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/* -*- Mode: C; tab-width: 8; indent-tabs-mode: nil; c-basic-offset: 4 -*- |
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* |
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* ***** BEGIN LICENSE BLOCK ***** |
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* Version: MPL 1.1/GPL 2.0/LGPL 2.1 |
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* |
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* The contents of this file are subject to the Mozilla Public License Version |
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* 1.1 (the "License"); you may not use this file except in compliance with |
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* the License. You may obtain a copy of the License at |
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* http://www.mozilla.org/MPL/ |
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* |
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* Software distributed under the License is distributed on an "AS IS" basis, |
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* WITHOUT WARRANTY OF ANY KIND, either express or implied. See the License |
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* for the specific language governing rights and limitations under the |
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* License. |
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* |
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* The Original Code is Mozilla Communicator client code, released |
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* March 31, 1998. |
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* |
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* The Initial Developer of the Original Code is |
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* Sun Microsystems, Inc. |
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* Portions created by the Initial Developer are Copyright (C) 1998 |
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* the Initial Developer. All Rights Reserved. |
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* |
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* Contributor(s): |
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* |
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* Alternatively, the contents of this file may be used under the terms of |
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* either of the GNU General Public License Version 2 or later (the "GPL"), |
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* or the GNU Lesser General Public License Version 2.1 or later (the "LGPL"), |
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* in which case the provisions of the GPL or the LGPL are applicable instead |
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* of those above. If you wish to allow use of your version of this file only |
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* under the terms of either the GPL or the LGPL, and not to allow others to |
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* use your version of this file under the terms of the MPL, indicate your |
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* decision by deleting the provisions above and replace them with the notice |
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* and other provisions required by the GPL or the LGPL. If you do not delete |
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* the provisions above, a recipient may use your version of this file under |
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* the terms of any one of the MPL, the GPL or the LGPL. |
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* |
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* ***** END LICENSE BLOCK ***** */ |
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|
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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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/* |
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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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|
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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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|
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#include "fdlibm.h" |
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|
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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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|
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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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|
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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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|
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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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|
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/* initialize jk*/ |
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jk = init_jk[prec]; |
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jp = jk; |
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|
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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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|
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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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|
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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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|
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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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|
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/* compute n */ |
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z = fd_scalbn(z,q0); /* actual value of z */ |
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z -= 8.0*fd_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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|
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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 -= fd_scalbn(one,q0); |
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} |
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} |
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|
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/* check if recomputation is needed */ |
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if(z==zero) { |
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j = 0; |
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for (i=jz-1;i>=jk;i--) j |= iq[i]; |
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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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|
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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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for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; |
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q[i] = fw; |
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} |
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jz += k; |
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goto recompute; |
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} |
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} |
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|
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/* chop off zero terms */ |
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if(z==0.0) { |
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jz -= 1; q0 -= 24; |
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while(iq[jz]==0) { jz--; q0-=24;} |
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} else { /* break z into 24-bit if necessary */ |
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z = fd_scalbn(z,-q0); |
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if(z>=two24) { |
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fw = (double)((int)(twon24*z)); |
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iq[jz] = (int)(z-two24*fw); |
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jz += 1; q0 += 24; |
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iq[jz] = (int) fw; |
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} else iq[jz] = (int) z ; |
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} |
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|
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/* convert integer "bit" chunk to floating-point value */ |
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fw = fd_scalbn(one,q0); |
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for(i=jz;i>=0;i--) { |
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q[i] = fw*(double)iq[i]; fw*=twon24; |
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} |
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|
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/* compute PIo2[0,...,jp]*q[jz,...,0] */ |
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for(i=jz;i>=0;i--) { |
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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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fq[jz-i] = fw; |
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} |
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|
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/* compress fq[] into y[] */ |
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switch(prec) { |
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case 0: |
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fw = 0.0; |
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for (i=jz;i>=0;i--) fw += fq[i]; |
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y[0] = (ih==0)? fw: -fw; |
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break; |
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case 1: |
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case 2: |
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fw = 0.0; |
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for (i=jz;i>=0;i--) fw += fq[i]; |
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y[0] = (ih==0)? fw: -fw; |
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fw = fq[0]-fw; |
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for (i=1;i<=jz;i++) fw += fq[i]; |
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y[1] = (ih==0)? fw: -fw; |
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break; |
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case 3: /* painful */ |
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for (i=jz;i>0;i--) { |
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fw = fq[i-1]+fq[i]; |
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fq[i] += fq[i-1]-fw; |
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fq[i-1] = fw; |
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} |
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for (i=jz;i>1;i--) { |
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fw = fq[i-1]+fq[i]; |
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fq[i] += fq[i-1]-fw; |
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fq[i-1] = fw; |
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} |
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for (fw=0.0,i=jz;i>=2;i--) fw += fq[i]; |
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if(ih==0) { |
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y[0] = fq[0]; y[1] = fq[1]; y[2] = fw; |
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} else { |
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y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw; |
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} |
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} |
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return n&7; |
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} |
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