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Annotation of 43BSD/usr.lib/libm/log__L.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[] = "@(#)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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