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Annotation of 43BSD/usr.lib/libm/IEEE/cbrt.c, revision 1.1

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[] = "@(#)cbrt.c     1.1 (Berkeley) 5/23/85";
        !            16: #endif not lint
        !            17: 
        !            18: /* kahan's cube root (53 bits IEEE double precision)
        !            19:  * for IEEE machines only
        !            20:  * coded in C by K.C. Ng, 4/30/85
        !            21:  *
        !            22:  * Accuracy:
        !            23:  *     better than 0.667 ulps according to an error analysis. Maximum
        !            24:  * error observed was 0.666 ulps in an 1,000,000 random arguments test.
        !            25:  *
        !            26:  * Warning: this code is semi machine dependent; the ordering of words in
        !            27:  * a floating point number must be known in advance. I assume that the
        !            28:  * long interger at the address of a floating point number will be the
        !            29:  * leading 32 bits of that floating point number (i.e., sign, exponent,
        !            30:  * and the 20 most significant bits).
        !            31:  * On a National machine, it has different ordering; therefore, this code 
        !            32:  * must be compiled with flag -DNATIONAL. 
        !            33:  */
        !            34: #ifndef VAX
        !            35: 
        !            36: static unsigned long B1 = 715094163, /* B1 = (682-0.03306235651)*2**20 */
        !            37:                     B2 = 696219795; /* B2 = (664-0.03306235651)*2**20 */
        !            38: static double
        !            39:            C= 19./35.,
        !            40:            D= -864./1225.,
        !            41:            E= 99./70.,
        !            42:            F= 45./28.,
        !            43:            G= 5./14.;
        !            44: 
        !            45: double cbrt(x) 
        !            46: double x;
        !            47: {
        !            48:        double r,s,t=0.0,w;
        !            49:        unsigned long *px = (unsigned long *) &x,
        !            50:                      *pt = (unsigned long *) &t,
        !            51:                      mexp,sign;
        !            52: 
        !            53: #ifdef NATIONAL /* ordering of words in a floating points number */
        !            54:        int n0=1,n1=0;
        !            55: #else
        !            56:        int n0=0,n1=1;
        !            57: #endif
        !            58: 
        !            59:        mexp=px[n0]&0x7ff00000;
        !            60:        if(mexp==0x7ff00000) return(x); /* cbrt(NaN,INF) is itself */
        !            61:        if(x==0.0) return(x);           /* cbrt(0) is itself */
        !            62: 
        !            63:        sign=px[n0]&0x80000000; /* sign= sign(x) */
        !            64:        px[n0] ^= sign;         /* x=|x| */
        !            65: 
        !            66: 
        !            67:     /* rough cbrt to 5 bits */
        !            68:        if(mexp==0)             /* subnormal number */
        !            69:          {pt[n0]=0x43500000;   /* set t= 2**54 */
        !            70:           t*=x; pt[n0]=pt[n0]/3+B2;
        !            71:          }
        !            72:        else
        !            73:          pt[n0]=px[n0]/3+B1;   
        !            74: 
        !            75: 
        !            76:     /* new cbrt to 23 bits, may be implemented in single precision */
        !            77:        r=t*t/x;
        !            78:        s=C+r*t;
        !            79:        t*=G+F/(s+E+D/s);       
        !            80: 
        !            81:     /* chopped to 20 bits and make it larger than cbrt(x) */ 
        !            82:        pt[n1]=0; pt[n0]+=0x00000001;
        !            83: 
        !            84: 
        !            85:     /* one step newton iteration to 53 bits with error less than 0.667 ulps */
        !            86:        s=t*t;          /* t*t is exact */
        !            87:        r=x/s;
        !            88:        w=t+t;
        !            89:        r=(r-t)/(w+r);  /* r-s is exact */
        !            90:        t=t+t*r;
        !            91: 
        !            92: 
        !            93:     /* retore the sign bit */
        !            94:        pt[n0] |= sign;
        !            95:        return(t);
        !            96: }
        !            97: #endif

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