134123Sbostic /*
2*61283Sbostic * Copyright (c) 1985, 1993
3*61283Sbostic * The Regents of the University of California. All rights reserved.
434123Sbostic *
542657Sbostic * %sccs.include.redist.c%
624580Szliu */
724580Szliu
824580Szliu #ifndef lint
9*61283Sbostic static char sccsid[] = "@(#)cbrt.c 8.1 (Berkeley) 06/04/93";
1034123Sbostic #endif /* not lint */
1124580Szliu
1246401Sbostic #include <sys/cdefs.h>
1342656Sbostic
1424580Szliu /* kahan's cube root (53 bits IEEE double precision)
1524580Szliu * for IEEE machines only
1624580Szliu * coded in C by K.C. Ng, 4/30/85
1724580Szliu *
1824580Szliu * Accuracy:
1924580Szliu * better than 0.667 ulps according to an error analysis. Maximum
2024580Szliu * error observed was 0.666 ulps in an 1,000,000 random arguments test.
2124580Szliu *
2224580Szliu * Warning: this code is semi machine dependent; the ordering of words in
2324580Szliu * a floating point number must be known in advance. I assume that the
2424580Szliu * long interger at the address of a floating point number will be the
2524580Szliu * leading 32 bits of that floating point number (i.e., sign, exponent,
2624580Szliu * and the 20 most significant bits).
2724580Szliu * On a National machine, it has different ordering; therefore, this code
2824580Szliu * must be compiled with flag -DNATIONAL.
2924580Szliu */
3031856Szliu #if !defined(vax)&&!defined(tahoe)
3124580Szliu
3235681Sbostic static const unsigned long
3335681Sbostic B1 = 715094163, /* B1 = (682-0.03306235651)*2**20 */
3424580Szliu B2 = 696219795; /* B2 = (664-0.03306235651)*2**20 */
3535681Sbostic static const double
3624580Szliu C= 19./35.,
3724580Szliu D= -864./1225.,
3824580Szliu E= 99./70.,
3924580Szliu F= 45./28.,
4024580Szliu G= 5./14.;
4124580Szliu
cbrt(x)4224580Szliu double cbrt(x)
4324580Szliu double x;
4424580Szliu {
4524580Szliu double r,s,t=0.0,w;
4624580Szliu unsigned long *px = (unsigned long *) &x,
4724580Szliu *pt = (unsigned long *) &t,
4824580Szliu mexp,sign;
4924580Szliu
5031856Szliu #ifdef national /* ordering of words in a floating points number */
5135681Sbostic const int n0=1,n1=0;
5231856Szliu #else /* national */
5335681Sbostic const int n0=0,n1=1;
5431856Szliu #endif /* national */
5524580Szliu
5624580Szliu mexp=px[n0]&0x7ff00000;
5724580Szliu if(mexp==0x7ff00000) return(x); /* cbrt(NaN,INF) is itself */
5824580Szliu if(x==0.0) return(x); /* cbrt(0) is itself */
5924580Szliu
6024580Szliu sign=px[n0]&0x80000000; /* sign= sign(x) */
6124580Szliu px[n0] ^= sign; /* x=|x| */
6224580Szliu
6324580Szliu
6424580Szliu /* rough cbrt to 5 bits */
6524580Szliu if(mexp==0) /* subnormal number */
6624580Szliu {pt[n0]=0x43500000; /* set t= 2**54 */
6724580Szliu t*=x; pt[n0]=pt[n0]/3+B2;
6824580Szliu }
6924580Szliu else
7024580Szliu pt[n0]=px[n0]/3+B1;
7124580Szliu
7224580Szliu
7324580Szliu /* new cbrt to 23 bits, may be implemented in single precision */
7424580Szliu r=t*t/x;
7524580Szliu s=C+r*t;
7624580Szliu t*=G+F/(s+E+D/s);
7724580Szliu
7824580Szliu /* chopped to 20 bits and make it larger than cbrt(x) */
7924580Szliu pt[n1]=0; pt[n0]+=0x00000001;
8024580Szliu
8124580Szliu
8224580Szliu /* one step newton iteration to 53 bits with error less than 0.667 ulps */
8324580Szliu s=t*t; /* t*t is exact */
8424580Szliu r=x/s;
8524580Szliu w=t+t;
8633938Sbostic r=(r-t)/(w+r); /* r-t is exact */
8724580Szliu t=t+t*r;
8824580Szliu
8924580Szliu
9024580Szliu /* retore the sign bit */
9124580Szliu pt[n0] |= sign;
9224580Szliu return(t);
9324580Szliu }
9424580Szliu #endif
95