4.3BSD - /usr/src/usr.lib/libm/exp.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.c	4.3 (Berkeley) 8/21/85";
  16: #endif not lint
  17: 
  18: /* EXP(X)
  19:  * RETURN THE EXPONENTIAL OF X
  20:  * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS)
  21:  * CODED IN C BY K.C. NG, 1/19/85;
  22:  * REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85.
  23:  *
  24:  * Required system supported functions:
  25:  *	scalb(x,n)
  26:  *	copysign(x,y)
  27:  *	finite(x)
  28:  *
  29:  * Kernel function:
  30:  *	exp__E(x,c)
  31:  *
  32:  * Method:
  33:  *	1. Argument Reduction: given the input x, find r and integer k such
  34:  *	   that
  35:  *	                   x = k*ln2 + r,  |r| <= 0.5*ln2 .
  36:  *	   r will be represented as r := z+c for better accuracy.
  37:  *
  38:  *	2. Compute expm1(r)=exp(r)-1 by
  39:  *
  40:  *			expm1(r=z+c) := z + exp__E(z,r)
  41:  *
  42:  *	3. exp(x) = 2^k * ( expm1(r) + 1 ).
  43:  *
  44:  * Special cases:
  45:  *	exp(INF) is INF, exp(NaN) is NaN;
  46:  *	exp(-INF)=  0;
  47:  *	for finite argument, only exp(0)=1 is exact.
  48:  *
  49:  * Accuracy:
  50:  *	exp(x) returns the exponential of x nearly rounded. In a test run
  51:  *	with 1,156,000 random arguments on a VAX, the maximum observed
  52:  *	error was .768 ulps (units in the last place).
  53:  *
  54:  * Constants:
  55:  * The hexadecimal values are the intended ones for the following constants.
  56:  * The decimal values may be used, provided that the compiler will convert
  57:  * from decimal to binary accurately enough to produce the hexadecimal values
  58:  * shown.
  59:  */
  60: 
  61: #ifdef VAX  /* VAX D format */
  62: /* double static */
  63: /* ln2hi  =  6.9314718055829871446E-1    , Hex  2^  0   *  .B17217F7D00000 */
  64: /* ln2lo  =  1.6465949582897081279E-12   , Hex  2^-39   *  .E7BCD5E4F1D9CC */
  65: /* lnhuge =  9.4961163736712506989E1     , Hex  2^  7   *  .BDEC1DA73E9010 */
  66: /* lntiny = -9.5654310917272452386E1     , Hex  2^  7   * -.BF4F01D72E33AF */
  67: /* invln2 =  1.4426950408889634148E0     ; Hex  2^  1   *  .B8AA3B295C17F1 */
  68: static long     ln2hix[] = { 0x72174031, 0x0000f7d0};
  69: static long     ln2lox[] = { 0xbcd52ce7, 0xd9cce4f1};
  70: static long    lnhugex[] = { 0xec1d43bd, 0x9010a73e};
  71: static long    lntinyx[] = { 0x4f01c3bf, 0x33afd72e};
  72: static long    invln2x[] = { 0xaa3b40b8, 0x17f1295c};
  73: #define    ln2hi    (*(double*)ln2hix)
  74: #define    ln2lo    (*(double*)ln2lox)
  75: #define   lnhuge    (*(double*)lnhugex)
  76: #define   lntiny    (*(double*)lntinyx)
  77: #define   invln2    (*(double*)invln2x)
  78: #else   /* IEEE double */
  79: double static
  80: ln2hi  =  6.9314718036912381649E-1    , /*Hex  2^ -1   *  1.62E42FEE00000 */
  81: ln2lo  =  1.9082149292705877000E-10   , /*Hex  2^-33   *  1.A39EF35793C76 */
  82: lnhuge =  7.1602103751842355450E2     , /*Hex  2^  9   *  1.6602B15B7ECF2 */
  83: lntiny = -7.5137154372698068983E2     , /*Hex  2^  9   * -1.77AF8EBEAE354 */
  84: invln2 =  1.4426950408889633870E0     ; /*Hex  2^  0   *  1.71547652B82FE */
  85: #endif
  86: 
  87: double exp(x)
  88: double x;
  89: {
  90:     double scalb(), copysign(), exp__E(), z,hi,lo,c;
  91:     int k,finite();
  92: 
  93: #ifndef VAX
  94:     if(x!=x) return(x); /* x is NaN */
  95: #endif
  96:     if( x <= lnhuge ) {
  97:         if( x >= lntiny ) {
  98: 
  99:             /* argument reduction : x --> x - k*ln2 */
 100: 
 101:             k=invln2*x+copysign(0.5,x); /* k=NINT(x/ln2) */
 102: 
 103:             /* express x-k*ln2 as z+c */
 104:             hi=x-k*ln2hi;
 105:             z=hi-(lo=k*ln2lo);
 106:             c=(hi-z)-lo;
 107: 
 108:             /* return 2^k*[expm1(x) + 1]  */
 109:             z += exp__E(z,c);
 110:             return (scalb(z+1.0,k));
 111:         }
 112:         /* end of x > lntiny */
 113: 
 114:         else
 115:              /* exp(-big#) underflows to zero */
 116:              if(finite(x))  return(scalb(1.0,-5000));
 117: 
 118:              /* exp(-INF) is zero */
 119:              else return(0.0);
 120:     }
 121:     /* end of x < lnhuge */
 122: 
 123:     else
 124:     /* exp(INF) is INF, exp(+big#) overflows to INF */
 125:         return( finite(x) ?  scalb(1.0,5000)  : x);
 126: }

Defined functions

Defined variables

Defined macros