/*
 * This file is part of the Independent JPEG Group's software.
 *
 * The authors make NO WARRANTY or representation, either express or implied,
 * with respect to this software, its quality, accuracy, merchantability, or
 * fitness for a particular purpose.  This software is provided "AS IS", and
 * you, its user, assume the entire risk as to its quality and accuracy.
 *
 * This software is copyright (C) 1994-1996, Thomas G. Lane.
 * All Rights Reserved except as specified below.
 *
 * Permission is hereby granted to use, copy, modify, and distribute this
 * software (or portions thereof) for any purpose, without fee, subject to
 * these conditions:
 * (1) If any part of the source code for this software is distributed, then
 * this README file must be included, with this copyright and no-warranty
 * notice unaltered; and any additions, deletions, or changes to the original
 * files must be clearly indicated in accompanying documentation.
 * (2) If only executable code is distributed, then the accompanying
 * documentation must state that "this software is based in part on the work
 * of the Independent JPEG Group".
 * (3) Permission for use of this software is granted only if the user accepts
 * full responsibility for any undesirable consequences; the authors accept
 * NO LIABILITY for damages of any kind.
 *
 * These conditions apply to any software derived from or based on the IJG
 * code, not just to the unmodified library.  If you use our work, you ought
 * to acknowledge us.
 *
 * Permission is NOT granted for the use of any IJG author's name or company
 * name in advertising or publicity relating to this software or products
 * derived from it.  This software may be referred to only as "the Independent
 * JPEG Group's software".
 *
 * We specifically permit and encourage the use of this software as the basis
 * of commercial products, provided that all warranty or liability claims are
 * assumed by the product vendor.
 *
 * This file contains a fast, not so accurate integer implementation of the
 * forward DCT (Discrete Cosine Transform).
 *
 * A 2-D DCT can be done by 1-D DCT on each row followed by 1-D DCT
 * on each column.  Direct algorithms are also available, but they are
 * much more complex and seem not to be any faster when reduced to code.
 *
 * This implementation is based on Arai, Agui, and Nakajima's algorithm for
 * scaled DCT.  Their original paper (Trans. IEICE E-71(11):1095) is in
 * Japanese, but the algorithm is described in the Pennebaker & Mitchell
 * JPEG textbook (see REFERENCES section in file README).  The following code
 * is based directly on figure 4-8 in P&M.
 * While an 8-point DCT cannot be done in less than 11 multiplies, it is
 * possible to arrange the computation so that many of the multiplies are
 * simple scalings of the final outputs.  These multiplies can then be
 * folded into the multiplications or divisions by the JPEG quantization
 * table entries.  The AA&N method leaves only 5 multiplies and 29 adds
 * to be done in the DCT itself.
 * The primary disadvantage of this method is that with fixed-point math,
 * accuracy is lost due to imprecise representation of the scaled
 * quantization values.  The smaller the quantization table entry, the less
 * precise the scaled value, so this implementation does worse with high-
 * quality-setting files than with low-quality ones.
 */