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 1.15 2010/04/23 19:17:07 drochner Exp $"); 19 #endif 20 21 #include "namespace.h" 22 #include "math.h" 23 #include "math_private.h" 24 25 static const float huge = 1.0e+30, tiny = 1.0e-30; 26 27 static const float 28 bp[] = {1.0, 1.5,}, 29 dp_h[] = { 0.0, 5.84960938e-01,}, /* 0x3f15c000 */ 30 dp_l[] = { 0.0, 1.56322085e-06,}, /* 0x35d1cfdc */ 31 zero = 0.0, 32 one = 1.0, 33 two = 2.0, 34 two24 = 16777216.0, /* 0x4b800000 */ 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 = 9.6179199219e-01, /* 0x3f763800 =head of cp */ 53 cp_l = 4.7017383622e-06, /* 0x369dc3a0 =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 yy1,t1,t2,r,s,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 /* +-NaN return x+y */ 74 if(ix > 0x7f800000 || 75 iy > 0x7f800000) 76 return x+y; 77 78 /* determine if y is an odd int when x < 0 79 * yisint = 0 ... y is not an integer 80 * yisint = 1 ... y is an odd int 81 * yisint = 2 ... y is an even int 82 */ 83 yisint = 0; 84 if(hx<0) { 85 if(iy>=0x4b800000) yisint = 2; /* even integer y */ 86 else if(iy>=0x3f800000) { 87 k = (iy>>23)-0x7f; /* exponent */ 88 j = iy>>(23-k); 89 if((j<<(23-k))==iy) yisint = 2-(j&1); 90 } 91 } 92 93 /* special value of y */ 94 if (iy==0x7f800000) { /* y is +-inf */ 95 if (ix==0x3f800000) 96 return y - y; /* inf**+-1 is NaN */ 97 else if (ix > 0x3f800000)/* (|x|>1)**+-inf = inf,0 */ 98 return (hy>=0)? y: zero; 99 else /* (|x|<1)**-,+inf = inf,0 */ 100 return (hy<0)?-y: zero; 101 } 102 if(iy==0x3f800000) { /* y is +-1 */ 103 if(hy<0) return one/x; else return x; 104 } 105 if(hy==0x40000000) return x*x; /* y is 2 */ 106 if(hy==0x3f000000) { /* y is 0.5 */ 107 if(hx>=0) /* x >= +0 */ 108 return __ieee754_sqrtf(x); 109 } 110 111 ax = fabsf(x); 112 /* special value of x */ 113 if(ix==0x7f800000||ix==0||ix==0x3f800000){ 114 z = ax; /*x is +-0,+-inf,+-1*/ 115 if(hy<0) z = one/z; /* z = (1/|x|) */ 116 if(hx<0) { 117 if(((ix-0x3f800000)|yisint)==0) { 118 z = (z-z)/(z-z); /* (-1)**non-int is NaN */ 119 } else if(yisint==1) 120 z = -z; /* (x<0)**odd = -(|x|**odd) */ 121 } 122 return z; 123 } 124 125 /* (x<0)**(non-int) is NaN */ 126 if(((((u_int32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x); 127 128 /* |y| is huge */ 129 if(iy>0x4d000000) { /* if |y| > 2**27 */ 130 /* over/underflow if x is not close to one */ 131 if(ix<0x3f7ffff8) return (hy<0)? huge*huge:tiny*tiny; 132 if(ix>0x3f800007) return (hy>0)? huge*huge:tiny*tiny; 133 /* now |1-x| is tiny <= 2**-20, suffice to compute 134 log(x) by x-x^2/2+x^3/3-x^4/4 */ 135 t = ax-one; /* t has 20 trailing zeros */ 136 w = (t*t)*((float)0.5-t*((float)0.333333333333-t*(float)0.25)); 137 u = ivln2_h*t; /* ivln2_h has 16 sig. bits */ 138 v = t*ivln2_l-w*ivln2; 139 t1 = u+v; 140 GET_FLOAT_WORD(is,t1); 141 SET_FLOAT_WORD(t1,is&0xfffff000); 142 t2 = v-(t1-u); 143 } else { 144 float s2,s_h,s_l,t_h,t_l; 145 n = 0; 146 /* take care subnormal number */ 147 if(ix<0x00800000) 148 {ax *= two24; n -= 24; GET_FLOAT_WORD(ix,ax); } 149 n += ((ix)>>23)-0x7f; 150 j = ix&0x007fffff; 151 /* determine interval */ 152 ix = j|0x3f800000; /* normalize ix */ 153 if(j<=0x1cc471) k=0; /* |x|<sqrt(3/2) */ 154 else if(j<0x5db3d7) k=1; /* |x|<sqrt(3) */ 155 else {k=0;n+=1;ix -= 0x00800000;} 156 SET_FLOAT_WORD(ax,ix); 157 158 /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ 159 u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ 160 v = one/(ax+bp[k]); 161 s = u*v; 162 s_h = s; 163 GET_FLOAT_WORD(is,s_h); 164 SET_FLOAT_WORD(s_h,is&0xfffff000); 165 /* t_h=ax+bp[k] High */ 166 SET_FLOAT_WORD(t_h,((ix>>1)|0x20000000)+0x0040000+(k<<21)); 167 t_l = ax - (t_h-bp[k]); 168 s_l = v*((u-s_h*t_h)-s_h*t_l); 169 /* compute log(ax) */ 170 s2 = s*s; 171 r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6))))); 172 r += s_l*(s_h+s); 173 s2 = s_h*s_h; 174 t_h = (float)3.0+s2+r; 175 GET_FLOAT_WORD(is,t_h); 176 SET_FLOAT_WORD(t_h,is&0xfffff000); 177 t_l = r-((t_h-(float)3.0)-s2); 178 /* u+v = s*(1+...) */ 179 u = s_h*t_h; 180 v = s_l*t_h+t_l*s; 181 /* 2/(3log2)*(s+...) */ 182 p_h = u+v; 183 GET_FLOAT_WORD(is,p_h); 184 SET_FLOAT_WORD(p_h,is&0xfffff000); 185 p_l = v-(p_h-u); 186 z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */ 187 z_l = cp_l*p_h+p_l*cp+dp_l[k]; 188 /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */ 189 t = (float)n; 190 t1 = (((z_h+z_l)+dp_h[k])+t); 191 GET_FLOAT_WORD(is,t1); 192 SET_FLOAT_WORD(t1,is&0xfffff000); 193 t2 = z_l-(((t1-t)-dp_h[k])-z_h); 194 } 195 196 s = one; /* s (sign of result -ve**odd) = -1 else = 1 */ 197 if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0) 198 s = -one; /* (-ve)**(odd int) */ 199 200 /* split up y into yy1+y2 and compute (yy1+y2)*(t1+t2) */ 201 GET_FLOAT_WORD(is,y); 202 SET_FLOAT_WORD(yy1,is&0xfffff000); 203 p_l = (y-yy1)*t1+y*t2; 204 p_h = yy1*t1; 205 z = p_l+p_h; 206 GET_FLOAT_WORD(j,z); 207 if (j>0x43000000) /* if z > 128 */ 208 return s*huge*huge; /* overflow */ 209 else if (j==0x43000000) { /* if z == 128 */ 210 if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */ 211 } 212 else if ((uint32_t)j==0xc3160000){ /* z == -150 */ 213 if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */ 214 } 215 else if ((j&0x7fffffff)>0x43160000) /* z <= -150 */ 216 return s*tiny*tiny; /* underflow */ 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&0xfffff000); 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 s*z; 247 } 248