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Annotation of 43BSD/usr.lib/libm/expm1.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[] = "@(#)expm1.c    1.2 (Berkeley) 8/21/85";
        !            16: #endif not lint
        !            17: 
        !            18: /* EXPM1(X)
        !            19:  * RETURN THE EXPONENTIAL OF X MINUS ONE
        !            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, 3/7/85, 3/21/85, 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,c)
        !            41:  *
        !            42:  *     3. EXPM1(x) =  2^k * ( EXPM1(r) + 1-2^-k ).
        !            43:  *
        !            44:  *     Remarks: 
        !            45:  *        1. When k=1 and z < -0.25, we use the following formula for
        !            46:  *           better accuracy:
        !            47:  *                     EXPM1(x) = 2 * ( (z+0.5) + exp__E(z,c) )
        !            48:  *        2. To avoid rounding error in 1-2^-k where k is large, we use
        !            49:  *                     EXPM1(x) = 2^k * { [z+(exp__E(z,c)-2^-k )] + 1 }
        !            50:  *           when k>56. 
        !            51:  *
        !            52:  * Special cases:
        !            53:  *     EXPM1(INF) is INF, EXPM1(NaN) is NaN;
        !            54:  *     EXPM1(-INF)= -1;
        !            55:  *     for finite argument, only EXPM1(0)=0 is exact.
        !            56:  *
        !            57:  * Accuracy:
        !            58:  *     EXPM1(x) returns the exact (exp(x)-1) nearly rounded. In a test run with
        !            59:  *     1,166,000 random arguments on a VAX, the maximum observed error was
        !            60:  *     .872 ulps (units of the last place).
        !            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: /* double static */
        !            71: /* ln2hi  =  6.9314718055829871446E-1    , Hex  2^  0   *  .B17217F7D00000 */
        !            72: /* ln2lo  =  1.6465949582897081279E-12   , Hex  2^-39   *  .E7BCD5E4F1D9CC */
        !            73: /* lnhuge =  9.4961163736712506989E1     , Hex  2^  7   *  .BDEC1DA73E9010 */
        !            74: /* invln2 =  1.4426950408889634148E0     ; Hex  2^  1   *  .B8AA3B295C17F1 */
        !            75: static long     ln2hix[] = { 0x72174031, 0x0000f7d0};
        !            76: static long     ln2lox[] = { 0xbcd52ce7, 0xd9cce4f1};
        !            77: static long    lnhugex[] = { 0xec1d43bd, 0x9010a73e};
        !            78: static long    invln2x[] = { 0xaa3b40b8, 0x17f1295c};
        !            79: #define    ln2hi    (*(double*)ln2hix)
        !            80: #define    ln2lo    (*(double*)ln2lox)
        !            81: #define   lnhuge    (*(double*)lnhugex)
        !            82: #define   invln2    (*(double*)invln2x)
        !            83: #else  /* IEEE double */
        !            84: double static
        !            85: ln2hi  =  6.9314718036912381649E-1    , /*Hex  2^ -1   *  1.62E42FEE00000 */
        !            86: ln2lo  =  1.9082149292705877000E-10   , /*Hex  2^-33   *  1.A39EF35793C76 */
        !            87: lnhuge =  7.1602103751842355450E2     , /*Hex  2^  9   *  1.6602B15B7ECF2 */
        !            88: invln2 =  1.4426950408889633870E0     ; /*Hex  2^  0   *  1.71547652B82FE */
        !            89: #endif
        !            90: 
        !            91: double expm1(x)
        !            92: double x;
        !            93: {
        !            94:        double static one=1.0, half=1.0/2.0; 
        !            95:        double scalb(), copysign(), exp__E(), z,hi,lo,c;
        !            96:        int k,finite();
        !            97: #ifdef VAX
        !            98:        static prec=56;
        !            99: #else  /* IEEE double */
        !           100:        static prec=53;
        !           101: #endif
        !           102: #ifndef VAX
        !           103:        if(x!=x) return(x);     /* x is NaN */
        !           104: #endif
        !           105: 
        !           106:        if( x <= lnhuge ) {
        !           107:                if( x >= -40.0 ) {
        !           108: 
        !           109:                    /* argument reduction : x - k*ln2 */
        !           110:                        k= invln2 *x+copysign(0.5,x);   /* k=NINT(x/ln2) */
        !           111:                        hi=x-k*ln2hi ; 
        !           112:                        z=hi-(lo=k*ln2lo);
        !           113:                        c=(hi-z)-lo;
        !           114: 
        !           115:                        if(k==0) return(z+exp__E(z,c));
        !           116:                        if(k==1)
        !           117:                            if(z< -0.25) 
        !           118:                                {x=z+half;x +=exp__E(z,c); return(x+x);}
        !           119:                            else
        !           120:                                {z+=exp__E(z,c); x=half+z; return(x+x);}
        !           121:                    /* end of k=1 */
        !           122: 
        !           123:                        else {
        !           124:                            if(k<=prec)
        !           125:                              { x=one-scalb(one,-k); z += exp__E(z,c);}
        !           126:                            else if(k<100)
        !           127:                              { x = exp__E(z,c)-scalb(one,-k); x+=z; z=one;}
        !           128:                            else 
        !           129:                              { x = exp__E(z,c)+z; z=one;}
        !           130: 
        !           131:                            return (scalb(x+z,k));  
        !           132:                        }
        !           133:                }
        !           134:                /* end of x > lnunfl */
        !           135: 
        !           136:                else 
        !           137:                     /* expm1(-big#) rounded to -1 (inexact) */
        !           138:                     if(finite(x))  
        !           139:                         { ln2hi+ln2lo; return(-one);}
        !           140: 
        !           141:                     /* expm1(-INF) is -1 */
        !           142:                     else return(-one);
        !           143:        }
        !           144:        /* end of x < lnhuge */
        !           145: 
        !           146:        else 
        !           147:        /*  expm1(INF) is INF, expm1(+big#) overflows to INF */
        !           148:            return( finite(x) ?  scalb(one,5000) : x);
        !           149: }

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