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 __FBSDID("$FreeBSD$"); 18 19 #include "math.h" 20 #include "math_private.h" 21 22 static const float 23 bp[] = {1.0, 1.5,}, 24 dp_h[] = { 0.0, 5.84960938e-01,}, /* 0x3f15c000 */ 25 dp_l[] = { 0.0, 1.56322085e-06,}, /* 0x35d1cfdc */ 26 zero = 0.0, 27 half = 0.5, 28 qrtr = 0.25, 29 thrd = 3.33333343e-01, /* 0x3eaaaaab */ 30 one = 1.0, 31 two = 2.0, 32 two24 = 16777216.0, /* 0x4b800000 */ 33 huge = 1.0e30, 34 tiny = 1.0e-30, 35 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */ 36 L1 = 6.0000002384e-01, /* 0x3f19999a */ 37 L2 = 4.2857143283e-01, /* 0x3edb6db7 */ 38 L3 = 3.3333334327e-01, /* 0x3eaaaaab */ 39 L4 = 2.7272811532e-01, /* 0x3e8ba305 */ 40 L5 = 2.3066075146e-01, /* 0x3e6c3255 */ 41 L6 = 2.0697501302e-01, /* 0x3e53f142 */ 42 P1 = 1.6666667163e-01, /* 0x3e2aaaab */ 43 P2 = -2.7777778450e-03, /* 0xbb360b61 */ 44 P3 = 6.6137559770e-05, /* 0x388ab355 */ 45 P4 = -1.6533901999e-06, /* 0xb5ddea0e */ 46 P5 = 4.1381369442e-08, /* 0x3331bb4c */ 47 lg2 = 6.9314718246e-01, /* 0x3f317218 */ 48 lg2_h = 6.93145752e-01, /* 0x3f317200 */ 49 lg2_l = 1.42860654e-06, /* 0x35bfbe8c */ 50 ovt = 4.2995665694e-08, /* -(128-log2(ovfl+.5ulp)) */ 51 cp = 9.6179670095e-01, /* 0x3f76384f =2/(3ln2) */ 52 cp_h = 9.6191406250e-01, /* 0x3f764000 =12b cp */ 53 cp_l = -1.1736857402e-04, /* 0xb8f623c6 =tail of cp_h */ 54 ivln2 = 1.4426950216e+00, /* 0x3fb8aa3b =1/ln2 */ 55 ivln2_h = 1.4426879883e+00, /* 0x3fb8aa00 =16b 1/ln2*/ 56 ivln2_l = 7.0526075433e-06; /* 0x36eca570 =1/ln2 tail*/ 57 58 float 59 __ieee754_powf(float x, float y) 60 { 61 float z,ax,z_h,z_l,p_h,p_l; 62 float y1,t1,t2,r,s,sn,t,u,v,w; 63 int32_t i,j,k,yisint,n; 64 int32_t hx,hy,ix,iy,is; 65 66 GET_FLOAT_WORD(hx,x); 67 GET_FLOAT_WORD(hy,y); 68 ix = hx&0x7fffffff; iy = hy&0x7fffffff; 69 70 /* y==zero: x**0 = 1 */ 71 if(iy==0) return one; 72 73 /* x==1: 1**y = 1, even if y is NaN */ 74 if (hx==0x3f800000) return one; 75 76 /* y!=zero: result is NaN if either arg is NaN */ 77 if(ix > 0x7f800000 || 78 iy > 0x7f800000) 79 return nan_mix(x, y); 80 81 /* determine if y is an odd int when x < 0 82 * yisint = 0 ... y is not an integer 83 * yisint = 1 ... y is an odd int 84 * yisint = 2 ... y is an even int 85 */ 86 yisint = 0; 87 if(hx<0) { 88 if(iy>=0x4b800000) yisint = 2; /* even integer y */ 89 else if(iy>=0x3f800000) { 90 k = (iy>>23)-0x7f; /* exponent */ 91 j = iy>>(23-k); 92 if((j<<(23-k))==iy) yisint = 2-(j&1); 93 } 94 } 95 96 /* special value of y */ 97 if (iy==0x7f800000) { /* y is +-inf */ 98 if (ix==0x3f800000) 99 return one; /* (-1)**+-inf is NaN */ 100 else if (ix > 0x3f800000)/* (|x|>1)**+-inf = inf,0 */ 101 return (hy>=0)? y: zero; 102 else /* (|x|<1)**-,+inf = inf,0 */ 103 return (hy<0)?-y: zero; 104 } 105 if(iy==0x3f800000) { /* y is +-1 */ 106 if(hy<0) return one/x; else return x; 107 } 108 if(hy==0x40000000) return x*x; /* y is 2 */ 109 if(hy==0x3f000000) { /* y is 0.5 */ 110 if(hx>=0) /* x >= +0 */ 111 return __ieee754_sqrtf(x); 112 } 113 114 ax = fabsf(x); 115 /* special value of x */ 116 if(ix==0x7f800000||ix==0||ix==0x3f800000){ 117 z = ax; /*x is +-0,+-inf,+-1*/ 118 if(hy<0) z = one/z; /* z = (1/|x|) */ 119 if(hx<0) { 120 if(((ix-0x3f800000)|yisint)==0) { 121 z = (z-z)/(z-z); /* (-1)**non-int is NaN */ 122 } else if(yisint==1) 123 z = -z; /* (x<0)**odd = -(|x|**odd) */ 124 } 125 return z; 126 } 127 128 n = ((u_int32_t)hx>>31)-1; 129 130 /* (x<0)**(non-int) is NaN */ 131 if((n|yisint)==0) return (x-x)/(x-x); 132 133 sn = one; /* s (sign of result -ve**odd) = -1 else = 1 */ 134 if((n|(yisint-1))==0) sn = -one;/* (-ve)**(odd int) */ 135 136 /* |y| is huge */ 137 if(iy>0x4d000000) { /* if |y| > 2**27 */ 138 /* over/underflow if x is not close to one */ 139 if(ix<0x3f7ffff7) return (hy<0)? sn*huge*huge:sn*tiny*tiny; 140 if(ix>0x3f800007) return (hy>0)? sn*huge*huge:sn*tiny*tiny; 141 /* now |1-x| is tiny <= 2**-20, suffice to compute 142 log(x) by x-x^2/2+x^3/3-x^4/4 */ 143 t = ax-1; /* t has 20 trailing zeros */ 144 w = (t*t)*(half-t*(thrd-t*qrtr)); 145 u = ivln2_h*t; /* ivln2_h has 16 sig. bits */ 146 v = t*ivln2_l-w*ivln2; 147 t1 = u+v; 148 GET_FLOAT_WORD(is,t1); 149 SET_FLOAT_WORD(t1,is&0xfffff000); 150 t2 = v-(t1-u); 151 } else { 152 float s2,s_h,s_l,t_h,t_l; 153 n = 0; 154 /* take care subnormal number */ 155 if(ix<0x00800000) 156 {ax *= two24; n -= 24; GET_FLOAT_WORD(ix,ax); } 157 n += ((ix)>>23)-0x7f; 158 j = ix&0x007fffff; 159 /* determine interval */ 160 ix = j|0x3f800000; /* normalize ix */ 161 if(j<=0x1cc471) k=0; /* |x|<sqrt(3/2) */ 162 else if(j<0x5db3d7) k=1; /* |x|<sqrt(3) */ 163 else {k=0;n+=1;ix -= 0x00800000;} 164 SET_FLOAT_WORD(ax,ix); 165 166 /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ 167 u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ 168 v = one/(ax+bp[k]); 169 s = u*v; 170 s_h = s; 171 GET_FLOAT_WORD(is,s_h); 172 SET_FLOAT_WORD(s_h,is&0xfffff000); 173 /* t_h=ax+bp[k] High */ 174 is = ((ix>>1)&0xfffff000)|0x20000000; 175 SET_FLOAT_WORD(t_h,is+0x00400000+(k<<21)); 176 t_l = ax - (t_h-bp[k]); 177 s_l = v*((u-s_h*t_h)-s_h*t_l); 178 /* compute log(ax) */ 179 s2 = s*s; 180 r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6))))); 181 r += s_l*(s_h+s); 182 s2 = s_h*s_h; 183 t_h = 3+s2+r; 184 GET_FLOAT_WORD(is,t_h); 185 SET_FLOAT_WORD(t_h,is&0xfffff000); 186 t_l = r-((t_h-3)-s2); 187 /* u+v = s*(1+...) */ 188 u = s_h*t_h; 189 v = s_l*t_h+t_l*s; 190 /* 2/(3log2)*(s+...) */ 191 p_h = u+v; 192 GET_FLOAT_WORD(is,p_h); 193 SET_FLOAT_WORD(p_h,is&0xfffff000); 194 p_l = v-(p_h-u); 195 z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */ 196 z_l = cp_l*p_h+p_l*cp+dp_l[k]; 197 /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */ 198 t = n; 199 t1 = (((z_h+z_l)+dp_h[k])+t); 200 GET_FLOAT_WORD(is,t1); 201 SET_FLOAT_WORD(t1,is&0xfffff000); 202 t2 = z_l-(((t1-t)-dp_h[k])-z_h); 203 } 204 205 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ 206 GET_FLOAT_WORD(is,y); 207 SET_FLOAT_WORD(y1,is&0xfffff000); 208 p_l = (y-y1)*t1+y*t2; 209 p_h = y1*t1; 210 z = p_l+p_h; 211 GET_FLOAT_WORD(j,z); 212 if (j>0x43000000) /* if z > 128 */ 213 return sn*huge*huge; /* overflow */ 214 else if (j==0x43000000) { /* if z == 128 */ 215 if(p_l+ovt>z-p_h) return sn*huge*huge; /* overflow */ 216 } 217 else if ((j&0x7fffffff)>0x43160000) /* z <= -150 */ 218 return sn*tiny*tiny; /* underflow */ 219 else if (j==0xc3160000){ /* z == -150 */ 220 if(p_l<=z-p_h) return sn*tiny*tiny; /* underflow */ 221 } 222 /* 223 * compute 2**(p_h+p_l) 224 */ 225 i = j&0x7fffffff; 226 k = (i>>23)-0x7f; 227 n = 0; 228 if(i>0x3f000000) { /* if |z| > 0.5, set n = [z+0.5] */ 229 n = j+(0x00800000>>(k+1)); 230 k = ((n&0x7fffffff)>>23)-0x7f; /* new k for n */ 231 SET_FLOAT_WORD(t,n&~(0x007fffff>>k)); 232 n = ((n&0x007fffff)|0x00800000)>>(23-k); 233 if(j<0) n = -n; 234 p_h -= t; 235 } 236 t = p_l+p_h; 237 GET_FLOAT_WORD(is,t); 238 SET_FLOAT_WORD(t,is&0xffff8000); 239 u = t*lg2_h; 240 v = (p_l-(t-p_h))*lg2+t*lg2_l; 241 z = u+v; 242 w = v-(z-u); 243 t = z*z; 244 t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); 245 r = (z*t1)/(t1-two)-(w+z*w); 246 z = one-(r-z); 247 GET_FLOAT_WORD(j,z); 248 j += (n<<23); 249 if((j>>23)<=0) z = scalbnf(z,n); /* subnormal output */ 250 else SET_FLOAT_WORD(z,j); 251 return sn*z; 252 } 253