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Annotation of 43BSDTahoe/usr.lib/libm/exp__E.c, revision 1.1

1.1     ! root        1: /*
        !             2:  * Copyright (c) 1985 Regents of the University of California.
        !             3:  * All rights reserved.
        !             4:  *
        !             5:  * Redistribution and use in source and binary forms are permitted
        !             6:  * provided that the above copyright notice and this paragraph are
        !             7:  * duplicated in all such forms and that any documentation,
        !             8:  * advertising materials, and other materials related to such
        !             9:  * distribution and use acknowledge that the software was developed
        !            10:  * by the University of California, Berkeley.  The name of the
        !            11:  * University may not be used to endorse or promote products derived
        !            12:  * from this software without specific prior written permission.
        !            13:  * THIS SOFTWARE IS PROVIDED ``AS IS'' AND WITHOUT ANY EXPRESS OR
        !            14:  * IMPLIED WARRANTIES, INCLUDING, WITHOUT LIMITATION, THE IMPLIED
        !            15:  * WARRANTIES OF MERCHANTIBILITY AND FITNESS FOR A PARTICULAR PURPOSE.
        !            16:  *
        !            17:  * All recipients should regard themselves as participants in an ongoing
        !            18:  * research project and hence should feel obligated to report their
        !            19:  * experiences (good or bad) with these elementary function codes, using
        !            20:  * the sendbug(8) program, to the authors.
        !            21:  */
        !            22: 
        !            23: #ifndef lint
        !            24: static char sccsid[] = "@(#)exp__E.c   5.3 (Berkeley) 6/30/88";
        !            25: #endif /* not lint */
        !            26: 
        !            27: /* exp__E(x,c)
        !            28:  * ASSUMPTION: c << x  SO THAT  fl(x+c)=x.
        !            29:  * (c is the correction term for x)
        !            30:  * exp__E RETURNS
        !            31:  *
        !            32:  *                      /  exp(x+c) - 1 - x ,  1E-19 < |x| < .3465736
        !            33:  *       exp__E(x,c) =         |                    
        !            34:  *                      \  0 ,  |x| < 1E-19.
        !            35:  *
        !            36:  * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS)
        !            37:  * KERNEL FUNCTION OF EXP, EXPM1, POW FUNCTIONS
        !            38:  * CODED IN C BY K.C. NG, 1/31/85;
        !            39:  * REVISED BY K.C. NG on 3/16/85, 4/16/85.
        !            40:  *
        !            41:  * Required system supported function:
        !            42:  *     copysign(x,y)   
        !            43:  *
        !            44:  * Method:
        !            45:  *     1. Rational approximation. Let r=x+c.
        !            46:  *        Based on
        !            47:  *                                   2 * sinh(r/2)     
        !            48:  *                exp(r) - 1 =   ----------------------   ,
        !            49:  *                               cosh(r/2) - sinh(r/2)
        !            50:  *        exp__E(r) is computed using
        !            51:  *                   x*x            (x/2)*W - ( Q - ( 2*P  + x*P ) )
        !            52:  *                   --- + (c + x*[---------------------------------- + c ])
        !            53:  *                    2                          1 - W
        !            54:  *        where  P := p1*x^2 + p2*x^4,
        !            55:  *               Q := q1*x^2 + q2*x^4 (for 56 bits precision, add q3*x^6)
        !            56:  *               W := x/2-(Q-x*P),
        !            57:  *
        !            58:  *        (See the listing below for the values of p1,p2,q1,q2,q3. The poly-
        !            59:  *         nomials P and Q may be regarded as the approximations to sinh
        !            60:  *         and cosh :
        !            61:  *             sinh(r/2) =  r/2 + r * P  ,  cosh(r/2) =  1 + Q . )
        !            62:  *
        !            63:  *         The coefficients were obtained by a special Remez algorithm.
        !            64:  *
        !            65:  * Approximation error:
        !            66:  *
        !            67:  *   | exp(x) - 1                         |        2**(-57),  (IEEE double)
        !            68:  *   | ------------  -  (exp__E(x,0)+x)/x  |  <= 
        !            69:  *   |      x                             |        2**(-69).  (VAX D)
        !            70:  *
        !            71:  * Constants:
        !            72:  * The hexadecimal values are the intended ones for the following constants.
        !            73:  * The decimal values may be used, provided that the compiler will convert
        !            74:  * from decimal to binary accurately enough to produce the hexadecimal values
        !            75:  * shown.
        !            76:  */
        !            77: 
        !            78: #if defined(vax)||defined(tahoe)       /* VAX D format */
        !            79: #ifdef vax
        !            80: #define _0x(A,B)       0x/**/A/**/B
        !            81: #else  /* vax */
        !            82: #define _0x(A,B)       0x/**/B/**/A
        !            83: #endif /* vax */
        !            84: /* static double */
        !            85: /* p1     =  1.5150724356786683059E-2    , Hex  2^ -6   *  .F83ABE67E1066A */
        !            86: /* p2     =  6.3112487873718332688E-5    , Hex  2^-13   *  .845B4248CD0173 */
        !            87: /* q1     =  1.1363478204690669916E-1    , Hex  2^ -3   *  .E8B95A44A2EC45 */
        !            88: /* q2     =  1.2624568129896839182E-3    , Hex  2^ -9   *  .A5790572E4F5E7 */
        !            89: /* q3     =  1.5021856115869022674E-6    ; Hex  2^-19   *  .C99EB4604AC395 */
        !            90: static long        p1x[] = { _0x(3abe,3d78), _0x(066a,67e1)};
        !            91: static long        p2x[] = { _0x(5b42,3984), _0x(0173,48cd)};
        !            92: static long        q1x[] = { _0x(b95a,3ee8), _0x(ec45,44a2)};
        !            93: static long        q2x[] = { _0x(7905,3ba5), _0x(f5e7,72e4)};
        !            94: static long        q3x[] = { _0x(9eb4,36c9), _0x(c395,604a)};
        !            95: #define       p1    (*(double*)p1x)
        !            96: #define       p2    (*(double*)p2x)
        !            97: #define       q1    (*(double*)q1x)
        !            98: #define       q2    (*(double*)q2x)
        !            99: #define       q3    (*(double*)q3x)
        !           100: #else  /* defined(vax)||defined(tahoe) */
        !           101: static double 
        !           102: p1     =  1.3887401997267371720E-2    , /*Hex  2^ -7   *  1.C70FF8B3CC2CF */
        !           103: p2     =  3.3044019718331897649E-5    , /*Hex  2^-15   *  1.15317DF4526C4 */
        !           104: q1     =  1.1110813732786649355E-1    , /*Hex  2^ -4   *  1.C719538248597 */
        !           105: q2     =  9.9176615021572857300E-4    ; /*Hex  2^-10   *  1.03FC4CB8C98E8 */
        !           106: #endif /* defined(vax)||defined(tahoe) */
        !           107: 
        !           108: double exp__E(x,c)
        !           109: double x,c;
        !           110: {
        !           111:        static double zero=0.0, one=1.0, half=1.0/2.0, small=1.0E-19;
        !           112:        double copysign(),z,p,q,xp,xh,w;
        !           113:        if(copysign(x,one)>small) {
        !           114:            z = x*x  ;
        !           115:           p = z*( p1 +z* p2 );
        !           116: #if defined(vax)||defined(tahoe)
        !           117:            q = z*( q1 +z*( q2 +z* q3 ));
        !           118: #else  /* defined(vax)||defined(tahoe) */
        !           119:            q = z*( q1 +z*  q2 );
        !           120: #endif /* defined(vax)||defined(tahoe) */
        !           121:            xp= x*p     ; 
        !           122:           xh= x*half  ;
        !           123:            w = xh-(q-xp)  ;
        !           124:           p = p+p;
        !           125:           c += x*((xh*w-(q-(p+xp)))/(one-w)+c);
        !           126:           return(z*half+c);
        !           127:        }
        !           128:        /* end of |x| > small */
        !           129: 
        !           130:        else {
        !           131:            if(x!=zero) one+small;      /* raise the inexact flag */
        !           132:            return(copysign(zero,x));
        !           133:        }
        !           134: }

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