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Annotation of 43BSDTahoe/usr.lib/libm/expm1.c, revision 1.1

1.1     ! root        1: /*
        !             2:  * Copyright (c) 1985 Regents of the University of California.
        !             3:  * All rights reserved.
        !             4:  *
        !             5:  * Redistribution and use in source and binary forms are permitted
        !             6:  * provided that the above copyright notice and this paragraph are
        !             7:  * duplicated in all such forms and that any documentation,
        !             8:  * advertising materials, and other materials related to such
        !             9:  * distribution and use acknowledge that the software was developed
        !            10:  * by the University of California, Berkeley.  The name of the
        !            11:  * University may not be used to endorse or promote products derived
        !            12:  * from this software without specific prior written permission.
        !            13:  * THIS SOFTWARE IS PROVIDED ``AS IS'' AND WITHOUT ANY EXPRESS OR
        !            14:  * IMPLIED WARRANTIES, INCLUDING, WITHOUT LIMITATION, THE IMPLIED
        !            15:  * WARRANTIES OF MERCHANTIBILITY AND FITNESS FOR A PARTICULAR PURPOSE.
        !            16:  *
        !            17:  * All recipients should regard themselves as participants in an ongoing
        !            18:  * research project and hence should feel obligated to report their
        !            19:  * experiences (good or bad) with these elementary function codes, using
        !            20:  * the sendbug(8) program, to the authors.
        !            21:  */
        !            22: 
        !            23: #ifndef lint
        !            24: static char sccsid[] = "@(#)expm1.c    5.3 (Berkeley) 6/30/88";
        !            25: #endif /* not lint */
        !            26: 
        !            27: /* EXPM1(X)
        !            28:  * RETURN THE EXPONENTIAL OF X MINUS ONE
        !            29:  * DOUBLE PRECISION (IEEE 53 BITS, VAX D FORMAT 56 BITS)
        !            30:  * CODED IN C BY K.C. NG, 1/19/85; 
        !            31:  * REVISED BY K.C. NG on 2/6/85, 3/7/85, 3/21/85, 4/16/85.
        !            32:  *
        !            33:  * Required system supported functions:
        !            34:  *     scalb(x,n)      
        !            35:  *     copysign(x,y)   
        !            36:  *     finite(x)
        !            37:  *
        !            38:  * Kernel function:
        !            39:  *     exp__E(x,c)
        !            40:  *
        !            41:  * Method:
        !            42:  *     1. Argument Reduction: given the input x, find r and integer k such 
        !            43:  *        that
        !            44:  *                        x = k*ln2 + r,  |r| <= 0.5*ln2 .  
        !            45:  *        r will be represented as r := z+c for better accuracy.
        !            46:  *
        !            47:  *     2. Compute EXPM1(r)=exp(r)-1 by 
        !            48:  *
        !            49:  *                     EXPM1(r=z+c) := z + exp__E(z,c)
        !            50:  *
        !            51:  *     3. EXPM1(x) =  2^k * ( EXPM1(r) + 1-2^-k ).
        !            52:  *
        !            53:  *     Remarks: 
        !            54:  *        1. When k=1 and z < -0.25, we use the following formula for
        !            55:  *           better accuracy:
        !            56:  *                     EXPM1(x) = 2 * ( (z+0.5) + exp__E(z,c) )
        !            57:  *        2. To avoid rounding error in 1-2^-k where k is large, we use
        !            58:  *                     EXPM1(x) = 2^k * { [z+(exp__E(z,c)-2^-k )] + 1 }
        !            59:  *           when k>56. 
        !            60:  *
        !            61:  * Special cases:
        !            62:  *     EXPM1(INF) is INF, EXPM1(NaN) is NaN;
        !            63:  *     EXPM1(-INF)= -1;
        !            64:  *     for finite argument, only EXPM1(0)=0 is exact.
        !            65:  *
        !            66:  * Accuracy:
        !            67:  *     EXPM1(x) returns the exact (exp(x)-1) nearly rounded. In a test run with
        !            68:  *     1,166,000 random arguments on a VAX, the maximum observed error was
        !            69:  *     .872 ulps (units of the last place).
        !            70:  *
        !            71:  * Constants:
        !            72:  * The hexadecimal values are the intended ones for the following constants.
        !            73:  * The decimal values may be used, provided that the compiler will convert
        !            74:  * from decimal to binary accurately enough to produce the hexadecimal values
        !            75:  * shown.
        !            76:  */
        !            77: 
        !            78: #if defined(vax)||defined(tahoe)       /* VAX D format */
        !            79: #ifdef vax
        !            80: #define _0x(A,B)       0x/**/A/**/B
        !            81: #else  /* vax */
        !            82: #define _0x(A,B)       0x/**/B/**/A
        !            83: #endif /* vax */
        !            84: /* static double */
        !            85: /* ln2hi  =  6.9314718055829871446E-1    , Hex  2^  0   *  .B17217F7D00000 */
        !            86: /* ln2lo  =  1.6465949582897081279E-12   , Hex  2^-39   *  .E7BCD5E4F1D9CC */
        !            87: /* lnhuge =  9.4961163736712506989E1     , Hex  2^  7   *  .BDEC1DA73E9010 */
        !            88: /* invln2 =  1.4426950408889634148E0     ; Hex  2^  1   *  .B8AA3B295C17F1 */
        !            89: static long     ln2hix[] = { _0x(7217,4031), _0x(0000,f7d0)};
        !            90: static long     ln2lox[] = { _0x(bcd5,2ce7), _0x(d9cc,e4f1)};
        !            91: static long    lnhugex[] = { _0x(ec1d,43bd), _0x(9010,a73e)};
        !            92: static long    invln2x[] = { _0x(aa3b,40b8), _0x(17f1,295c)};
        !            93: #define    ln2hi    (*(double*)ln2hix)
        !            94: #define    ln2lo    (*(double*)ln2lox)
        !            95: #define   lnhuge    (*(double*)lnhugex)
        !            96: #define   invln2    (*(double*)invln2x)
        !            97: #else  /* defined(vax)||defined(tahoe) */
        !            98: static double
        !            99: ln2hi  =  6.9314718036912381649E-1    , /*Hex  2^ -1   *  1.62E42FEE00000 */
        !           100: ln2lo  =  1.9082149292705877000E-10   , /*Hex  2^-33   *  1.A39EF35793C76 */
        !           101: lnhuge =  7.1602103751842355450E2     , /*Hex  2^  9   *  1.6602B15B7ECF2 */
        !           102: invln2 =  1.4426950408889633870E0     ; /*Hex  2^  0   *  1.71547652B82FE */
        !           103: #endif /* defined(vax)||defined(tahoe) */
        !           104: 
        !           105: double expm1(x)
        !           106: double x;
        !           107: {
        !           108:        static double one=1.0, half=1.0/2.0; 
        !           109:        double scalb(), copysign(), exp__E(), z,hi,lo,c;
        !           110:        int k,finite();
        !           111: #if defined(vax)||defined(tahoe)
        !           112:        static prec=56;
        !           113: #else  /* defined(vax)||defined(tahoe) */
        !           114:        static prec=53;
        !           115: #endif /* defined(vax)||defined(tahoe) */
        !           116: #if !defined(vax)&&!defined(tahoe)
        !           117:        if(x!=x) return(x);     /* x is NaN */
        !           118: #endif /* !defined(vax)&&!defined(tahoe) */
        !           119: 
        !           120:        if( x <= lnhuge ) {
        !           121:                if( x >= -40.0 ) {
        !           122: 
        !           123:                    /* argument reduction : x - k*ln2 */
        !           124:                        k= invln2 *x+copysign(0.5,x);   /* k=NINT(x/ln2) */
        !           125:                        hi=x-k*ln2hi ; 
        !           126:                        z=hi-(lo=k*ln2lo);
        !           127:                        c=(hi-z)-lo;
        !           128: 
        !           129:                        if(k==0) return(z+exp__E(z,c));
        !           130:                        if(k==1)
        !           131:                            if(z< -0.25) 
        !           132:                                {x=z+half;x +=exp__E(z,c); return(x+x);}
        !           133:                            else
        !           134:                                {z+=exp__E(z,c); x=half+z; return(x+x);}
        !           135:                    /* end of k=1 */
        !           136: 
        !           137:                        else {
        !           138:                            if(k<=prec)
        !           139:                              { x=one-scalb(one,-k); z += exp__E(z,c);}
        !           140:                            else if(k<100)
        !           141:                              { x = exp__E(z,c)-scalb(one,-k); x+=z; z=one;}
        !           142:                            else 
        !           143:                              { x = exp__E(z,c)+z; z=one;}
        !           144: 
        !           145:                            return (scalb(x+z,k));  
        !           146:                        }
        !           147:                }
        !           148:                /* end of x > lnunfl */
        !           149: 
        !           150:                else 
        !           151:                     /* expm1(-big#) rounded to -1 (inexact) */
        !           152:                     if(finite(x))  
        !           153:                         { ln2hi+ln2lo; return(-one);}
        !           154: 
        !           155:                     /* expm1(-INF) is -1 */
        !           156:                     else return(-one);
        !           157:        }
        !           158:        /* end of x < lnhuge */
        !           159: 
        !           160:        else 
        !           161:        /*  expm1(INF) is INF, expm1(+big#) overflows to INF */
        !           162:            return( finite(x) ?  scalb(one,5000) : x);
        !           163: }

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