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| 9b525f39 | 1 | /* |
| 9f4a7cc1 | 2 | * Copyright (c) 1985 Regents of the University of California. |
| 9b525f39 KB |
3 | * All rights reserved. |
| 4 | * | |
| af359dea C |
5 | * Redistribution and use in source and binary forms, with or without |
| 6 | * modification, are permitted provided that the following conditions | |
| 7 | * are met: | |
| 8 | * 1. Redistributions of source code must retain the above copyright | |
| 9 | * notice, this list of conditions and the following disclaimer. | |
| 10 | * 2. Redistributions in binary form must reproduce the above copyright | |
| 11 | * notice, this list of conditions and the following disclaimer in the | |
| 12 | * documentation and/or other materials provided with the distribution. | |
| 13 | * 3. All advertising materials mentioning features or use of this software | |
| 14 | * must display the following acknowledgement: | |
| 15 | * This product includes software developed by the University of | |
| 16 | * California, Berkeley and its contributors. | |
| 17 | * 4. Neither the name of the University nor the names of its contributors | |
| 18 | * may be used to endorse or promote products derived from this software | |
| 19 | * without specific prior written permission. | |
| 9b525f39 | 20 | * |
| af359dea C |
21 | * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND |
| 22 | * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE | |
| 23 | * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE | |
| 24 | * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE | |
| 25 | * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL | |
| 26 | * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS | |
| 27 | * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) | |
| 28 | * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT | |
| 29 | * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY | |
| 30 | * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF | |
| 31 | * SUCH DAMAGE. | |
| 9f4a7cc1 ZAL |
32 | */ |
| 33 | ||
| 34 | #ifndef lint | |
| af359dea | 35 | static char sccsid[] = "@(#)pow.c 5.7 (Berkeley) 10/9/90"; |
| 9b525f39 | 36 | #endif /* not lint */ |
| 9f4a7cc1 ZAL |
37 | |
| 38 | /* POW(X,Y) | |
| 39 | * RETURN X**Y | |
| 40 | * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS) | |
| 41 | * CODED IN C BY K.C. NG, 1/8/85; | |
| 42 | * REVISED BY K.C. NG on 7/10/85. | |
| 43 | * | |
| 44 | * Required system supported functions: | |
| 45 | * scalb(x,n) | |
| 46 | * logb(x) | |
| 47 | * copysign(x,y) | |
| 48 | * finite(x) | |
| 49 | * drem(x,y) | |
| 50 | * | |
| 51 | * Required kernel functions: | |
| 52 | * exp__E(a,c) ...return exp(a+c) - 1 - a*a/2 | |
| 53 | * log__L(x) ...return (log(1+x) - 2s)/s, s=x/(2+x) | |
| 54 | * pow_p(x,y) ...return +(anything)**(finite non zero) | |
| 55 | * | |
| 56 | * Method | |
| 57 | * 1. Compute and return log(x) in three pieces: | |
| 58 | * log(x) = n*ln2 + hi + lo, | |
| 59 | * where n is an integer. | |
| 60 | * 2. Perform y*log(x) by simulating muti-precision arithmetic and | |
| 61 | * return the answer in three pieces: | |
| 62 | * y*log(x) = m*ln2 + hi + lo, | |
| 63 | * where m is an integer. | |
| 64 | * 3. Return x**y = exp(y*log(x)) | |
| 65 | * = 2^m * ( exp(hi+lo) ). | |
| 66 | * | |
| 67 | * Special cases: | |
| 68 | * (anything) ** 0 is 1 ; | |
| 69 | * (anything) ** 1 is itself; | |
| 70 | * (anything) ** NaN is NaN; | |
| 71 | * NaN ** (anything except 0) is NaN; | |
| 72 | * +-(anything > 1) ** +INF is +INF; | |
| 73 | * +-(anything > 1) ** -INF is +0; | |
| 74 | * +-(anything < 1) ** +INF is +0; | |
| 75 | * +-(anything < 1) ** -INF is +INF; | |
| 76 | * +-1 ** +-INF is NaN and signal INVALID; | |
| 77 | * +0 ** +(anything except 0, NaN) is +0; | |
| 78 | * -0 ** +(anything except 0, NaN, odd integer) is +0; | |
| 79 | * +0 ** -(anything except 0, NaN) is +INF and signal DIV-BY-ZERO; | |
| 80 | * -0 ** -(anything except 0, NaN, odd integer) is +INF with signal; | |
| 81 | * -0 ** (odd integer) = -( +0 ** (odd integer) ); | |
| 82 | * +INF ** +(anything except 0,NaN) is +INF; | |
| 83 | * +INF ** -(anything except 0,NaN) is +0; | |
| 84 | * -INF ** (odd integer) = -( +INF ** (odd integer) ); | |
| 85 | * -INF ** (even integer) = ( +INF ** (even integer) ); | |
| 86 | * -INF ** -(anything except integer,NaN) is NaN with signal; | |
| 87 | * -(x=anything) ** (k=integer) is (-1)**k * (x ** k); | |
| 88 | * -(anything except 0) ** (non-integer) is NaN with signal; | |
| 89 | * | |
| 90 | * Accuracy: | |
| 91 | * pow(x,y) returns x**y nearly rounded. In particular, on a SUN, a VAX, | |
| 92 | * and a Zilog Z8000, | |
| 93 | * pow(integer,integer) | |
| 94 | * always returns the correct integer provided it is representable. | |
| 95 | * In a test run with 100,000 random arguments with 0 < x, y < 20.0 | |
| 96 | * on a VAX, the maximum observed error was 1.79 ulps (units in the | |
| 97 | * last place). | |
| 98 | * | |
| 99 | * Constants : | |
| 100 | * The hexadecimal values are the intended ones for the following constants. | |
| 101 | * The decimal values may be used, provided that the compiler will convert | |
| 102 | * from decimal to binary accurately enough to produce the hexadecimal values | |
| 103 | * shown. | |
| 104 | */ | |
| 105 | ||
| 9f4a7cc1 | 106 | #include <errno.h> |
| 9eda3584 | 107 | #include "mathimpl.h" |
| 9f4a7cc1 | 108 | |
| 9eda3584 KB |
109 | vc(ln2hi, 6.9314718055829871446E-1 ,7217,4031,0000,f7d0, 0, .B17217F7D00000) |
| 110 | vc(ln2lo, 1.6465949582897081279E-12 ,bcd5,2ce7,d9cc,e4f1, -39, .E7BCD5E4F1D9CC) | |
| 111 | vc(invln2, 1.4426950408889634148E0 ,aa3b,40b8,17f1,295c, 1, .B8AA3B295C17F1) | |
| 112 | vc(sqrt2, 1.4142135623730950622E0 ,04f3,40b5,de65,33f9, 1, .B504F333F9DE65) | |
| 113 | ||
| 114 | ic(ln2hi, 6.9314718036912381649E-1, -1, 1.62E42FEE00000) | |
| 115 | ic(ln2lo, 1.9082149292705877000E-10, -33, 1.A39EF35793C76) | |
| 116 | ic(invln2, 1.4426950408889633870E0, 0, 1.71547652B82FE) | |
| 117 | ic(sqrt2, 1.4142135623730951455E0, 0, 1.6A09E667F3BCD) | |
| 118 | ||
| 119 | #ifdef vccast | |
| 120 | #define ln2hi vccast(ln2hi) | |
| 121 | #define ln2lo vccast(ln2lo) | |
| 122 | #define invln2 vccast(invln2) | |
| 123 | #define sqrt2 vccast(sqrt2) | |
| 124 | #endif | |
| 125 | ||
| 126 | const static double zero=0.0, half=1.0/2.0, one=1.0, two=2.0, negone= -1.0; | |
| 127 | ||
| 128 | static double pow_p(); | |
| 9f4a7cc1 ZAL |
129 | |
| 130 | double pow(x,y) | |
| 131 | double x,y; | |
| 132 | { | |
| 9eda3584 | 133 | double t; |
| 9f4a7cc1 ZAL |
134 | |
| 135 | if (y==zero) return(one); | |
| 136 | else if(y==one | |
| 859dc438 | 137 | #if !defined(vax)&&!defined(tahoe) |
| 9f4a7cc1 | 138 | ||x!=x |
| 859dc438 | 139 | #endif /* !defined(vax)&&!defined(tahoe) */ |
| 9f4a7cc1 | 140 | ) return( x ); /* if x is NaN or y=1 */ |
| 859dc438 | 141 | #if !defined(vax)&&!defined(tahoe) |
| 9f4a7cc1 | 142 | else if(y!=y) return( y ); /* if y is NaN */ |
| 859dc438 | 143 | #endif /* !defined(vax)&&!defined(tahoe) */ |
| 9f4a7cc1 ZAL |
144 | else if(!finite(y)) /* if y is INF */ |
| 145 | if((t=copysign(x,one))==one) return(zero/zero); | |
| 146 | else if(t>one) return((y>zero)?y:zero); | |
| 147 | else return((y<zero)?-y:zero); | |
| 148 | else if(y==two) return(x*x); | |
| 149 | else if(y==negone) return(one/x); | |
| 150 | ||
| 151 | /* sign(x) = 1 */ | |
| 152 | else if(copysign(one,x)==one) return(pow_p(x,y)); | |
| 153 | ||
| 154 | /* sign(x)= -1 */ | |
| 155 | /* if y is an even integer */ | |
| 156 | else if ( (t=drem(y,two)) == zero) return( pow_p(-x,y) ); | |
| 157 | ||
| 158 | /* if y is an odd integer */ | |
| 159 | else if (copysign(t,one) == one) return( -pow_p(-x,y) ); | |
| 160 | ||
| 161 | /* Henceforth y is not an integer */ | |
| 162 | else if(x==zero) /* x is -0 */ | |
| 163 | return((y>zero)?-x:one/(-x)); | |
| 164 | else { /* return NaN */ | |
| 859dc438 | 165 | #if defined(vax)||defined(tahoe) |
| 9f4a7cc1 | 166 | return (infnan(EDOM)); /* NaN */ |
| 859dc438 | 167 | #else /* defined(vax)||defined(tahoe) */ |
| 9f4a7cc1 | 168 | return(zero/zero); |
| 859dc438 | 169 | #endif /* defined(vax)||defined(tahoe) */ |
| 9f4a7cc1 ZAL |
170 | } |
| 171 | } | |
| 172 | ||
| 6229246a | 173 | #ifndef mc68881 |
| 9f4a7cc1 ZAL |
174 | /* pow_p(x,y) return x**y for x with sign=1 and finite y */ |
| 175 | static double pow_p(x,y) | |
| 176 | double x,y; | |
| 177 | { | |
| 9f4a7cc1 | 178 | double c,s,t,z,tx,ty; |
| 859dc438 | 179 | #ifdef tahoe |
| 471c7555 | 180 | double tahoe_tmp; |
| 859dc438 | 181 | #endif /* tahoe */ |
| 9f4a7cc1 ZAL |
182 | float sx,sy; |
| 183 | long k=0; | |
| 184 | int n,m; | |
| 185 | ||
| 186 | if(x==zero||!finite(x)) { /* if x is +INF or +0 */ | |
| 859dc438 | 187 | #if defined(vax)||defined(tahoe) |
| 9f4a7cc1 | 188 | return((y>zero)?x:infnan(ERANGE)); /* if y<zero, return +INF */ |
| 859dc438 | 189 | #else /* defined(vax)||defined(tahoe) */ |
| 9f4a7cc1 | 190 | return((y>zero)?x:one/x); |
| 859dc438 | 191 | #endif /* defined(vax)||defined(tahoe) */ |
| 9f4a7cc1 ZAL |
192 | } |
| 193 | if(x==1.0) return(x); /* if x=1.0, return 1 since y is finite */ | |
| 194 | ||
| 195 | /* reduce x to z in [sqrt(1/2)-1, sqrt(2)-1] */ | |
| 196 | z=scalb(x,-(n=logb(x))); | |
| 859dc438 | 197 | #if !defined(vax)&&!defined(tahoe) /* IEEE double; subnormal number */ |
| 9f4a7cc1 | 198 | if(n <= -1022) {n += (m=logb(z)); z=scalb(z,-m);} |
| 859dc438 | 199 | #endif /* !defined(vax)&&!defined(tahoe) */ |
| 9f4a7cc1 ZAL |
200 | if(z >= sqrt2 ) {n += 1; z *= half;} z -= one ; |
| 201 | ||
| 202 | /* log(x) = nlog2+log(1+z) ~ nlog2 + t + tx */ | |
| 203 | s=z/(two+z); c=z*z*half; tx=s*(c+log__L(s*s)); | |
| 204 | t= z-(c-tx); tx += (z-t)-c; | |
| 205 | ||
| 206 | /* if y*log(x) is neither too big nor too small */ | |
| 207 | if((s=logb(y)+logb(n+t)) < 12.0) | |
| 208 | if(s>-60.0) { | |
| 209 | ||
| 210 | /* compute y*log(x) ~ mlog2 + t + c */ | |
| 211 | s=y*(n+invln2*t); | |
| 212 | m=s+copysign(half,s); /* m := nint(y*log(x)) */ | |
| 213 | k=y; | |
| 214 | if((double)k==y) { /* if y is an integer */ | |
| 215 | k = m-k*n; | |
| 216 | sx=t; tx+=(t-sx); } | |
| 217 | else { /* if y is not an integer */ | |
| 218 | k =m; | |
| 219 | tx+=n*ln2lo; | |
| 220 | sx=(c=n*ln2hi)+t; tx+=(c-sx)+t; } | |
| 221 | /* end of checking whether k==y */ | |
| 222 | ||
| 223 | sy=y; ty=y-sy; /* y ~ sy + ty */ | |
| 859dc438 | 224 | #ifdef tahoe |
| 471c7555 | 225 | s = (tahoe_tmp = sx)*sy-k*ln2hi; |
| 859dc438 | 226 | #else /* tahoe */ |
| 9f4a7cc1 | 227 | s=(double)sx*sy-k*ln2hi; /* (sy+ty)*(sx+tx)-kln2 */ |
| 859dc438 | 228 | #endif /* tahoe */ |
| 9f4a7cc1 ZAL |
229 | z=(tx*ty-k*ln2lo); |
| 230 | tx=tx*sy; ty=sx*ty; | |
| 231 | t=ty+z; t+=tx; t+=s; | |
| 232 | c= -((((t-s)-tx)-ty)-z); | |
| 233 | ||
| 234 | /* return exp(y*log(x)) */ | |
| 235 | t += exp__E(t,c); return(scalb(one+t,m)); | |
| 236 | } | |
| 237 | /* end of if log(y*log(x)) > -60.0 */ | |
| 238 | ||
| 239 | else | |
| 240 | /* exp(+- tiny) = 1 with inexact flag */ | |
| 241 | {ln2hi+ln2lo; return(one);} | |
| 242 | else if(copysign(one,y)*(n+invln2*t) <zero) | |
| 243 | /* exp(-(big#)) underflows to zero */ | |
| 244 | return(scalb(one,-5000)); | |
| 245 | else | |
| 246 | /* exp(+(big#)) overflows to INF */ | |
| 247 | return(scalb(one, 5000)); | |
| 248 | ||
| 249 | } | |
| 6229246a | 250 | #endif /* mc68881 */ |