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jeremybenn |
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/* @(#)k_rem_pio2.c 1.3 95/01/18 */
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/*
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* ====================================================
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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*
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* Developed at SunSoft, a Sun Microsystems, Inc. business.
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* Permission to use, copy, modify, and distribute this
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* software is freely granted, provided that this notice
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* is preserved.
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* ====================================================
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*/
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/*
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* __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
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* double x[],y[]; int e0,nx,prec; int ipio2[];
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*
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* __kernel_rem_pio2 return the last three digits of N with
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* y = x - N*pi/2
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* so that |y| < pi/2.
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*
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* The method is to compute the integer (mod 8) and fraction parts of
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* (2/pi)*x without doing the full multiplication. In general we
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* skip the part of the product that are known to be a huge integer (
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* more accurately, = 0 mod 8 ). Thus the number of operations are
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* independent of the exponent of the input.
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*
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* (2/pi) is represented by an array of 24-bit integers in ipio2[].
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*
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* Input parameters:
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* x[] The input value (must be positive) is broken into nx
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* pieces of 24-bit integers in double precision format.
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* x[i] will be the i-th 24 bit of x. The scaled exponent
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* of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
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* match x's up to 24 bits.
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*
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* Example of breaking a double positive z into x[0]+x[1]+x[2]:
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* e0 = ilogb(z)-23
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* z = scalbn(z,-e0)
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* for i = 0,1,2
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* x[i] = floor(z)
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* z = (z-x[i])*2**24
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*
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*
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* y[] ouput result in an array of double precision numbers.
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* The dimension of y[] is:
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* 24-bit precision 1
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* 53-bit precision 2
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* 64-bit precision 2
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* 113-bit precision 3
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* The actual value is the sum of them. Thus for 113-bit
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* precison, one may have to do something like:
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*
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* long double t,w,r_head, r_tail;
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* t = (long double)y[2] + (long double)y[1];
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* w = (long double)y[0];
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* r_head = t+w;
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* r_tail = w - (r_head - t);
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*
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* e0 The exponent of x[0]
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*
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* nx dimension of x[]
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*
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* prec an integer indicating the precision:
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* 0 24 bits (single)
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* 1 53 bits (double)
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* 2 64 bits (extended)
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* 3 113 bits (quad)
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*
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* ipio2[]
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* integer array, contains the (24*i)-th to (24*i+23)-th
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* bit of 2/pi after binary point. The corresponding
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* floating value is
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*
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* ipio2[i] * 2^(-24(i+1)).
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*
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* External function:
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* double scalbn(), floor();
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*
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*
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* Here is the description of some local variables:
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*
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* jk jk+1 is the initial number of terms of ipio2[] needed
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* in the computation. The recommended value is 2,3,4,
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* 6 for single, double, extended,and quad.
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*
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* jz local integer variable indicating the number of
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* terms of ipio2[] used.
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*
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* jx nx - 1
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*
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* jv index for pointing to the suitable ipio2[] for the
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* computation. In general, we want
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* ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
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* is an integer. Thus
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* e0-3-24*jv >= 0 or (e0-3)/24 >= jv
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* Hence jv = max(0,(e0-3)/24).
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*
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* jp jp+1 is the number of terms in PIo2[] needed, jp = jk.
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*
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* q[] double array with integral value, representing the
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* 24-bits chunk of the product of x and 2/pi.
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*
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* q0 the corresponding exponent of q[0]. Note that the
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* exponent for q[i] would be q0-24*i.
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*
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* PIo2[] double precision array, obtained by cutting pi/2
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* into 24 bits chunks.
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*
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* f[] ipio2[] in floating point
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*
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* iq[] integer array by breaking up q[] in 24-bits chunk.
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*
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* fq[] final product of x*(2/pi) in fq[0],..,fq[jk]
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*
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* ih integer. If >0 it indicates q[] is >= 0.5, hence
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* it also indicates the *sign* of the result.
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*
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*/
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/*
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* Constants:
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* The hexadecimal values are the intended ones for the following
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* constants. The decimal values may be used, provided that the
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* compiler will convert from decimal to binary accurately enough
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* to produce the hexadecimal values shown.
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*/
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#include "fdlibm.h"
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#ifdef __STDC__
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static const int init_jk[] = {2,3,4,6}; /* initial value for jk */
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#else
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static int init_jk[] = {2,3,4,6};
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#endif
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#ifdef __STDC__
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static const double PIo2[] = {
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#else
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static double PIo2[] = {
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#endif
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1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
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7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
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5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
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3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
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1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
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1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
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2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
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2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
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};
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#ifdef __STDC__
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static const double
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#else
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static double
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#endif
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zero = 0.0,
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one = 1.0,
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two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
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twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
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#ifdef __STDC__
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int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int *ipio2)
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#else
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int __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
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double x[], y[]; int e0,nx,prec; int ipio2[];
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#endif
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{
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int jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;
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double z,fw,f[20],fq[20],q[20];
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/* initialize jk*/
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jk = init_jk[prec];
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jp = jk;
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/* determine jx,jv,q0, note that 3>q0 */
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jx = nx-1;
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jv = (e0-3)/24; if(jv<0) jv=0;
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q0 = e0-24*(jv+1);
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/* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
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j = jv-jx; m = jx+jk;
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for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j];
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/* compute q[0],q[1],...q[jk] */
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for (i=0;i<=jk;i++) {
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for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw;
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}
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jz = jk;
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recompute:
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/* distill q[] into iq[] reversingly */
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for(i=0,j=jz,z=q[jz];j>0;i++,j--) {
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fw = (double)((int)(twon24* z));
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iq[i] = (int)(z-two24*fw);
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z = q[j-1]+fw;
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}
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/* compute n */
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z = scalbn(z,q0); /* actual value of z */
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z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */
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n = (int) z;
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z -= (double)n;
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ih = 0;
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if(q0>0) { /* need iq[jz-1] to determine n */
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i = (iq[jz-1]>>(24-q0)); n += i;
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iq[jz-1] -= i<<(24-q0);
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ih = iq[jz-1]>>(23-q0);
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}
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else if(q0==0) ih = iq[jz-1]>>23;
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else if(z>=0.5) ih=2;
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if(ih>0) { /* q > 0.5 */
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n += 1; carry = 0;
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for(i=0;i<jz ;i++) { /* compute 1-q */
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j = iq[i];
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if(carry==0) {
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if(j!=0) {
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carry = 1; iq[i] = 0x1000000- j;
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}
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} else iq[i] = 0xffffff - j;
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}
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if(q0>0) { /* rare case: chance is 1 in 12 */
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switch(q0) {
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case 1:
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iq[jz-1] &= 0x7fffff; break;
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case 2:
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iq[jz-1] &= 0x3fffff; break;
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}
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}
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if(ih==2) {
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z = one - z;
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if(carry!=0) z -= scalbn(one,q0);
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}
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}
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| 238 |
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/* check if recomputation is needed */
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| 239 |
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if(z==zero) {
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j = 0;
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| 241 |
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for (i=jz-1;i>=jk;i--) j |= iq[i];
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| 242 |
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if(j==0) { /* need recomputation */
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for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */
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| 244 |
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| 245 |
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for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */
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f[jx+i] = (double) ipio2[jv+i];
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| 247 |
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for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
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| 248 |
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q[i] = fw;
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| 249 |
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}
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| 250 |
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jz += k;
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| 251 |
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goto recompute;
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| 252 |
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}
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| 253 |
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}
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| 254 |
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| 255 |
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/* chop off zero terms */
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| 256 |
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if(z==0.0) {
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| 257 |
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jz -= 1; q0 -= 24;
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| 258 |
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while(iq[jz]==0) { jz--; q0-=24;}
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| 259 |
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} else { /* break z into 24-bit if necessary */
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| 260 |
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z = scalbn(z,-q0);
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| 261 |
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if(z>=two24) {
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| 262 |
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fw = (double)((int)(twon24*z));
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| 263 |
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iq[jz] = (int)(z-two24*fw);
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| 264 |
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jz += 1; q0 += 24;
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| 265 |
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iq[jz] = (int) fw;
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| 266 |
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} else iq[jz] = (int) z ;
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| 267 |
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}
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| 268 |
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| 269 |
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/* convert integer "bit" chunk to floating-point value */
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| 270 |
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fw = scalbn(one,q0);
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| 271 |
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for(i=jz;i>=0;i--) {
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| 272 |
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q[i] = fw*(double)iq[i]; fw*=twon24;
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| 273 |
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}
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| 274 |
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| 275 |
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/* compute PIo2[0,...,jp]*q[jz,...,0] */
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| 276 |
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for(i=jz;i>=0;i--) {
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| 277 |
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for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
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| 278 |
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fq[jz-i] = fw;
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| 279 |
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}
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| 280 |
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| 281 |
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/* compress fq[] into y[] */
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| 282 |
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switch(prec) {
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| 283 |
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case 0:
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| 284 |
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fw = 0.0;
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| 285 |
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for (i=jz;i>=0;i--) fw += fq[i];
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| 286 |
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y[0] = (ih==0)? fw: -fw;
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| 287 |
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break;
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| 288 |
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case 1:
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| 289 |
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case 2:
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| 290 |
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fw = 0.0;
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| 291 |
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for (i=jz;i>=0;i--) fw += fq[i];
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| 292 |
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y[0] = (ih==0)? fw: -fw;
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| 293 |
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fw = fq[0]-fw;
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| 294 |
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for (i=1;i<=jz;i++) fw += fq[i];
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| 295 |
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y[1] = (ih==0)? fw: -fw;
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| 296 |
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break;
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| 297 |
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case 3: /* painful */
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| 298 |
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for (i=jz;i>0;i--) {
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| 299 |
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fw = fq[i-1]+fq[i];
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| 300 |
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fq[i] += fq[i-1]-fw;
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| 301 |
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fq[i-1] = fw;
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| 302 |
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}
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| 303 |
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for (i=jz;i>1;i--) {
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| 304 |
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fw = fq[i-1]+fq[i];
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| 305 |
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fq[i] += fq[i-1]-fw;
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| 306 |
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fq[i-1] = fw;
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| 307 |
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}
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| 308 |
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for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];
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| 309 |
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if(ih==0) {
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| 310 |
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y[0] = fq[0]; y[1] = fq[1]; y[2] = fw;
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| 311 |
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} else {
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| 312 |
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|
y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
|
| 313 |
|
|
}
|
| 314 |
|
|
}
|
| 315 |
|
|
return n&7;
|
| 316 |
|
|
}
|
