1 /* e_powf.c -- float version of e_pow.c. 2 * Conversion to float by Ian Lance Taylor, Cygnus Support, ian (at) 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 #ifndef lint 17 static char rcsid[] = "$FreeBSD: src/lib/msun/src/e_powf.c,v 1.12 2004/06/01 19:33:30 bde Exp $"; 18 #endif 19 20 #include "math.h" 21 #include "math_private.h" 22 23 static const float 24 bp[] = {1.0, 1.5,}, 25 dp_h[] = { 0.0, 5.84960938e-01,}, /* 0x3f15c000 */ 26 dp_l[] = { 0.0, 1.56322085e-06,}, /* 0x35d1cfdc */ 27 zero = 0.0, 28 one = 1.0, 29 two = 2.0, 30 two24 = 16777216.0, /* 0x4b800000 */ 31 huge = 1.0e30, 32 tiny = 1.0e-30, 33 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */ 34 L1 = 6.0000002384e-01, /* 0x3f19999a */ 35 L2 = 4.2857143283e-01, /* 0x3edb6db7 */ 36 L3 = 3.3333334327e-01, /* 0x3eaaaaab */ 37 L4 = 2.7272811532e-01, /* 0x3e8ba305 */ 38 L5 = 2.3066075146e-01, /* 0x3e6c3255 */ 39 L6 = 2.0697501302e-01, /* 0x3e53f142 */ 40 P1 = 1.6666667163e-01, /* 0x3e2aaaab */ 41 P2 = -2.7777778450e-03, /* 0xbb360b61 */ 42 P3 = 6.6137559770e-05, /* 0x388ab355 */ 43 P4 = -1.6533901999e-06, /* 0xb5ddea0e */ 44 P5 = 4.1381369442e-08, /* 0x3331bb4c */ 45 lg2 = 6.9314718246e-01, /* 0x3f317218 */ 46 lg2_h = 6.93145752e-01, /* 0x3f317200 */ 47 lg2_l = 1.42860654e-06, /* 0x35bfbe8c */ 48 ovt = 4.2995665694e-08, /* -(128-log2(ovfl+.5ulp)) */ 49 cp = 9.6179670095e-01, /* 0x3f76384f =2/(3ln2) */ 50 cp_h = 9.6179199219e-01, /* 0x3f763800 =head of cp */ 51 cp_l = 4.7017383622e-06, /* 0x369dc3a0 =tail of cp_h */ 52 ivln2 = 1.4426950216e+00, /* 0x3fb8aa3b =1/ln2 */ 53 ivln2_h = 1.4426879883e+00, /* 0x3fb8aa00 =16b 1/ln2*/ 54 ivln2_l = 7.0526075433e-06; /* 0x36eca570 =1/ln2 tail*/ 55 56 float 57 __ieee754_powf(float x, float y) 58 { 59 float z,ax,z_h,z_l,p_h,p_l; 60 float y1,t1,t2,r,s,sn,t,u,v,w; 61 int32_t i,j,k,yisint,n; 62 int32_t hx,hy,ix,iy,is; 63 64 GET_FLOAT_WORD(hx,x); 65 GET_FLOAT_WORD(hy,y); 66 ix = hx&0x7fffffff; iy = hy&0x7fffffff; 67 68 /* y==zero: x**0 = 1 */ 69 if(iy==0) return one; 70 71 /* +-NaN return x+y */ 72 if(ix > 0x7f800000 || 73 iy > 0x7f800000) 74 return x+y; 75 76 /* determine if y is an odd int when x < 0 77 * yisint = 0 ... y is not an integer 78 * yisint = 1 ... y is an odd int 79 * yisint = 2 ... y is an even int 80 */ 81 yisint = 0; 82 if(hx<0) { 83 if(iy>=0x4b800000) yisint = 2; /* even integer y */ 84 else if(iy>=0x3f800000) { 85 k = (iy>>23)-0x7f; /* exponent */ 86 j = iy>>(23-k); 87 if((j<<(23-k))==iy) yisint = 2-(j&1); 88 } 89 } 90 91 /* special value of y */ 92 if (iy==0x7f800000) { /* y is +-inf */ 93 if (ix==0x3f800000) 94 return y - y; /* inf**+-1 is NaN */ 95 else if (ix > 0x3f800000)/* (|x|>1)**+-inf = inf,0 */ 96 return (hy>=0)? y: zero; 97 else /* (|x|<1)**-,+inf = inf,0 */ 98 return (hy<0)?-y: zero; 99 } 100 if(iy==0x3f800000) { /* y is +-1 */ 101 if(hy<0) return one/x; else return x; 102 } 103 if(hy==0x40000000) return x*x; /* y is 2 */ 104 if(hy==0x3f000000) { /* y is 0.5 */ 105 if(hx>=0) /* x >= +0 */ 106 return __ieee754_sqrtf(x); 107 } 108 109 ax = fabsf(x); 110 /* special value of x */ 111 if(ix==0x7f800000||ix==0||ix==0x3f800000){ 112 z = ax; /*x is +-0,+-inf,+-1*/ 113 if(hy<0) z = one/z; /* z = (1/|x|) */ 114 if(hx<0) { 115 if(((ix-0x3f800000)|yisint)==0) { 116 z = (z-z)/(z-z); /* (-1)**non-int is NaN */ 117 } else if(yisint==1) 118 z = -z; /* (x<0)**odd = -(|x|**odd) */ 119 } 120 return z; 121 } 122 123 n = ((u_int32_t)hx>>31)-1; 124 125 /* (x<0)**(non-int) is NaN */ 126 if((n|yisint)==0) return (x-x)/(x-x); 127 128 sn = one; /* s (sign of result -ve**odd) = -1 else = 1 */ 129 if((n|(yisint-1))==0) sn = -one;/* (-ve)**(odd int) */ 130 131 /* |y| is huge */ 132 if(iy>0x4d000000) { /* if |y| > 2**27 */ 133 /* over/underflow if x is not close to one */ 134 if(ix<0x3f7ffff8) return (hy<0)? sn*huge*huge:sn*tiny*tiny; 135 if(ix>0x3f800007) return (hy>0)? sn*huge*huge:sn*tiny*tiny; 136 /* now |1-x| is tiny <= 2**-20, suffice to compute 137 log(x) by x-x^2/2+x^3/3-x^4/4 */ 138 t = ax-1; /* t has 20 trailing zeros */ 139 w = (t*t)*((float)0.5-t*((float)0.333333333333-t*(float)0.25)); 140 u = ivln2_h*t; /* ivln2_h has 16 sig. bits */ 141 v = t*ivln2_l-w*ivln2; 142 t1 = u+v; 143 GET_FLOAT_WORD(is,t1); 144 SET_FLOAT_WORD(t1,is&0xfffff000); 145 t2 = v-(t1-u); 146 } else { 147 float s2,s_h,s_l,t_h,t_l; 148 n = 0; 149 /* take care subnormal number */ 150 if(ix<0x00800000) 151 {ax *= two24; n -= 24; GET_FLOAT_WORD(ix,ax); } 152 n += ((ix)>>23)-0x7f; 153 j = ix&0x007fffff; 154 /* determine interval */ 155 ix = j|0x3f800000; /* normalize ix */ 156 if(j<=0x1cc471) k=0; /* |x|<sqrt(3/2) */ 157 else if(j<0x5db3d7) k=1; /* |x|<sqrt(3) */ 158 else {k=0;n+=1;ix -= 0x00800000;} 159 SET_FLOAT_WORD(ax,ix); 160 161 /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ 162 u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ 163 v = one/(ax+bp[k]); 164 s = u*v; 165 s_h = s; 166 GET_FLOAT_WORD(is,s_h); 167 SET_FLOAT_WORD(s_h,is&0xfffff000); 168 /* t_h=ax+bp[k] High */ 169 is = ((ix>>1)&0xfffff000)|0x20000000; 170 SET_FLOAT_WORD(t_h,is+0x00400000+(k<<21)); 171 t_l = ax - (t_h-bp[k]); 172 s_l = v*((u-s_h*t_h)-s_h*t_l); 173 /* compute log(ax) */ 174 s2 = s*s; 175 r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6))))); 176 r += s_l*(s_h+s); 177 s2 = s_h*s_h; 178 t_h = (float)3.0+s2+r; 179 GET_FLOAT_WORD(is,t_h); 180 SET_FLOAT_WORD(t_h,is&0xfffff000); 181 t_l = r-((t_h-(float)3.0)-s2); 182 /* u+v = s*(1+...) */ 183 u = s_h*t_h; 184 v = s_l*t_h+t_l*s; 185 /* 2/(3log2)*(s+...) */ 186 p_h = u+v; 187 GET_FLOAT_WORD(is,p_h); 188 SET_FLOAT_WORD(p_h,is&0xfffff000); 189 p_l = v-(p_h-u); 190 z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */ 191 z_l = cp_l*p_h+p_l*cp+dp_l[k]; 192 /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */ 193 t = (float)n; 194 t1 = (((z_h+z_l)+dp_h[k])+t); 195 GET_FLOAT_WORD(is,t1); 196 SET_FLOAT_WORD(t1,is&0xfffff000); 197 t2 = z_l-(((t1-t)-dp_h[k])-z_h); 198 } 199 200 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ 201 GET_FLOAT_WORD(is,y); 202 SET_FLOAT_WORD(y1,is&0xfffff000); 203 p_l = (y-y1)*t1+y*t2; 204 p_h = y1*t1; 205 z = p_l+p_h; 206 GET_FLOAT_WORD(j,z); 207 if (j>0x43000000) /* if z > 128 */ 208 return sn*huge*huge; /* overflow */ 209 else if (j==0x43000000) { /* if z == 128 */ 210 if(p_l+ovt>z-p_h) return sn*huge*huge; /* overflow */ 211 } 212 else if ((j&0x7fffffff)>0x43160000) /* z <= -150 */ 213 return sn*tiny*tiny; /* underflow */ 214 else if (j==0xc3160000){ /* z == -150 */ 215 if(p_l<=z-p_h) return sn*tiny*tiny; /* underflow */ 216 } 217 /* 218 * compute 2**(p_h+p_l) 219 */ 220 i = j&0x7fffffff; 221 k = (i>>23)-0x7f; 222 n = 0; 223 if(i>0x3f000000) { /* if |z| > 0.5, set n = [z+0.5] */ 224 n = j+(0x00800000>>(k+1)); 225 k = ((n&0x7fffffff)>>23)-0x7f; /* new k for n */ 226 SET_FLOAT_WORD(t,n&~(0x007fffff>>k)); 227 n = ((n&0x007fffff)|0x00800000)>>(23-k); 228 if(j<0) n = -n; 229 p_h -= t; 230 } 231 t = p_l+p_h; 232 GET_FLOAT_WORD(is,t); 233 SET_FLOAT_WORD(t,is&0xffff8000); 234 u = t*lg2_h; 235 v = (p_l-(t-p_h))*lg2+t*lg2_l; 236 z = u+v; 237 w = v-(z-u); 238 t = z*z; 239 t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); 240 r = (z*t1)/(t1-two)-(w+z*w); 241 z = one-(r-z); 242 GET_FLOAT_WORD(j,z); 243 j += (n<<23); 244 if((j>>23)<=0) z = scalbnf(z,n); /* subnormal output */ 245 else SET_FLOAT_WORD(z,j); 246 return sn*z; 247 } 248
