_h); 175 SET_FLOAT_WORD(t_h,is&0xfffff000); 176 t_l = r-((t_h-(float)3.0)-s2); 177 /* u+v = s*(1+...) */ 178 u = s_h*t_h; 179 v = s_l*t_h+t_l*s; 180 /* 2/(3log2)*(s+...) */ 181 p_h = u+v; 182 GET_FLOAT_WORD(is,p_h); 183 SET_FLOAT_WORD(p_h,is&0xfffff000); 184 p_l = v-(p_h-u); 185 z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */ 186 z_l = cp_l*p_h+p_l*cp+dp_l[k]; 187 /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */ 188 t = (float)n; 189 t1 = (((z_h+z_l)+dp_h[k])+t); 190 GET_FLOAT_WORD(is,t1); 191 SET_FLOAT_WORD(t1,is&0xfffff000); 192 t2 = z_l-(((t1-t)-dp_h[k])-z_h); 193 } 194 195 s = one; /* s (sign of result -ve**odd) = -1 else = 1 */ 196 if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0) 197 s = -one; /* (-ve)**(odd int) */ 198 199 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ 200 GET_FLOAT_WORD(is,y); 201 SET_FLOAT_WORD(y1,is&0xfffff000); 202 p_l = (y-y1)*t1+y*t2; 203 p_h = y1*t1; 204 z = p_l+p_h; 205 GET_FLOAT_WORD(j,z); 206 if (j>0x43000000) /* if z > 128 */ 207 return s*huge*huge; /* overflow */ 208 else if (j==0x43000000) { /* if z == 128 */ 209 if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */ 210 } 211 else if ((j&0x7fffffff)>0x43160000) /* z <= -150 */ 212 return s*tiny*tiny; /* underflow */ 213 else if (j==0xc3160000){ /* z == -150 */ 214 if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */ 215 } 216 /* 217 * compute 2**(p_h+p_l) 218 */ 219 i = j&0x7fffffff; 220 k = (i>>23)-0x7f; 221 n = 0; 222 if(i>0x3f000000) { /* if |z| > 0.5, set n = [z+0.5] */ 223 n = j+(0x00800000>>(k+1)); 224 k = ((n&0x7fffffff)>>23)-0x7f; /* new k for n */ 225 SET_FLOAT_WORD(t,n&~(0x007fffff>>k)); 226 n = ((n&0x007fffff)|0x00800000)>>(23-k); 227 if(j<0) n = -n; 228 p_h -= t; 229 } 230 t = p_l+p_h; 231 GET_FLOAT_WORD(is,t); 232 SET_FLOAT_WORD(t,is&0xfffff000); 233 u = t*lg2_h; 234 v = (p_l-(t-p_h))*lg2+t*lg2_l; 235 z = u+v; 236 w = v-(z-u); 237 t = z*z; 238 t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); 239 r = (z*t1)/(t1-two)-(w+z*w); 240 z = one-(r-z); 241 GET_FLOAT_WORD(j,z); 242 j += (n<<23); 243 if((j>>23)<=0) z = scalbnf(z,n); /* subnormal output */ 244 else SET_FLOAT_WORD(z,j); 245 return s*z; 246 } 247