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1.1 root 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 4.5 (Berkeley) 8/21/85";
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: }
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