1/* origin: FreeBSD /usr/src/lib/msun/src/e_powf.c */ 2/* 3 * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. 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 "libm.h" 17 18static const float 19bp[] = {1.0, 1.5,}, 20dp_h[] = { 0.0, 5.84960938e-01,}, /* 0x3f15c000 */ 21dp_l[] = { 0.0, 1.56322085e-06,}, /* 0x35d1cfdc */ 22two24 = 16777216.0, /* 0x4b800000 */ 23huge = 1.0e30, 24tiny = 1.0e-30, 25/* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */ 26L1 = 6.0000002384e-01, /* 0x3f19999a */ 27L2 = 4.2857143283e-01, /* 0x3edb6db7 */ 28L3 = 3.3333334327e-01, /* 0x3eaaaaab */ 29L4 = 2.7272811532e-01, /* 0x3e8ba305 */ 30L5 = 2.3066075146e-01, /* 0x3e6c3255 */ 31L6 = 2.0697501302e-01, /* 0x3e53f142 */ 32P1 = 1.6666667163e-01, /* 0x3e2aaaab */ 33P2 = -2.7777778450e-03, /* 0xbb360b61 */ 34P3 = 6.6137559770e-05, /* 0x388ab355 */ 35P4 = -1.6533901999e-06, /* 0xb5ddea0e */ 36P5 = 4.1381369442e-08, /* 0x3331bb4c */ 37lg2 = 6.9314718246e-01, /* 0x3f317218 */ 38lg2_h = 6.93145752e-01, /* 0x3f317200 */ 39lg2_l = 1.42860654e-06, /* 0x35bfbe8c */ 40ovt = 4.2995665694e-08, /* -(128-log2(ovfl+.5ulp)) */ 41cp = 9.6179670095e-01, /* 0x3f76384f =2/(3ln2) */ 42cp_h = 9.6191406250e-01, /* 0x3f764000 =12b cp */ 43cp_l = -1.1736857402e-04, /* 0xb8f623c6 =tail of cp_h */ 44ivln2 = 1.4426950216e+00, /* 0x3fb8aa3b =1/ln2 */ 45ivln2_h = 1.4426879883e+00, /* 0x3fb8aa00 =16b 1/ln2*/ 46ivln2_l = 7.0526075433e-06; /* 0x36eca570 =1/ln2 tail*/ 47 48float powf(float x, float y) 49{ 50 float z,ax,z_h,z_l,p_h,p_l; 51 float y1,t1,t2,r,s,sn,t,u,v,w; 52 int32_t i,j,k,yisint,n; 53 int32_t hx,hy,ix,iy,is; 54 55 GET_FLOAT_WORD(hx, x); 56 GET_FLOAT_WORD(hy, y); 57 ix = hx & 0x7fffffff; 58 iy = hy & 0x7fffffff; 59 60 /* x**0 = 1, even if x is NaN */ 61 if (iy == 0) 62 return 1.0f; 63 /* 1**y = 1, even if y is NaN */ 64 if (hx == 0x3f800000) 65 return 1.0f; 66 /* NaN if either arg is NaN */ 67 if (ix > 0x7f800000 || iy > 0x7f800000) 68 return x + y; 69 70 /* determine if y is an odd int when x < 0 71 * yisint = 0 ... y is not an integer 72 * yisint = 1 ... y is an odd int 73 * yisint = 2 ... y is an even int 74 */ 75 yisint = 0; 76 if (hx < 0) { 77 if (iy >= 0x4b800000) 78 yisint = 2; /* even integer y */ 79 else if (iy >= 0x3f800000) { 80 k = (iy>>23) - 0x7f; /* exponent */ 81 j = iy>>(23-k); 82 if ((j<<(23-k)) == iy) 83 yisint = 2 - (j & 1); 84 } 85 } 86 87 /* special value of y */ 88 if (iy == 0x7f800000) { /* y is +-inf */ 89 if (ix == 0x3f800000) /* (-1)**+-inf is 1 */ 90 return 1.0f; 91 else if (ix > 0x3f800000) /* (|x|>1)**+-inf = inf,0 */ 92 return hy >= 0 ? y : 0.0f; 93 else /* (|x|<1)**+-inf = 0,inf */ 94 return hy >= 0 ? 0.0f: -y; 95 } 96 if (iy == 0x3f800000) /* y is +-1 */ 97 return hy >= 0 ? x : 1.0f/x; 98 if (hy == 0x40000000) /* y is 2 */ 99 return x*x; 100 if (hy == 0x3f000000) { /* y is 0.5 */ 101 if (hx >= 0) /* x >= +0 */ 102 return sqrtf(x); 103 } 104 105 ax = fabsf(x); 106 /* special value of x */ 107 if (ix == 0x7f800000 || ix == 0 || ix == 0x3f800000) { /* x is +-0,+-inf,+-1 */ 108 z = ax; 109 if (hy < 0) /* z = (1/|x|) */ 110 z = 1.0f/z; 111 if (hx < 0) { 112 if (((ix-0x3f800000)|yisint) == 0) { 113 z = (z-z)/(z-z); /* (-1)**non-int is NaN */ 114 } else if (yisint == 1) 115 z = -z; /* (x<0)**odd = -(|x|**odd) */ 116 } 117 return z; 118 } 119 120 sn = 1.0f; /* sign of result */ 121 if (hx < 0) { 122 if (yisint == 0) /* (x<0)**(non-int) is NaN */ 123 return (x-x)/(x-x); 124 if (yisint == 1) /* (x<0)**(odd int) */ 125 sn = -1.0f; 126 } 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) 132 return hy < 0 ? sn*huge*huge : sn*tiny*tiny; 133 if (ix > 0x3f800007) 134 return hy > 0 ? sn*huge*huge : sn*tiny*tiny; 135 /* now |1-x| is tiny <= 2**-20, suffice to compute 136 log(x) by x-x^2/2+x^3/3-x^4/4 */ 137 t = ax - 1; /* t has 20 trailing zeros */ 138 w = (t*t)*(0.5f - t*(0.333333333333f - t*0.25f)); 139 u = ivln2_h*t; /* ivln2_h has 16 sig. bits */ 140 v = t*ivln2_l - w*ivln2; 141 t1 = u + v; 142 GET_FLOAT_WORD(is, t1); 143 SET_FLOAT_WORD(t1, is & 0xfffff000); 144 t2 = v - (t1-u); 145 } else { 146 float s2,s_h,s_l,t_h,t_l; 147 n = 0; 148 /* take care subnormal number */ 149 if (ix < 0x00800000) { 150 ax *= two24; 151 n -= 24; 152 GET_FLOAT_WORD(ix, ax); 153 } 154 n += ((ix)>>23) - 0x7f; 155 j = ix & 0x007fffff; 156 /* determine interval */ 157 ix = j | 0x3f800000; /* normalize ix */ 158 if (j <= 0x1cc471) /* |x|<sqrt(3/2) */ 159 k = 0; 160 else if (j < 0x5db3d7) /* |x|<sqrt(3) */ 161 k = 1; 162 else { 163 k = 0; 164 n += 1; 165 ix -= 0x00800000; 166 } 167 SET_FLOAT_WORD(ax, ix); 168 169 /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ 170 u = ax - bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ 171 v = 1.0f/(ax+bp[k]); 172 s = u*v; 173 s_h = s; 174 GET_FLOAT_WORD(is, s_h); 175 SET_FLOAT_WORD(s_h, is & 0xfffff000); 176 /* t_h=ax+bp[k] High */ 177 is = ((ix>>1) & 0xfffff000) | 0x20000000; 178 SET_FLOAT_WORD(t_h, is + 0x00400000 + (k<<21)); 179 t_l = ax - (t_h - bp[k]); 180 s_l = v*((u - s_h*t_h) - s_h*t_l); 181 /* compute log(ax) */ 182 s2 = s*s; 183 r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6))))); 184 r += s_l*(s_h+s); 185 s2 = s_h*s_h; 186 t_h = 3.0f + s2 + r; 187 GET_FLOAT_WORD(is, t_h); 188 SET_FLOAT_WORD(t_h, is & 0xfffff000); 189 t_l = r - ((t_h - 3.0f) - s2); 190 /* u+v = s*(1+...) */ 191 u = s_h*t_h; 192 v = s_l*t_h + t_l*s; 193 /* 2/(3log2)*(s+...) */ 194 p_h = u + v; 195 GET_FLOAT_WORD(is, p_h); 196 SET_FLOAT_WORD(p_h, is & 0xfffff000); 197 p_l = v - (p_h - u); 198 z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */ 199 z_l = cp_l*p_h + p_l*cp+dp_l[k]; 200 /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */ 201 t = (float)n; 202 t1 = (((z_h + z_l) + dp_h[k]) + t); 203 GET_FLOAT_WORD(is, t1); 204 SET_FLOAT_WORD(t1, is & 0xfffff000); 205 t2 = z_l - (((t1 - t) - dp_h[k]) - z_h); 206 } 207 208 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ 209 GET_FLOAT_WORD(is, y); 210 SET_FLOAT_WORD(y1, is & 0xfffff000); 211 p_l = (y-y1)*t1 + y*t2; 212 p_h = y1*t1; 213 z = p_l + p_h; 214 GET_FLOAT_WORD(j, z); 215 if (j > 0x43000000) /* if z > 128 */ 216 return sn*huge*huge; /* overflow */ 217 else if (j == 0x43000000) { /* if z == 128 */ 218 if (p_l + ovt > z - p_h) 219 return sn*huge*huge; /* overflow */ 220 } else if ((j&0x7fffffff) > 0x43160000) /* z < -150 */ // FIXME: check should be (uint32_t)j > 0xc3160000 221 return sn*tiny*tiny; /* underflow */ 222 else if (j == 0xc3160000) { /* z == -150 */ 223 if (p_l <= z-p_h) 224 return sn*tiny*tiny; /* underflow */ 225 } 226 /* 227 * compute 2**(p_h+p_l) 228 */ 229 i = j & 0x7fffffff; 230 k = (i>>23) - 0x7f; 231 n = 0; 232 if (i > 0x3f000000) { /* if |z| > 0.5, set n = [z+0.5] */ 233 n = j + (0x00800000>>(k+1)); 234 k = ((n&0x7fffffff)>>23) - 0x7f; /* new k for n */ 235 SET_FLOAT_WORD(t, n & ~(0x007fffff>>k)); 236 n = ((n&0x007fffff)|0x00800000)>>(23-k); 237 if (j < 0) 238 n = -n; 239 p_h -= t; 240 } 241 t = p_l + p_h; 242 GET_FLOAT_WORD(is, t); 243 SET_FLOAT_WORD(t, is & 0xffff8000); 244 u = t*lg2_h; 245 v = (p_l-(t-p_h))*lg2 + t*lg2_l; 246 z = u + v; 247 w = v - (z - u); 248 t = z*z; 249 t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); 250 r = (z*t1)/(t1-2.0f) - (w+z*w); 251 z = 1.0f - (r - z); 252 GET_FLOAT_WORD(j, z); 253 j += n<<23; 254 if ((j>>23) <= 0) /* subnormal output */ 255 z = scalbnf(z, n); 256 else 257 SET_FLOAT_WORD(z, j); 258 return sn*z; 259} 260