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Annotation of 43BSD/usr.lib/libm/log__L.c, revision 1.1.1.1

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[] = "@(#)log__L.c   1.2 (Berkeley) 8/21/85";
                     16: #endif not lint
                     17: 
                     18: /* log__L(Z)
                     19:  *             LOG(1+X) - 2S                          X
                     20:  * RETURN      ---------------  WHERE Z = S*S,  S = ------- , 0 <= Z <= .0294...
                     21:  *                   S                              2 + X
                     22:  *                  
                     23:  * DOUBLE PRECISION (VAX D FORMAT 56 bits or IEEE DOUBLE 53 BITS)
                     24:  * KERNEL FUNCTION FOR LOG; TO BE USED IN LOG1P, LOG, AND POW FUNCTIONS
                     25:  * CODED IN C BY K.C. NG, 1/19/85; 
                     26:  * REVISED BY K.C. Ng, 2/3/85, 4/16/85.
                     27:  *
                     28:  * Method :
                     29:  *     1. Polynomial approximation: let s = x/(2+x). 
                     30:  *        Based on log(1+x) = log(1+s) - log(1-s)
                     31:  *              = 2s + 2/3 s**3 + 2/5 s**5 + .....,
                     32:  *
                     33:  *        (log(1+x) - 2s)/s is computed by
                     34:  *
                     35:  *            z*(L1 + z*(L2 + z*(... (L7 + z*L8)...)))
                     36:  *
                     37:  *        where z=s*s. (See the listing below for Lk's values.) The 
                     38:  *        coefficients are obtained by a special Remez algorithm. 
                     39:  *
                     40:  * Accuracy:
                     41:  *     Assuming no rounding error, the maximum magnitude of the approximation 
                     42:  *     error (absolute) is 2**(-58.49) for IEEE double, and 2**(-63.63)
                     43:  *     for VAX D format.
                     44:  *
                     45:  * Constants:
                     46:  * The hexadecimal values are the intended ones for the following constants.
                     47:  * The decimal values may be used, provided that the compiler will convert
                     48:  * from decimal to binary accurately enough to produce the hexadecimal values
                     49:  * shown.
                     50:  */
                     51: 
                     52: #ifdef VAX     /* VAX D format (56 bits) */
                     53: /* static double */
                     54: /* L1     =  6.6666666666666703212E-1    , Hex  2^  0   *  .AAAAAAAAAAAAC5 */
                     55: /* L2     =  3.9999999999970461961E-1    , Hex  2^ -1   *  .CCCCCCCCCC2684 */
                     56: /* L3     =  2.8571428579395698188E-1    , Hex  2^ -1   *  .92492492F85782 */
                     57: /* L4     =  2.2222221233634724402E-1    , Hex  2^ -2   *  .E38E3839B7AF2C */
                     58: /* L5     =  1.8181879517064680057E-1    , Hex  2^ -2   *  .BA2EB4CC39655E */
                     59: /* L6     =  1.5382888777946145467E-1    , Hex  2^ -2   *  .9D8551E8C5781D */
                     60: /* L7     =  1.3338356561139403517E-1    , Hex  2^ -2   *  .8895B3907FCD92 */
                     61: /* L8     =  1.2500000000000000000E-1    , Hex  2^ -2   *  .80000000000000 */
                     62: static long        L1x[] = { 0xaaaa402a, 0xaac5aaaa};
                     63: static long        L2x[] = { 0xcccc3fcc, 0x2684cccc};
                     64: static long        L3x[] = { 0x49243f92, 0x578292f8};
                     65: static long        L4x[] = { 0x8e383f63, 0xaf2c39b7};
                     66: static long        L5x[] = { 0x2eb43f3a, 0x655ecc39};
                     67: static long        L6x[] = { 0x85513f1d, 0x781de8c5};
                     68: static long        L7x[] = { 0x95b33f08, 0xcd92907f};
                     69: static long        L8x[] = { 0x00003f00, 0x00000000};
                     70: #define       L1    (*(double*)L1x)
                     71: #define       L2    (*(double*)L2x)
                     72: #define       L3    (*(double*)L3x)
                     73: #define       L4    (*(double*)L4x)
                     74: #define       L5    (*(double*)L5x)
                     75: #define       L6    (*(double*)L6x)
                     76: #define       L7    (*(double*)L7x)
                     77: #define       L8    (*(double*)L8x)
                     78: #else  /* IEEE double */
                     79: static double
                     80: L1     =  6.6666666666667340202E-1    , /*Hex  2^ -1   *  1.5555555555592 */
                     81: L2     =  3.9999999999416702146E-1    , /*Hex  2^ -2   *  1.999999997FF24 */
                     82: L3     =  2.8571428742008753154E-1    , /*Hex  2^ -2   *  1.24924941E07B4 */
                     83: L4     =  2.2222198607186277597E-1    , /*Hex  2^ -3   *  1.C71C52150BEA6 */
                     84: L5     =  1.8183562745289935658E-1    , /*Hex  2^ -3   *  1.74663CC94342F */
                     85: L6     =  1.5314087275331442206E-1    , /*Hex  2^ -3   *  1.39A1EC014045B */
                     86: L7     =  1.4795612545334174692E-1    ; /*Hex  2^ -3   *  1.2F039F0085122 */
                     87: #endif
                     88: 
                     89: double log__L(z)
                     90: double z;
                     91: {
                     92: #ifdef VAX
                     93:     return(z*(L1+z*(L2+z*(L3+z*(L4+z*(L5+z*(L6+z*(L7+z*L8))))))));
                     94: #else  /* IEEE double */
                     95:     return(z*(L1+z*(L2+z*(L3+z*(L4+z*(L5+z*(L6+z*L7)))))));
                     96: #endif
                     97: }

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