1 /* 2 * Copyright (c) 1985 Regents of the University of California. 3 * All rights reserved. 4 * 5 * %sccs.include.redist.c% 6 * 7 * All recipients should regard themselves as participants in an ongoing 8 * research project and hence should feel obligated to report their 9 * experiences (good or bad) with these elementary function codes, using 10 * the sendbug(8) program, to the authors. 11 */ 12 13 #ifndef lint 14 static char sccsid[] = "@(#)pow.c 5.6 (Berkeley) 06/01/90"; 15 #endif /* not lint */ 16 17 /* POW(X,Y) 18 * RETURN X**Y 19 * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS) 20 * CODED IN C BY K.C. NG, 1/8/85; 21 * REVISED BY K.C. NG on 7/10/85. 22 * 23 * Required system supported functions: 24 * scalb(x,n) 25 * logb(x) 26 * copysign(x,y) 27 * finite(x) 28 * drem(x,y) 29 * 30 * Required kernel functions: 31 * exp__E(a,c) ...return exp(a+c) - 1 - a*a/2 32 * log__L(x) ...return (log(1+x) - 2s)/s, s=x/(2+x) 33 * pow_p(x,y) ...return +(anything)**(finite non zero) 34 * 35 * Method 36 * 1. Compute and return log(x) in three pieces: 37 * log(x) = n*ln2 + hi + lo, 38 * where n is an integer. 39 * 2. Perform y*log(x) by simulating muti-precision arithmetic and 40 * return the answer in three pieces: 41 * y*log(x) = m*ln2 + hi + lo, 42 * where m is an integer. 43 * 3. Return x**y = exp(y*log(x)) 44 * = 2^m * ( exp(hi+lo) ). 45 * 46 * Special cases: 47 * (anything) ** 0 is 1 ; 48 * (anything) ** 1 is itself; 49 * (anything) ** NaN is NaN; 50 * NaN ** (anything except 0) is NaN; 51 * +-(anything > 1) ** +INF is +INF; 52 * +-(anything > 1) ** -INF is +0; 53 * +-(anything < 1) ** +INF is +0; 54 * +-(anything < 1) ** -INF is +INF; 55 * +-1 ** +-INF is NaN and signal INVALID; 56 * +0 ** +(anything except 0, NaN) is +0; 57 * -0 ** +(anything except 0, NaN, odd integer) is +0; 58 * +0 ** -(anything except 0, NaN) is +INF and signal DIV-BY-ZERO; 59 * -0 ** -(anything except 0, NaN, odd integer) is +INF with signal; 60 * -0 ** (odd integer) = -( +0 ** (odd integer) ); 61 * +INF ** +(anything except 0,NaN) is +INF; 62 * +INF ** -(anything except 0,NaN) is +0; 63 * -INF ** (odd integer) = -( +INF ** (odd integer) ); 64 * -INF ** (even integer) = ( +INF ** (even integer) ); 65 * -INF ** -(anything except integer,NaN) is NaN with signal; 66 * -(x=anything) ** (k=integer) is (-1)**k * (x ** k); 67 * -(anything except 0) ** (non-integer) is NaN with signal; 68 * 69 * Accuracy: 70 * pow(x,y) returns x**y nearly rounded. In particular, on a SUN, a VAX, 71 * and a Zilog Z8000, 72 * pow(integer,integer) 73 * always returns the correct integer provided it is representable. 74 * In a test run with 100,000 random arguments with 0 < x, y < 20.0 75 * on a VAX, the maximum observed error was 1.79 ulps (units in the 76 * last place). 77 * 78 * Constants : 79 * The hexadecimal values are the intended ones for the following constants. 80 * The decimal values may be used, provided that the compiler will convert 81 * from decimal to binary accurately enough to produce the hexadecimal values 82 * shown. 83 */ 84 85 #include <errno.h> 86 #include "mathimpl.h" 87 88 vc(ln2hi, 6.9314718055829871446E-1 ,7217,4031,0000,f7d0, 0, .B17217F7D00000) 89 vc(ln2lo, 1.6465949582897081279E-12 ,bcd5,2ce7,d9cc,e4f1, -39, .E7BCD5E4F1D9CC) 90 vc(invln2, 1.4426950408889634148E0 ,aa3b,40b8,17f1,295c, 1, .B8AA3B295C17F1) 91 vc(sqrt2, 1.4142135623730950622E0 ,04f3,40b5,de65,33f9, 1, .B504F333F9DE65) 92 93 ic(ln2hi, 6.9314718036912381649E-1, -1, 1.62E42FEE00000) 94 ic(ln2lo, 1.9082149292705877000E-10, -33, 1.A39EF35793C76) 95 ic(invln2, 1.4426950408889633870E0, 0, 1.71547652B82FE) 96 ic(sqrt2, 1.4142135623730951455E0, 0, 1.6A09E667F3BCD) 97 98 #ifdef vccast 99 #define ln2hi vccast(ln2hi) 100 #define ln2lo vccast(ln2lo) 101 #define invln2 vccast(invln2) 102 #define sqrt2 vccast(sqrt2) 103 #endif 104 105 const static double zero=0.0, half=1.0/2.0, one=1.0, two=2.0, negone= -1.0; 106 107 static double pow_p(); 108 109 double pow(x,y) 110 double x,y; 111 { 112 double t; 113 114 if (y==zero) return(one); 115 else if(y==one 116 #if !defined(vax)&&!defined(tahoe) 117 ||x!=x 118 #endif /* !defined(vax)&&!defined(tahoe) */ 119 ) return( x ); /* if x is NaN or y=1 */ 120 #if !defined(vax)&&!defined(tahoe) 121 else if(y!=y) return( y ); /* if y is NaN */ 122 #endif /* !defined(vax)&&!defined(tahoe) */ 123 else if(!finite(y)) /* if y is INF */ 124 if((t=copysign(x,one))==one) return(zero/zero); 125 else if(t>one) return((y>zero)?y:zero); 126 else return((y<zero)?-y:zero); 127 else if(y==two) return(x*x); 128 else if(y==negone) return(one/x); 129 130 /* sign(x) = 1 */ 131 else if(copysign(one,x)==one) return(pow_p(x,y)); 132 133 /* sign(x)= -1 */ 134 /* if y is an even integer */ 135 else if ( (t=drem(y,two)) == zero) return( pow_p(-x,y) ); 136 137 /* if y is an odd integer */ 138 else if (copysign(t,one) == one) return( -pow_p(-x,y) ); 139 140 /* Henceforth y is not an integer */ 141 else if(x==zero) /* x is -0 */ 142 return((y>zero)?-x:one/(-x)); 143 else { /* return NaN */ 144 #if defined(vax)||defined(tahoe) 145 return (infnan(EDOM)); /* NaN */ 146 #else /* defined(vax)||defined(tahoe) */ 147 return(zero/zero); 148 #endif /* defined(vax)||defined(tahoe) */ 149 } 150 } 151 152 #ifndef mc68881 153 /* pow_p(x,y) return x**y for x with sign=1 and finite y */ 154 static double pow_p(x,y) 155 double x,y; 156 { 157 double c,s,t,z,tx,ty; 158 #ifdef tahoe 159 double tahoe_tmp; 160 #endif /* tahoe */ 161 float sx,sy; 162 long k=0; 163 int n,m; 164 165 if(x==zero||!finite(x)) { /* if x is +INF or +0 */ 166 #if defined(vax)||defined(tahoe) 167 return((y>zero)?x:infnan(ERANGE)); /* if y<zero, return +INF */ 168 #else /* defined(vax)||defined(tahoe) */ 169 return((y>zero)?x:one/x); 170 #endif /* defined(vax)||defined(tahoe) */ 171 } 172 if(x==1.0) return(x); /* if x=1.0, return 1 since y is finite */ 173 174 /* reduce x to z in [sqrt(1/2)-1, sqrt(2)-1] */ 175 z=scalb(x,-(n=logb(x))); 176 #if !defined(vax)&&!defined(tahoe) /* IEEE double; subnormal number */ 177 if(n <= -1022) {n += (m=logb(z)); z=scalb(z,-m);} 178 #endif /* !defined(vax)&&!defined(tahoe) */ 179 if(z >= sqrt2 ) {n += 1; z *= half;} z -= one ; 180 181 /* log(x) = nlog2+log(1+z) ~ nlog2 + t + tx */ 182 s=z/(two+z); c=z*z*half; tx=s*(c+log__L(s*s)); 183 t= z-(c-tx); tx += (z-t)-c; 184 185 /* if y*log(x) is neither too big nor too small */ 186 if((s=logb(y)+logb(n+t)) < 12.0) 187 if(s>-60.0) { 188 189 /* compute y*log(x) ~ mlog2 + t + c */ 190 s=y*(n+invln2*t); 191 m=s+copysign(half,s); /* m := nint(y*log(x)) */ 192 k=y; 193 if((double)k==y) { /* if y is an integer */ 194 k = m-k*n; 195 sx=t; tx+=(t-sx); } 196 else { /* if y is not an integer */ 197 k =m; 198 tx+=n*ln2lo; 199 sx=(c=n*ln2hi)+t; tx+=(c-sx)+t; } 200 /* end of checking whether k==y */ 201 202 sy=y; ty=y-sy; /* y ~ sy + ty */ 203 #ifdef tahoe 204 s = (tahoe_tmp = sx)*sy-k*ln2hi; 205 #else /* tahoe */ 206 s=(double)sx*sy-k*ln2hi; /* (sy+ty)*(sx+tx)-kln2 */ 207 #endif /* tahoe */ 208 z=(tx*ty-k*ln2lo); 209 tx=tx*sy; ty=sx*ty; 210 t=ty+z; t+=tx; t+=s; 211 c= -((((t-s)-tx)-ty)-z); 212 213 /* return exp(y*log(x)) */ 214 t += exp__E(t,c); return(scalb(one+t,m)); 215 } 216 /* end of if log(y*log(x)) > -60.0 */ 217 218 else 219 /* exp(+- tiny) = 1 with inexact flag */ 220 {ln2hi+ln2lo; return(one);} 221 else if(copysign(one,y)*(n+invln2*t) <zero) 222 /* exp(-(big#)) underflows to zero */ 223 return(scalb(one,-5000)); 224 else 225 /* exp(+(big#)) overflows to INF */ 226 return(scalb(one, 5000)); 227 228 } 229 #endif /* mc68881 */ 230