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

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