1 /* 2 * Copyright (c) 1985 Regents of the University of California. 3 * All rights reserved. 4 * 5 * %sccs.include.redist.c% 6 */ 7 8 #ifndef lint 9 static char sccsid[] = "@(#)pow.c 5.7 (Berkeley) 10/09/90"; 10 #endif /* not lint */ 11 12 /* POW(X,Y) 13 * RETURN X**Y 14 * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS) 15 * CODED IN C BY K.C. NG, 1/8/85; 16 * REVISED BY K.C. NG on 7/10/85. 17 * 18 * Required system supported functions: 19 * scalb(x,n) 20 * logb(x) 21 * copysign(x,y) 22 * finite(x) 23 * drem(x,y) 24 * 25 * Required kernel functions: 26 * exp__E(a,c) ...return exp(a+c) - 1 - a*a/2 27 * log__L(x) ...return (log(1+x) - 2s)/s, s=x/(2+x) 28 * pow_p(x,y) ...return +(anything)**(finite non zero) 29 * 30 * Method 31 * 1. Compute and return log(x) in three pieces: 32 * log(x) = n*ln2 + hi + lo, 33 * where n is an integer. 34 * 2. Perform y*log(x) by simulating muti-precision arithmetic and 35 * return the answer in three pieces: 36 * y*log(x) = m*ln2 + hi + lo, 37 * where m is an integer. 38 * 3. Return x**y = exp(y*log(x)) 39 * = 2^m * ( exp(hi+lo) ). 40 * 41 * Special cases: 42 * (anything) ** 0 is 1 ; 43 * (anything) ** 1 is itself; 44 * (anything) ** NaN is NaN; 45 * NaN ** (anything except 0) is NaN; 46 * +-(anything > 1) ** +INF is +INF; 47 * +-(anything > 1) ** -INF is +0; 48 * +-(anything < 1) ** +INF is +0; 49 * +-(anything < 1) ** -INF is +INF; 50 * +-1 ** +-INF is NaN and signal INVALID; 51 * +0 ** +(anything except 0, NaN) is +0; 52 * -0 ** +(anything except 0, NaN, odd integer) is +0; 53 * +0 ** -(anything except 0, NaN) is +INF and signal DIV-BY-ZERO; 54 * -0 ** -(anything except 0, NaN, odd integer) is +INF with signal; 55 * -0 ** (odd integer) = -( +0 ** (odd integer) ); 56 * +INF ** +(anything except 0,NaN) is +INF; 57 * +INF ** -(anything except 0,NaN) is +0; 58 * -INF ** (odd integer) = -( +INF ** (odd integer) ); 59 * -INF ** (even integer) = ( +INF ** (even integer) ); 60 * -INF ** -(anything except integer,NaN) is NaN with signal; 61 * -(x=anything) ** (k=integer) is (-1)**k * (x ** k); 62 * -(anything except 0) ** (non-integer) is NaN with signal; 63 * 64 * Accuracy: 65 * pow(x,y) returns x**y nearly rounded. In particular, on a SUN, a VAX, 66 * and a Zilog Z8000, 67 * pow(integer,integer) 68 * always returns the correct integer provided it is representable. 69 * In a test run with 100,000 random arguments with 0 < x, y < 20.0 70 * on a VAX, the maximum observed error was 1.79 ulps (units in the 71 * last place). 72 * 73 * Constants : 74 * The hexadecimal values are the intended ones for the following constants. 75 * The decimal values may be used, provided that the compiler will convert 76 * from decimal to binary accurately enough to produce the hexadecimal values 77 * shown. 78 */ 79 80 #include <errno.h> 81 #include "mathimpl.h" 82 83 vc(ln2hi, 6.9314718055829871446E-1 ,7217,4031,0000,f7d0, 0, .B17217F7D00000) 84 vc(ln2lo, 1.6465949582897081279E-12 ,bcd5,2ce7,d9cc,e4f1, -39, .E7BCD5E4F1D9CC) 85 vc(invln2, 1.4426950408889634148E0 ,aa3b,40b8,17f1,295c, 1, .B8AA3B295C17F1) 86 vc(sqrt2, 1.4142135623730950622E0 ,04f3,40b5,de65,33f9, 1, .B504F333F9DE65) 87 88 ic(ln2hi, 6.9314718036912381649E-1, -1, 1.62E42FEE00000) 89 ic(ln2lo, 1.9082149292705877000E-10, -33, 1.A39EF35793C76) 90 ic(invln2, 1.4426950408889633870E0, 0, 1.71547652B82FE) 91 ic(sqrt2, 1.4142135623730951455E0, 0, 1.6A09E667F3BCD) 92 93 #ifdef vccast 94 #define ln2hi vccast(ln2hi) 95 #define ln2lo vccast(ln2lo) 96 #define invln2 vccast(invln2) 97 #define sqrt2 vccast(sqrt2) 98 #endif 99 100 const static double zero=0.0, half=1.0/2.0, one=1.0, two=2.0, negone= -1.0; 101 102 static double pow_p(); 103 104 double pow(x,y) 105 double x,y; 106 { 107 double t; 108 109 if (y==zero) return(one); 110 else if(y==one 111 #if !defined(vax)&&!defined(tahoe) 112 ||x!=x 113 #endif /* !defined(vax)&&!defined(tahoe) */ 114 ) return( x ); /* if x is NaN or y=1 */ 115 #if !defined(vax)&&!defined(tahoe) 116 else if(y!=y) return( y ); /* if y is NaN */ 117 #endif /* !defined(vax)&&!defined(tahoe) */ 118 else if(!finite(y)) /* if y is INF */ 119 if((t=copysign(x,one))==one) return(zero/zero); 120 else if(t>one) return((y>zero)?y:zero); 121 else return((y<zero)?-y:zero); 122 else if(y==two) return(x*x); 123 else if(y==negone) return(one/x); 124 125 /* sign(x) = 1 */ 126 else if(copysign(one,x)==one) return(pow_p(x,y)); 127 128 /* sign(x)= -1 */ 129 /* if y is an even integer */ 130 else if ( (t=drem(y,two)) == zero) return( pow_p(-x,y) ); 131 132 /* if y is an odd integer */ 133 else if (copysign(t,one) == one) return( -pow_p(-x,y) ); 134 135 /* Henceforth y is not an integer */ 136 else if(x==zero) /* x is -0 */ 137 return((y>zero)?-x:one/(-x)); 138 else { /* return NaN */ 139 #if defined(vax)||defined(tahoe) 140 return (infnan(EDOM)); /* NaN */ 141 #else /* defined(vax)||defined(tahoe) */ 142 return(zero/zero); 143 #endif /* defined(vax)||defined(tahoe) */ 144 } 145 } 146 147 #ifndef mc68881 148 /* pow_p(x,y) return x**y for x with sign=1 and finite y */ 149 static double pow_p(x,y) 150 double x,y; 151 { 152 double c,s,t,z,tx,ty; 153 #ifdef tahoe 154 double tahoe_tmp; 155 #endif /* tahoe */ 156 float sx,sy; 157 long k=0; 158 int n,m; 159 160 if(x==zero||!finite(x)) { /* if x is +INF or +0 */ 161 #if defined(vax)||defined(tahoe) 162 return((y>zero)?x:infnan(ERANGE)); /* if y<zero, return +INF */ 163 #else /* defined(vax)||defined(tahoe) */ 164 return((y>zero)?x:one/x); 165 #endif /* defined(vax)||defined(tahoe) */ 166 } 167 if(x==1.0) return(x); /* if x=1.0, return 1 since y is finite */ 168 169 /* reduce x to z in [sqrt(1/2)-1, sqrt(2)-1] */ 170 z=scalb(x,-(n=logb(x))); 171 #if !defined(vax)&&!defined(tahoe) /* IEEE double; subnormal number */ 172 if(n <= -1022) {n += (m=logb(z)); z=scalb(z,-m);} 173 #endif /* !defined(vax)&&!defined(tahoe) */ 174 if(z >= sqrt2 ) {n += 1; z *= half;} z -= one ; 175 176 /* log(x) = nlog2+log(1+z) ~ nlog2 + t + tx */ 177 s=z/(two+z); c=z*z*half; tx=s*(c+log__L(s*s)); 178 t= z-(c-tx); tx += (z-t)-c; 179 180 /* if y*log(x) is neither too big nor too small */ 181 if((s=logb(y)+logb(n+t)) < 12.0) 182 if(s>-60.0) { 183 184 /* compute y*log(x) ~ mlog2 + t + c */ 185 s=y*(n+invln2*t); 186 m=s+copysign(half,s); /* m := nint(y*log(x)) */ 187 k=y; 188 if((double)k==y) { /* if y is an integer */ 189 k = m-k*n; 190 sx=t; tx+=(t-sx); } 191 else { /* if y is not an integer */ 192 k =m; 193 tx+=n*ln2lo; 194 sx=(c=n*ln2hi)+t; tx+=(c-sx)+t; } 195 /* end of checking whether k==y */ 196 197 sy=y; ty=y-sy; /* y ~ sy + ty */ 198 #ifdef tahoe 199 s = (tahoe_tmp = sx)*sy-k*ln2hi; 200 #else /* tahoe */ 201 s=(double)sx*sy-k*ln2hi; /* (sy+ty)*(sx+tx)-kln2 */ 202 #endif /* tahoe */ 203 z=(tx*ty-k*ln2lo); 204 tx=tx*sy; ty=sx*ty; 205 t=ty+z; t+=tx; t+=s; 206 c= -((((t-s)-tx)-ty)-z); 207 208 /* return exp(y*log(x)) */ 209 t += exp__E(t,c); return(scalb(one+t,m)); 210 } 211 /* end of if log(y*log(x)) > -60.0 */ 212 213 else 214 /* exp(+- tiny) = 1 with inexact flag */ 215 {ln2hi+ln2lo; return(one);} 216 else if(copysign(one,y)*(n+invln2*t) <zero) 217 /* exp(-(big#)) underflows to zero */ 218 return(scalb(one,-5000)); 219 else 220 /* exp(+(big#)) overflows to INF */ 221 return(scalb(one, 5000)); 222 223 } 224 #endif /* mc68881 */ 225