1509 lines
54 KiB
Java
1509 lines
54 KiB
Java
/*
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* Copyright (c) 2013, Oracle and/or its affiliates. All rights reserved.
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* DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
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*
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* This code is free software; you can redistribute it and/or modify it
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* under the terms of the GNU General Public License version 2 only, as
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* published by the Free Software Foundation. Oracle designates this
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* particular file as subject to the "Classpath" exception as provided
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* by Oracle in the LICENSE file that accompanied this code.
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*
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* This code is distributed in the hope that it will be useful, but WITHOUT
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* ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
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* FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
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* version 2 for more details (a copy is included in the LICENSE file that
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* accompanied this code).
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*
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* You should have received a copy of the GNU General Public License version
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* 2 along with this work; if not, write to the Free Software Foundation,
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* Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
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*
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* Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
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* or visit www.oracle.com if you need additional information or have any
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* questions.
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*/
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package jdk.internal.math;
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import java.math.BigInteger;
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import java.util.Arrays;
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//@ model import org.jmlspecs.models.JMLMath;
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/**
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* A simple big integer package specifically for floating point base conversion.
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*/
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public /*@ spec_bigint_math @*/ class FDBigInteger {
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//
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// This class contains many comments that start with "/*@" mark.
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// They are behavourial specification in
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// the Java Modelling Language (JML):
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// http://www.eecs.ucf.edu/~leavens/JML//index.shtml
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//
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/*@
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@ public pure model static \bigint UNSIGNED(int v) {
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@ return v >= 0 ? v : v + (((\bigint)1) << 32);
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@ }
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@
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@ public pure model static \bigint UNSIGNED(long v) {
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@ return v >= 0 ? v : v + (((\bigint)1) << 64);
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@ }
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@
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@ public pure model static \bigint AP(int[] data, int len) {
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@ return (\sum int i; 0 <= 0 && i < len; UNSIGNED(data[i]) << (i*32));
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@ }
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@
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@ public pure model static \bigint pow52(int p5, int p2) {
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@ ghost \bigint v = 1;
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@ for (int i = 0; i < p5; i++) v *= 5;
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@ return v << p2;
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@ }
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@
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@ public pure model static \bigint pow10(int p10) {
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@ return pow52(p10, p10);
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@ }
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@*/
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static final int[] SMALL_5_POW = {
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1,
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5,
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5 * 5,
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5 * 5 * 5,
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5 * 5 * 5 * 5,
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5 * 5 * 5 * 5 * 5,
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5 * 5 * 5 * 5 * 5 * 5,
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5 * 5 * 5 * 5 * 5 * 5 * 5,
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5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
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5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
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5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
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5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
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5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
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5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5
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};
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static final long[] LONG_5_POW = {
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1L,
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5L,
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5L * 5,
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5L * 5 * 5,
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5L * 5 * 5 * 5,
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5L * 5 * 5 * 5 * 5,
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5L * 5 * 5 * 5 * 5 * 5,
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5L * 5 * 5 * 5 * 5 * 5 * 5,
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5L * 5 * 5 * 5 * 5 * 5 * 5 * 5,
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5L * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
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5L * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
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5L * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
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5L * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
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5L * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
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5L * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
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5L * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
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5L * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
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5L * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
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5L * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
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5L * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
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5L * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
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5L * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
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5L * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
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5L * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
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5L * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
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5L * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
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5L * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
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};
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// Maximum size of cache of powers of 5 as FDBigIntegers.
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private static final int MAX_FIVE_POW = 340;
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// Cache of big powers of 5 as FDBigIntegers.
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private static final FDBigInteger POW_5_CACHE[];
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// Initialize FDBigInteger cache of powers of 5.
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static {
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POW_5_CACHE = new FDBigInteger[MAX_FIVE_POW];
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int i = 0;
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while (i < SMALL_5_POW.length) {
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FDBigInteger pow5 = new FDBigInteger(new int[]{SMALL_5_POW[i]}, 0);
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pow5.makeImmutable();
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POW_5_CACHE[i] = pow5;
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i++;
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}
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FDBigInteger prev = POW_5_CACHE[i - 1];
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while (i < MAX_FIVE_POW) {
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POW_5_CACHE[i] = prev = prev.mult(5);
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prev.makeImmutable();
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i++;
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}
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}
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// Zero as an FDBigInteger.
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public static final FDBigInteger ZERO = new FDBigInteger(new int[0], 0);
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// Ensure ZERO is immutable.
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static {
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ZERO.makeImmutable();
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}
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// Constant for casting an int to a long via bitwise AND.
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private static final long LONG_MASK = 0xffffffffL;
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//@ spec_public non_null;
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private int data[]; // value: data[0] is least significant
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//@ spec_public;
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private int offset; // number of least significant zero padding ints
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//@ spec_public;
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private int nWords; // data[nWords-1]!=0, all values above are zero
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// if nWords==0 -> this FDBigInteger is zero
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//@ spec_public;
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private boolean isImmutable = false;
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/*@
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@ public invariant 0 <= nWords && nWords <= data.length && offset >= 0;
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@ public invariant nWords == 0 ==> offset == 0;
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@ public invariant nWords > 0 ==> data[nWords - 1] != 0;
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@ public invariant (\forall int i; nWords <= i && i < data.length; data[i] == 0);
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@ public pure model \bigint value() {
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@ return AP(data, nWords) << (offset*32);
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@ }
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@*/
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/**
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* Constructs an <code>FDBigInteger</code> from data and padding. The
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* <code>data</code> parameter has the least significant <code>int</code> at
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* the zeroth index. The <code>offset</code> parameter gives the number of
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* zero <code>int</code>s to be inferred below the least significant element
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* of <code>data</code>.
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*
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* @param data An array containing all non-zero <code>int</code>s of the value.
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* @param offset An offset indicating the number of zero <code>int</code>s to pad
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* below the least significant element of <code>data</code>.
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*/
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/*@
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@ requires data != null && offset >= 0;
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@ ensures this.value() == \old(AP(data, data.length) << (offset*32));
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@ ensures this.data == \old(data);
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@*/
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private FDBigInteger(int[] data, int offset) {
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this.data = data;
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this.offset = offset;
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this.nWords = data.length;
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trimLeadingZeros();
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}
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/**
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* Constructs an <code>FDBigInteger</code> from a starting value and some
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* decimal digits.
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*
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* @param lValue The starting value.
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* @param digits The decimal digits.
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* @param kDigits The initial index into <code>digits</code>.
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* @param nDigits The final index into <code>digits</code>.
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*/
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/*@
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@ requires digits != null;
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@ requires 0 <= kDigits && kDigits <= nDigits && nDigits <= digits.length;
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@ requires (\forall int i; 0 <= i && i < nDigits; '0' <= digits[i] && digits[i] <= '9');
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@ ensures this.value() == \old(lValue * pow10(nDigits - kDigits) + (\sum int i; kDigits <= i && i < nDigits; (digits[i] - '0') * pow10(nDigits - i - 1)));
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@*/
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public FDBigInteger(long lValue, char[] digits, int kDigits, int nDigits) {
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int n = Math.max((nDigits + 8) / 9, 2); // estimate size needed.
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data = new int[n]; // allocate enough space
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data[0] = (int) lValue; // starting value
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data[1] = (int) (lValue >>> 32);
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offset = 0;
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nWords = 2;
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int i = kDigits;
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int limit = nDigits - 5; // slurp digits 5 at a time.
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int v;
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while (i < limit) {
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int ilim = i + 5;
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v = (int) digits[i++] - (int) '0';
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while (i < ilim) {
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v = 10 * v + (int) digits[i++] - (int) '0';
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}
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multAddMe(100000, v); // ... where 100000 is 10^5.
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}
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int factor = 1;
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v = 0;
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while (i < nDigits) {
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v = 10 * v + (int) digits[i++] - (int) '0';
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factor *= 10;
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}
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if (factor != 1) {
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multAddMe(factor, v);
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}
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trimLeadingZeros();
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}
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/**
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* Returns an <code>FDBigInteger</code> with the numerical value
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* <code>5<sup>p5</sup> * 2<sup>p2</sup></code>.
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*
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* @param p5 The exponent of the power-of-five factor.
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* @param p2 The exponent of the power-of-two factor.
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* @return <code>5<sup>p5</sup> * 2<sup>p2</sup></code>
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*/
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/*@
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@ requires p5 >= 0 && p2 >= 0;
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@ assignable \nothing;
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@ ensures \result.value() == \old(pow52(p5, p2));
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@*/
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public static FDBigInteger valueOfPow52(int p5, int p2) {
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if (p5 != 0) {
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if (p2 == 0) {
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return big5pow(p5);
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} else if (p5 < SMALL_5_POW.length) {
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int pow5 = SMALL_5_POW[p5];
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int wordcount = p2 >> 5;
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int bitcount = p2 & 0x1f;
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if (bitcount == 0) {
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return new FDBigInteger(new int[]{pow5}, wordcount);
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} else {
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return new FDBigInteger(new int[]{
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pow5 << bitcount,
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pow5 >>> (32 - bitcount)
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}, wordcount);
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}
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} else {
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return big5pow(p5).leftShift(p2);
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}
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} else {
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return valueOfPow2(p2);
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}
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}
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/**
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* Returns an <code>FDBigInteger</code> with the numerical value
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* <code>value * 5<sup>p5</sup> * 2<sup>p2</sup></code>.
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*
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* @param value The constant factor.
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* @param p5 The exponent of the power-of-five factor.
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* @param p2 The exponent of the power-of-two factor.
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* @return <code>value * 5<sup>p5</sup> * 2<sup>p2</sup></code>
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*/
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/*@
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@ requires p5 >= 0 && p2 >= 0;
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@ assignable \nothing;
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@ ensures \result.value() == \old(UNSIGNED(value) * pow52(p5, p2));
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@*/
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public static FDBigInteger valueOfMulPow52(long value, int p5, int p2) {
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assert p5 >= 0 : p5;
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assert p2 >= 0 : p2;
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int v0 = (int) value;
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int v1 = (int) (value >>> 32);
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int wordcount = p2 >> 5;
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int bitcount = p2 & 0x1f;
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if (p5 != 0) {
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if (p5 < SMALL_5_POW.length) {
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long pow5 = SMALL_5_POW[p5] & LONG_MASK;
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long carry = (v0 & LONG_MASK) * pow5;
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v0 = (int) carry;
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carry >>>= 32;
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carry = (v1 & LONG_MASK) * pow5 + carry;
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v1 = (int) carry;
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int v2 = (int) (carry >>> 32);
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if (bitcount == 0) {
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return new FDBigInteger(new int[]{v0, v1, v2}, wordcount);
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} else {
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return new FDBigInteger(new int[]{
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v0 << bitcount,
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(v1 << bitcount) | (v0 >>> (32 - bitcount)),
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(v2 << bitcount) | (v1 >>> (32 - bitcount)),
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v2 >>> (32 - bitcount)
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}, wordcount);
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}
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} else {
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FDBigInteger pow5 = big5pow(p5);
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int[] r;
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if (v1 == 0) {
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r = new int[pow5.nWords + 1 + ((p2 != 0) ? 1 : 0)];
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mult(pow5.data, pow5.nWords, v0, r);
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} else {
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r = new int[pow5.nWords + 2 + ((p2 != 0) ? 1 : 0)];
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mult(pow5.data, pow5.nWords, v0, v1, r);
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}
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return (new FDBigInteger(r, pow5.offset)).leftShift(p2);
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}
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} else if (p2 != 0) {
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if (bitcount == 0) {
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return new FDBigInteger(new int[]{v0, v1}, wordcount);
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} else {
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return new FDBigInteger(new int[]{
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v0 << bitcount,
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(v1 << bitcount) | (v0 >>> (32 - bitcount)),
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v1 >>> (32 - bitcount)
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}, wordcount);
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}
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}
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return new FDBigInteger(new int[]{v0, v1}, 0);
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}
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/**
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* Returns an <code>FDBigInteger</code> with the numerical value
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* <code>2<sup>p2</sup></code>.
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*
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* @param p2 The exponent of 2.
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* @return <code>2<sup>p2</sup></code>
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*/
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/*@
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@ requires p2 >= 0;
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@ assignable \nothing;
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@ ensures \result.value() == pow52(0, p2);
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@*/
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private static FDBigInteger valueOfPow2(int p2) {
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int wordcount = p2 >> 5;
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int bitcount = p2 & 0x1f;
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return new FDBigInteger(new int[]{1 << bitcount}, wordcount);
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}
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/**
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* Removes all leading zeros from this <code>FDBigInteger</code> adjusting
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* the offset and number of non-zero leading words accordingly.
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*/
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/*@
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@ requires data != null;
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@ requires 0 <= nWords && nWords <= data.length && offset >= 0;
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@ requires nWords == 0 ==> offset == 0;
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@ ensures nWords == 0 ==> offset == 0;
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@ ensures nWords > 0 ==> data[nWords - 1] != 0;
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@*/
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private /*@ helper @*/ void trimLeadingZeros() {
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int i = nWords;
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if (i > 0 && (data[--i] == 0)) {
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//for (; i > 0 && data[i - 1] == 0; i--) ;
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while(i > 0 && data[i - 1] == 0) {
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i--;
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}
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this.nWords = i;
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if (i == 0) { // all words are zero
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this.offset = 0;
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}
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}
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}
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/**
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* Retrieves the normalization bias of the <code>FDBigIntger</code>. The
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* normalization bias is a left shift such that after it the highest word
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* of the value will have the 4 highest bits equal to zero:
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* {@code (highestWord & 0xf0000000) == 0}, but the next bit should be 1
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* {@code (highestWord & 0x08000000) != 0}.
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*
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* @return The normalization bias.
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*/
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/*@
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@ requires this.value() > 0;
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@*/
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public /*@ pure @*/ int getNormalizationBias() {
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if (nWords == 0) {
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throw new IllegalArgumentException("Zero value cannot be normalized");
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}
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int zeros = Integer.numberOfLeadingZeros(data[nWords - 1]);
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return (zeros < 4) ? 28 + zeros : zeros - 4;
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}
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// TODO: Why is anticount param needed if it is always 32 - bitcount?
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/**
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* Left shifts the contents of one int array into another.
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*
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* @param src The source array.
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* @param idx The initial index of the source array.
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* @param result The destination array.
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* @param bitcount The left shift.
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* @param anticount The left anti-shift, e.g., <code>32-bitcount</code>.
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* @param prev The prior source value.
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*/
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/*@
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@ requires 0 < bitcount && bitcount < 32 && anticount == 32 - bitcount;
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@ requires src.length >= idx && result.length > idx;
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@ assignable result[*];
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@ ensures AP(result, \old(idx + 1)) == \old((AP(src, idx) + UNSIGNED(prev) << (idx*32)) << bitcount);
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@*/
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private static void leftShift(int[] src, int idx, int result[], int bitcount, int anticount, int prev){
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for (; idx > 0; idx--) {
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int v = (prev << bitcount);
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prev = src[idx - 1];
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v |= (prev >>> anticount);
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result[idx] = v;
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}
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int v = prev << bitcount;
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result[0] = v;
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}
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/**
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* Shifts this <code>FDBigInteger</code> to the left. The shift is performed
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* in-place unless the <code>FDBigInteger</code> is immutable in which case
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* a new instance of <code>FDBigInteger</code> is returned.
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*
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* @param shift The number of bits to shift left.
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* @return The shifted <code>FDBigInteger</code>.
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*/
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/*@
|
|
@ requires this.value() == 0 || shift == 0;
|
|
@ assignable \nothing;
|
|
@ ensures \result == this;
|
|
@
|
|
@ also
|
|
@
|
|
@ requires this.value() > 0 && shift > 0 && this.isImmutable;
|
|
@ assignable \nothing;
|
|
@ ensures \result.value() == \old(this.value() << shift);
|
|
@
|
|
@ also
|
|
@
|
|
@ requires this.value() > 0 && shift > 0 && this.isImmutable;
|
|
@ assignable \nothing;
|
|
@ ensures \result == this;
|
|
@ ensures \result.value() == \old(this.value() << shift);
|
|
@*/
|
|
public FDBigInteger leftShift(int shift) {
|
|
if (shift == 0 || nWords == 0) {
|
|
return this;
|
|
}
|
|
int wordcount = shift >> 5;
|
|
int bitcount = shift & 0x1f;
|
|
if (this.isImmutable) {
|
|
if (bitcount == 0) {
|
|
return new FDBigInteger(Arrays.copyOf(data, nWords), offset + wordcount);
|
|
} else {
|
|
int anticount = 32 - bitcount;
|
|
int idx = nWords - 1;
|
|
int prev = data[idx];
|
|
int hi = prev >>> anticount;
|
|
int[] result;
|
|
if (hi != 0) {
|
|
result = new int[nWords + 1];
|
|
result[nWords] = hi;
|
|
} else {
|
|
result = new int[nWords];
|
|
}
|
|
leftShift(data,idx,result,bitcount,anticount,prev);
|
|
return new FDBigInteger(result, offset + wordcount);
|
|
}
|
|
} else {
|
|
if (bitcount != 0) {
|
|
int anticount = 32 - bitcount;
|
|
if ((data[0] << bitcount) == 0) {
|
|
int idx = 0;
|
|
int prev = data[idx];
|
|
for (; idx < nWords - 1; idx++) {
|
|
int v = (prev >>> anticount);
|
|
prev = data[idx + 1];
|
|
v |= (prev << bitcount);
|
|
data[idx] = v;
|
|
}
|
|
int v = prev >>> anticount;
|
|
data[idx] = v;
|
|
if(v==0) {
|
|
nWords--;
|
|
}
|
|
offset++;
|
|
} else {
|
|
int idx = nWords - 1;
|
|
int prev = data[idx];
|
|
int hi = prev >>> anticount;
|
|
int[] result = data;
|
|
int[] src = data;
|
|
if (hi != 0) {
|
|
if(nWords == data.length) {
|
|
data = result = new int[nWords + 1];
|
|
}
|
|
result[nWords++] = hi;
|
|
}
|
|
leftShift(src,idx,result,bitcount,anticount,prev);
|
|
}
|
|
}
|
|
offset += wordcount;
|
|
return this;
|
|
}
|
|
}
|
|
|
|
/**
|
|
* Returns the number of <code>int</code>s this <code>FDBigInteger</code> represents.
|
|
*
|
|
* @return Number of <code>int</code>s required to represent this <code>FDBigInteger</code>.
|
|
*/
|
|
/*@
|
|
@ requires this.value() == 0;
|
|
@ ensures \result == 0;
|
|
@
|
|
@ also
|
|
@
|
|
@ requires this.value() > 0;
|
|
@ ensures ((\bigint)1) << (\result - 1) <= this.value() && this.value() <= ((\bigint)1) << \result;
|
|
@*/
|
|
private /*@ pure @*/ int size() {
|
|
return nWords + offset;
|
|
}
|
|
|
|
|
|
/**
|
|
* Computes
|
|
* <pre>
|
|
* q = (int)( this / S )
|
|
* this = 10 * ( this mod S )
|
|
* Return q.
|
|
* </pre>
|
|
* This is the iteration step of digit development for output.
|
|
* We assume that S has been normalized, as above, and that
|
|
* "this" has been left-shifted accordingly.
|
|
* Also assumed, of course, is that the result, q, can be expressed
|
|
* as an integer, {@code 0 <= q < 10}.
|
|
*
|
|
* @param S The divisor of this <code>FDBigInteger</code>.
|
|
* @return <code>q = (int)(this / S)</code>.
|
|
*/
|
|
/*@
|
|
@ requires !this.isImmutable;
|
|
@ requires this.size() <= S.size();
|
|
@ requires this.data.length + this.offset >= S.size();
|
|
@ requires S.value() >= ((\bigint)1) << (S.size()*32 - 4);
|
|
@ assignable this.nWords, this.offset, this.data, this.data[*];
|
|
@ ensures \result == \old(this.value() / S.value());
|
|
@ ensures this.value() == \old(10 * (this.value() % S.value()));
|
|
@*/
|
|
public int quoRemIteration(FDBigInteger S) throws IllegalArgumentException {
|
|
assert !this.isImmutable : "cannot modify immutable value";
|
|
// ensure that this and S have the same number of
|
|
// digits. If S is properly normalized and q < 10 then
|
|
// this must be so.
|
|
int thSize = this.size();
|
|
int sSize = S.size();
|
|
if (thSize < sSize) {
|
|
// this value is significantly less than S, result of division is zero.
|
|
// just mult this by 10.
|
|
int p = multAndCarryBy10(this.data, this.nWords, this.data);
|
|
if(p!=0) {
|
|
this.data[nWords++] = p;
|
|
} else {
|
|
trimLeadingZeros();
|
|
}
|
|
return 0;
|
|
} else if (thSize > sSize) {
|
|
throw new IllegalArgumentException("disparate values");
|
|
}
|
|
// estimate q the obvious way. We will usually be
|
|
// right. If not, then we're only off by a little and
|
|
// will re-add.
|
|
long q = (this.data[this.nWords - 1] & LONG_MASK) / (S.data[S.nWords - 1] & LONG_MASK);
|
|
long diff = multDiffMe(q, S);
|
|
if (diff != 0L) {
|
|
//@ assert q != 0;
|
|
//@ assert this.offset == \old(Math.min(this.offset, S.offset));
|
|
//@ assert this.offset <= S.offset;
|
|
|
|
// q is too big.
|
|
// add S back in until this turns +. This should
|
|
// not be very many times!
|
|
long sum = 0L;
|
|
int tStart = S.offset - this.offset;
|
|
//@ assert tStart >= 0;
|
|
int[] sd = S.data;
|
|
int[] td = this.data;
|
|
while (sum == 0L) {
|
|
for (int sIndex = 0, tIndex = tStart; tIndex < this.nWords; sIndex++, tIndex++) {
|
|
sum += (td[tIndex] & LONG_MASK) + (sd[sIndex] & LONG_MASK);
|
|
td[tIndex] = (int) sum;
|
|
sum >>>= 32; // Signed or unsigned, answer is 0 or 1
|
|
}
|
|
//
|
|
// Originally the following line read
|
|
// "if ( sum !=0 && sum != -1 )"
|
|
// but that would be wrong, because of the
|
|
// treatment of the two values as entirely unsigned,
|
|
// it would be impossible for a carry-out to be interpreted
|
|
// as -1 -- it would have to be a single-bit carry-out, or +1.
|
|
//
|
|
assert sum == 0 || sum == 1 : sum; // carry out of division correction
|
|
q -= 1;
|
|
}
|
|
}
|
|
// finally, we can multiply this by 10.
|
|
// it cannot overflow, right, as the high-order word has
|
|
// at least 4 high-order zeros!
|
|
int p = multAndCarryBy10(this.data, this.nWords, this.data);
|
|
assert p == 0 : p; // Carry out of *10
|
|
trimLeadingZeros();
|
|
return (int) q;
|
|
}
|
|
|
|
/**
|
|
* Multiplies this <code>FDBigInteger</code> by 10. The operation will be
|
|
* performed in place unless the <code>FDBigInteger</code> is immutable in
|
|
* which case a new <code>FDBigInteger</code> will be returned.
|
|
*
|
|
* @return The <code>FDBigInteger</code> multiplied by 10.
|
|
*/
|
|
/*@
|
|
@ requires this.value() == 0;
|
|
@ assignable \nothing;
|
|
@ ensures \result == this;
|
|
@
|
|
@ also
|
|
@
|
|
@ requires this.value() > 0 && this.isImmutable;
|
|
@ assignable \nothing;
|
|
@ ensures \result.value() == \old(this.value() * 10);
|
|
@
|
|
@ also
|
|
@
|
|
@ requires this.value() > 0 && !this.isImmutable;
|
|
@ assignable this.nWords, this.data, this.data[*];
|
|
@ ensures \result == this;
|
|
@ ensures \result.value() == \old(this.value() * 10);
|
|
@*/
|
|
public FDBigInteger multBy10() {
|
|
if (nWords == 0) {
|
|
return this;
|
|
}
|
|
if (isImmutable) {
|
|
int[] res = new int[nWords + 1];
|
|
res[nWords] = multAndCarryBy10(data, nWords, res);
|
|
return new FDBigInteger(res, offset);
|
|
} else {
|
|
int p = multAndCarryBy10(this.data, this.nWords, this.data);
|
|
if (p != 0) {
|
|
if (nWords == data.length) {
|
|
if (data[0] == 0) {
|
|
System.arraycopy(data, 1, data, 0, --nWords);
|
|
offset++;
|
|
} else {
|
|
data = Arrays.copyOf(data, data.length + 1);
|
|
}
|
|
}
|
|
data[nWords++] = p;
|
|
} else {
|
|
trimLeadingZeros();
|
|
}
|
|
return this;
|
|
}
|
|
}
|
|
|
|
/**
|
|
* Multiplies this <code>FDBigInteger</code> by
|
|
* <code>5<sup>p5</sup> * 2<sup>p2</sup></code>. The operation will be
|
|
* performed in place if possible, otherwise a new <code>FDBigInteger</code>
|
|
* will be returned.
|
|
*
|
|
* @param p5 The exponent of the power-of-five factor.
|
|
* @param p2 The exponent of the power-of-two factor.
|
|
* @return The multiplication result.
|
|
*/
|
|
/*@
|
|
@ requires this.value() == 0 || p5 == 0 && p2 == 0;
|
|
@ assignable \nothing;
|
|
@ ensures \result == this;
|
|
@
|
|
@ also
|
|
@
|
|
@ requires this.value() > 0 && (p5 > 0 && p2 >= 0 || p5 == 0 && p2 > 0 && this.isImmutable);
|
|
@ assignable \nothing;
|
|
@ ensures \result.value() == \old(this.value() * pow52(p5, p2));
|
|
@
|
|
@ also
|
|
@
|
|
@ requires this.value() > 0 && p5 == 0 && p2 > 0 && !this.isImmutable;
|
|
@ assignable this.nWords, this.data, this.data[*];
|
|
@ ensures \result == this;
|
|
@ ensures \result.value() == \old(this.value() * pow52(p5, p2));
|
|
@*/
|
|
public FDBigInteger multByPow52(int p5, int p2) {
|
|
if (this.nWords == 0) {
|
|
return this;
|
|
}
|
|
FDBigInteger res = this;
|
|
if (p5 != 0) {
|
|
int[] r;
|
|
int extraSize = (p2 != 0) ? 1 : 0;
|
|
if (p5 < SMALL_5_POW.length) {
|
|
r = new int[this.nWords + 1 + extraSize];
|
|
mult(this.data, this.nWords, SMALL_5_POW[p5], r);
|
|
res = new FDBigInteger(r, this.offset);
|
|
} else {
|
|
FDBigInteger pow5 = big5pow(p5);
|
|
r = new int[this.nWords + pow5.size() + extraSize];
|
|
mult(this.data, this.nWords, pow5.data, pow5.nWords, r);
|
|
res = new FDBigInteger(r, this.offset + pow5.offset);
|
|
}
|
|
}
|
|
return res.leftShift(p2);
|
|
}
|
|
|
|
/**
|
|
* Multiplies two big integers represented as int arrays.
|
|
*
|
|
* @param s1 The first array factor.
|
|
* @param s1Len The number of elements of <code>s1</code> to use.
|
|
* @param s2 The second array factor.
|
|
* @param s2Len The number of elements of <code>s2</code> to use.
|
|
* @param dst The product array.
|
|
*/
|
|
/*@
|
|
@ requires s1 != dst && s2 != dst;
|
|
@ requires s1.length >= s1Len && s2.length >= s2Len && dst.length >= s1Len + s2Len;
|
|
@ assignable dst[0 .. s1Len + s2Len - 1];
|
|
@ ensures AP(dst, s1Len + s2Len) == \old(AP(s1, s1Len) * AP(s2, s2Len));
|
|
@*/
|
|
private static void mult(int[] s1, int s1Len, int[] s2, int s2Len, int[] dst) {
|
|
for (int i = 0; i < s1Len; i++) {
|
|
long v = s1[i] & LONG_MASK;
|
|
long p = 0L;
|
|
for (int j = 0; j < s2Len; j++) {
|
|
p += (dst[i + j] & LONG_MASK) + v * (s2[j] & LONG_MASK);
|
|
dst[i + j] = (int) p;
|
|
p >>>= 32;
|
|
}
|
|
dst[i + s2Len] = (int) p;
|
|
}
|
|
}
|
|
|
|
/**
|
|
* Subtracts the supplied <code>FDBigInteger</code> subtrahend from this
|
|
* <code>FDBigInteger</code>. Assert that the result is positive.
|
|
* If the subtrahend is immutable, store the result in this(minuend).
|
|
* If this(minuend) is immutable a new <code>FDBigInteger</code> is created.
|
|
*
|
|
* @param subtrahend The <code>FDBigInteger</code> to be subtracted.
|
|
* @return This <code>FDBigInteger</code> less the subtrahend.
|
|
*/
|
|
/*@
|
|
@ requires this.isImmutable;
|
|
@ requires this.value() >= subtrahend.value();
|
|
@ assignable \nothing;
|
|
@ ensures \result.value() == \old(this.value() - subtrahend.value());
|
|
@
|
|
@ also
|
|
@
|
|
@ requires !subtrahend.isImmutable;
|
|
@ requires this.value() >= subtrahend.value();
|
|
@ assignable this.nWords, this.offset, this.data, this.data[*];
|
|
@ ensures \result == this;
|
|
@ ensures \result.value() == \old(this.value() - subtrahend.value());
|
|
@*/
|
|
public FDBigInteger leftInplaceSub(FDBigInteger subtrahend) {
|
|
assert this.size() >= subtrahend.size() : "result should be positive";
|
|
FDBigInteger minuend;
|
|
if (this.isImmutable) {
|
|
minuend = new FDBigInteger(this.data.clone(), this.offset);
|
|
} else {
|
|
minuend = this;
|
|
}
|
|
int offsetDiff = subtrahend.offset - minuend.offset;
|
|
int[] sData = subtrahend.data;
|
|
int[] mData = minuend.data;
|
|
int subLen = subtrahend.nWords;
|
|
int minLen = minuend.nWords;
|
|
if (offsetDiff < 0) {
|
|
// need to expand minuend
|
|
int rLen = minLen - offsetDiff;
|
|
if (rLen < mData.length) {
|
|
System.arraycopy(mData, 0, mData, -offsetDiff, minLen);
|
|
Arrays.fill(mData, 0, -offsetDiff, 0);
|
|
} else {
|
|
int[] r = new int[rLen];
|
|
System.arraycopy(mData, 0, r, -offsetDiff, minLen);
|
|
minuend.data = mData = r;
|
|
}
|
|
minuend.offset = subtrahend.offset;
|
|
minuend.nWords = minLen = rLen;
|
|
offsetDiff = 0;
|
|
}
|
|
long borrow = 0L;
|
|
int mIndex = offsetDiff;
|
|
for (int sIndex = 0; sIndex < subLen && mIndex < minLen; sIndex++, mIndex++) {
|
|
long diff = (mData[mIndex] & LONG_MASK) - (sData[sIndex] & LONG_MASK) + borrow;
|
|
mData[mIndex] = (int) diff;
|
|
borrow = diff >> 32; // signed shift
|
|
}
|
|
for (; borrow != 0 && mIndex < minLen; mIndex++) {
|
|
long diff = (mData[mIndex] & LONG_MASK) + borrow;
|
|
mData[mIndex] = (int) diff;
|
|
borrow = diff >> 32; // signed shift
|
|
}
|
|
assert borrow == 0L : borrow; // borrow out of subtract,
|
|
// result should be positive
|
|
minuend.trimLeadingZeros();
|
|
return minuend;
|
|
}
|
|
|
|
/**
|
|
* Subtracts the supplied <code>FDBigInteger</code> subtrahend from this
|
|
* <code>FDBigInteger</code>. Assert that the result is positive.
|
|
* If the this(minuend) is immutable, store the result in subtrahend.
|
|
* If subtrahend is immutable a new <code>FDBigInteger</code> is created.
|
|
*
|
|
* @param subtrahend The <code>FDBigInteger</code> to be subtracted.
|
|
* @return This <code>FDBigInteger</code> less the subtrahend.
|
|
*/
|
|
/*@
|
|
@ requires subtrahend.isImmutable;
|
|
@ requires this.value() >= subtrahend.value();
|
|
@ assignable \nothing;
|
|
@ ensures \result.value() == \old(this.value() - subtrahend.value());
|
|
@
|
|
@ also
|
|
@
|
|
@ requires !subtrahend.isImmutable;
|
|
@ requires this.value() >= subtrahend.value();
|
|
@ assignable subtrahend.nWords, subtrahend.offset, subtrahend.data, subtrahend.data[*];
|
|
@ ensures \result == subtrahend;
|
|
@ ensures \result.value() == \old(this.value() - subtrahend.value());
|
|
@*/
|
|
public FDBigInteger rightInplaceSub(FDBigInteger subtrahend) {
|
|
assert this.size() >= subtrahend.size() : "result should be positive";
|
|
FDBigInteger minuend = this;
|
|
if (subtrahend.isImmutable) {
|
|
subtrahend = new FDBigInteger(subtrahend.data.clone(), subtrahend.offset);
|
|
}
|
|
int offsetDiff = minuend.offset - subtrahend.offset;
|
|
int[] sData = subtrahend.data;
|
|
int[] mData = minuend.data;
|
|
int subLen = subtrahend.nWords;
|
|
int minLen = minuend.nWords;
|
|
if (offsetDiff < 0) {
|
|
int rLen = minLen;
|
|
if (rLen < sData.length) {
|
|
System.arraycopy(sData, 0, sData, -offsetDiff, subLen);
|
|
Arrays.fill(sData, 0, -offsetDiff, 0);
|
|
} else {
|
|
int[] r = new int[rLen];
|
|
System.arraycopy(sData, 0, r, -offsetDiff, subLen);
|
|
subtrahend.data = sData = r;
|
|
}
|
|
subtrahend.offset = minuend.offset;
|
|
subLen -= offsetDiff;
|
|
offsetDiff = 0;
|
|
} else {
|
|
int rLen = minLen + offsetDiff;
|
|
if (rLen >= sData.length) {
|
|
subtrahend.data = sData = Arrays.copyOf(sData, rLen);
|
|
}
|
|
}
|
|
//@ assert minuend == this && minuend.value() == \old(this.value());
|
|
//@ assert mData == minuend.data && minLen == minuend.nWords;
|
|
//@ assert subtrahend.offset + subtrahend.data.length >= minuend.size();
|
|
//@ assert sData == subtrahend.data;
|
|
//@ assert AP(subtrahend.data, subtrahend.data.length) << subtrahend.offset == \old(subtrahend.value());
|
|
//@ assert subtrahend.offset == Math.min(\old(this.offset), minuend.offset);
|
|
//@ assert offsetDiff == minuend.offset - subtrahend.offset;
|
|
//@ assert 0 <= offsetDiff && offsetDiff + minLen <= sData.length;
|
|
int sIndex = 0;
|
|
long borrow = 0L;
|
|
for (; sIndex < offsetDiff; sIndex++) {
|
|
long diff = 0L - (sData[sIndex] & LONG_MASK) + borrow;
|
|
sData[sIndex] = (int) diff;
|
|
borrow = diff >> 32; // signed shift
|
|
}
|
|
//@ assert sIndex == offsetDiff;
|
|
for (int mIndex = 0; mIndex < minLen; sIndex++, mIndex++) {
|
|
//@ assert sIndex == offsetDiff + mIndex;
|
|
long diff = (mData[mIndex] & LONG_MASK) - (sData[sIndex] & LONG_MASK) + borrow;
|
|
sData[sIndex] = (int) diff;
|
|
borrow = diff >> 32; // signed shift
|
|
}
|
|
assert borrow == 0L : borrow; // borrow out of subtract,
|
|
// result should be positive
|
|
subtrahend.nWords = sIndex;
|
|
subtrahend.trimLeadingZeros();
|
|
return subtrahend;
|
|
|
|
}
|
|
|
|
/**
|
|
* Determines whether all elements of an array are zero for all indices less
|
|
* than a given index.
|
|
*
|
|
* @param a The array to be examined.
|
|
* @param from The index strictly below which elements are to be examined.
|
|
* @return Zero if all elements in range are zero, 1 otherwise.
|
|
*/
|
|
/*@
|
|
@ requires 0 <= from && from <= a.length;
|
|
@ ensures \result == (AP(a, from) == 0 ? 0 : 1);
|
|
@*/
|
|
private /*@ pure @*/ static int checkZeroTail(int[] a, int from) {
|
|
while (from > 0) {
|
|
if (a[--from] != 0) {
|
|
return 1;
|
|
}
|
|
}
|
|
return 0;
|
|
}
|
|
|
|
/**
|
|
* Compares the parameter with this <code>FDBigInteger</code>. Returns an
|
|
* integer accordingly as:
|
|
* <pre>{@code
|
|
* > 0: this > other
|
|
* 0: this == other
|
|
* < 0: this < other
|
|
* }</pre>
|
|
*
|
|
* @param other The <code>FDBigInteger</code> to compare.
|
|
* @return A negative value, zero, or a positive value according to the
|
|
* result of the comparison.
|
|
*/
|
|
/*@
|
|
@ ensures \result == (this.value() < other.value() ? -1 : this.value() > other.value() ? +1 : 0);
|
|
@*/
|
|
public /*@ pure @*/ int cmp(FDBigInteger other) {
|
|
int aSize = nWords + offset;
|
|
int bSize = other.nWords + other.offset;
|
|
if (aSize > bSize) {
|
|
return 1;
|
|
} else if (aSize < bSize) {
|
|
return -1;
|
|
}
|
|
int aLen = nWords;
|
|
int bLen = other.nWords;
|
|
while (aLen > 0 && bLen > 0) {
|
|
int a = data[--aLen];
|
|
int b = other.data[--bLen];
|
|
if (a != b) {
|
|
return ((a & LONG_MASK) < (b & LONG_MASK)) ? -1 : 1;
|
|
}
|
|
}
|
|
if (aLen > 0) {
|
|
return checkZeroTail(data, aLen);
|
|
}
|
|
if (bLen > 0) {
|
|
return -checkZeroTail(other.data, bLen);
|
|
}
|
|
return 0;
|
|
}
|
|
|
|
/**
|
|
* Compares this <code>FDBigInteger</code> with
|
|
* <code>5<sup>p5</sup> * 2<sup>p2</sup></code>.
|
|
* Returns an integer accordingly as:
|
|
* <pre>{@code
|
|
* > 0: this > other
|
|
* 0: this == other
|
|
* < 0: this < other
|
|
* }</pre>
|
|
* @param p5 The exponent of the power-of-five factor.
|
|
* @param p2 The exponent of the power-of-two factor.
|
|
* @return A negative value, zero, or a positive value according to the
|
|
* result of the comparison.
|
|
*/
|
|
/*@
|
|
@ requires p5 >= 0 && p2 >= 0;
|
|
@ ensures \result == (this.value() < pow52(p5, p2) ? -1 : this.value() > pow52(p5, p2) ? +1 : 0);
|
|
@*/
|
|
public /*@ pure @*/ int cmpPow52(int p5, int p2) {
|
|
if (p5 == 0) {
|
|
int wordcount = p2 >> 5;
|
|
int bitcount = p2 & 0x1f;
|
|
int size = this.nWords + this.offset;
|
|
if (size > wordcount + 1) {
|
|
return 1;
|
|
} else if (size < wordcount + 1) {
|
|
return -1;
|
|
}
|
|
int a = this.data[this.nWords -1];
|
|
int b = 1 << bitcount;
|
|
if (a != b) {
|
|
return ( (a & LONG_MASK) < (b & LONG_MASK)) ? -1 : 1;
|
|
}
|
|
return checkZeroTail(this.data, this.nWords - 1);
|
|
}
|
|
return this.cmp(big5pow(p5).leftShift(p2));
|
|
}
|
|
|
|
/**
|
|
* Compares this <code>FDBigInteger</code> with <code>x + y</code>. Returns a
|
|
* value according to the comparison as:
|
|
* <pre>{@code
|
|
* -1: this < x + y
|
|
* 0: this == x + y
|
|
* 1: this > x + y
|
|
* }</pre>
|
|
* @param x The first addend of the sum to compare.
|
|
* @param y The second addend of the sum to compare.
|
|
* @return -1, 0, or 1 according to the result of the comparison.
|
|
*/
|
|
/*@
|
|
@ ensures \result == (this.value() < x.value() + y.value() ? -1 : this.value() > x.value() + y.value() ? +1 : 0);
|
|
@*/
|
|
public /*@ pure @*/ int addAndCmp(FDBigInteger x, FDBigInteger y) {
|
|
FDBigInteger big;
|
|
FDBigInteger small;
|
|
int xSize = x.size();
|
|
int ySize = y.size();
|
|
int bSize;
|
|
int sSize;
|
|
if (xSize >= ySize) {
|
|
big = x;
|
|
small = y;
|
|
bSize = xSize;
|
|
sSize = ySize;
|
|
} else {
|
|
big = y;
|
|
small = x;
|
|
bSize = ySize;
|
|
sSize = xSize;
|
|
}
|
|
int thSize = this.size();
|
|
if (bSize == 0) {
|
|
return thSize == 0 ? 0 : 1;
|
|
}
|
|
if (sSize == 0) {
|
|
return this.cmp(big);
|
|
}
|
|
if (bSize > thSize) {
|
|
return -1;
|
|
}
|
|
if (bSize + 1 < thSize) {
|
|
return 1;
|
|
}
|
|
long top = (big.data[big.nWords - 1] & LONG_MASK);
|
|
if (sSize == bSize) {
|
|
top += (small.data[small.nWords - 1] & LONG_MASK);
|
|
}
|
|
if ((top >>> 32) == 0) {
|
|
if (((top + 1) >>> 32) == 0) {
|
|
// good case - no carry extension
|
|
if (bSize < thSize) {
|
|
return 1;
|
|
}
|
|
// here sum.nWords == this.nWords
|
|
long v = (this.data[this.nWords - 1] & LONG_MASK);
|
|
if (v < top) {
|
|
return -1;
|
|
}
|
|
if (v > top + 1) {
|
|
return 1;
|
|
}
|
|
}
|
|
} else { // (top>>>32)!=0 guaranteed carry extension
|
|
if (bSize + 1 > thSize) {
|
|
return -1;
|
|
}
|
|
// here sum.nWords == this.nWords
|
|
top >>>= 32;
|
|
long v = (this.data[this.nWords - 1] & LONG_MASK);
|
|
if (v < top) {
|
|
return -1;
|
|
}
|
|
if (v > top + 1) {
|
|
return 1;
|
|
}
|
|
}
|
|
return this.cmp(big.add(small));
|
|
}
|
|
|
|
/**
|
|
* Makes this <code>FDBigInteger</code> immutable.
|
|
*/
|
|
/*@
|
|
@ assignable this.isImmutable;
|
|
@ ensures this.isImmutable;
|
|
@*/
|
|
public void makeImmutable() {
|
|
this.isImmutable = true;
|
|
}
|
|
|
|
/**
|
|
* Multiplies this <code>FDBigInteger</code> by an integer.
|
|
*
|
|
* @param i The factor by which to multiply this <code>FDBigInteger</code>.
|
|
* @return This <code>FDBigInteger</code> multiplied by an integer.
|
|
*/
|
|
/*@
|
|
@ requires this.value() == 0;
|
|
@ assignable \nothing;
|
|
@ ensures \result == this;
|
|
@
|
|
@ also
|
|
@
|
|
@ requires this.value() != 0;
|
|
@ assignable \nothing;
|
|
@ ensures \result.value() == \old(this.value() * UNSIGNED(i));
|
|
@*/
|
|
private FDBigInteger mult(int i) {
|
|
if (this.nWords == 0) {
|
|
return this;
|
|
}
|
|
int[] r = new int[nWords + 1];
|
|
mult(data, nWords, i, r);
|
|
return new FDBigInteger(r, offset);
|
|
}
|
|
|
|
/**
|
|
* Multiplies this <code>FDBigInteger</code> by another <code>FDBigInteger</code>.
|
|
*
|
|
* @param other The <code>FDBigInteger</code> factor by which to multiply.
|
|
* @return The product of this and the parameter <code>FDBigInteger</code>s.
|
|
*/
|
|
/*@
|
|
@ requires this.value() == 0;
|
|
@ assignable \nothing;
|
|
@ ensures \result == this;
|
|
@
|
|
@ also
|
|
@
|
|
@ requires this.value() != 0 && other.value() == 0;
|
|
@ assignable \nothing;
|
|
@ ensures \result == other;
|
|
@
|
|
@ also
|
|
@
|
|
@ requires this.value() != 0 && other.value() != 0;
|
|
@ assignable \nothing;
|
|
@ ensures \result.value() == \old(this.value() * other.value());
|
|
@*/
|
|
private FDBigInteger mult(FDBigInteger other) {
|
|
if (this.nWords == 0) {
|
|
return this;
|
|
}
|
|
if (this.size() == 1) {
|
|
return other.mult(data[0]);
|
|
}
|
|
if (other.nWords == 0) {
|
|
return other;
|
|
}
|
|
if (other.size() == 1) {
|
|
return this.mult(other.data[0]);
|
|
}
|
|
int[] r = new int[nWords + other.nWords];
|
|
mult(this.data, this.nWords, other.data, other.nWords, r);
|
|
return new FDBigInteger(r, this.offset + other.offset);
|
|
}
|
|
|
|
/**
|
|
* Adds another <code>FDBigInteger</code> to this <code>FDBigInteger</code>.
|
|
*
|
|
* @param other The <code>FDBigInteger</code> to add.
|
|
* @return The sum of the <code>FDBigInteger</code>s.
|
|
*/
|
|
/*@
|
|
@ assignable \nothing;
|
|
@ ensures \result.value() == \old(this.value() + other.value());
|
|
@*/
|
|
private FDBigInteger add(FDBigInteger other) {
|
|
FDBigInteger big, small;
|
|
int bigLen, smallLen;
|
|
int tSize = this.size();
|
|
int oSize = other.size();
|
|
if (tSize >= oSize) {
|
|
big = this;
|
|
bigLen = tSize;
|
|
small = other;
|
|
smallLen = oSize;
|
|
} else {
|
|
big = other;
|
|
bigLen = oSize;
|
|
small = this;
|
|
smallLen = tSize;
|
|
}
|
|
int[] r = new int[bigLen + 1];
|
|
int i = 0;
|
|
long carry = 0L;
|
|
for (; i < smallLen; i++) {
|
|
carry += (i < big.offset ? 0L : (big.data[i - big.offset] & LONG_MASK) )
|
|
+ ((i < small.offset ? 0L : (small.data[i - small.offset] & LONG_MASK)));
|
|
r[i] = (int) carry;
|
|
carry >>= 32; // signed shift.
|
|
}
|
|
for (; i < bigLen; i++) {
|
|
carry += (i < big.offset ? 0L : (big.data[i - big.offset] & LONG_MASK) );
|
|
r[i] = (int) carry;
|
|
carry >>= 32; // signed shift.
|
|
}
|
|
r[bigLen] = (int) carry;
|
|
return new FDBigInteger(r, 0);
|
|
}
|
|
|
|
|
|
/**
|
|
* Multiplies a <code>FDBigInteger</code> by an int and adds another int. The
|
|
* result is computed in place. This method is intended only to be invoked
|
|
* from
|
|
* <code>
|
|
* FDBigInteger(long lValue, char[] digits, int kDigits, int nDigits)
|
|
* </code>.
|
|
*
|
|
* @param iv The factor by which to multiply this <code>FDBigInteger</code>.
|
|
* @param addend The value to add to the product of this
|
|
* <code>FDBigInteger</code> and <code>iv</code>.
|
|
*/
|
|
/*@
|
|
@ requires this.value()*UNSIGNED(iv) + UNSIGNED(addend) < ((\bigint)1) << ((this.data.length + this.offset)*32);
|
|
@ assignable this.data[*];
|
|
@ ensures this.value() == \old(this.value()*UNSIGNED(iv) + UNSIGNED(addend));
|
|
@*/
|
|
private /*@ helper @*/ void multAddMe(int iv, int addend) {
|
|
long v = iv & LONG_MASK;
|
|
// unroll 0th iteration, doing addition.
|
|
long p = v * (data[0] & LONG_MASK) + (addend & LONG_MASK);
|
|
data[0] = (int) p;
|
|
p >>>= 32;
|
|
for (int i = 1; i < nWords; i++) {
|
|
p += v * (data[i] & LONG_MASK);
|
|
data[i] = (int) p;
|
|
p >>>= 32;
|
|
}
|
|
if (p != 0L) {
|
|
data[nWords++] = (int) p; // will fail noisily if illegal!
|
|
}
|
|
}
|
|
|
|
//
|
|
// original doc:
|
|
//
|
|
// do this -=q*S
|
|
// returns borrow
|
|
//
|
|
/**
|
|
* Multiplies the parameters and subtracts them from this
|
|
* <code>FDBigInteger</code>.
|
|
*
|
|
* @param q The integer parameter.
|
|
* @param S The <code>FDBigInteger</code> parameter.
|
|
* @return <code>this - q*S</code>.
|
|
*/
|
|
/*@
|
|
@ ensures nWords == 0 ==> offset == 0;
|
|
@ ensures nWords > 0 ==> data[nWords - 1] != 0;
|
|
@*/
|
|
/*@
|
|
@ requires 0 < q && q <= (1L << 31);
|
|
@ requires data != null;
|
|
@ requires 0 <= nWords && nWords <= data.length && offset >= 0;
|
|
@ requires !this.isImmutable;
|
|
@ requires this.size() == S.size();
|
|
@ requires this != S;
|
|
@ assignable this.nWords, this.offset, this.data, this.data[*];
|
|
@ ensures -q <= \result && \result <= 0;
|
|
@ ensures this.size() == \old(this.size());
|
|
@ ensures this.value() + (\result << (this.size()*32)) == \old(this.value() - q*S.value());
|
|
@ ensures this.offset == \old(Math.min(this.offset, S.offset));
|
|
@ ensures \old(this.offset <= S.offset) ==> this.nWords == \old(this.nWords);
|
|
@ ensures \old(this.offset <= S.offset) ==> this.offset == \old(this.offset);
|
|
@ ensures \old(this.offset <= S.offset) ==> this.data == \old(this.data);
|
|
@
|
|
@ also
|
|
@
|
|
@ requires q == 0;
|
|
@ assignable \nothing;
|
|
@ ensures \result == 0;
|
|
@*/
|
|
private /*@ helper @*/ long multDiffMe(long q, FDBigInteger S) {
|
|
long diff = 0L;
|
|
if (q != 0) {
|
|
int deltaSize = S.offset - this.offset;
|
|
if (deltaSize >= 0) {
|
|
int[] sd = S.data;
|
|
int[] td = this.data;
|
|
for (int sIndex = 0, tIndex = deltaSize; sIndex < S.nWords; sIndex++, tIndex++) {
|
|
diff += (td[tIndex] & LONG_MASK) - q * (sd[sIndex] & LONG_MASK);
|
|
td[tIndex] = (int) diff;
|
|
diff >>= 32; // N.B. SIGNED shift.
|
|
}
|
|
} else {
|
|
deltaSize = -deltaSize;
|
|
int[] rd = new int[nWords + deltaSize];
|
|
int sIndex = 0;
|
|
int rIndex = 0;
|
|
int[] sd = S.data;
|
|
for (; rIndex < deltaSize && sIndex < S.nWords; sIndex++, rIndex++) {
|
|
diff -= q * (sd[sIndex] & LONG_MASK);
|
|
rd[rIndex] = (int) diff;
|
|
diff >>= 32; // N.B. SIGNED shift.
|
|
}
|
|
int tIndex = 0;
|
|
int[] td = this.data;
|
|
for (; sIndex < S.nWords; sIndex++, tIndex++, rIndex++) {
|
|
diff += (td[tIndex] & LONG_MASK) - q * (sd[sIndex] & LONG_MASK);
|
|
rd[rIndex] = (int) diff;
|
|
diff >>= 32; // N.B. SIGNED shift.
|
|
}
|
|
this.nWords += deltaSize;
|
|
this.offset -= deltaSize;
|
|
this.data = rd;
|
|
}
|
|
}
|
|
return diff;
|
|
}
|
|
|
|
|
|
/**
|
|
* Multiplies by 10 a big integer represented as an array. The final carry
|
|
* is returned.
|
|
*
|
|
* @param src The array representation of the big integer.
|
|
* @param srcLen The number of elements of <code>src</code> to use.
|
|
* @param dst The product array.
|
|
* @return The final carry of the multiplication.
|
|
*/
|
|
/*@
|
|
@ requires src.length >= srcLen && dst.length >= srcLen;
|
|
@ assignable dst[0 .. srcLen - 1];
|
|
@ ensures 0 <= \result && \result < 10;
|
|
@ ensures AP(dst, srcLen) + (\result << (srcLen*32)) == \old(AP(src, srcLen) * 10);
|
|
@*/
|
|
private static int multAndCarryBy10(int[] src, int srcLen, int[] dst) {
|
|
long carry = 0;
|
|
for (int i = 0; i < srcLen; i++) {
|
|
long product = (src[i] & LONG_MASK) * 10L + carry;
|
|
dst[i] = (int) product;
|
|
carry = product >>> 32;
|
|
}
|
|
return (int) carry;
|
|
}
|
|
|
|
/**
|
|
* Multiplies by a constant value a big integer represented as an array.
|
|
* The constant factor is an <code>int</code>.
|
|
*
|
|
* @param src The array representation of the big integer.
|
|
* @param srcLen The number of elements of <code>src</code> to use.
|
|
* @param value The constant factor by which to multiply.
|
|
* @param dst The product array.
|
|
*/
|
|
/*@
|
|
@ requires src.length >= srcLen && dst.length >= srcLen + 1;
|
|
@ assignable dst[0 .. srcLen];
|
|
@ ensures AP(dst, srcLen + 1) == \old(AP(src, srcLen) * UNSIGNED(value));
|
|
@*/
|
|
private static void mult(int[] src, int srcLen, int value, int[] dst) {
|
|
long val = value & LONG_MASK;
|
|
long carry = 0;
|
|
for (int i = 0; i < srcLen; i++) {
|
|
long product = (src[i] & LONG_MASK) * val + carry;
|
|
dst[i] = (int) product;
|
|
carry = product >>> 32;
|
|
}
|
|
dst[srcLen] = (int) carry;
|
|
}
|
|
|
|
/**
|
|
* Multiplies by a constant value a big integer represented as an array.
|
|
* The constant factor is a long represent as two <code>int</code>s.
|
|
*
|
|
* @param src The array representation of the big integer.
|
|
* @param srcLen The number of elements of <code>src</code> to use.
|
|
* @param v0 The lower 32 bits of the long factor.
|
|
* @param v1 The upper 32 bits of the long factor.
|
|
* @param dst The product array.
|
|
*/
|
|
/*@
|
|
@ requires src != dst;
|
|
@ requires src.length >= srcLen && dst.length >= srcLen + 2;
|
|
@ assignable dst[0 .. srcLen + 1];
|
|
@ ensures AP(dst, srcLen + 2) == \old(AP(src, srcLen) * (UNSIGNED(v0) + (UNSIGNED(v1) << 32)));
|
|
@*/
|
|
private static void mult(int[] src, int srcLen, int v0, int v1, int[] dst) {
|
|
long v = v0 & LONG_MASK;
|
|
long carry = 0;
|
|
for (int j = 0; j < srcLen; j++) {
|
|
long product = v * (src[j] & LONG_MASK) + carry;
|
|
dst[j] = (int) product;
|
|
carry = product >>> 32;
|
|
}
|
|
dst[srcLen] = (int) carry;
|
|
v = v1 & LONG_MASK;
|
|
carry = 0;
|
|
for (int j = 0; j < srcLen; j++) {
|
|
long product = (dst[j + 1] & LONG_MASK) + v * (src[j] & LONG_MASK) + carry;
|
|
dst[j + 1] = (int) product;
|
|
carry = product >>> 32;
|
|
}
|
|
dst[srcLen + 1] = (int) carry;
|
|
}
|
|
|
|
// Fails assertion for negative exponent.
|
|
/**
|
|
* Computes <code>5</code> raised to a given power.
|
|
*
|
|
* @param p The exponent of 5.
|
|
* @return <code>5<sup>p</sup></code>.
|
|
*/
|
|
private static FDBigInteger big5pow(int p) {
|
|
assert p >= 0 : p; // negative power of 5
|
|
if (p < MAX_FIVE_POW) {
|
|
return POW_5_CACHE[p];
|
|
}
|
|
return big5powRec(p);
|
|
}
|
|
|
|
// slow path
|
|
/**
|
|
* Computes <code>5</code> raised to a given power.
|
|
*
|
|
* @param p The exponent of 5.
|
|
* @return <code>5<sup>p</sup></code>.
|
|
*/
|
|
private static FDBigInteger big5powRec(int p) {
|
|
if (p < MAX_FIVE_POW) {
|
|
return POW_5_CACHE[p];
|
|
}
|
|
// construct the value.
|
|
// recursively.
|
|
int q, r;
|
|
// in order to compute 5^p,
|
|
// compute its square root, 5^(p/2) and square.
|
|
// or, let q = p / 2, r = p -q, then
|
|
// 5^p = 5^(q+r) = 5^q * 5^r
|
|
q = p >> 1;
|
|
r = p - q;
|
|
FDBigInteger bigq = big5powRec(q);
|
|
if (r < SMALL_5_POW.length) {
|
|
return bigq.mult(SMALL_5_POW[r]);
|
|
} else {
|
|
return bigq.mult(big5powRec(r));
|
|
}
|
|
}
|
|
|
|
// for debugging ...
|
|
/**
|
|
* Converts this <code>FDBigInteger</code> to a hexadecimal string.
|
|
*
|
|
* @return The hexadecimal string representation.
|
|
*/
|
|
public String toHexString(){
|
|
if(nWords ==0) {
|
|
return "0";
|
|
}
|
|
StringBuilder sb = new StringBuilder((nWords +offset)*8);
|
|
for(int i= nWords -1; i>=0; i--) {
|
|
String subStr = Integer.toHexString(data[i]);
|
|
for(int j = subStr.length(); j<8; j++) {
|
|
sb.append('0');
|
|
}
|
|
sb.append(subStr);
|
|
}
|
|
for(int i=offset; i>0; i--) {
|
|
sb.append("00000000");
|
|
}
|
|
return sb.toString();
|
|
}
|
|
|
|
// for debugging ...
|
|
/**
|
|
* Converts this <code>FDBigInteger</code> to a <code>BigInteger</code>.
|
|
*
|
|
* @return The <code>BigInteger</code> representation.
|
|
*/
|
|
public BigInteger toBigInteger() {
|
|
byte[] magnitude = new byte[nWords * 4 + 1];
|
|
for (int i = 0; i < nWords; i++) {
|
|
int w = data[i];
|
|
magnitude[magnitude.length - 4 * i - 1] = (byte) w;
|
|
magnitude[magnitude.length - 4 * i - 2] = (byte) (w >> 8);
|
|
magnitude[magnitude.length - 4 * i - 3] = (byte) (w >> 16);
|
|
magnitude[magnitude.length - 4 * i - 4] = (byte) (w >> 24);
|
|
}
|
|
return new BigInteger(magnitude).shiftLeft(offset * 32);
|
|
}
|
|
|
|
// for debugging ...
|
|
/**
|
|
* Converts this <code>FDBigInteger</code> to a string.
|
|
*
|
|
* @return The string representation.
|
|
*/
|
|
@Override
|
|
public String toString(){
|
|
return toBigInteger().toString();
|
|
}
|
|
}
|