4.3BSD - /usr/src/usr.lib/libm/exp__E.c

   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: }

Defined functions

Defined variables

Defined macros