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Annotation of 43BSD/usr.lib/libm/exp__E.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[] = "@(#)exp__E.c   1.2 (Berkeley) 8/21/85";
                     16: #endif not lint
                     17: 
                     18: /* exp__E(x,c)
                     19:  * ASSUMPTION: c << x  SO THAT  fl(x+c)=x.
                     20:  * (c is the correction term for x)
                     21:  * exp__E RETURNS
                     22:  *
                     23:  *                      /  exp(x+c) - 1 - x ,  1E-19 < |x| < .3465736
                     24:  *       exp__E(x,c) =         |                    
                     25:  *                      \  0 ,  |x| < 1E-19.
                     26:  *
                     27:  * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS)
                     28:  * KERNEL FUNCTION OF EXP, EXPM1, POW FUNCTIONS
                     29:  * CODED IN C BY K.C. NG, 1/31/85;
                     30:  * REVISED BY K.C. NG on 3/16/85, 4/16/85.
                     31:  *
                     32:  * Required system supported function:
                     33:  *     copysign(x,y)   
                     34:  *
                     35:  * Method:
                     36:  *     1. Rational approximation. Let r=x+c.
                     37:  *        Based on
                     38:  *                                   2 * sinh(r/2)     
                     39:  *                exp(r) - 1 =   ----------------------   ,
                     40:  *                               cosh(r/2) - sinh(r/2)
                     41:  *        exp__E(r) is computed using
                     42:  *                   x*x            (x/2)*W - ( Q - ( 2*P  + x*P ) )
                     43:  *                   --- + (c + x*[---------------------------------- + c ])
                     44:  *                    2                          1 - W
                     45:  *        where  P := p1*x^2 + p2*x^4,
                     46:  *               Q := q1*x^2 + q2*x^4 (for 56 bits precision, add q3*x^6)
                     47:  *               W := x/2-(Q-x*P),
                     48:  *
                     49:  *        (See the listing below for the values of p1,p2,q1,q2,q3. The poly-
                     50:  *         nomials P and Q may be regarded as the approximations to sinh
                     51:  *         and cosh :
                     52:  *             sinh(r/2) =  r/2 + r * P  ,  cosh(r/2) =  1 + Q . )
                     53:  *
                     54:  *         The coefficients were obtained by a special Remez algorithm.
                     55:  *
                     56:  * Approximation error:
                     57:  *
                     58:  *   | exp(x) - 1                         |        2**(-57),  (IEEE double)
                     59:  *   | ------------  -  (exp__E(x,0)+x)/x  |  <= 
                     60:  *   |      x                             |        2**(-69).  (VAX D)
                     61:  *
                     62:  * Constants:
                     63:  * The hexadecimal values are the intended ones for the following constants.
                     64:  * The decimal values may be used, provided that the compiler will convert
                     65:  * from decimal to binary accurately enough to produce the hexadecimal values
                     66:  * shown.
                     67:  */
                     68: 
                     69: #ifdef VAX     /* VAX D format */
                     70: /* static double */
                     71: /* p1     =  1.5150724356786683059E-2    , Hex  2^ -6   *  .F83ABE67E1066A */
                     72: /* p2     =  6.3112487873718332688E-5    , Hex  2^-13   *  .845B4248CD0173 */
                     73: /* q1     =  1.1363478204690669916E-1    , Hex  2^ -3   *  .E8B95A44A2EC45 */
                     74: /* q2     =  1.2624568129896839182E-3    , Hex  2^ -9   *  .A5790572E4F5E7 */
                     75: /* q3     =  1.5021856115869022674E-6    ; Hex  2^-19   *  .C99EB4604AC395 */
                     76: static long        p1x[] = { 0x3abe3d78, 0x066a67e1};
                     77: static long        p2x[] = { 0x5b423984, 0x017348cd};
                     78: static long        q1x[] = { 0xb95a3ee8, 0xec4544a2};
                     79: static long        q2x[] = { 0x79053ba5, 0xf5e772e4};
                     80: static long        q3x[] = { 0x9eb436c9, 0xc395604a};
                     81: #define       p1    (*(double*)p1x)
                     82: #define       p2    (*(double*)p2x)
                     83: #define       q1    (*(double*)q1x)
                     84: #define       q2    (*(double*)q2x)
                     85: #define       q3    (*(double*)q3x)
                     86: #else  /* IEEE double */
                     87: static double 
                     88: p1     =  1.3887401997267371720E-2    , /*Hex  2^ -7   *  1.C70FF8B3CC2CF */
                     89: p2     =  3.3044019718331897649E-5    , /*Hex  2^-15   *  1.15317DF4526C4 */
                     90: q1     =  1.1110813732786649355E-1    , /*Hex  2^ -4   *  1.C719538248597 */
                     91: q2     =  9.9176615021572857300E-4    ; /*Hex  2^-10   *  1.03FC4CB8C98E8 */
                     92: #endif
                     93: 
                     94: double exp__E(x,c)
                     95: double x,c;
                     96: {
                     97:        double static zero=0.0, one=1.0, half=1.0/2.0, small=1.0E-19;
                     98:        double copysign(),z,p,q,xp,xh,w;
                     99:        if(copysign(x,one)>small) {
                    100:            z = x*x  ;
                    101:           p = z*( p1 +z* p2 );
                    102: #ifdef VAX
                    103:            q = z*( q1 +z*( q2 +z* q3 ));
                    104: #else  /* IEEE double */
                    105:            q = z*( q1 +z*  q2 );
                    106: #endif
                    107:            xp= x*p     ; 
                    108:           xh= x*half  ;
                    109:            w = xh-(q-xp)  ;
                    110:           p = p+p;
                    111:           c += x*((xh*w-(q-(p+xp)))/(one-w)+c);
                    112:           return(z*half+c);
                    113:        }
                    114:        /* end of |x| > small */
                    115: 
                    116:        else {
                    117:            if(x!=zero) one+small;      /* raise the inexact flag */
                    118:            return(copysign(zero,x));
                    119:        }
                    120: }

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