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	/*
	* Copyright (c) 2013, Oracle and/or its affiliates. All rights reserved.
	* DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
	*
	* This code is free software; you can redistribute it and/or modify it
	* under the terms of the GNU General Public License version 2 only, as
	* published by the Free Software Foundation. Oracle designates this
	* particular file as subject to the "Classpath" exception as provided
	* by Oracle in the LICENSE file that accompanied this code.
	*
	* This code is distributed in the hope that it will be useful, but WITHOUT
	* ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
	* FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
	* version 2 for more details (a copy is included in the LICENSE file that
	* accompanied this code).
	*
	* You should have received a copy of the GNU General Public License version
	* 2 along with this work; if not, write to the Free Software Foundation,
	* Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
	*
	* Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
	* or visit www.oracle.com if you need additional information or have any
	* questions.
	*/
	package sun.misc;

	import java.math.BigInteger;
	import java.util.Arrays;
	//@ model import org.jmlspecs.models.JMLMath;

	/**
	* A simple big integer package specifically for floating point base conversion.
	*/
	public /@ spec_bigint_math @/ class FDBigInteger {

	//
	// This class contains many comments that start with "/*@" mark.
	// They are behavourial specification in
	// the Java Modelling Language (JML):
	// http://www.eecs.ucf.edu/~leavens/JML//index.shtml
	//

	/*@
	@ public pure model static \bigint UNSIGNED(int v) {
	@ return v >= 0 ? v : v + (((\bigint)1) << 32);
	@ }
	@
	@ public pure model static \bigint UNSIGNED(long v) {
	@ return v >= 0 ? v : v + (((\bigint)1) << 64);
	@ }
	@
	@ public pure model static \bigint AP(int[] data, int len) {
	@ return (\sum int i; 0 <= 0 && i < len; UNSIGNED(data[i]) << (i*32));
	@ }
	@
	@ public pure model static \bigint pow52(int p5, int p2) {
	@ ghost \bigint v = 1;
	@ for (int i = 0; i < p5; i++) v *= 5;
	@ return v << p2;
	@ }
	@
	@ public pure model static \bigint pow10(int p10) {
	@ return pow52(p10, p10);
	@ }
	@*/

	static final int[] SMALL_5_POW = {
	1,
	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 * 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 * 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 * 5 * 5 * 5,
	5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5
	};

	static final long[] LONG_5_POW = {
	1L,
	5L,
	5L * 5,
	5L * 5 * 5,
	5L * 5 * 5 * 5,
	5L * 5 * 5 * 5 * 5,
	5L * 5 * 5 * 5 * 5 * 5,
	5L * 5 * 5 * 5 * 5 * 5 * 5,
	5L * 5 * 5 * 5 * 5 * 5 * 5 * 5,
	5L * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
	5L * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
	5L * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
	5L * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
	5L * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
	5L * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
	5L * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
	5L * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
	5L * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
	5L * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
	5L * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
	5L * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
	5L * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
	5L * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
	5L * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
	5L * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5,
	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,
	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,
	};

	// Maximum size of cache of powers of 5 as FDBigIntegers.
	private static final int MAX_FIVE_POW = 340;

	// Cache of big powers of 5 as FDBigIntegers.
	private static final FDBigInteger POW_5_CACHE[];

	// Initialize FDBigInteger cache of powers of 5.
	static {
	POW_5_CACHE = new FDBigInteger[MAX_FIVE_POW];
	int i = 0;
	while (i < SMALL_5_POW.length) {
	FDBigInteger pow5 = new FDBigInteger(new int[]{SMALL_5_POW[i]}, 0);
	pow5.makeImmutable();
	POW_5_CACHE[i] = pow5;
	i++;
	}
	FDBigInteger prev = POW_5_CACHE[i - 1];
	while (i < MAX_FIVE_POW) {
	POW_5_CACHE[i] = prev = prev.mult(5);
	prev.makeImmutable();
	i++;
	}
	}

	// Zero as an FDBigInteger.
	public static final FDBigInteger ZERO = new FDBigInteger(new int[0], 0);

	// Ensure ZERO is immutable.
	static {
	ZERO.makeImmutable();
	}

	// Constant for casting an int to a long via bitwise AND.
	private final static long LONG_MASK = 0xffffffffL;

	//@ spec_public non_null;
	private int data[]; // value: data[0] is least significant
	//@ spec_public;
	private int offset; // number of least significant zero padding ints
	//@ spec_public;
	private int nWords; // data[nWords-1]!=0, all values above are zero
	// if nWords==0 -> this FDBigInteger is zero
	//@ spec_public;
	private boolean isImmutable = false;

	/*@
	@ public invariant 0 <= nWords && nWords <= data.length && offset >= 0;
	@ public invariant nWords == 0 ==> offset == 0;
	@ public invariant nWords > 0 ==> data[nWords - 1] != 0;
	@ public invariant (\forall int i; nWords <= i && i < data.length; data[i] == 0);
	@ public pure model \bigint value() {
	@ return AP(data, nWords) << (offset*32);
	@ }
	@*/

	/**
	* Constructs an <code>FDBigInteger</code> from data and padding. The
	* <code>data</code> parameter has the least significant <code>int</code> at
	* the zeroth index. The <code>offset</code> parameter gives the number of
	* zero <code>int</code>s to be inferred below the least significant element
	* of <code>data</code>.
	*
	* @param data An array containing all non-zero <code>int</code>s of the value.
	* @param offset An offset indicating the number of zero <code>int</code>s to pad
	* below the least significant element of <code>data</code>.
	*/
	/*@
	@ requires data != null && offset >= 0;
	@ ensures this.value() == \old(AP(data, data.length) << (offset*32));
	@ ensures this.data == \old(data);
	@*/
	private FDBigInteger(int[] data, int offset) {
	this.data = data;
	this.offset = offset;
	this.nWords = data.length;
	trimLeadingZeros();
	}

	/**
	* Constructs an <code>FDBigInteger</code> from a starting value and some
	* decimal digits.
	*
	* @param lValue The starting value.
	* @param digits The decimal digits.
	* @param kDigits The initial index into <code>digits</code>.
	* @param nDigits The final index into <code>digits</code>.
	*/
	/*@
	@ requires digits != null;
	@ requires 0 <= kDigits && kDigits <= nDigits && nDigits <= digits.length;
	@ requires (\forall int i; 0 <= i && i < nDigits; '0' <= digits[i] && digits[i] <= '9');
	@ ensures this.value() == \old(lValue * pow10(nDigits - kDigits) + (\sum int i; kDigits <= i && i < nDigits; (digits[i] - '0') * pow10(nDigits - i - 1)));
	@*/
	public FDBigInteger(long lValue, char[] digits, int kDigits, int nDigits) {
	int n = Math.max((nDigits + 8) / 9, 2); // estimate size needed.
	data = new int[n]; // allocate enough space
	data[0] = (int) lValue; // starting value
	data[1] = (int) (lValue >>> 32);
	offset = 0;
	nWords = 2;
	int i = kDigits;
	int limit = nDigits - 5; // slurp digits 5 at a time.
	int v;
	while (i < limit) {
	int ilim = i + 5;
	v = (int) digits[i++] - (int) '0';
	while (i < ilim) {
	v = 10 * v + (int) digits[i++] - (int) '0';
	}
	multAddMe(100000, v); // ... where 100000 is 10^5.
	}
	int factor = 1;
	v = 0;
	while (i < nDigits) {
	v = 10 * v + (int) digits[i++] - (int) '0';
	factor *= 10;
	}
	if (factor != 1) {
	multAddMe(factor, v);
	}
	trimLeadingZeros();
	}

	/**
	* Returns an <code>FDBigInteger</code> with the numerical value
	* <code>5<sup>p5</sup> * 2<sup>p2</sup></code>.
	*
	* @param p5 The exponent of the power-of-five factor.
	* @param p2 The exponent of the power-of-two factor.
	* @return <code>5<sup>p5</sup> * 2<sup>p2</sup></code>
	*/
	/*@
	@ requires p5 >= 0 && p2 >= 0;
	@ assignable \nothing;
	@ ensures \result.value() == \old(pow52(p5, p2));
	@*/
	public static FDBigInteger valueOfPow52(int p5, int p2) {
	if (p5 != 0) {
	if (p2 == 0) {
	return big5pow(p5);
	} else if (p5 < SMALL_5_POW.length) {
	int pow5 = SMALL_5_POW[p5];
	int wordcount = p2 >> 5;
	int bitcount = p2 & 0x1f;
	if (bitcount == 0) {
	return new FDBigInteger(new int[]{pow5}, wordcount);
	} else {
	return new FDBigInteger(new int[]{
	pow5 << bitcount,
	pow5 >>> (32 - bitcount)
	}, wordcount);
	}
	} else {
	return big5pow(p5).leftShift(p2);
	}
	} else {
	return valueOfPow2(p2);
	}
	}

	/**
	* Returns an <code>FDBigInteger</code> with the numerical value
	* <code>value * 5<sup>p5</sup> * 2<sup>p2</sup></code>.
	*
	* @param value The constant factor.
	* @param p5 The exponent of the power-of-five factor.
	* @param p2 The exponent of the power-of-two factor.
	* @return <code>value * 5<sup>p5</sup> * 2<sup>p2</sup></code>
	*/
	/*@
	@ requires p5 >= 0 && p2 >= 0;
	@ assignable \nothing;
	@ ensures \result.value() == \old(UNSIGNED(value) * pow52(p5, p2));
	@*/
	public static FDBigInteger valueOfMulPow52(long value, int p5, int p2) {
	assert p5 >= 0 : p5;
	assert p2 >= 0 : p2;
	int v0 = (int) value;
	int v1 = (int) (value >>> 32);
	int wordcount = p2 >> 5;
	int bitcount = p2 & 0x1f;
	if (p5 != 0) {
	if (p5 < SMALL_5_POW.length) {
	long pow5 = SMALL_5_POW[p5] & LONG_MASK;
	long carry = (v0 & LONG_MASK) * pow5;
	v0 = (int) carry;
	carry >>>= 32;
	carry = (v1 & LONG_MASK) * pow5 + carry;
	v1 = (int) carry;
	int v2 = (int) (carry >>> 32);
	if (bitcount == 0) {
	return new FDBigInteger(new int[]{v0, v1, v2}, wordcount);
	} else {
	return new FDBigInteger(new int[]{
	v0 << bitcount,
	(v1 << bitcount) \| (v0 >>> (32 - bitcount)),
	(v2 << bitcount) \| (v1 >>> (32 - bitcount)),
	v2 >>> (32 - bitcount)
	}, wordcount);
	}
	} else {
	FDBigInteger pow5 = big5pow(p5);
	int[] r;
	if (v1 == 0) {
	r = new int[pow5.nWords + 1 + ((p2 != 0) ? 1 : 0)];
	mult(pow5.data, pow5.nWords, v0, r);
	} else {
	r = new int[pow5.nWords + 2 + ((p2 != 0) ? 1 : 0)];
	mult(pow5.data, pow5.nWords, v0, v1, r);
	}
	return (new FDBigInteger(r, pow5.offset)).leftShift(p2);
	}
	} else if (p2 != 0) {
	if (bitcount == 0) {
	return new FDBigInteger(new int[]{v0, v1}, wordcount);
	} else {
	return new FDBigInteger(new int[]{
	v0 << bitcount,
	(v1 << bitcount) \| (v0 >>> (32 - bitcount)),
	v1 >>> (32 - bitcount)
	}, wordcount);
	}
	}
	return new FDBigInteger(new int[]{v0, v1}, 0);
	}

	/**
	* Returns an <code>FDBigInteger</code> with the numerical value
	* <code>2<sup>p2</sup></code>.
	*
	* @param p2 The exponent of 2.
	* @return <code>2<sup>p2</sup></code>
	*/
	/*@
	@ requires p2 >= 0;
	@ assignable \nothing;
	@ ensures \result.value() == pow52(0, p2);
	@*/
	private static FDBigInteger valueOfPow2(int p2) {
	int wordcount = p2 >> 5;
	int bitcount = p2 & 0x1f;
	return new FDBigInteger(new int[]{1 << bitcount}, wordcount);
	}

	/**
	* Removes all leading zeros from this <code>FDBigInteger</code> adjusting
	* the offset and number of non-zero leading words accordingly.
	*/
	/*@
	@ requires data != null;
	@ requires 0 <= nWords && nWords <= data.length && offset >= 0;
	@ requires nWords == 0 ==> offset == 0;
	@ ensures nWords == 0 ==> offset == 0;
	@ ensures nWords > 0 ==> data[nWords - 1] != 0;
	@*/
	private /@ helper @/ void trimLeadingZeros() {
	int i = nWords;
	if (i > 0 && (data[--i] == 0)) {
	//for (; i > 0 && data[i - 1] == 0; i--) ;
	while(i > 0 && data[i - 1] == 0) {
	i--;
	}
	this.nWords = i;
	if (i == 0) { // all words are zero
	this.offset = 0;
	}
	}
	}

	/**
	* Retrieves the normalization bias of the <code>FDBigIntger</code>. The
	* normalization bias is a left shift such that after it the highest word
	* of the value will have the 4 highest bits equal to zero:
	* <code>(highestWord & 0xf0000000) == 0</code>, but the next bit should be 1
	* <code>(highestWord & 0x08000000) != 0</code>.
	*
	* @return The normalization bias.
	*/
	/*@
	@ requires this.value() > 0;
	@*/
	public /@ pure @/ int getNormalizationBias() {
	if (nWords == 0) {
	throw new IllegalArgumentException("Zero value cannot be normalized");
	}
	int zeros = Integer.numberOfLeadingZeros(data[nWords - 1]);
	return (zeros < 4) ? 28 + zeros : zeros - 4;
	}

	// TODO: Why is anticount param needed if it is always 32 - bitcount?
	/**
	* Left shifts the contents of one int array into another.
	*
	* @param src The source array.
	* @param idx The initial index of the source array.
	* @param result The destination array.
	* @param bitcount The left shift.
	* @param anticount The left anti-shift, e.g., <code>32-bitcount</code>.
	* @param prev The prior source value.
	*/
	/*@
	@ requires 0 < bitcount && bitcount < 32 && anticount == 32 - bitcount;
	@ requires src.length >= idx && result.length > idx;
	@ assignable result[*];
	@ ensures AP(result, \old(idx + 1)) == \old((AP(src, idx) + UNSIGNED(prev) << (idx*32)) << bitcount);
	@*/
	private static void leftShift(int[] src, int idx, int result[], int bitcount, int anticount, int prev){
	for (; idx > 0; idx--) {
	int v = (prev << bitcount);
	prev = src[idx - 1];
	v \|= (prev >>> anticount);
	result[idx] = v;
	}
	int v = prev << bitcount;
	result[0] = v;
	}

	/**
	* Shifts this <code>FDBigInteger</code> to the left. The shift is performed
	* in-place unless the <code>FDBigInteger</code> is immutable in which case
	* a new instance of <code>FDBigInteger</code> is returned.
	*
	* @param shift The number of bits to shift left.
	* @return The shifted <code>FDBigInteger</code>.
	*/
	/*@
	@ 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, 0 <= q < 10.
	*
	* @param 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
	*/
	/*@
	@ 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>
	* >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>
	* >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>
	* -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() - qS.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 << (srcLen32)) == \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();
	}
	}

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