1 /* 2 * Copyright (c) 1985 Regents of the University of California. 3 * All rights reserved. 4 * 5 * Redistribution and use in source and binary forms are permitted 6 * provided that the above copyright notice and this paragraph are 7 * duplicated in all such forms and that any documentation, 8 * advertising materials, and other materials related to such 9 * distribution and use acknowledge that the software was developed 10 * by the University of California, Berkeley. The name of the 11 * University may not be used to endorse or promote products derived 12 * from this software without specific prior written permission. 13 * THIS SOFTWARE IS PROVIDED ``AS IS'' AND WITHOUT ANY EXPRESS OR 14 * IMPLIED WARRANTIES, INCLUDING, WITHOUT LIMITATION, THE IMPLIED 15 * WARRANTIES OF MERCHANTIBILITY AND FITNESS FOR A PARTICULAR PURPOSE. 16 * 17 * All recipients should regard themselves as participants in an ongoing 18 * research project and hence should feel obligated to report their 19 * experiences (good or bad) with these elementary function codes, using 20 * the sendbug(8) program, to the authors. 21 */ 22 23 #ifndef lint 24 static char sccsid[] = "@(#)pow.c 5.4 (Berkeley) 09/22/88"; 25 #endif /* not lint */ 26 27 /* POW(X,Y) 28 * RETURN X**Y 29 * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS) 30 * CODED IN C BY K.C. NG, 1/8/85; 31 * REVISED BY K.C. NG on 7/10/85. 32 * 33 * Required system supported functions: 34 * scalb(x,n) 35 * logb(x) 36 * copysign(x,y) 37 * finite(x) 38 * drem(x,y) 39 * 40 * Required kernel functions: 41 * exp__E(a,c) ...return exp(a+c) - 1 - a*a/2 42 * log__L(x) ...return (log(1+x) - 2s)/s, s=x/(2+x) 43 * pow_p(x,y) ...return +(anything)**(finite non zero) 44 * 45 * Method 46 * 1. Compute and return log(x) in three pieces: 47 * log(x) = n*ln2 + hi + lo, 48 * where n is an integer. 49 * 2. Perform y*log(x) by simulating muti-precision arithmetic and 50 * return the answer in three pieces: 51 * y*log(x) = m*ln2 + hi + lo, 52 * where m is an integer. 53 * 3. Return x**y = exp(y*log(x)) 54 * = 2^m * ( exp(hi+lo) ). 55 * 56 * Special cases: 57 * (anything) ** 0 is 1 ; 58 * (anything) ** 1 is itself; 59 * (anything) ** NaN is NaN; 60 * NaN ** (anything except 0) is NaN; 61 * +-(anything > 1) ** +INF is +INF; 62 * +-(anything > 1) ** -INF is +0; 63 * +-(anything < 1) ** +INF is +0; 64 * +-(anything < 1) ** -INF is +INF; 65 * +-1 ** +-INF is NaN and signal INVALID; 66 * +0 ** +(anything except 0, NaN) is +0; 67 * -0 ** +(anything except 0, NaN, odd integer) is +0; 68 * +0 ** -(anything except 0, NaN) is +INF and signal DIV-BY-ZERO; 69 * -0 ** -(anything except 0, NaN, odd integer) is +INF with signal; 70 * -0 ** (odd integer) = -( +0 ** (odd integer) ); 71 * +INF ** +(anything except 0,NaN) is +INF; 72 * +INF ** -(anything except 0,NaN) is +0; 73 * -INF ** (odd integer) = -( +INF ** (odd integer) ); 74 * -INF ** (even integer) = ( +INF ** (even integer) ); 75 * -INF ** -(anything except integer,NaN) is NaN with signal; 76 * -(x=anything) ** (k=integer) is (-1)**k * (x ** k); 77 * -(anything except 0) ** (non-integer) is NaN with signal; 78 * 79 * Accuracy: 80 * pow(x,y) returns x**y nearly rounded. In particular, on a SUN, a VAX, 81 * and a Zilog Z8000, 82 * pow(integer,integer) 83 * always returns the correct integer provided it is representable. 84 * In a test run with 100,000 random arguments with 0 < x, y < 20.0 85 * on a VAX, the maximum observed error was 1.79 ulps (units in the 86 * last place). 87 * 88 * Constants : 89 * The hexadecimal values are the intended ones for the following constants. 90 * The decimal values may be used, provided that the compiler will convert 91 * from decimal to binary accurately enough to produce the hexadecimal values 92 * shown. 93 */ 94 95 #include <errno.h> 96 #include "mathimpl.h" 97 98 vc(ln2hi, 6.9314718055829871446E-1 ,7217,4031,0000,f7d0, 0, .B17217F7D00000) 99 vc(ln2lo, 1.6465949582897081279E-12 ,bcd5,2ce7,d9cc,e4f1, -39, .E7BCD5E4F1D9CC) 100 vc(invln2, 1.4426950408889634148E0 ,aa3b,40b8,17f1,295c, 1, .B8AA3B295C17F1) 101 vc(sqrt2, 1.4142135623730950622E0 ,04f3,40b5,de65,33f9, 1, .B504F333F9DE65) 102 103 ic(ln2hi, 6.9314718036912381649E-1, -1, 1.62E42FEE00000) 104 ic(ln2lo, 1.9082149292705877000E-10, -33, 1.A39EF35793C76) 105 ic(invln2, 1.4426950408889633870E0, 0, 1.71547652B82FE) 106 ic(sqrt2, 1.4142135623730951455E0, 0, 1.6A09E667F3BCD) 107 108 #ifdef vccast 109 #define ln2hi vccast(ln2hi) 110 #define ln2lo vccast(ln2lo) 111 #define invln2 vccast(invln2) 112 #define sqrt2 vccast(sqrt2) 113 #endif 114 115 const static double zero=0.0, half=1.0/2.0, one=1.0, two=2.0, negone= -1.0; 116 117 static double pow_p(); 118 119 double pow(x,y) 120 double x,y; 121 { 122 double t; 123 124 if (y==zero) return(one); 125 else if(y==one 126 #if !defined(vax)&&!defined(tahoe) 127 ||x!=x 128 #endif /* !defined(vax)&&!defined(tahoe) */ 129 ) return( x ); /* if x is NaN or y=1 */ 130 #if !defined(vax)&&!defined(tahoe) 131 else if(y!=y) return( y ); /* if y is NaN */ 132 #endif /* !defined(vax)&&!defined(tahoe) */ 133 else if(!finite(y)) /* if y is INF */ 134 if((t=copysign(x,one))==one) return(zero/zero); 135 else if(t>one) return((y>zero)?y:zero); 136 else return((y<zero)?-y:zero); 137 else if(y==two) return(x*x); 138 else if(y==negone) return(one/x); 139 140 /* sign(x) = 1 */ 141 else if(copysign(one,x)==one) return(pow_p(x,y)); 142 143 /* sign(x)= -1 */ 144 /* if y is an even integer */ 145 else if ( (t=drem(y,two)) == zero) return( pow_p(-x,y) ); 146 147 /* if y is an odd integer */ 148 else if (copysign(t,one) == one) return( -pow_p(-x,y) ); 149 150 /* Henceforth y is not an integer */ 151 else if(x==zero) /* x is -0 */ 152 return((y>zero)?-x:one/(-x)); 153 else { /* return NaN */ 154 #if defined(vax)||defined(tahoe) 155 return (infnan(EDOM)); /* NaN */ 156 #else /* defined(vax)||defined(tahoe) */ 157 return(zero/zero); 158 #endif /* defined(vax)||defined(tahoe) */ 159 } 160 } 161 162 /* pow_p(x,y) return x**y for x with sign=1 and finite y */ 163 static double pow_p(x,y) 164 double x,y; 165 { 166 double c,s,t,z,tx,ty; 167 #ifdef tahoe 168 double tahoe_tmp; 169 #endif /* tahoe */ 170 float sx,sy; 171 long k=0; 172 int n,m; 173 174 if(x==zero||!finite(x)) { /* if x is +INF or +0 */ 175 #if defined(vax)||defined(tahoe) 176 return((y>zero)?x:infnan(ERANGE)); /* if y<zero, return +INF */ 177 #else /* defined(vax)||defined(tahoe) */ 178 return((y>zero)?x:one/x); 179 #endif /* defined(vax)||defined(tahoe) */ 180 } 181 if(x==1.0) return(x); /* if x=1.0, return 1 since y is finite */ 182 183 /* reduce x to z in [sqrt(1/2)-1, sqrt(2)-1] */ 184 z=scalb(x,-(n=logb(x))); 185 #if !defined(vax)&&!defined(tahoe) /* IEEE double; subnormal number */ 186 if(n <= -1022) {n += (m=logb(z)); z=scalb(z,-m);} 187 #endif /* !defined(vax)&&!defined(tahoe) */ 188 if(z >= sqrt2 ) {n += 1; z *= half;} z -= one ; 189 190 /* log(x) = nlog2+log(1+z) ~ nlog2 + t + tx */ 191 s=z/(two+z); c=z*z*half; tx=s*(c+log__L(s*s)); 192 t= z-(c-tx); tx += (z-t)-c; 193 194 /* if y*log(x) is neither too big nor too small */ 195 if((s=logb(y)+logb(n+t)) < 12.0) 196 if(s>-60.0) { 197 198 /* compute y*log(x) ~ mlog2 + t + c */ 199 s=y*(n+invln2*t); 200 m=s+copysign(half,s); /* m := nint(y*log(x)) */ 201 k=y; 202 if((double)k==y) { /* if y is an integer */ 203 k = m-k*n; 204 sx=t; tx+=(t-sx); } 205 else { /* if y is not an integer */ 206 k =m; 207 tx+=n*ln2lo; 208 sx=(c=n*ln2hi)+t; tx+=(c-sx)+t; } 209 /* end of checking whether k==y */ 210 211 sy=y; ty=y-sy; /* y ~ sy + ty */ 212 #ifdef tahoe 213 s = (tahoe_tmp = sx)*sy-k*ln2hi; 214 #else /* tahoe */ 215 s=(double)sx*sy-k*ln2hi; /* (sy+ty)*(sx+tx)-kln2 */ 216 #endif /* tahoe */ 217 z=(tx*ty-k*ln2lo); 218 tx=tx*sy; ty=sx*ty; 219 t=ty+z; t+=tx; t+=s; 220 c= -((((t-s)-tx)-ty)-z); 221 222 /* return exp(y*log(x)) */ 223 t += exp__E(t,c); return(scalb(one+t,m)); 224 } 225 /* end of if log(y*log(x)) > -60.0 */ 226 227 else 228 /* exp(+- tiny) = 1 with inexact flag */ 229 {ln2hi+ln2lo; return(one);} 230 else if(copysign(one,y)*(n+invln2*t) <zero) 231 /* exp(-(big#)) underflows to zero */ 232 return(scalb(one,-5000)); 233 else 234 /* exp(+(big#)) overflows to INF */ 235 return(scalb(one, 5000)); 236 237 } 238