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 1.8 1997/10/09 11:29:42 lukem 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 #ifdef __STDC__ 27 static const float 28 #else 29 static float 30 #endif 31 bp[] = {1.0, 1.5,}, 32 dp_h[] = { 0.0, 5.84960938e-01,}, /* 0x3f15c000 */ 33 dp_l[] = { 0.0, 1.56322085e-06,}, /* 0x35d1cfdc */ 34 zero = 0.0, 35 one = 1.0, 36 two = 2.0, 37 two24 = 16777216.0, /* 0x4b800000 */ 38 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */ 39 L1 = 6.0000002384e-01, /* 0x3f19999a */ 40 L2 = 4.2857143283e-01, /* 0x3edb6db7 */ 41 L3 = 3.3333334327e-01, /* 0x3eaaaaab */ 42 L4 = 2.7272811532e-01, /* 0x3e8ba305 */ 43 L5 = 2.3066075146e-01, /* 0x3e6c3255 */ 44 L6 = 2.0697501302e-01, /* 0x3e53f142 */ 45 P1 = 1.6666667163e-01, /* 0x3e2aaaab */ 46 P2 = -2.7777778450e-03, /* 0xbb360b61 */ 47 P3 = 6.6137559770e-05, /* 0x388ab355 */ 48 P4 = -1.6533901999e-06, /* 0xb5ddea0e */ 49 P5 = 4.1381369442e-08, /* 0x3331bb4c */ 50 lg2 = 6.9314718246e-01, /* 0x3f317218 */ 51 lg2_h = 6.93145752e-01, /* 0x3f317200 */ 52 lg2_l = 1.42860654e-06, /* 0x35bfbe8c */ 53 ovt = 4.2995665694e-08, /* -(128-log2(ovfl+.5ulp)) */ 54 cp = 9.6179670095e-01, /* 0x3f76384f =2/(3ln2) */ 55 cp_h = 9.6179199219e-01, /* 0x3f763800 =head of cp */ 56 cp_l = 4.7017383622e-06, /* 0x369dc3a0 =tail of cp_h */ 57 ivln2 = 1.4426950216e+00, /* 0x3fb8aa3b =1/ln2 */ 58 ivln2_h = 1.4426879883e+00, /* 0x3fb8aa00 =16b 1/ln2*/ 59 ivln2_l = 7.0526075433e-06; /* 0x36eca570 =1/ln2 tail*/ 60 61 #ifdef __STDC__ 62 float __ieee754_powf(float x, float y) 63 #else 64 float __ieee754_powf(x,y) 65 float x, y; 66 #endif 67 { 68 float z,ax,z_h,z_l,p_h,p_l; 69 float y1,t1,t2,r,s,t,u,v,w; 70 int32_t i,j,k,yisint,n; 71 int32_t hx,hy,ix,iy,is; 72 73 GET_FLOAT_WORD(hx,x); 74 GET_FLOAT_WORD(hy,y); 75 ix = hx&0x7fffffff; iy = hy&0x7fffffff; 76 77 /* y==zero: x**0 = 1 */ 78 if(iy==0) return one; 79 80 /* +-NaN return x+y */ 81 if(ix > 0x7f800000 || 82 iy > 0x7f800000) 83 return x+y; 84 85 /* determine if y is an odd int when x < 0 86 * yisint = 0 ... y is not an integer 87 * yisint = 1 ... y is an odd int 88 * yisint = 2 ... y is an even int 89 */ 90 yisint = 0; 91 if(hx<0) { 92 if(iy>=0x4b800000) yisint = 2; /* even integer y */ 93 else if(iy>=0x3f800000) { 94 k = (iy>>23)-0x7f; /* exponent */ 95 j = iy>>(23-k); 96 if((j<<(23-k))==iy) yisint = 2-(j&1); 97 } 98 } 99 100 /* special value of y */ 101 if (iy==0x7f800000) { /* y is +-inf */ 102 if (ix==0x3f800000) 103 return y - y; /* inf**+-1 is NaN */ 104 else if (ix > 0x3f800000)/* (|x|>1)**+-inf = inf,0 */ 105 return (hy>=0)? y: zero; 106 else /* (|x|<1)**-,+inf = inf,0 */ 107 return (hy<0)?-y: zero; 108 } 109 if(iy==0x3f800000) { /* y is +-1 */ 110 if(hy<0) return one/x; else return x; 111 } 112 if(hy==0x40000000) return x*x; /* y is 2 */ 113 if(hy==0x3f000000) { /* y is 0.5 */ 114 if(hx>=0) /* x >= +0 */ 115 return __ieee754_sqrtf(x); 116 } 117 118 ax = fabsf(x); 119 /* special value of x */ 120 if(ix==0x7f800000||ix==0||ix==0x3f800000){ 121 z = ax; /*x is +-0,+-inf,+-1*/ 122 if(hy<0) z = one/z; /* z = (1/|x|) */ 123 if(hx<0) { 124 if(((ix-0x3f800000)|yisint)==0) { 125 z = (z-z)/(z-z); /* (-1)**non-int is NaN */ 126 } else if(yisint==1) 127 z = -z; /* (x<0)**odd = -(|x|**odd) */ 128 } 129 return z; 130 } 131 132 /* (x<0)**(non-int) is NaN */ 133 if(((((u_int32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x); 134 135 /* |y| is huge */ 136 if(iy>0x4d000000) { /* if |y| > 2**27 */ 137 /* over/underflow if x is not close to one */ 138 if(ix<0x3f7ffff8) return (hy<0)? huge*huge:tiny*tiny; 139 if(ix>0x3f800007) return (hy>0)? huge*huge:tiny*tiny; 140 /* now |1-x| is tiny <= 2**-20, suffice to compute 141 log(x) by x-x^2/2+x^3/3-x^4/4 */ 142 t = x-1; /* t has 20 trailing zeros */ 143 w = (t*t)*((float)0.5-t*((float)0.333333333333-t*(float)0.25)); 144 u = ivln2_h*t; /* ivln2_h has 16 sig. bits */ 145 v = t*ivln2_l-w*ivln2; 146 t1 = u+v; 147 GET_FLOAT_WORD(is,t1); 148 SET_FLOAT_WORD(t1,is&0xfffff000); 149 t2 = v-(t1-u); 150 } else { 151 float s2,s_h,s_l,t_h,t_l; 152 n = 0; 153 /* take care subnormal number */ 154 if(ix<0x00800000) 155 {ax *= two24; n -= 24; GET_FLOAT_WORD(ix,ax); } 156 n += ((ix)>>23)-0x7f; 157 j = ix&0x007fffff; 158 /* determine interval */ 159 ix = j|0x3f800000; /* normalize ix */ 160 if(j<=0x1cc471) k=0; /* |x|<sqrt(3/2) */ 161 else if(j<0x5db3d7) k=1; /* |x|<sqrt(3) */ 162 else {k=0;n+=1;ix -= 0x00800000;} 163 SET_FLOAT_WORD(ax,ix); 164 165 /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ 166 u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ 167 v = one/(ax+bp[k]); 168 s = u*v; 169 s_h = s; 170 GET_FLOAT_WORD(is,s_h); 171 SET_FLOAT_WORD(s_h,is&0xfffff000); 172 /* t_h=ax+bp[k] High */ 173 SET_FLOAT_WORD(t_h,((ix>>1)|0x20000000)+0x0040000+(k<<21)); 174 t_l = ax - (t_h-bp[k]); 175 s_l = v*((u-s_h*t_h)-s_h*t_l); 176 /* compute log(ax) */ 177 s2 = s*s; 178 r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6))))); 179 r += s_l*(s_h+s); 180 s2 = s_h*s_h; 181 t_h = (float)3.0+s2+r; 182 GET_FLOAT_WORD(is,t_h); 183 SET_FLOAT_WORD(t_h,is&0xfffff000); 184 t_l = r-((t_h-(float)3.0)-s2); 185 /* u+v = s*(1+...) */ 186 u = s_h*t_h; 187 v = s_l*t_h+t_l*s; 188 /* 2/(3log2)*(s+...) */ 189 p_h = u+v; 190 GET_FLOAT_WORD(is,p_h); 191 SET_FLOAT_WORD(p_h,is&0xfffff000); 192 p_l = v-(p_h-u); 193 z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */ 194 z_l = cp_l*p_h+p_l*cp+dp_l[k]; 195 /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */ 196 t = (float)n; 197 t1 = (((z_h+z_l)+dp_h[k])+t); 198 GET_FLOAT_WORD(is,t1); 199 SET_FLOAT_WORD(t1,is&0xfffff000); 200 t2 = z_l-(((t1-t)-dp_h[k])-z_h); 201 } 202 203 s = one; /* s (sign of result -ve**odd) = -1 else = 1 */ 204 if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0) 205 s = -one; /* (-ve)**(odd int) */ 206 207 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ 208 GET_FLOAT_WORD(is,y); 209 SET_FLOAT_WORD(y1,is&0xfffff000); 210 p_l = (y-y1)*t1+y*t2; 211 p_h = y1*t1; 212 z = p_l+p_h; 213 GET_FLOAT_WORD(j,z); 214 if (j>0x43000000) /* if z > 128 */ 215 return s*huge*huge; /* overflow */ 216 else if (j==0x43000000) { /* if z == 128 */ 217 if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */ 218 } 219 else if ((j&0x7fffffff)>0x43160000) /* z <= -150 */ 220 return s*tiny*tiny; /* underflow */ 221 else if (j==0xc3160000){ /* z == -150 */ 222 if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */ 223 } 224 /* 225 * compute 2**(p_h+p_l) 226 */ 227 i = j&0x7fffffff; 228 k = (i>>23)-0x7f; 229 n = 0; 230 if(i>0x3f000000) { /* if |z| > 0.5, set n = [z+0.5] */ 231 n = j+(0x00800000>>(k+1)); 232 k = ((n&0x7fffffff)>>23)-0x7f; /* new k for n */ 233 SET_FLOAT_WORD(t,n&~(0x007fffff>>k)); 234 n = ((n&0x007fffff)|0x00800000)>>(23-k); 235 if(j<0) n = -n; 236 p_h -= t; 237 } 238 t = p_l+p_h; 239 GET_FLOAT_WORD(is,t); 240 SET_FLOAT_WORD(t,is&0xfffff000); 241 u = t*lg2_h; 242 v = (p_l-(t-p_h))*lg2+t*lg2_l; 243 z = u+v; 244 w = v-(z-u); 245 t = z*z; 246 t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); 247 r = (z*t1)/(t1-two)-(w+z*w); 248 z = one-(r-z); 249 GET_FLOAT_WORD(j,z); 250 j += (n<<23); 251 if((j>>23)<=0) z = scalbnf(z,n); /* subnormal output */ 252 else SET_FLOAT_WORD(z,j); 253 return s*z; 254 } 255