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365 lines
10 KiB
365 lines
10 KiB
/***
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*pow.c - raise to a power
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*
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* Copyright (c) 1991-2001, Microsoft Corporation. All rights reserved.
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*
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*Purpose:
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*
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*Revision History:
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* 8-15-91 GDP written
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* 12-20-91 GDP support IEEE exceptions & denormals
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* 1-11-92 GDP special handling of small powers
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* special handling of u1, u2 when cancellation occurs
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* 3-22-92 GDP changed handling of int exponents, pow(0, neg)
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* added check to avoid internal overflow due to large y
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* 6-23-92 GDP adjusted special return values according to NCEG spec
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* 02-06-95 JWM Mac merge
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* 02-07-95 JWM powhlp() usage restored to Intel version.
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* 10-07-97 RDL Added IA64.
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*
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*******************************************************************************/
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#include <math.h>
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#include <trans.h>
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#include <float.h>
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#if defined(_M_IA64)
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#pragma function(pow)
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#endif
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static double _reduce(double);
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static double const a1[18] = {
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0.00000000000000000000e+000, /* dummy element */
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1.00000000000000000000e+000,
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9.57603280698573646910e-001,
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9.17004043204671231754e-001,
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8.78126080186649741555e-001,
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8.40896415253714543073e-001,
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8.05245165974627154042e-001,
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7.71105412703970411793e-001,
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7.38413072969749655712e-001,
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7.07106781186547524436e-001,
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6.77127773468446364133e-001,
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6.48419777325504832961e-001,
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6.20928906036742024317e-001,
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5.94603557501360533344e-001,
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5.69394317378345826849e-001,
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5.45253866332628829604e-001,
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5.22136891213706920173e-001,
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5.00000000000000000000e-001
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};
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static double const a2[9] = {
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0.00000000000000000000e+000, /* dummy element */
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-5.31259064517897172664e-017,
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1.47993596544271355242e-017,
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1.23056946577104753260e-017,
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-1.74014448683923461658e-017,
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3.84891771232354074073e-017,
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2.33103467084383453312e-017,
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4.45607092891542322377e-017,
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4.27717757045531499216e-017
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};
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static double const log2inv = 1.44269504088896340739e+0; // 1/log(2)
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static double const K = 0.44269504088896340736e+0;
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static double const p1 = 0.83333333333333211405e-1;
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static double const p2 = 0.12500000000503799174e-1;
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static double const p3 = 0.22321421285924258967e-2;
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static double const p4 = 0.43445775672163119635e-3;
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#define P(v) (((p4 * v + p3) * v + p2) * v + p1)
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static double const q1 = 0.69314718055994529629e+0;
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static double const q2 = 0.24022650695909537056e+0;
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static double const q3 = 0.55504108664085595326e-1;
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static double const q4 = 0.96181290595172416964e-2;
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static double const q5 = 0.13333541313585784703e-2;
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static double const q6 = 0.15400290440989764601e-3;
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static double const q7 = 0.14928852680595608186e-4;
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#define Q(w) ((((((q7 * w + q6) * w + q5) * w + q4) * w + \
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q3) * w + q2) * w + q1)
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/*
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* Thresholds for over/underflow that results in an adjusted value
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* too big/small to be represented as a double. An infinity or 0
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* is delivered to the trap handler instead
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*/
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static _dbl const _ovfx ={SET_DBL(0x40e40000,0)}; // 16*log2(XMAX*2^IEEE_ADJ)
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static _dbl const _uflx ={SET_DBL(0xc0e3fc00,0)}; // 16*log2(XMIN*2^(-IEEE_ADJ))
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#define OVFX _ovfx.dbl
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#define UFLX _uflx.dbl
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#define INT_POW_LIMIT 128.0
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static double ymax = 1e20;
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static double _reduce(double x)
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{
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return 0.0625 * _frnd( 16.0 * x);
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}
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/***
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*double pow(double x, double y) - x raised to the power of y
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*
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*Purpose:
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* Calculate x^y
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* Algorithm from Cody & Waite
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*
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*Entry:
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*
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*Exit:
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*
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*Exceptions:
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*
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* All 5 IEEE exceptions may occur
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*
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*******************************************************************************/
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double pow(double x, double y)
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{
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uintptr_t savedcw;
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int m,mprim;
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int p,pprim;
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int i,iw1;
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int iy;
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int newexp;
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double diw1;
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double sign;
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double g,z,bigz,v,rz,result;
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double u1,u2,y1,y2,w,w1,w2;
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double savedx;
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/* save user fp control word */
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savedcw = _maskfp();
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savedx = x; // save original value of first argument
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if (_fpclass(y) & (_FPCLASS_NZ | _FPCLASS_PZ)) {
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RETURN(savedcw, 1.0);
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}
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/* Check for zero^y */
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if (_fpclass(x) & (_FPCLASS_NZ | _FPCLASS_PZ)) { /* x==0? */
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int type;
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type = _d_inttype(y);
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if (y < 0.0) {
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result = (type == _D_ODD ? _copysign(D_INF,x) : D_INF);
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return _except2(FP_Z,OP_POW,savedx,y,result,savedcw|ISW_ZERODIVIDE);
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}
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else if (y > 0.0) {
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result = (type == _D_ODD ? x : 0.0);
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RETURN(savedcw, result);
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}
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}
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/* check for infinity or NAN */
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if (IS_D_SPECIAL(x) || IS_D_SPECIAL(y)) {
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double absx = fabs(x);
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if (IS_D_SNAN(x) || IS_D_SNAN(y)) {
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return _except2(FP_I,OP_POW,savedx,y,_d_snan2(x,y),savedcw | (ISW_INVALID>>5) );
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}
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if (IS_D_QNAN(x) || IS_D_QNAN(y)){
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return _handle_qnan2(OP_POW,x,y,savedcw | (ISW_INVALID>>5) );
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}
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/* there is at least one infinite argument ... */
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if (_powhlp(x, y, &result)) { /* removed "<" 0. */
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return _except2(FP_I,OP_POW,savedx,y,result,savedcw | (ISW_INVALID>>5) );
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}
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RETURN(savedcw, result);
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}
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sign = 1.0;
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if (x < 0) {
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switch (_d_inttype(y)) {
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case _D_ODD: /* y is an odd integral value */
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sign = -1.0;
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/* NO BREAK */
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case _D_EVEN:
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x = -x;
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break;
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default: /* y is not an integral value */
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return _except2(FP_I,OP_POW,savedx,y,D_IND,savedcw|(ISW_INVALID>>5));
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}
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}
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//
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// This is here in order to prevent internal overflows
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// due to a large value of y
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// The following relation holds on overflow with a scaled
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// result out of range
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// (lg stands for log base 2)
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// |y| * |lg(x)| > MAXEXP + IEEE_ADJUST <=>
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// |y| > 2560 / |lg(x)|
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// The values of lg(x) closer to 0 are:
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// x lg(x)
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// 3fefffffffffffff (0,99...9) -1.601e-16
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// 3ff0000000000000 (1.0) 0.0
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// 3ff0000000000001 (1.00...1) 3.203e-16
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//
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// So if |y| > 2560/1.6e-16 = 1.6e19 overflow occurs
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// We set ymax to 1e20 in order to have a safety margin
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//
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if (ABS(y) > ymax) {
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if (y < 0) {
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y = -y;
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//
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// this may cause an underflow
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// there is no problem with fp sw pollution because
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// a FP_U exception is going to be raised anyway.
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//
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x = 1.0 / x;
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}
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if (x > 1.0) {
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return _except2(FP_O | FP_P,OP_POW,savedx,y,sign*D_INF,savedcw|ISW_OVERFLOW);
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}
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else if (x < 1.0){
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return _except2(FP_U | FP_P,OP_POW,savedx,y,sign*0.0,savedcw|ISW_UNDERFLOW);
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}
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else {
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RETURN(savedcw, sign*1.0);
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}
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}
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/* determine m, g */
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g = _decomp(x, &m);
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/* handle small integer powers
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* for small integer powers this is faster that Cody&Waite's
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* algorithm, and yields better precision
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* Without this piece of code there was not enough precision
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* to satisfy all requirements of the 'paranoia' test.
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* We choose INT_POW_LIMIT such that (1) no overflow or underflow
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* occurs while computing bigz (g is in the range
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* [0.5, 1.0) or (1.0, 2.0] so INT_POW_LIMIT should be less than
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* approximately 10^3) and (2) no extraordinary loss of precision
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* occurs because of repeated multiplications (this practically
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* restricts the maximum INT_POW_LIMIT to 128).
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*/
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if (y <= INT_POW_LIMIT &&
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_d_inttype(x) != _D_NOINT &&
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_d_inttype(y) != _D_NOINT &&
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y > 0.0 ) {
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iy = (int)y;
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mprim = m * iy;
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for (bigz=1 ; iy ; iy >>= 1, g *= g) {
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if (iy & 0x1)
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bigz *= g;
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}
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newexp = _get_exp(bigz) + mprim;
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if (newexp > MAXEXP + IEEE_ADJUST) {
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return _except2(FP_O | FP_P, OP_POW, savedx, y, sign*bigz*D_INF, savedcw);
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}
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if (newexp < MINEXP - IEEE_ADJUST) {
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return _except2(FP_U | FP_P, OP_POW, savedx, y, sign*bigz*0.0, savedcw);
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}
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}
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else {
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/* determine p using binary search */
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p = 1;
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if (g <= a1[9])
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p = 9;
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if (g <= a1[p+4])
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p += 4;
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if (g <= a1[p+2])
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p += 2;
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/* C&W's algorithm is not very accurate when m*16-p == 1,
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* because there is cancellation between u1 and u2.
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* Handle this separately.
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*/
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if (ABS(m*16-p) == 1) {
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u1 = log(x) * log2inv;
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u2 = 0.0;
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}
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else {
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/* determine z */
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z = ( (g - a1[p+1]) - a2[(p+1)/2] ) / ( g + a1[p+1] );
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z += z;
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/* determine u2 */
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v = z * z;
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rz = P(v) * v * z;
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rz += K * rz;
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u2 = (rz + z * K) + z;
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u1 = (m * 16 - p) * 0.0625;
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}
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/* determine w1, w2 */
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y1 = _reduce(y);
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y2 = y - y1;
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w = u2 * y + u1 * y2;
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w1 = _reduce(w);
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w2 = w - w1;
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w = w1 + u1 * y1;
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w1 = _reduce(w);
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w2 += w - w1;
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w = _reduce(w2);
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diw1 = 16 * (w1 + w); /* iw1 might overflow here, so use diw1 */
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w2 -= w;
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if (diw1 > OVFX) {
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return _except2(FP_O | FP_P,OP_POW,savedx,y,sign*D_INF,savedcw | ISW_OVERFLOW);
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}
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if (diw1 < UFLX) {
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return _except2(FP_U | FP_P,OP_POW,savedx,y,sign*0.0,savedcw | ISW_UNDERFLOW);
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}
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iw1 = (int) diw1; /* now it is safe to cast to int */
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/* make sure w2 <= 0 */
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if (w2 > 0) {
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iw1 += 1;
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w2 -= 0.0625;
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}
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/* determine mprim, pprim */
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i = iw1 < 0 ? 0 : 1;
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mprim = iw1 / 16 + i;
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pprim = 16 * mprim - iw1;
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/* determine 2^w2 */
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bigz = Q(w2) * w2;
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/* determine final result */
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bigz = a1[pprim + 1] + a1[pprim + 1] * bigz;
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newexp = _get_exp(bigz) + mprim;
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}
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if (newexp > MAXEXP) {
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result = sign * _set_exp(bigz, newexp - IEEE_ADJUST);
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return _except2(FP_O | FP_P, OP_POW, savedx, y, sign*D_INF, savedcw|ISW_OVERFLOW);
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}
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if (newexp < MINEXP) {
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result = sign * _set_exp(bigz, newexp + IEEE_ADJUST);
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return _except2(FP_U | FP_P, OP_POW, savedx, y, sign*0.0, savedcw|ISW_UNDERFLOW);
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}
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result = sign * _set_exp(bigz, newexp);
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RETURN_INEXACT2(OP_POW, savedx, y, result, savedcw|ISW_INEXACT);
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}
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