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1.1 root 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 5.3 (Berkeley) 6/30/88";
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: #if defined(vax)||defined(tahoe) /* VAX D format */
96: #include <errno.h>
97: extern double infnan();
98: #ifdef vax
99: #define _0x(A,B) 0x/**/A/**/B
100: #else /* vax */
101: #define _0x(A,B) 0x/**/B/**/A
102: #endif /* vax */
103: /* static double */
104: /* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */
105: /* ln2lo = 1.6465949582897081279E-12 , Hex 2^-39 * .E7BCD5E4F1D9CC */
106: /* invln2 = 1.4426950408889634148E0 , Hex 2^ 1 * .B8AA3B295C17F1 */
107: /* sqrt2 = 1.4142135623730950622E0 ; Hex 2^ 1 * .B504F333F9DE65 */
108: static long ln2hix[] = { _0x(7217,4031), _0x(0000,f7d0)};
109: static long ln2lox[] = { _0x(bcd5,2ce7), _0x(d9cc,e4f1)};
110: static long invln2x[] = { _0x(aa3b,40b8), _0x(17f1,295c)};
111: static long sqrt2x[] = { _0x(04f3,40b5), _0x(de65,33f9)};
112: #define ln2hi (*(double*)ln2hix)
113: #define ln2lo (*(double*)ln2lox)
114: #define invln2 (*(double*)invln2x)
115: #define sqrt2 (*(double*)sqrt2x)
116: #else /* defined(vax)||defined(tahoe) */
117: static double
118: ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */
119: ln2lo = 1.9082149292705877000E-10 , /*Hex 2^-33 * 1.A39EF35793C76 */
120: invln2 = 1.4426950408889633870E0 , /*Hex 2^ 0 * 1.71547652B82FE */
121: sqrt2 = 1.4142135623730951455E0 ; /*Hex 2^ 0 * 1.6A09E667F3BCD */
122: #endif /* defined(vax)||defined(tahoe) */
123:
124: static double zero=0.0, half=1.0/2.0, one=1.0, two=2.0, negone= -1.0;
125:
126: double pow(x,y)
127: double x,y;
128: {
129: double drem(),pow_p(),copysign(),t;
130: int finite();
131:
132: if (y==zero) return(one);
133: else if(y==one
134: #if !defined(vax)&&!defined(tahoe)
135: ||x!=x
136: #endif /* !defined(vax)&&!defined(tahoe) */
137: ) return( x ); /* if x is NaN or y=1 */
138: #if !defined(vax)&&!defined(tahoe)
139: else if(y!=y) return( y ); /* if y is NaN */
140: #endif /* !defined(vax)&&!defined(tahoe) */
141: else if(!finite(y)) /* if y is INF */
142: if((t=copysign(x,one))==one) return(zero/zero);
143: else if(t>one) return((y>zero)?y:zero);
144: else return((y<zero)?-y:zero);
145: else if(y==two) return(x*x);
146: else if(y==negone) return(one/x);
147:
148: /* sign(x) = 1 */
149: else if(copysign(one,x)==one) return(pow_p(x,y));
150:
151: /* sign(x)= -1 */
152: /* if y is an even integer */
153: else if ( (t=drem(y,two)) == zero) return( pow_p(-x,y) );
154:
155: /* if y is an odd integer */
156: else if (copysign(t,one) == one) return( -pow_p(-x,y) );
157:
158: /* Henceforth y is not an integer */
159: else if(x==zero) /* x is -0 */
160: return((y>zero)?-x:one/(-x));
161: else { /* return NaN */
162: #if defined(vax)||defined(tahoe)
163: return (infnan(EDOM)); /* NaN */
164: #else /* defined(vax)||defined(tahoe) */
165: return(zero/zero);
166: #endif /* defined(vax)||defined(tahoe) */
167: }
168: }
169:
170: /* pow_p(x,y) return x**y for x with sign=1 and finite y */
171: static double pow_p(x,y)
172: double x,y;
173: {
174: double logb(),scalb(),copysign(),log__L(),exp__E();
175: double c,s,t,z,tx,ty;
176: #ifdef tahoe
177: double tahoe_tmp;
178: #endif /* tahoe */
179: float sx,sy;
180: long k=0;
181: int n,m;
182:
183: if(x==zero||!finite(x)) { /* if x is +INF or +0 */
184: #if defined(vax)||defined(tahoe)
185: return((y>zero)?x:infnan(ERANGE)); /* if y<zero, return +INF */
186: #else /* defined(vax)||defined(tahoe) */
187: return((y>zero)?x:one/x);
188: #endif /* defined(vax)||defined(tahoe) */
189: }
190: if(x==1.0) return(x); /* if x=1.0, return 1 since y is finite */
191:
192: /* reduce x to z in [sqrt(1/2)-1, sqrt(2)-1] */
193: z=scalb(x,-(n=logb(x)));
194: #if !defined(vax)&&!defined(tahoe) /* IEEE double; subnormal number */
195: if(n <= -1022) {n += (m=logb(z)); z=scalb(z,-m);}
196: #endif /* !defined(vax)&&!defined(tahoe) */
197: if(z >= sqrt2 ) {n += 1; z *= half;} z -= one ;
198:
199: /* log(x) = nlog2+log(1+z) ~ nlog2 + t + tx */
200: s=z/(two+z); c=z*z*half; tx=s*(c+log__L(s*s));
201: t= z-(c-tx); tx += (z-t)-c;
202:
203: /* if y*log(x) is neither too big nor too small */
204: if((s=logb(y)+logb(n+t)) < 12.0)
205: if(s>-60.0) {
206:
207: /* compute y*log(x) ~ mlog2 + t + c */
208: s=y*(n+invln2*t);
209: m=s+copysign(half,s); /* m := nint(y*log(x)) */
210: k=y;
211: if((double)k==y) { /* if y is an integer */
212: k = m-k*n;
213: sx=t; tx+=(t-sx); }
214: else { /* if y is not an integer */
215: k =m;
216: tx+=n*ln2lo;
217: sx=(c=n*ln2hi)+t; tx+=(c-sx)+t; }
218: /* end of checking whether k==y */
219:
220: sy=y; ty=y-sy; /* y ~ sy + ty */
221: #ifdef tahoe
222: s = (tahoe_tmp = sx)*sy-k*ln2hi;
223: #else /* tahoe */
224: s=(double)sx*sy-k*ln2hi; /* (sy+ty)*(sx+tx)-kln2 */
225: #endif /* tahoe */
226: z=(tx*ty-k*ln2lo);
227: tx=tx*sy; ty=sx*ty;
228: t=ty+z; t+=tx; t+=s;
229: c= -((((t-s)-tx)-ty)-z);
230:
231: /* return exp(y*log(x)) */
232: t += exp__E(t,c); return(scalb(one+t,m));
233: }
234: /* end of if log(y*log(x)) > -60.0 */
235:
236: else
237: /* exp(+- tiny) = 1 with inexact flag */
238: {ln2hi+ln2lo; return(one);}
239: else if(copysign(one,y)*(n+invln2*t) <zero)
240: /* exp(-(big#)) underflows to zero */
241: return(scalb(one,-5000));
242: else
243: /* exp(+(big#)) overflows to INF */
244: return(scalb(one, 5000));
245:
246: }
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