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207 lines
5.4 KiB
C
207 lines
5.4 KiB
C
/* exp.c
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*
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* Exponential function
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*
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*
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*
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* SYNOPSIS:
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*
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* double x, y, exp();
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*
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* y = exp( x );
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*
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*
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*
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* DESCRIPTION:
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*
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* Returns e (2.71828...) raised to the x power.
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*
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* Range reduction is accomplished by separating the argument
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* into an integer k and fraction f such that
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*
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* x k f
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* e = 2 e.
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*
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* A Pade' form 1 + 2x P(x**2)/( Q(x**2) - P(x**2) )
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* of degree 2/3 is used to approximate exp(f) in the basic
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* interval [-0.5, 0.5].
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*
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*
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* ACCURACY:
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*
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* Relative error:
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* arithmetic domain # trials peak rms
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* DEC 0, MAXLOG 38000 3.0e-17 6.2e-18
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* IEEE +- 708 40000 2.0e-16 5.6e-17
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*
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*
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* Error amplification in the exponential function can be
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* a serious matter. The error propagation involves
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* exp( X(1+delta) ) = exp(X) ( 1 + X*delta + ... ),
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* which shows that a 1 lsb error in representing X produces
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* a relative error of X times 1 lsb in the function.
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* While the routine gives an accurate result for arguments
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* that are exactly represented by a double precision
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* computer number, the result contains amplified roundoff
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* error for large arguments not exactly represented.
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*
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*
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* ERROR MESSAGES:
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*
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* message condition value returned
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* exp underflow x < MINLOG 0.0
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* exp overflow x > MAXLOG MAXNUM
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*
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*/
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/*
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Cephes Math Library Release 2.2: January, 1991
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Copyright 1984, 1991 by Stephen L. Moshier
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Redistribution and use in source and binary forms, with or without
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modification, are permitted provided that the following conditions are met:
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1. Redistributions of source code must retain the above copyright notice,
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this list of conditions and the following disclaimer.
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2. Redistributions in binary form must reproduce the above copyright
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notice, this list of conditions and the following disclaimer in the
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documentation and/or other materials provided with the distribution.
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3. Neither the name of the <ORGANIZATION> nor the names of its
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contributors may be used to endorse or promote products derived from
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this software without specific prior written permission.
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THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
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AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
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ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
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LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
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CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
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SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
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INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
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CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
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ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
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POSSIBILITY OF SUCH DAMAGE.
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Direct inquiries to 30 Frost Street, Cambridge, MA 02140
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*/
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/* Exponential function */
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#include "mconf.h"
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static char fname[] = {"exp"};
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#ifdef UNK
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static double P[] = {
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1.26177193074810590878E-4,
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3.02994407707441961300E-2,
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9.99999999999999999910E-1,
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};
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static double Q[] = {
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3.00198505138664455042E-6,
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2.52448340349684104192E-3,
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2.27265548208155028766E-1,
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2.00000000000000000009E0,
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};
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static double C1 = 6.93145751953125E-1;
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static double C2 = 1.42860682030941723212E-6;
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#endif
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#ifdef DEC
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static short P[] = {
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0035004,0047156,0127442,0057502,
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0036770,0033210,0063121,0061764,
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0040200,0000000,0000000,0000000,
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};
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static short Q[] = {
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0033511,0072665,0160662,0176377,
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0036045,0070715,0124105,0132777,
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0037550,0134114,0142077,0001637,
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0040400,0000000,0000000,0000000,
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};
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static short sc1[] = {0040061,0071000,0000000,0000000};
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#define C1 (*(double *)sc1)
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static short sc2[] = {0033277,0137216,0075715,0057117};
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#define C2 (*(double *)sc2)
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#endif
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#ifdef IBMPC
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static short P[] = {
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0x4be8,0xd5e4,0x89cd,0x3f20,
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0x2c7e,0x0cca,0x06d1,0x3f9f,
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0x0000,0x0000,0x0000,0x3ff0,
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};
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static short Q[] = {
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0x5fa0,0xbc36,0x2eb6,0x3ec9,
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0xb6c0,0xb508,0xae39,0x3f64,
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0xe074,0x9887,0x1709,0x3fcd,
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0x0000,0x0000,0x0000,0x4000,
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};
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static short sc1[] = {0x0000,0x0000,0x2e40,0x3fe6};
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#define C1 (*(double *)sc1)
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static short sc2[] = {0xabca,0xcf79,0xf7d1,0x3eb7};
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#define C2 (*(double *)sc2)
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#endif
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#ifdef MIEEE
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static short P[] = {
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0x3f20,0x89cd,0xd5e4,0x4be8,
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0x3f9f,0x06d1,0x0cca,0x2c7e,
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0x3ff0,0x0000,0x0000,0x0000,
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};
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static short Q[] = {
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0x3ec9,0x2eb6,0xbc36,0x5fa0,
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0x3f64,0xae39,0xb508,0xb6c0,
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0x3fcd,0x1709,0x9887,0xe074,
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0x4000,0x0000,0x0000,0x0000,
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};
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static short sc1[] = {0x3fe6,0x2e40,0x0000,0x0000};
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#define C1 (*(double *)sc1)
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static short sc2[] = {0x3eb7,0xf7d1,0xcf79,0xabca};
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#define C2 (*(double *)sc2)
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#endif
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extern double LOGE2, LOG2E, MAXLOG, MINLOG, MAXNUM;
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double c_exp(x)
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double x;
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{
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double px, xx;
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int n;
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double polevl(), floor(), ldexp();
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if( x > MAXLOG)
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{
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mtherr( fname, OVERFLOW );
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return( MAXNUM );
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}
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if( x < MINLOG )
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{
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mtherr( fname, UNDERFLOW );
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return(0.0);
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}
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/* Express e**x = e**g 2**n
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* = e**g e**( n loge(2) )
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* = e**( g + n loge(2) )
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*/
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px = floor( LOG2E * x + 0.5 ); /* floor() truncates toward -infinity. */
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n = (int)px;
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x -= px * C1;
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x -= px * C2;
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/* rational approximation for exponential
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* of the fractional part:
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* e**x = 1 + 2x P(x**2)/( Q(x**2) - P(x**2) )
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*/
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xx = x * x;
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px = x * polevl( xx, P, 2 );
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x = px/( polevl( xx, Q, 3 ) - px );
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x = 1.0 + ldexp( x, 1 );
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/* multiply by power of 2 */
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x = ldexp( x, n );
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return(x);
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}
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