RTAI API: base/math/e_pow.c Source File

00001 /* @(#)e_pow.c 5.1 93/09/24 */ 00002 /* 00003 * ==================================================== 00004 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 00005 * 00006 * Developed at SunPro, a Sun Microsystems, Inc. business. 00007 * Permission to use, copy, modify, and distribute this 00008 * software is freely granted, provided that this notice 00009 * is preserved. 00010 * ==================================================== 00011 */ 00012 00013 #if defined(LIBM_SCCS) && !defined(lint) 00014 static char rcsid[] = "$NetBSD: e_pow.c,v 1.9 1995/05/12 04:57:32 jtc Exp $"; 00015 #endif 00016 00017 /* __ieee754_pow(x,y) return x**y 00018 * 00019 * n 00020 * Method: Let x = 2 * (1+f) 00021 * 1. Compute and return log2(x) in two pieces: 00022 * log2(x) = w1 + w2, 00023 * where w1 has 53-24 = 29 bit trailing zeros. 00024 * 2. Perform y*log2(x) = n+y' by simulating muti-precision 00025 * arithmetic, where |y'|<=0.5. 00026 * 3. Return x**y = 2**n*exp(y'*log2) 00027 * 00028 * Special cases: 00029 * 1. (anything) ** 0 is 1 00030 * 2. (anything) ** 1 is itself 00031 * 3. (anything) ** NAN is NAN 00032 * 4. NAN ** (anything except 0) is NAN 00033 * 5. +-(|x| > 1) ** +INF is +INF 00034 * 6. +-(|x| > 1) ** -INF is +0 00035 * 7. +-(|x| < 1) ** +INF is +0 00036 * 8. +-(|x| < 1) ** -INF is +INF 00037 * 9. +-1 ** +-INF is NAN 00038 * 10. +0 ** (+anything except 0, NAN) is +0 00039 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0 00040 * 12. +0 ** (-anything except 0, NAN) is +INF 00041 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF 00042 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) ) 00043 * 15. +INF ** (+anything except 0,NAN) is +INF 00044 * 16. +INF ** (-anything except 0,NAN) is +0 00045 * 17. -INF ** (anything) = -0 ** (-anything) 00046 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) 00047 * 19. (-anything except 0 and inf) ** (non-integer) is NAN 00048 * 00049 * Accuracy: 00050 * pow(x,y) returns x**y nearly rounded. In particular 00051 * pow(integer,integer) 00052 * always returns the correct integer provided it is 00053 * representable. 00054 * 00055 * Constants : 00056 * The hexadecimal values are the intended ones for the following 00057 * constants. The decimal values may be used, provided that the 00058 * compiler will convert from decimal to binary accurately enough 00059 * to produce the hexadecimal values shown. 00060 */ 00061 00062 #include "math.h" 00063 #include "mathP.h" 00064 00065 #ifdef __STDC__ 00066 static const double 00067 #else 00068 static double 00069 #endif 00070 bp[] = {1.0, 1.5,}, 00071 dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */ 00072 dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */ 00073 zero = 0.0, 00074 one = 1.0, 00075 two = 2.0, 00076 two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */ 00077 huge = 1.0e300, 00078 tiny = 1.0e-300, 00079 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */ 00080 L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */ 00081 L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */ 00082 L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */ 00083 L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */ 00084 L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */ 00085 L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */ 00086 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ 00087 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ 00088 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ 00089 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ 00090 P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */ 00091 lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */ 00092 lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */ 00093 lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */ 00094 ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */ 00095 cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */ 00096 cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */ 00097 cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/ 00098 ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */ 00099 ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/ 00100 ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/ 00101 00102 #ifdef __STDC__ 00103 double __ieee754_pow(double x, double y) 00104 #else 00105 double __ieee754_pow(x,y) 00106 double x, y; 00107 #endif 00108 { 00109 double z,ax,z_h,z_l,p_h,p_l; 00110 double y1,t1,t2,r,s,t,u,v,w; 00111 int32_t i,j,k,yisint,n; 00112 int32_t hx,hy,ix,iy; 00113 u_int32_t lx,ly; 00114 00115 EXTRACT_WORDS(hx,lx,x); 00116 EXTRACT_WORDS(hy,ly,y); 00117 ix = hx&0x7fffffff; iy = hy&0x7fffffff; 00118 00119 /* y==zero: x**0 = 1 */ 00120 if((iy|ly)==0) return one; 00121 00122 /* +-NaN return x+y */ 00123 if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) || 00124 iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0))) 00125 return x+y; 00126 00127 /* determine if y is an odd int when x < 0 00128 * yisint = 0 ... y is not an integer 00129 * yisint = 1 ... y is an odd int 00130 * yisint = 2 ... y is an even int 00131 */ 00132 yisint = 0; 00133 if(hx<0) { 00134 if(iy>=0x43400000) yisint = 2; /* even integer y */ 00135 else if(iy>=0x3ff00000) { 00136 k = (iy>>20)-0x3ff; /* exponent */ 00137 if(k>20) { 00138 j = ly>>(52-k); 00139 if((j<<(52-k))==ly) yisint = 2-(j&1); 00140 } else if(ly==0) { 00141 j = iy>>(20-k); 00142 if((j<<(20-k))==iy) yisint = 2-(j&1); 00143 } 00144 } 00145 } 00146 00147 /* special value of y */ 00148 if(ly==0) { 00149 if (iy==0x7ff00000) { /* y is +-inf */ 00150 if(((ix-0x3ff00000)|lx)==0) 00151 return y - y; /* inf**+-1 is NaN */ 00152 else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */ 00153 return (hy>=0)? y: zero; 00154 else /* (|x|<1)**-,+inf = inf,0 */ 00155 return (hy<0)?-y: zero; 00156 } 00157 if(iy==0x3ff00000) { /* y is +-1 */ 00158 if(hy<0) return one/x; else return x; 00159 } 00160 if(hy==0x40000000) return x*x; /* y is 2 */ 00161 if(hy==0x3fe00000) { /* y is 0.5 */ 00162 if(hx>=0) /* x >= +0 */ 00163 return __ieee754_sqrt(x); 00164 } 00165 } 00166 00167 ax = fabs(x); 00168 /* special value of x */ 00169 if(lx==0) { 00170 if(ix==0x7ff00000||ix==0||ix==0x3ff00000){ 00171 z = ax; /*x is +-0,+-inf,+-1*/ 00172 if(hy<0) z = one/z; /* z = (1/|x|) */ 00173 if(hx<0) { 00174 if(((ix-0x3ff00000)|yisint)==0) { 00175 z = (z-z)/(z-z); /* (-1)**non-int is NaN */ 00176 } else if(yisint==1) 00177 z = -z; /* (x<0)**odd = -(|x|**odd) */ 00178 } 00179 return z; 00180 } 00181 } 00182 00183 /* (x<0)**(non-int) is NaN */ 00184 if(((((u_int32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x); 00185 00186 /* |y| is huge */ 00187 if(iy>0x41e00000) { /* if |y| > 2**31 */ 00188 if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */ 00189 if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny; 00190 if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny; 00191 } 00192 /* over/underflow if x is not close to one */ 00193 if(ix<0x3fefffff) return (hy<0)? huge*huge:tiny*tiny; 00194 if(ix>0x3ff00000) return (hy>0)? huge*huge:tiny*tiny; 00195 /* now |1-x| is tiny <= 2**-20, suffice to compute 00196 log(x) by x-x^2/2+x^3/3-x^4/4 */ 00197 t = x-1; /* t has 20 trailing zeros */ 00198 w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25)); 00199 u = ivln2_h*t; /* ivln2_h has 21 sig. bits */ 00200 v = t*ivln2_l-w*ivln2; 00201 t1 = u+v; 00202 SET_LOW_WORD(t1,0); 00203 t2 = v-(t1-u); 00204 } else { 00205 double s2,s_h,s_l,t_h,t_l; 00206 n = 0; 00207 /* take care subnormal number */ 00208 if(ix<0x00100000) 00209 {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); } 00210 n += ((ix)>>20)-0x3ff; 00211 j = ix&0x000fffff; 00212 /* determine interval */ 00213 ix = j|0x3ff00000; /* normalize ix */ 00214 if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */ 00215 else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */ 00216 else {k=0;n+=1;ix -= 0x00100000;} 00217 SET_HIGH_WORD(ax,ix); 00218 00219 /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ 00220 u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ 00221 v = one/(ax+bp[k]); 00222 s = u*v; 00223 s_h = s; 00224 SET_LOW_WORD(s_h,0); 00225 /* t_h=ax+bp[k] High */ 00226 t_h = zero; 00227 SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18)); 00228 t_l = ax - (t_h-bp[k]); 00229 s_l = v*((u-s_h*t_h)-s_h*t_l); 00230 /* compute log(ax) */ 00231 s2 = s*s; 00232 r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6))))); 00233 r += s_l*(s_h+s); 00234 s2 = s_h*s_h; 00235 t_h = 3.0+s2+r; 00236 SET_LOW_WORD(t_h,0); 00237 t_l = r-((t_h-3.0)-s2); 00238 /* u+v = s*(1+...) */ 00239 u = s_h*t_h; 00240 v = s_l*t_h+t_l*s; 00241 /* 2/(3log2)*(s+...) */ 00242 p_h = u+v; 00243 SET_LOW_WORD(p_h,0); 00244 p_l = v-(p_h-u); 00245 z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */ 00246 z_l = cp_l*p_h+p_l*cp+dp_l[k]; 00247 /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */ 00248 t = (double)n; 00249 t1 = (((z_h+z_l)+dp_h[k])+t); 00250 SET_LOW_WORD(t1,0); 00251 t2 = z_l-(((t1-t)-dp_h[k])-z_h); 00252 } 00253 00254 s = one; /* s (sign of result -ve**odd) = -1 else = 1 */ 00255 if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0) 00256 s = -one;/* (-ve)**(odd int) */ 00257 00258 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ 00259 y1 = y; 00260 SET_LOW_WORD(y1,0); 00261 p_l = (y-y1)*t1+y*t2; 00262 p_h = y1*t1; 00263 z = p_l+p_h; 00264 EXTRACT_WORDS(j,i,z); 00265 if (j>=0x40900000) { /* z >= 1024 */ 00266 if(((j-0x40900000)|i)!=0) /* if z > 1024 */ 00267 return s*huge*huge; /* overflow */ 00268 else { 00269 if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */ 00270 } 00271 } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */ 00272 if(((j-0xc090cc00)|i)!=0) /* z < -1075 */ 00273 return s*tiny*tiny; /* underflow */ 00274 else { 00275 if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */ 00276 } 00277 } 00278 /* 00279 * compute 2**(p_h+p_l) 00280 */ 00281 i = j&0x7fffffff; 00282 k = (i>>20)-0x3ff; 00283 n = 0; 00284 if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */ 00285 n = j+(0x00100000>>(k+1)); 00286 k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */ 00287 t = zero; 00288 SET_HIGH_WORD(t,n&~(0x000fffff>>k)); 00289 n = ((n&0x000fffff)|0x00100000)>>(20-k); 00290 if(j<0) n = -n; 00291 p_h -= t; 00292 } 00293 t = p_l+p_h; 00294 SET_LOW_WORD(t,0); 00295 u = t*lg2_h; 00296 v = (p_l-(t-p_h))*lg2+t*lg2_l; 00297 z = u+v; 00298 w = v-(z-u); 00299 t = z*z; 00300 t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); 00301 r = (z*t1)/(t1-two)-(w+z*w); 00302 z = one-(r-z); 00303 GET_HIGH_WORD(j,z); 00304 j += (n<<20); 00305 if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */ 00306 else SET_HIGH_WORD(z,j); 00307 return s*z; 00308 }