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Annotation of 43BSDReno/lib/libm/common_source/exp.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: (1) source distributions retain this entire copyright
        !             7:  * notice and comment, and (2) distributions including binaries display
        !             8:  * the following acknowledgement:  ``This product includes software
        !             9:  * developed by the University of California, Berkeley and its contributors''
        !            10:  * in the documentation or other materials provided with the distribution
        !            11:  * and in all advertising materials mentioning features or use of this
        !            12:  * software. Neither the name of the University nor the names of its
        !            13:  * contributors may be used to endorse or promote products derived
        !            14:  * from this software without specific prior written permission.
        !            15:  * THIS SOFTWARE IS PROVIDED ``AS IS'' AND WITHOUT ANY EXPRESS OR
        !            16:  * IMPLIED WARRANTIES, INCLUDING, WITHOUT LIMITATION, THE IMPLIED
        !            17:  * WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.
        !            18:  *
        !            19:  * All recipients should regard themselves as participants in an ongoing
        !            20:  * research project and hence should feel obligated to report their
        !            21:  * experiences (good or bad) with these elementary function codes, using
        !            22:  * the sendbug(8) program, to the authors.
        !            23:  */
        !            24: 
        !            25: #ifndef lint
        !            26: static char sccsid[] = "@(#)exp.c      5.5 (Berkeley) 6/1/90";
        !            27: #endif /* not lint */
        !            28: 
        !            29: /* EXP(X)
        !            30:  * RETURN THE EXPONENTIAL OF X
        !            31:  * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS)
        !            32:  * CODED IN C BY K.C. NG, 1/19/85; 
        !            33:  * REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86.
        !            34:  *
        !            35:  * Required system supported functions:
        !            36:  *     scalb(x,n)      
        !            37:  *     copysign(x,y)   
        !            38:  *     finite(x)
        !            39:  *
        !            40:  * Method:
        !            41:  *     1. Argument Reduction: given the input x, find r and integer k such 
        !            42:  *        that
        !            43:  *                        x = k*ln2 + r,  |r| <= 0.5*ln2 .  
        !            44:  *        r will be represented as r := z+c for better accuracy.
        !            45:  *
        !            46:  *     2. Compute exp(r) by 
        !            47:  *
        !            48:  *             exp(r) = 1 + r + r*R1/(2-R1),
        !            49:  *        where
        !            50:  *             R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))).
        !            51:  *
        !            52:  *     3. exp(x) = 2^k * exp(r) .
        !            53:  *
        !            54:  * Special cases:
        !            55:  *     exp(INF) is INF, exp(NaN) is NaN;
        !            56:  *     exp(-INF)=  0;
        !            57:  *     for finite argument, only exp(0)=1 is exact.
        !            58:  *
        !            59:  * Accuracy:
        !            60:  *     exp(x) returns the exponential of x nearly rounded. In a test run
        !            61:  *     with 1,156,000 random arguments on a VAX, the maximum observed
        !            62:  *     error was 0.869 ulps (units in the last place).
        !            63:  *
        !            64:  * Constants:
        !            65:  * The hexadecimal values are the intended ones for the following constants.
        !            66:  * The decimal values may be used, provided that the compiler will convert
        !            67:  * from decimal to binary accurately enough to produce the hexadecimal values
        !            68:  * shown.
        !            69:  */
        !            70: 
        !            71: #include "mathimpl.h"
        !            72: 
        !            73: vc(ln2hi,  6.9314718055829871446E-1  ,7217,4031,0000,f7d0,   0, .B17217F7D00000)
        !            74: vc(ln2lo,  1.6465949582897081279E-12 ,bcd5,2ce7,d9cc,e4f1, -39, .E7BCD5E4F1D9CC)
        !            75: vc(lnhuge, 9.4961163736712506989E1   ,ec1d,43bd,9010,a73e,   7, .BDEC1DA73E9010)
        !            76: vc(lntiny,-9.5654310917272452386E1   ,4f01,c3bf,33af,d72e,   7,-.BF4F01D72E33AF)
        !            77: vc(invln2, 1.4426950408889634148E0   ,aa3b,40b8,17f1,295c,   1, .B8AA3B295C17F1)
        !            78: vc(p1,     1.6666666666666602251E-1  ,aaaa,3f2a,a9f1,aaaa,  -2, .AAAAAAAAAAA9F1)
        !            79: vc(p2,    -2.7777777777015591216E-3  ,0b60,bc36,ec94,b5f5,  -8,-.B60B60B5F5EC94)
        !            80: vc(p3,     6.6137563214379341918E-5  ,b355,398a,f15f,792e, -13, .8AB355792EF15F)
        !            81: vc(p4,    -1.6533902205465250480E-6  ,ea0e,b6dd,5f84,2e93, -19,-.DDEA0E2E935F84)
        !            82: vc(p5,     4.1381367970572387085E-8  ,bb4b,3431,2683,95f5, -24, .B1BB4B95F52683)
        !            83: 
        !            84: #ifdef vccast
        !            85: #define    ln2hi    vccast(ln2hi)
        !            86: #define    ln2lo    vccast(ln2lo)
        !            87: #define   lnhuge    vccast(lnhuge)
        !            88: #define   lntiny    vccast(lntiny)
        !            89: #define   invln2    vccast(invln2)
        !            90: #define       p1    vccast(p1)
        !            91: #define       p2    vccast(p2)
        !            92: #define       p3    vccast(p3)
        !            93: #define       p4    vccast(p4)
        !            94: #define       p5    vccast(p5)
        !            95: #endif
        !            96: 
        !            97: ic(p1,     1.6666666666666601904E-1,  -3,  1.555555555553E)
        !            98: ic(p2,    -2.7777777777015593384E-3,  -9, -1.6C16C16BEBD93)
        !            99: ic(p3,     6.6137563214379343612E-5, -14,  1.1566AAF25DE2C)
        !           100: ic(p4,    -1.6533902205465251539E-6, -20, -1.BBD41C5D26BF1)
        !           101: ic(p5,     4.1381367970572384604E-8, -25,  1.6376972BEA4D0)
        !           102: ic(ln2hi,  6.9314718036912381649E-1,  -1,  1.62E42FEE00000)
        !           103: ic(ln2lo,  1.9082149292705877000E-10,-33,  1.A39EF35793C76)
        !           104: ic(lnhuge, 7.1602103751842355450E2,    9,  1.6602B15B7ECF2)
        !           105: ic(lntiny,-7.5137154372698068983E2,    9, -1.77AF8EBEAE354)
        !           106: ic(invln2, 1.4426950408889633870E0,    0,  1.71547652B82FE)
        !           107: 
        !           108: double exp(x)
        !           109: double x;
        !           110: {
        !           111:        double  z,hi,lo,c;
        !           112:        int k;
        !           113: 
        !           114: #if !defined(vax)&&!defined(tahoe)
        !           115:        if(x!=x) return(x);     /* x is NaN */
        !           116: #endif /* !defined(vax)&&!defined(tahoe) */
        !           117:        if( x <= lnhuge ) {
        !           118:                if( x >= lntiny ) {
        !           119: 
        !           120:                    /* argument reduction : x --> x - k*ln2 */
        !           121: 
        !           122:                        k=invln2*x+copysign(0.5,x);     /* k=NINT(x/ln2) */
        !           123: 
        !           124:                    /* express x-k*ln2 as hi-lo and let x=hi-lo rounded */
        !           125: 
        !           126:                        hi=x-k*ln2hi;
        !           127:                        x=hi-(lo=k*ln2lo);
        !           128: 
        !           129:                    /* return 2^k*[1+x+x*c/(2+c)]  */
        !           130:                        z=x*x;
        !           131:                        c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
        !           132:                        return  scalb(1.0+(hi-(lo-(x*c)/(2.0-c))),k);
        !           133: 
        !           134:                }
        !           135:                /* end of x > lntiny */
        !           136: 
        !           137:                else 
        !           138:                     /* exp(-big#) underflows to zero */
        !           139:                     if(finite(x))  return(scalb(1.0,-5000));
        !           140: 
        !           141:                     /* exp(-INF) is zero */
        !           142:                     else return(0.0);
        !           143:        }
        !           144:        /* end of x < lnhuge */
        !           145: 
        !           146:        else 
        !           147:        /* exp(INF) is INF, exp(+big#) overflows to INF */
        !           148:            return( finite(x) ?  scalb(1.0,5000)  : x);
        !           149: }

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