projects / unix-history / blame
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| 9f4a7cc1 ZAL |
1 | /* |
| 2 | * Copyright (c) 1985 Regents of the University of California. | |
| 3 | * | |
| 4 | * Use and reproduction of this software are granted in accordance with | |
| 5 | * the terms and conditions specified in the Berkeley Software License | |
| 6 | * Agreement (in particular, this entails acknowledgement of the programs' | |
| 7 | * source, and inclusion of this notice) with the additional understanding | |
| 8 | * that all recipients should regard themselves as participants in an | |
| 9 | * ongoing research project and hence should feel obligated to report | |
| 10 | * their experiences (good or bad) with these elementary function codes, | |
| 11 | * using "sendbug 4bsd-bugs@BERKELEY", to the authors. | |
| 12 | */ | |
| 13 | ||
| 14 | #ifndef lint | |
| 15 | static char sccsid[] = "@(#)pow.c 1.1 (ELEFUNT) %G%"; | |
| 16 | #endif not lint | |
| 17 | ||
| 18 | /* POW(X,Y) | |
| 19 | * RETURN X**Y | |
| 20 | * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS) | |
| 21 | * CODED IN C BY K.C. NG, 1/8/85; | |
| 22 | * REVISED BY K.C. NG on 7/10/85. | |
| 23 | * | |
| 24 | * Required system supported functions: | |
| 25 | * scalb(x,n) | |
| 26 | * logb(x) | |
| 27 | * copysign(x,y) | |
| 28 | * finite(x) | |
| 29 | * drem(x,y) | |
| 30 | * | |
| 31 | * Required kernel functions: | |
| 32 | * exp__E(a,c) ...return exp(a+c) - 1 - a*a/2 | |
| 33 | * log__L(x) ...return (log(1+x) - 2s)/s, s=x/(2+x) | |
| 34 | * pow_p(x,y) ...return +(anything)**(finite non zero) | |
| 35 | * | |
| 36 | * Method | |
| 37 | * 1. Compute and return log(x) in three pieces: | |
| 38 | * log(x) = n*ln2 + hi + lo, | |
| 39 | * where n is an integer. | |
| 40 | * 2. Perform y*log(x) by simulating muti-precision arithmetic and | |
| 41 | * return the answer in three pieces: | |
| 42 | * y*log(x) = m*ln2 + hi + lo, | |
| 43 | * where m is an integer. | |
| 44 | * 3. Return x**y = exp(y*log(x)) | |
| 45 | * = 2^m * ( exp(hi+lo) ). | |
| 46 | * | |
| 47 | * Special cases: | |
| 48 | * (anything) ** 0 is 1 ; | |
| 49 | * (anything) ** 1 is itself; | |
| 50 | * (anything) ** NaN is NaN; | |
| 51 | * NaN ** (anything except 0) is NaN; | |
| 52 | * +-(anything > 1) ** +INF is +INF; | |
| 53 | * +-(anything > 1) ** -INF is +0; | |
| 54 | * +-(anything < 1) ** +INF is +0; | |
| 55 | * +-(anything < 1) ** -INF is +INF; | |
| 56 | * +-1 ** +-INF is NaN and signal INVALID; | |
| 57 | * +0 ** +(anything except 0, NaN) is +0; | |
| 58 | * -0 ** +(anything except 0, NaN, odd integer) is +0; | |
| 59 | * +0 ** -(anything except 0, NaN) is +INF and signal DIV-BY-ZERO; | |
| 60 | * -0 ** -(anything except 0, NaN, odd integer) is +INF with signal; | |
| 61 | * -0 ** (odd integer) = -( +0 ** (odd integer) ); | |
| 62 | * +INF ** +(anything except 0,NaN) is +INF; | |
| 63 | * +INF ** -(anything except 0,NaN) is +0; | |
| 64 | * -INF ** (odd integer) = -( +INF ** (odd integer) ); | |
| 65 | * -INF ** (even integer) = ( +INF ** (even integer) ); | |
| 66 | * -INF ** -(anything except integer,NaN) is NaN with signal; | |
| 67 | * -(x=anything) ** (k=integer) is (-1)**k * (x ** k); | |
| 68 | * -(anything except 0) ** (non-integer) is NaN with signal; | |
| 69 | * | |
| 70 | * Accuracy: | |
| 71 | * pow(x,y) returns x**y nearly rounded. In particular, on a SUN, a VAX, | |
| 72 | * and a Zilog Z8000, | |
| 73 | * pow(integer,integer) | |
| 74 | * always returns the correct integer provided it is representable. | |
| 75 | * In a test run with 100,000 random arguments with 0 < x, y < 20.0 | |
| 76 | * on a VAX, the maximum observed error was 1.79 ulps (units in the | |
| 77 | * last place). | |
| 78 | * | |
| 79 | * Constants : | |
| 80 | * The hexadecimal values are the intended ones for the following constants. | |
| 81 | * The decimal values may be used, provided that the compiler will convert | |
| 82 | * from decimal to binary accurately enough to produce the hexadecimal values | |
| 83 | * shown. | |
| 84 | */ | |
| 85 | ||
| 86 | #ifdef VAX /* VAX D format */ | |
| 87 | #include <errno.h> | |
| 88 | extern double infnan(); | |
| 89 | ||
| 90 | /* double static */ | |
| 91 | /* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */ | |
| 92 | /* ln2lo = 1.6465949582897081279E-12 , Hex 2^-39 * .E7BCD5E4F1D9CC */ | |
| 93 | /* invln2 = 1.4426950408889634148E0 , Hex 2^ 1 * .B8AA3B295C17F1 */ | |
| 94 | /* sqrt2 = 1.4142135623730950622E0 ; Hex 2^ 1 * .B504F333F9DE65 */ | |
| 95 | static long ln2hix[] = { 0x72174031, 0x0000f7d0}; | |
| 96 | static long ln2lox[] = { 0xbcd52ce7, 0xd9cce4f1}; | |
| 97 | static long invln2x[] = { 0xaa3b40b8, 0x17f1295c}; | |
| 98 | static long sqrt2x[] = { 0x04f340b5, 0xde6533f9}; | |
| 99 | #define ln2hi (*(double*)ln2hix) | |
| 100 | #define ln2lo (*(double*)ln2lox) | |
| 101 | #define invln2 (*(double*)invln2x) | |
| 102 | #define sqrt2 (*(double*)sqrt2x) | |
| 103 | #else /* IEEE double */ | |
| 104 | double static | |
| 105 | ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */ | |
| 106 | ln2lo = 1.9082149292705877000E-10 , /*Hex 2^-33 * 1.A39EF35793C76 */ | |
| 107 | invln2 = 1.4426950408889633870E0 , /*Hex 2^ 0 * 1.71547652B82FE */ | |
| 108 | sqrt2 = 1.4142135623730951455E0 ; /*Hex 2^ 0 * 1.6A09E667F3BCD */ | |
| 109 | #endif | |
| 110 | ||
| 111 | double static zero=0.0, half=1.0/2.0, one=1.0, two=2.0, negone= -1.0; | |
| 112 | ||
| 113 | double pow(x,y) | |
| 114 | double x,y; | |
| 115 | { | |
| 116 | double drem(),pow_p(),copysign(),t; | |
| 117 | int finite(); | |
| 118 | ||
| 119 | if (y==zero) return(one); | |
| 120 | else if(y==one | |
| 121 | #ifndef VAX | |
| 122 | ||x!=x | |
| 123 | #endif | |
| 124 | ) return( x ); /* if x is NaN or y=1 */ | |
| 125 | #ifndef VAX | |
| 126 | else if(y!=y) return( y ); /* if y is NaN */ | |
| 127 | #endif | |
| 128 | else if(!finite(y)) /* if y is INF */ | |
| 129 | if((t=copysign(x,one))==one) return(zero/zero); | |
| 130 | else if(t>one) return((y>zero)?y:zero); | |
| 131 | else return((y<zero)?-y:zero); | |
| 132 | else if(y==two) return(x*x); | |
| 133 | else if(y==negone) return(one/x); | |
| 134 | ||
| 135 | /* sign(x) = 1 */ | |
| 136 | else if(copysign(one,x)==one) return(pow_p(x,y)); | |
| 137 | ||
| 138 | /* sign(x)= -1 */ | |
| 139 | /* if y is an even integer */ | |
| 140 | else if ( (t=drem(y,two)) == zero) return( pow_p(-x,y) ); | |
| 141 | ||
| 142 | /* if y is an odd integer */ | |
| 143 | else if (copysign(t,one) == one) return( -pow_p(-x,y) ); | |
| 144 | ||
| 145 | /* Henceforth y is not an integer */ | |
| 146 | else if(x==zero) /* x is -0 */ | |
| 147 | return((y>zero)?-x:one/(-x)); | |
| 148 | else { /* return NaN */ | |
| 149 | #ifdef VAX | |
| 150 | return (infnan(EDOM)); /* NaN */ | |
| 151 | #else /* IEEE double */ | |
| 152 | return(zero/zero); | |
| 153 | #endif | |
| 154 | } | |
| 155 | } | |
| 156 | ||
| 157 | /* pow_p(x,y) return x**y for x with sign=1 and finite y */ | |
| 158 | static double pow_p(x,y) | |
| 159 | double x,y; | |
| 160 | { | |
| 161 | double logb(),scalb(),copysign(),log__L(),exp__E(); | |
| 162 | double c,s,t,z,tx,ty; | |
| 163 | float sx,sy; | |
| 164 | long k=0; | |
| 165 | int n,m; | |
| 166 | ||
| 167 | if(x==zero||!finite(x)) { /* if x is +INF or +0 */ | |
| 168 | #ifdef VAX | |
| 169 | return((y>zero)?x:infnan(ERANGE)); /* if y<zero, return +INF */ | |
| 170 | #else | |
| 171 | return((y>zero)?x:one/x); | |
| 172 | #endif | |
| 173 | } | |
| 174 | if(x==1.0) return(x); /* if x=1.0, return 1 since y is finite */ | |
| 175 | ||
| 176 | /* reduce x to z in [sqrt(1/2)-1, sqrt(2)-1] */ | |
| 177 | z=scalb(x,-(n=logb(x))); | |
| 178 | #ifndef VAX /* IEEE double */ /* subnormal number */ | |
| 179 | if(n <= -1022) {n += (m=logb(z)); z=scalb(z,-m);} | |
| 180 | #endif | |
| 181 | if(z >= sqrt2 ) {n += 1; z *= half;} z -= one ; | |
| 182 | ||
| 183 | /* log(x) = nlog2+log(1+z) ~ nlog2 + t + tx */ | |
| 184 | s=z/(two+z); c=z*z*half; tx=s*(c+log__L(s*s)); | |
| 185 | t= z-(c-tx); tx += (z-t)-c; | |
| 186 | ||
| 187 | /* if y*log(x) is neither too big nor too small */ | |
| 188 | if((s=logb(y)+logb(n+t)) < 12.0) | |
| 189 | if(s>-60.0) { | |
| 190 | ||
| 191 | /* compute y*log(x) ~ mlog2 + t + c */ | |
| 192 | s=y*(n+invln2*t); | |
| 193 | m=s+copysign(half,s); /* m := nint(y*log(x)) */ | |
| 194 | k=y; | |
| 195 | if((double)k==y) { /* if y is an integer */ | |
| 196 | k = m-k*n; | |
| 197 | sx=t; tx+=(t-sx); } | |
| 198 | else { /* if y is not an integer */ | |
| 199 | k =m; | |
| 200 | tx+=n*ln2lo; | |
| 201 | sx=(c=n*ln2hi)+t; tx+=(c-sx)+t; } | |
| 202 | /* end of checking whether k==y */ | |
| 203 | ||
| 204 | sy=y; ty=y-sy; /* y ~ sy + ty */ | |
| 205 | s=(double)sx*sy-k*ln2hi; /* (sy+ty)*(sx+tx)-kln2 */ | |
| 206 | z=(tx*ty-k*ln2lo); | |
| 207 | tx=tx*sy; ty=sx*ty; | |
| 208 | t=ty+z; t+=tx; t+=s; | |
| 209 | c= -((((t-s)-tx)-ty)-z); | |
| 210 | ||
| 211 | /* return exp(y*log(x)) */ | |
| 212 | t += exp__E(t,c); return(scalb(one+t,m)); | |
| 213 | } | |
| 214 | /* end of if log(y*log(x)) > -60.0 */ | |
| 215 | ||
| 216 | else | |
| 217 | /* exp(+- tiny) = 1 with inexact flag */ | |
| 218 | {ln2hi+ln2lo; return(one);} | |
| 219 | else if(copysign(one,y)*(n+invln2*t) <zero) | |
| 220 | /* exp(-(big#)) underflows to zero */ | |
| 221 | return(scalb(one,-5000)); | |
| 222 | else | |
| 223 | /* exp(+(big#)) overflows to INF */ | |
| 224 | return(scalb(one, 5000)); | |
| 225 | ||
| 226 | } |