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dart / sdk.git / refs/tags/2.0.0-dev.34.0 / . / runtime / lib / bigint_patch.dart
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// Copyright (c) 2017, the Dart project authors. Please see the AUTHORS file
// for details. All rights reserved. Use of this source code is governed by a
// BSD-style license that can be found in the LICENSE file.
// part of dart.core;
// Copyright 2009 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
/*
* Copyright (c) 2003-2005 Tom Wu
* Copyright (c) 2012 Adam Singer (adam@solvr.io)
* All Rights Reserved.
*
* Permission is hereby granted, free of charge, to any person obtaining
* a copy of this software and associated documentation files (the
* "Software"), to deal in the Software without restriction, including
* without limitation the rights to use, copy, modify, merge, publish,
* distribute, sublicense, and/or sell copies of the Software, and to
* permit persons to whom the Software is furnished to do so, subject to
* the following conditions:
*
* The above copyright notice and this permission notice shall be
* included in all copies or substantial portions of the Software.
*
* THE SOFTWARE IS PROVIDED "AS-IS" AND WITHOUT WARRANTY OF ANY KIND,
* EXPRESS, IMPLIED OR OTHERWISE, INCLUDING WITHOUT LIMITATION, ANY
* WARRANTY OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE.
*
* IN NO EVENT SHALL TOM WU BE LIABLE FOR ANY SPECIAL, INCIDENTAL,
* INDIRECT OR CONSEQUENTIAL DAMAGES OF ANY KIND, OR ANY DAMAGES WHATSOEVER
* RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER OR NOT ADVISED OF
* THE POSSIBILITY OF DAMAGE, AND ON ANY THEORY OF LIABILITY, ARISING OUT
* OF OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
*
* In addition, the following condition applies:
*
* All redistributions must retain an intact copy of this copyright notice
* and disclaimer.
*/
@patch
class BigInt implements Comparable<BigInt> {
@patch
static BigInt get zero => _BigIntImpl.zero;
@patch
static BigInt get one => _BigIntImpl.one;
@patch
static BigInt get two => _BigIntImpl.two;
@patch
static BigInt parse(String source, {int radix}) =>
_BigIntImpl.parse(source, radix: radix);
@patch
factory BigInt.from(num value) => new _BigIntImpl.from(value);
}
int _max(int a, int b) => a > b ? a : b;
int _min(int a, int b) => a < b ? a : b;
/// Allocate a new digits list of even length.
Uint32List _newDigits(int length) => new Uint32List(length + (length & 1));
/**
* An implementation for the arbitrarily large integer.
*
* The integer number is represented by a sign, an array of 32-bit unsigned
* integers in little endian format, and a number of used digits in that array.
*/
class _BigIntImpl implements BigInt {
// Bits per digit.
static const int _digitBits = 32;
static const int _digitBase = 1 << _digitBits;
static const int _digitMask = (1 << _digitBits) - 1;
// Bits per half digit.
static const int _halfDigitBits = _digitBits >> 1;
static const int _halfDigitMask = (1 << _halfDigitBits) - 1;
static final _BigIntImpl zero = new _BigIntImpl._fromInt(0);
static final _BigIntImpl one = new _BigIntImpl._fromInt(1);
static final _BigIntImpl two = new _BigIntImpl._fromInt(2);
static final _BigIntImpl _minusOne = -one;
static final _BigIntImpl _oneDigitMask = new _BigIntImpl._fromInt(_digitMask);
static final _BigIntImpl _twoDigitMask = (one << (2 * _digitBits)) - one;
static final _BigIntImpl _oneBillion = new _BigIntImpl._fromInt(1000000000);
static const int _minInt = -0x8000000000000000;
static const int _maxInt = 0x7fffffffffffffff;
// Result cache for last _divRem call.
// Result cache for last _divRem call.
static Uint32List _lastDividendDigits;
static int _lastDividendUsed;
static Uint32List _lastDivisorDigits;
static int _lastDivisorUsed;
static Uint32List _lastQuoRemDigits;
static int _lastQuoRemUsed;
static int _lastRemUsed;
static int _lastRem_nsh;
/// Whether this bigint is negative.
final bool _isNegative;
/// The unsigned digits of this bigint.
///
/// The least significant digit is in slot 0.
/// The list may have more digits than needed. That is, `_digits.length` may
/// be strictly greater than `_used`.
/// Also, `_digits.length` must always be even, because intrinsics on 64-bit
/// platforms may process a digit pair as a 64-bit value.
final Uint32List _digits;
/// The number of used entries in [_digits].
///
/// To avoid reallocating [Uint32List]s, lists that are too big are not
/// replaced, but `_used` reflects the smaller number of digits actually used.
///
/// Note that functions shortening an existing list of digits to a smaller
/// `_used` number of digits must ensure that the highermost pair of digits
/// is correct when read as a 64-bit value by intrinsics. Therefore, if the
/// smaller '_used' number is odd, the high digit of that pair must be
/// explicitly cleared, i.e. _digits[_used] = 0, which cannot result in an
/// out of bounds access, since the length of the list is guaranteed to be
/// even.
final int _used;
/**
* Parses [source] as a, possibly signed, integer literal and returns its
* value.
*
* The [source] must be a non-empty sequence of base-[radix] digits,
* optionally prefixed with a minus or plus sign ('-' or '+').
*
* The [radix] must be in the range 2..36. The digits used are
* first the decimal digits 0..9, and then the letters 'a'..'z' with
* values 10 through 35. Also accepts upper-case letters with the same
* values as the lower-case ones.
*
* If no [radix] is given then it defaults to 10. In this case, the [source]
* digits may also start with `0x`, in which case the number is interpreted
* as a hexadecimal literal, which effectively means that the `0x` is ignored
* and the radix is instead set to 16.
*
* For any int `n` and radix `r`, it is guaranteed that
* `n == int.parse(n.toRadixString(r), radix: r)`.
*
* Throws a [FormatException] if the [source] is not a valid integer literal,
* optionally prefixed by a sign.
*/
static _BigIntImpl parse(String source, {int radix}) {
var result = _tryParse(source, radix: radix);
if (result == null) {
throw new FormatException("Could not parse BigInt", source);
}
return result;
}
/// Parses a decimal bigint literal.
///
/// The [source] must not contain leading or trailing whitespace.
static _BigIntImpl _parseDecimal(String source, bool isNegative) {
const _0 = 48;
int part = 0;
_BigIntImpl result = zero;
// Read in the source 9 digits at a time.
// The first part may have a few leading virtual '0's to make the remaining
// parts all have exactly 9 digits.
int digitInPartCount = 9 - source.length.remainder(9);
if (digitInPartCount == 9) digitInPartCount = 0;
for (int i = 0; i < source.length; i++) {
part = part * 10 + source.codeUnitAt(i) - _0;
if (++digitInPartCount == 9) {
result = result * _oneBillion + new _BigIntImpl._fromInt(part);
part = 0;
digitInPartCount = 0;
}
}
if (isNegative) return -result;
return result;
}
/// Returns the value of a given source digit.
///
/// Source digits between "0" and "9" (inclusive) return their decimal value.
///
/// Source digits between "a" and "z", or "A" and "Z" (inclusive) return
/// 10 + their position in the ASCII alphabet.
///
/// The incoming [codeUnit] must be an ASCII code-unit.
static int _codeUnitToRadixValue(int codeUnit) {
// We know that the characters must be ASCII as otherwise the
// regexp wouldn't have matched. Lowercasing by doing `| 0x20` is thus
// guaranteed to be a safe operation, since it preserves digits
// and lower-cases ASCII letters.
const int _0 = 48;
const int _9 = 57;
const int _a = 97;
if (_0 <= codeUnit && codeUnit <= _9) return codeUnit - _0;
codeUnit |= 0x20;
var result = codeUnit - _a + 10;
return result;
}
/// Parses the given [source] string, starting at [startPos], as a hex
/// literal.
///
/// If [isNegative] is true, negates the result before returning it.
///
/// The [source] (substring) must be a valid hex literal.
static _BigIntImpl _parseHex(String source, int startPos, bool isNegative) {
int hexCharsPerDigit = _digitBits ~/ 4;
int sourceLength = source.length - startPos;
int used = (sourceLength + hexCharsPerDigit - 1) ~/ hexCharsPerDigit;
var digits = _newDigits(used);
int lastDigitLength = sourceLength - (used - 1) * hexCharsPerDigit;
int digitIndex = used - 1;
int i = startPos;
int digit = 0;
for (int j = 0; j < lastDigitLength; j++) {
var value = _codeUnitToRadixValue(source.codeUnitAt(i++));
if (value >= 16) return null;
digit = digit * 16 + value;
}
digits[digitIndex--] = digit;
while (i < source.length) {
digit = 0;
for (int j = 0; j < hexCharsPerDigit; j++) {
var value = _codeUnitToRadixValue(source.codeUnitAt(i++));
if (value >= 16) return null;
digit = digit * 16 + value;
}
digits[digitIndex--] = digit;
}
if (used == 1 && digits[0] == 0) return zero;
return new _BigIntImpl._(isNegative, used, digits);
}
/// Parses the given [source] as a [radix] literal.
///
/// The [source] will be checked for invalid characters. If it is invalid,
/// this function returns `null`.
static _BigIntImpl _parseRadix(String source, int radix, bool isNegative) {
var result = zero;
var base = new _BigIntImpl._fromInt(radix);
for (int i = 0; i < source.length; i++) {
var value = _codeUnitToRadixValue(source.codeUnitAt(i));
if (value >= radix) return null;
result = result * base + new _BigIntImpl._fromInt(value);
}
if (isNegative) return -result;
return result;
}
/// Tries to parse the given [source] as a [radix] literal.
///
/// Returns the parsed big integer, or `null` if it failed.
///
/// If the [radix] is `null` accepts decimal literals or `0x` hex literals.
static _BigIntImpl _tryParse(String source, {int radix}) {
if (source == "") return null;
var re = new RegExp(r'^\s*([+-]?)((0x[a-f0-9]+)|(\d+)|([a-z0-9]+))\s*$',
caseSensitive: false);
var match = re.firstMatch(source);
int signIndex = 1;
int hexIndex = 3;
int decimalIndex = 4;
int nonDecimalHexIndex = 5;
if (match == null) return null;
bool isNegative = match[signIndex] == "-";
String decimalMatch = match[decimalIndex];
String hexMatch = match[hexIndex];
String nonDecimalMatch = match[nonDecimalHexIndex];
if (radix == null) {
if (decimalMatch != null) {
// Cannot fail because we know that the digits are all decimal.
return _parseDecimal(decimalMatch, isNegative);
}
if (hexMatch != null) {
// Cannot fail because we know that the digits are all hex.
return _parseHex(hexMatch, 2, isNegative);
}
return null;
}
if (radix is! int) {
throw new ArgumentError.value(radix, 'radix', 'is not an integer');
}
if (radix < 2 || radix > 36) {
throw new RangeError.range(radix, 2, 36, 'radix');
}
if (radix == 10 && decimalMatch != null) {
return _parseDecimal(decimalMatch, isNegative);
}
if (radix == 16 && (decimalMatch != null || nonDecimalMatch != null)) {
return _parseHex(decimalMatch ?? nonDecimalMatch, 0, isNegative);
}
return _parseRadix(
decimalMatch ?? nonDecimalMatch ?? hexMatch, radix, isNegative);
}
/// Finds the amount significant digits in the provided [digits] array.
static int _normalize(int used, Uint32List digits) {
while (used > 0 && digits[used - 1] == 0) used--;
return used;
}
/// Factory returning an instance initialized with the given field values.
/// If the [digits] array contains leading 0s, the [used] value is adjusted
/// accordingly. The [digits] array is not modified.
_BigIntImpl._(bool isNegative, int used, Uint32List digits)
: this._normalized(isNegative, _normalize(used, digits), digits);
_BigIntImpl._normalized(bool isNegative, this._used, this._digits)
: _isNegative = _used == 0 ? false : isNegative {
assert(_digits.length.isEven);
assert(_used.isEven || _digits[_used] == 0); // Leading zero for 64-bit.
}
/// Whether this big integer is zero.
bool get _isZero => _used == 0;
/// Allocates an array of the given [length] and copies the [digits] in the
/// range [from] to [to-1], starting at index 0, followed by leading zero
/// digits.
static Uint32List _cloneDigits(
Uint32List digits, int from, int to, int length) {
var resultDigits = _newDigits(length);
var n = to - from;
for (var i = 0; i < n; i++) {
resultDigits[i] = digits[from + i];
}
return resultDigits;
}
/// Allocates a big integer from the provided [value] number.
factory _BigIntImpl.from(num value) {
if (value == 0) return zero;
if (value == 1) return one;
if (value == 2) return two;
if (value.abs() < 0x100000000)
return new _BigIntImpl._fromInt(value.toInt());
if (value is double) return new _BigIntImpl._fromDouble(value);
return new _BigIntImpl._fromInt(value);
}
factory _BigIntImpl._fromInt(int value) {
bool isNegative = value < 0;
assert(_digitBits == 32);
var digits = _newDigits(2);
if (isNegative) {
// Handle the min 64-bit value differently, since its negation is not
// positive.
if (value == _minInt) {
digits[1] = 0x80000000;
return new _BigIntImpl._(true, 2, digits);
}
value = -value;
}
if (value < _digitBase) {
digits[0] = value;
return new _BigIntImpl._(isNegative, 1, digits);
}
digits[0] = value & _digitMask;
digits[1] = value >> _digitBits;
return new _BigIntImpl._(isNegative, 2, digits);
}
/// An 8-byte Uint8List we can reuse for [_fromDouble] to avoid generating
/// garbage.
static final Uint8List _bitsForFromDouble = new Uint8List(8);
factory _BigIntImpl._fromDouble(double value) {
const int exponentBias = 1075;
if (value.isNaN || value.isInfinite) {
throw new ArgumentError("Value must be finite: $value");
}
bool isNegative = value < 0;
if (isNegative) value = -value;
value = value.floorToDouble();
if (value == 0) return zero;
var bits = _bitsForFromDouble;
for (int i = 0; i < 8; i++) {
bits[i] = 0;
}
bits.buffer.asByteData().setFloat64(0, value, Endian.little);
// The exponent is in bits 53..63.
var biasedExponent = (bits[7] << 4) + (bits[6] >> 4);
var exponent = biasedExponent - exponentBias;
assert(_digitBits == 32);
// The significant bits are in 0 .. 52.
var unshiftedDigits = _newDigits(2);
unshiftedDigits[0] =
(bits[3] << 24) + (bits[2] << 16) + (bits[1] << 8) + bits[0];
// Don't forget to add the hidden bit.
unshiftedDigits[1] =
((0x10 | (bits[6] & 0xF)) << 16) + (bits[5] << 8) + bits[4];
var unshiftedBig = new _BigIntImpl._normalized(false, 2, unshiftedDigits);
_BigIntImpl absResult;
if (exponent < 0) {
absResult = unshiftedBig >> -exponent;
} else if (exponent > 0) {
absResult = unshiftedBig << exponent;
}
if (isNegative) return -absResult;
return absResult;
}
/**
* Return the negative value of this integer.
*
* The result of negating an integer always has the opposite sign, except
* for zero, which is its own negation.
*/
_BigIntImpl operator -() {
if (_used == 0) return this;
return new _BigIntImpl._(!_isNegative, _used, _digits);
}
/**
* Returns the absolute value of this integer.
*
* For any integer `x`, the result is the same as `x < 0 ? -x : x`.
*/
_BigIntImpl abs() => _isNegative ? -this : this;
/// Returns this << n*_digitBits.
_BigIntImpl _dlShift(int n) {
final used = _used;
if (used == 0) {
return zero;
}
final resultUsed = used + n;
final digits = _digits;
final resultDigits = _newDigits(resultUsed);
for (int i = used - 1; i >= 0; i--) {
resultDigits[i + n] = digits[i];
}
return new _BigIntImpl._(_isNegative, resultUsed, resultDigits);
}
/// Same as [_dlShift] but works on the decomposed big integers.
///
/// Returns `resultUsed`.
///
/// `resultDigits[0..resultUsed-1] = xDigits[0..xUsed-1] << n*_digitBits`.
static int _dlShiftDigits(
Uint32List xDigits, int xUsed, int n, Uint32List resultDigits) {
if (xUsed == 0) {
return 0;
}
if (n == 0 && identical(resultDigits, xDigits)) {
return xUsed;
}
final resultUsed = xUsed + n;
assert(resultDigits.length >= resultUsed + (resultUsed & 1));
for (int i = xUsed - 1; i >= 0; i--) {
resultDigits[i + n] = xDigits[i];
}
for (int i = n - 1; i >= 0; i--) {
resultDigits[i] = 0;
}
if (resultUsed.isOdd) {
resultDigits[resultUsed] = 0;
}
return resultUsed;
}
/// Returns `this >> n*_digitBits`.
_BigIntImpl _drShift(int n) {
final used = _used;
if (used == 0) {
return zero;
}
final resultUsed = used - n;
if (resultUsed <= 0) {
return _isNegative ? _minusOne : zero;
}
final digits = _digits;
final resultDigits = _newDigits(resultUsed);
for (var i = n; i < used; i++) {
resultDigits[i - n] = digits[i];
}
final result = new _BigIntImpl._(_isNegative, resultUsed, resultDigits);
if (_isNegative) {
// Round down if any bit was shifted out.
for (var i = 0; i < n; i++) {
if (digits[i] != 0) {
return result - one;
}
}
}
return result;
}
/// Same as [_drShift] but works on the decomposed big integers.
///
/// Returns `resultUsed`.
///
/// `resultDigits[0..resultUsed-1] = xDigits[0..xUsed-1] >> n*_digitBits`.
static int _drShiftDigits(
Uint32List xDigits, int xUsed, int n, Uint32List resultDigits) {
final resultUsed = xUsed - n;
if (resultUsed <= 0) {
return 0;
}
assert(resultDigits.length >= resultUsed + (resultUsed & 1));
for (var i = n; i < xUsed; i++) {
resultDigits[i - n] = xDigits[i];
}
if (resultUsed.isOdd) {
resultDigits[resultUsed] = 0;
}
return resultUsed;
}
/// Shifts the digits of [xDigits] into the right place in [resultDigits].
///
/// `resultDigits[ds..xUsed+ds] = xDigits[0..xUsed-1] << (n % _digitBits)`
/// where `ds = ceil(n / _digitBits)`
///
/// Does *not* clear digits below ds.
///
/// Note: This function may be intrinsified.
static void _lsh(
Uint32List xDigits, int xUsed, int n, Uint32List resultDigits) {
final digitShift = n ~/ _digitBits;
final bitShift = n % _digitBits;
final carryBitShift = _digitBits - bitShift;
final bitMask = (1 << carryBitShift) - 1;
var carry = 0;
for (int i = xUsed - 1; i >= 0; i--) {
final digit = xDigits[i];
resultDigits[i + digitShift + 1] = (digit >> carryBitShift) | carry;
carry = (digit & bitMask) << bitShift;
}
resultDigits[digitShift] = carry;
}
/**
* Shift the bits of this integer to the left by [shiftAmount].
*
* Shifting to the left makes the number larger, effectively multiplying
* the number by `pow(2, shiftIndex)`.
*
* There is no limit on the size of the result. It may be relevant to
* limit intermediate values by using the "and" operator with a suitable
* mask.
*
* It is an error if [shiftAmount] is negative.
*/
_BigIntImpl operator <<(int shiftAmount) {
if (shiftAmount < 0) {
throw new ArgumentError("shift-amount must be positive $shiftAmount");
}
final digitShift = shiftAmount ~/ _digitBits;
final bitShift = shiftAmount % _digitBits;
if (bitShift == 0) {
return _dlShift(digitShift);
}
// Need one extra digit to hold bits shifted by bitShift.
var resultUsed = _used + digitShift + 1;
var resultDigits = _newDigits(resultUsed);
_lsh(_digits, _used, shiftAmount, resultDigits);
return new _BigIntImpl._(_isNegative, resultUsed, resultDigits);
}
/// resultDigits[0..resultUsed-1] = xDigits[0..xUsed-1] << n.
/// Returns resultUsed.
static int _lShiftDigits(
Uint32List xDigits, int xUsed, int n, Uint32List resultDigits) {
final digitsShift = n ~/ _digitBits;
final bitShift = n % _digitBits;
if (bitShift == 0) {
return _dlShiftDigits(xDigits, xUsed, digitsShift, resultDigits);
}
// Need one extra digit to hold bits shifted by bitShift.
var resultUsed = xUsed + digitsShift + 1;
assert(resultDigits.length >= resultUsed + (resultUsed & 1));
_lsh(xDigits, xUsed, n, resultDigits);
var i = digitsShift;
while (--i >= 0) {
resultDigits[i] = 0;
}
if (resultDigits[resultUsed - 1] == 0) {
resultUsed--; // Clamp result.
} else if (resultUsed.isOdd) {
resultDigits[resultUsed] = 0;
}
return resultUsed;
}
/// resultDigits[0..resultUsed-1] = xDigits[0..xUsed-1] >> n.
///
/// Note: This function may be intrinsified.
static void _rsh(
Uint32List xDigits, int xUsed, int n, Uint32List resultDigits) {
final digitsShift = n ~/ _digitBits;
final bitShift = n % _digitBits;
final carryBitShift = _digitBits - bitShift;
final bitMask = (1 << bitShift) - 1;
var carry = xDigits[digitsShift] >> bitShift;
final last = xUsed - digitsShift - 1;
for (var i = 0; i < last; i++) {
final digit = xDigits[i + digitsShift + 1];
resultDigits[i] = ((digit & bitMask) << carryBitShift) | carry;
carry = digit >> bitShift;
}
resultDigits[last] = carry;
}
/**
* Shift the bits of this integer to the right by [shiftAmount].
*
* Shifting to the right makes the number smaller and drops the least
* significant bits, effectively doing an integer division by
*`pow(2, shiftIndex)`.
*
* It is an error if [shiftAmount] is negative.
*/
_BigIntImpl operator >>(int shiftAmount) {
if (shiftAmount < 0) {
throw new ArgumentError("shift-amount must be positive $shiftAmount");
}
final digitShift = shiftAmount ~/ _digitBits;
final bitShift = shiftAmount % _digitBits;
if (bitShift == 0) {
return _drShift(digitShift);
}
final used = _used;
final resultUsed = used - digitShift;
if (resultUsed <= 0) {
return _isNegative ? _minusOne : zero;
}
final digits = _digits;
final resultDigits = _newDigits(resultUsed);
_rsh(digits, used, shiftAmount, resultDigits);
final result = new _BigIntImpl._(_isNegative, resultUsed, resultDigits);
if (_isNegative) {
// Round down if any bit was shifted out.
if ((digits[digitShift] & ((1 << bitShift) - 1)) != 0) {
return result - one;
}
for (var i = 0; i < digitShift; i++) {
if (digits[i] != 0) {
return result - one;
}
}
}
return result;
}
/// resultDigits[0..resultUsed-1] = xDigits[0..xUsed-1] >> n.
/// Returns resultUsed.
static int _rShiftDigits(
Uint32List xDigits, int xUsed, int n, Uint32List resultDigits) {
final digitShift = n ~/ _digitBits;
final bitShift = n % _digitBits;
if (bitShift == 0) {
return _drShiftDigits(xDigits, xUsed, digitShift, resultDigits);
}
var resultUsed = xUsed - digitShift;
if (resultUsed <= 0) {
return 0;
}
assert(resultDigits.length >= resultUsed + (resultUsed & 1));
_rsh(xDigits, xUsed, n, resultDigits);
if (resultDigits[resultUsed - 1] == 0) {
resultUsed--; // Clamp result.
} else if (resultUsed.isOdd) {
resultDigits[resultUsed] = 0;
}
return resultUsed;
}
/// Compares this to [other] taking the absolute value of both operands.
///
/// Returns 0 if abs(this) == abs(other); a positive number if
/// abs(this) > abs(other); and a negative number if abs(this) < abs(other).
int _absCompare(BigInt bigInt) {
_BigIntImpl other = bigInt;
return _compareDigits(_digits, _used, other._digits, other._used);
}
/**
* Compares this to `other`.
*
* Returns a negative number if `this` is less than `other`, zero if they are
* equal, and a positive number if `this` is greater than `other`.
*/
int compareTo(BigInt bigInt) {
_BigIntImpl other = bigInt;
if (_isNegative == other._isNegative) {
var result = _absCompare(other);
// Use 0 - result to avoid negative zero in JavaScript.
return _isNegative ? 0 - result : result;
}
return _isNegative ? -1 : 1;
}
/// Compares `digits[0..used-1]` with `otherDigits[0..otherUsed-1]`.
///
/// Returns 0 if equal; a positive number if larger;
/// and a negative number if smaller.
static int _compareDigits(
Uint32List digits, int used, Uint32List otherDigits, int otherUsed) {
var result = used - otherUsed;
if (result == 0) {
for (int i = used - 1; i >= 0; i--) {
result = digits[i] - otherDigits[i];
if (result != 0) return result;
}
}
return result;
}
/// resultDigits[0..used] = digits[0..used-1] + otherDigits[0..otherUsed-1].
/// used >= otherUsed > 0.
///
/// Note: This function may be intrinsified.
static void _absAdd(Uint32List digits, int used, Uint32List otherDigits,
int otherUsed, Uint32List resultDigits) {
assert(used >= otherUsed && otherUsed > 0);
var carry = 0;
for (var i = 0; i < otherUsed; i++) {
carry += digits[i] + otherDigits[i];
resultDigits[i] = carry & _digitMask;
carry >>= _digitBits;
}
for (var i = otherUsed; i < used; i++) {
carry += digits[i];
resultDigits[i] = carry & _digitMask;
carry >>= _digitBits;
}
resultDigits[used] = carry;
}
/// resultDigits[0..used-1] = digits[0..used-1] - otherDigits[0..otherUsed-1].
/// used >= otherUsed > 0.
///
/// Note: This function may be intrinsified.
static void _absSub(Uint32List digits, int used, Uint32List otherDigits,
int otherUsed, Uint32List resultDigits) {
assert(used >= otherUsed && otherUsed > 0);
var carry = 0;
for (var i = 0; i < otherUsed; i++) {
carry += digits[i] - otherDigits[i];
resultDigits[i] = carry & _digitMask;
carry >>= _digitBits;
}
for (var i = otherUsed; i < used; i++) {
carry += digits[i];
resultDigits[i] = carry & _digitMask;
carry >>= _digitBits;
}
}
/// Returns `abs(this) + abs(other)` with sign set according to [isNegative].
_BigIntImpl _absAddSetSign(_BigIntImpl other, bool isNegative) {
var used = _used;
var otherUsed = other._used;
if (used < otherUsed) {
return other._absAddSetSign(this, isNegative);
}
if (used == 0) {
assert(!isNegative);
return zero;
}
if (otherUsed == 0) {
return _isNegative == isNegative ? this : -this;
}
var resultUsed = used + 1;
var resultDigits = _newDigits(resultUsed);
_absAdd(_digits, used, other._digits, otherUsed, resultDigits);
return new _BigIntImpl._(isNegative, resultUsed, resultDigits);
}
/// Returns `abs(this) - abs(other)` with sign set according to [isNegative].
///
/// Requirement: `abs(this) >= abs(other)`.
_BigIntImpl _absSubSetSign(_BigIntImpl other, bool isNegative) {
assert(_absCompare(other) >= 0);
var used = _used;
if (used == 0) {
assert(!isNegative);
return zero;
}
var otherUsed = other._used;
if (otherUsed == 0) {
return _isNegative == isNegative ? this : -this;
}
var resultDigits = _newDigits(used);
_absSub(_digits, used, other._digits, otherUsed, resultDigits);
return new _BigIntImpl._(isNegative, used, resultDigits);
}
/// Returns `abs(this) & abs(other)` with sign set according to [isNegative].
_BigIntImpl _absAndSetSign(_BigIntImpl other, bool isNegative) {
var resultUsed = _min(_used, other._used);
var digits = _digits;
var otherDigits = other._digits;
var resultDigits = _newDigits(resultUsed);
for (var i = 0; i < resultUsed; i++) {
resultDigits[i] = digits[i] & otherDigits[i];
}
return new _BigIntImpl._(isNegative, resultUsed, resultDigits);
}
/// Returns `abs(this) &~ abs(other)` with sign set according to [isNegative].
_BigIntImpl _absAndNotSetSign(_BigIntImpl other, bool isNegative) {
var resultUsed = _used;
var digits = _digits;
var otherDigits = other._digits;
var resultDigits = _newDigits(resultUsed);
var m = _min(resultUsed, other._used);
for (var i = 0; i < m; i++) {
resultDigits[i] = digits[i] & ~otherDigits[i];
}
for (var i = m; i < resultUsed; i++) {
resultDigits[i] = digits[i];
}
return new _BigIntImpl._(isNegative, resultUsed, resultDigits);
}
/// Returns `abs(this) | abs(other)` with sign set according to [isNegative].
_BigIntImpl _absOrSetSign(_BigIntImpl other, bool isNegative) {
var used = _used;
var otherUsed = other._used;
var resultUsed = _max(used, otherUsed);
var digits = _digits;
var otherDigits = other._digits;
var resultDigits = _newDigits(resultUsed);
var l, m;
if (used < otherUsed) {
l = other;
m = used;
} else {
l = this;
m = otherUsed;
}
for (var i = 0; i < m; i++) {
resultDigits[i] = digits[i] | otherDigits[i];
}
var lDigits = l._digits;
for (var i = m; i < resultUsed; i++) {
resultDigits[i] = lDigits[i];
}
return new _BigIntImpl._(isNegative, resultUsed, resultDigits);
}
/// Returns `abs(this) ^ abs(other)` with sign set according to [isNegative].
_BigIntImpl _absXorSetSign(_BigIntImpl other, bool isNegative) {
var used = _used;
var otherUsed = other._used;
var resultUsed = _max(used, otherUsed);
var digits = _digits;
var otherDigits = other._digits;
var resultDigits = _newDigits(resultUsed);
var l, m;
if (used < otherUsed) {
l = other;
m = used;
} else {
l = this;
m = otherUsed;
}
for (var i = 0; i < m; i++) {
resultDigits[i] = digits[i] ^ otherDigits[i];
}
var lDigits = l._digits;
for (var i = m; i < resultUsed; i++) {
resultDigits[i] = lDigits[i];
}
return new _BigIntImpl._(isNegative, resultUsed, resultDigits);
}
/**
* Bit-wise and operator.
*
* Treating both `this` and [other] as sufficiently large two's component
* integers, the result is a number with only the bits set that are set in
* both `this` and [other]
*
* Of both operands are negative, the result is negative, otherwise
* the result is non-negative.
*/
_BigIntImpl operator &(BigInt bigInt) {
_BigIntImpl other = bigInt;
if (_isNegative == other._isNegative) {
if (_isNegative) {
// (-this) & (-other) == ~(this-1) & ~(other-1)
// == ~((this-1) | (other-1))
// == -(((this-1) | (other-1)) + 1)
_BigIntImpl this1 = _absSubSetSign(one, true);
_BigIntImpl other1 = other._absSubSetSign(one, true);
// Result cannot be zero if this and other are negative.
return this1._absOrSetSign(other1, true)._absAddSetSign(one, true);
}
return _absAndSetSign(other, false);
}
// _isNegative != other._isNegative
var p, n;
if (_isNegative) {
p = other;
n = this;
} else {
// & is symmetric.
p = this;
n = other;
}
// p & (-n) == p & ~(n-1) == p &~ (n-1)
var n1 = n._absSubSetSign(one, false);
return p._absAndNotSetSign(n1, false);
}
/**
* Bit-wise or operator.
*
* Treating both `this` and [other] as sufficiently large two's component
* integers, the result is a number with the bits set that are set in either
* of `this` and [other]
*
* If both operands are non-negative, the result is non-negative,
* otherwise the result us negative.
*/
_BigIntImpl operator |(BigInt bigInt) {
_BigIntImpl other = bigInt;
if (_isNegative == other._isNegative) {
if (_isNegative) {
// (-this) | (-other) == ~(this-1) | ~(other-1)
// == ~((this-1) & (other-1))
// == -(((this-1) & (other-1)) + 1)
var this1 = _absSubSetSign(one, true);
var other1 = other._absSubSetSign(one, true);
// Result cannot be zero if this and a are negative.
return this1._absAndSetSign(other1, true)._absAddSetSign(one, true);
}
return _absOrSetSign(other, false);
}
// _neg != a._neg
var p, n;
if (_isNegative) {
p = other;
n = this;
} else {
// | is symmetric.
p = this;
n = other;
}
// p | (-n) == p | ~(n-1) == ~((n-1) &~ p) == -(~((n-1) &~ p) + 1)
var n1 = n._absSubSetSign(one, true);
// Result cannot be zero if only one of this or a is negative.
return n1._absAndNotSetSign(p, true)._absAddSetSign(one, true);
}
/**
* Bit-wise exclusive-or operator.
*
* Treating both `this` and [other] as sufficiently large two's component
* integers, the result is a number with the bits set that are set in one,
* but not both, of `this` and [other]
*
* If the operands have the same sign, the result is non-negative,
* otherwise the result is negative.
*/
_BigIntImpl operator ^(BigInt bigInt) {
_BigIntImpl other = bigInt;
if (_isNegative == other._isNegative) {
if (_isNegative) {
// (-this) ^ (-other) == ~(this-1) ^ ~(other-1) == (this-1) ^ (other-1)
var this1 = _absSubSetSign(one, true);
var other1 = other._absSubSetSign(one, true);
return this1._absXorSetSign(other1, false);
}
return _absXorSetSign(other, false);
}
// _isNegative != a._isNegative
var p, n;
if (_isNegative) {
p = other;
n = this;
} else {
// ^ is symmetric.
p = this;
n = other;
}
// p ^ (-n) == p ^ ~(n-1) == ~(p ^ (n-1)) == -((p ^ (n-1)) + 1)
var n1 = n._absSubSetSign(one, true);
// Result cannot be zero if only one of this or a is negative.
return p._absXorSetSign(n1, true)._absAddSetSign(one, true);
}
/**
* The bit-wise negate operator.
*
* Treating `this` as a sufficiently large two's component integer,
* the result is a number with the opposite bits set.
*
* This maps any integer `x` to `-x - 1`.
*/
_BigIntImpl operator ~() {
if (_isNegative) {
// ~(-this) == ~(~(this-1)) == this-1
return _absSubSetSign(one, false);
}
// ~this == -this-1 == -(this+1)
// Result cannot be zero if this is positive.
return _absAddSetSign(one, true);
}
/// Addition operator.
_BigIntImpl operator +(BigInt bigInt) {
_BigIntImpl other = bigInt;
var isNegative = _isNegative;
if (isNegative == other._isNegative) {
// this + other == this + other
// (-this) + (-other) == -(this + other)
return _absAddSetSign(other, isNegative);
}
// this + (-other) == this - other == -(this - other)
// (-this) + other == other - this == -(this - other)
if (_absCompare(other) >= 0) {
return _absSubSetSign(other, isNegative);
}
return other._absSubSetSign(this, !isNegative);
}
/// Subtraction operator.
_BigIntImpl operator -(BigInt bigInt) {
_BigIntImpl other = bigInt;
var isNegative = _isNegative;
if (isNegative != other._isNegative) {
// this - (-other) == this + other
// (-this) - other == -(this + other)
return _absAddSetSign(other, isNegative);
}
// this - other == this - a == -(this - other)
// (-this) - (-other) == other - this == -(this - other)
if (_absCompare(other) >= 0) {
return _absSubSetSign(other, isNegative);
}
return other._absSubSetSign(this, !isNegative);
}
/// Multiplies `xDigits[xIndex]` with `multiplicandDigits` and adds the result
/// to `accumulatorDigits`.
///
/// The `multiplicandDigits` in the range `i` to `i`+`n`-1 are the
/// multiplicand digits.
///
/// The `acculumatorDigits` in the range `j` to `j`+`n`-1 are the accumulator
/// digits.
///
/// Concretely:
/// `accumulatorDigits[j..j+n] += xDigits[xIndex] * m_digits[i..i+n-1]`.
/// Returns 1.
///
/// Note: This function may be intrinsified. Intrinsics on 64-bit platforms
/// process digit pairs at even indices and returns 2.
static int _mulAdd(
Uint32List xDigits,
int xIndex,
Uint32List multiplicandDigits,
int i,
Uint32List accumulatorDigits,
int j,
int n) {
int x = xDigits[xIndex];
if (x == 0) {
// No-op if x is 0.
return 1;
}
int carry = 0;
int xl = x & _halfDigitMask;
int xh = x >> _halfDigitBits;
while (--n >= 0) {
int ml = multiplicandDigits[i] & _halfDigitMask;
int mh = multiplicandDigits[i++] >> _halfDigitBits;
int ph = xh * ml + mh * xl;
int pl = xl * ml +
((ph & _halfDigitMask) << _halfDigitBits) +
accumulatorDigits[j] +
carry;
carry = (pl >> _digitBits) + (ph >> _halfDigitBits) + xh * mh;
accumulatorDigits[j++] = pl & _digitMask;
}
while (carry != 0) {
int l = accumulatorDigits[j] + carry;
carry = l >> _digitBits;
accumulatorDigits[j++] = l & _digitMask;
}
return 1;
}
/// Multiplies `xDigits[i]` with `xDigits` and adds the result to
/// `accumulatorDigits`.
///
/// The `xDigits` in the range `i` to `used`-1 are the multiplicand digits.
///
/// The `acculumatorDigits` in the range 2*`i` to `i`+`used`-1 are the
/// accumulator digits.
///
/// Concretely:
/// `accumulatorDigits[2*i..i+used-1] += xDigits[i]*xDigits[i] +
/// 2*xDigits[i]*xDigits[i+1..used-1]`.
/// Returns 1.
///
/// Note: This function may be intrinsified. Intrinsics on 64-bit platforms
/// process digit pairs at even indices and returns 2.
static int _sqrAdd(
Uint32List xDigits, int i, Uint32List acculumatorDigits, int used) {
int x = xDigits[i];
if (x == 0) return 1;
int j = 2 * i;
int carry = 0;
int xl = x & _halfDigitMask;
int xh = x >> _halfDigitBits;
int ph = 2 * xh * xl;
int pl = xl * xl +
((ph & _halfDigitMask) << _halfDigitBits) +
acculumatorDigits[j];
carry = (pl >> _digitBits) + (ph >> _halfDigitBits) + xh * xh;
acculumatorDigits[j] = pl & _digitMask;
x <<= 1;
xl = x & _halfDigitMask;
xh = x >> _halfDigitBits;
int n = used - i - 1;
int k = i + 1;
j++;
while (--n >= 0) {
int l = xDigits[k] & _halfDigitMask;
int h = xDigits[k++] >> _halfDigitBits;
int ph = xh * l + h * xl;
int pl = xl * l +
((ph & _halfDigitMask) << _halfDigitBits) +
acculumatorDigits[j] +
carry;
carry = (pl >> _digitBits) + (ph >> _halfDigitBits) + xh * h;
acculumatorDigits[j++] = pl & _digitMask;
}
carry += acculumatorDigits[i + used];
if (carry >= _digitBase) {
acculumatorDigits[i + used] = carry - _digitBase;
acculumatorDigits[i + used + 1] = 1;
} else {
acculumatorDigits[i + used] = carry;
}
return 1;
}
/// Multiplication operator.
_BigIntImpl operator *(BigInt bigInt) {
_BigIntImpl other = bigInt;
var used = _used;
var otherUsed = other._used;
if (used == 0 || otherUsed == 0) {
return zero;
}
var resultUsed = used + otherUsed;
var digits = _digits;
var otherDigits = other._digits;
var resultDigits = _newDigits(resultUsed);
var i = 0;
while (i < otherUsed) {
i += _mulAdd(otherDigits, i, digits, 0, resultDigits, i, used);
}
return new _BigIntImpl._(
_isNegative != other._isNegative, resultUsed, resultDigits);
}
// resultDigits[0..resultUsed-1] =
// xDigits[0..xUsed-1]*otherDigits[0..otherUsed-1].
// Returns resultUsed = xUsed + otherUsed.
static int _mulDigits(Uint32List xDigits, int xUsed, Uint32List otherDigits,
int otherUsed, Uint32List resultDigits) {
var resultUsed = xUsed + otherUsed;
var i = resultUsed + (resultUsed & 1);
assert(resultDigits.length >= i);
while (--i >= 0) {
resultDigits[i] = 0;
}
i = 0;
while (i < otherUsed) {
i += _mulAdd(otherDigits, i, xDigits, 0, resultDigits, i, xUsed);
}
return resultUsed;
}
// resultDigits[0..resultUsed-1] = xDigits[0..xUsed-1]^2.
// Returns resultUsed = 2*xUsed.
static int _sqrDigits(
Uint32List xDigits, int xUsed, Uint32List resultDigits) {
var resultUsed = 2 * xUsed;
assert(resultDigits.length >= resultUsed);
// Since resultUsed is even, no need for a leading zero for
// 64-bit processing.
var i = resultUsed;
while (--i >= 0) {
resultDigits[i] = 0;
}
i = 0;
while (i < xUsed - 1) {
i += _sqrAdd(xDigits, i, resultDigits, xUsed);
}
// The last step is already done if digit pairs were processed above.
if (i < xUsed) {
_mulAdd(xDigits, i, xDigits, i, resultDigits, 2 * i, 1);
}
return resultUsed;
}
// Indices of the arguments of _estimateQuotientDigit.
// For 64-bit processing by intrinsics on 64-bit platforms, the top digit pair
// of the divisor is provided in the args array, and a 64-bit estimated
// quotient is returned. However, on 32-bit platforms, the low 32-bit digit is
// ignored and only one 32-bit digit is returned as the estimated quotient.
static const int _divisorLowTopDigit = 0; // Low digit of top pair of divisor.
static const int _divisorTopDigit = 1; // Top digit of divisor.
static const int _quotientDigit = 2; // Estimated quotient.
static const int _quotientHighDigit = 3; // High digit of estimated quotient.
/// Estimate `args[_quotientDigit] = digits[i-1..i] ~/ args[_divisorTopDigit]`
/// Returns 1.
///
/// Note: This function may be intrinsified. Intrinsics on 64-bit platforms
/// process a digit pair (i always odd):
/// Estimate `args[_quotientDigit.._quotientHighDigit] = digits[i-3..i] ~/
/// args[_divisorLowTopDigit.._divisorTopDigit]`.
/// Returns 2.
static int _estimateQuotientDigit(Uint32List args, Uint32List digits, int i) {
// Verify that digit pairs are accessible for 64-bit processing.
assert(digits.length >= 4);
if (digits[i] == args[_divisorTopDigit]) {
args[_quotientDigit] = _digitMask;
} else {
// Chop off one bit, since a Mint cannot hold 2 digits.
var quotientDigit =
((digits[i] << (_digitBits - 1)) | (digits[i - 1] >> 1)) ~/
(args[_divisorTopDigit] >> 1);
if (quotientDigit > _digitMask) {
args[_quotientDigit] = _digitMask;
} else {
args[_quotientDigit] = quotientDigit;
}
}
return 1;
}
/// Returns `trunc(this / other)`, with `other != 0`.
_BigIntImpl _div(BigInt bigInt) {
_BigIntImpl other = bigInt;
assert(other._used > 0);
if (_used < other._used) {
return zero;
}
_divRem(other);
// Return quotient, i.e.
// _lastQuoRem_digits[_lastRem_used.._lastQuoRem_used-1] with proper sign.
var lastQuo_used = _lastQuoRemUsed - _lastRemUsed;
var quo_digits = _cloneDigits(
_lastQuoRemDigits, _lastRemUsed, _lastQuoRemUsed, lastQuo_used);
var quo = new _BigIntImpl._(false, lastQuo_used, quo_digits);
if ((_isNegative != other._isNegative) && (quo._used > 0)) {
quo = -quo;
}
return quo;
}
/// Returns `this - other * trunc(this / other)`, with `other != 0`.
_BigIntImpl _rem(BigInt bigInt) {
_BigIntImpl other = bigInt;
assert(other._used > 0);
if (_used < other._used) {
return this;
}
_divRem(other);
// Return remainder, i.e.
// denormalized _lastQuoRem_digits[0.._lastRem_used-1] with proper sign.
var remDigits =
_cloneDigits(_lastQuoRemDigits, 0, _lastRemUsed, _lastRemUsed);
var rem = new _BigIntImpl._(false, _lastRemUsed, remDigits);
if (_lastRem_nsh > 0) {
rem = rem >> _lastRem_nsh; // Denormalize remainder.
}
if (_isNegative && (rem._used > 0)) {
rem = -rem;
}
return rem;
}
/// Computes this ~/ other and this.remainder(other).
///
/// Stores the result in [_lastQuoRemDigits], [_lastQuoRemUsed] and
/// [_lastRemUsed]. The [_lastQuoRemDigits] contains the digits of *both*, the
/// quotient and the remainder.
///
/// Caches the input to avoid doing the work again when users write
/// `a ~/ b` followed by a `a % b`.
void _divRem(_BigIntImpl other) {
// Check if result is already cached.
if ((this._used == _lastDividendUsed) &&
(other._used == _lastDivisorUsed) &&
identical(this._digits, _lastDividendDigits) &&
identical(other._digits, _lastDivisorDigits)) {
return;
}
assert(_used >= other._used);
var nsh = _digitBits - other._digits[other._used - 1].bitLength;
// For 64-bit processing, make sure other has an even number of digits.
if (other._used.isOdd) {
nsh += _digitBits;
}
// Concatenated positive quotient and normalized positive remainder.
// The resultDigits can have at most one more digit than the dividend.
Uint32List resultDigits;
int resultUsed;
// Normalized positive divisor (referred to as 'y').
// The normalized divisor has the most-significant bit of its most
// significant digit set.
// This makes estimating the quotient easier.
Uint32List yDigits;
int yUsed;
if (nsh > 0) {
// Extra digits for normalization.
yDigits = _newDigits(other._used + (nsh % _digitBits) + 1);
yUsed = _lShiftDigits(other._digits, other._used, nsh, yDigits);
// Extra digits for normalization, also used for possible _mulAdd carry.
resultDigits = _newDigits(_used + (nsh % _digitBits) + 1);
resultUsed = _lShiftDigits(_digits, _used, nsh, resultDigits);
} else {
yDigits = other._digits;
yUsed = other._used;
// Extra digit to hold possible _mulAdd carry.
resultDigits = _cloneDigits(_digits, 0, _used, _used + 1);
resultUsed = _used;
}
Uint32List args = _newDigits(4);
args[_divisorLowTopDigit] = yDigits[yUsed - 2];
args[_divisorTopDigit] = yDigits[yUsed - 1];
// For 64-bit processing, make sure yUsed, i, and j are even.
assert(yUsed.isEven);
var i = resultUsed + (resultUsed & 1);
var j = i - yUsed;
// tmpDigits is a temporary array of i (even resultUsed) digits.
var tmpDigits = _newDigits(i);
var tmpUsed = _dlShiftDigits(yDigits, yUsed, j, tmpDigits);
// Explicit first division step in case normalized dividend is larger or
// equal to shifted normalized divisor.
if (_compareDigits(resultDigits, resultUsed, tmpDigits, tmpUsed) >= 0) {
assert(i == resultUsed);
resultDigits[resultUsed++] = 1; // Quotient = 1.
// Subtract divisor from remainder.
_absSub(resultDigits, resultUsed, tmpDigits, tmpUsed, resultDigits);
} else {
// Account for possible carry in _mulAdd step.
resultDigits[resultUsed++] = 0;
}
if (resultUsed.isOdd) {
resultDigits[resultUsed] = 0; // Leading zero for 64-bit processing.
}
// Negate y so we can later use _mulAdd instead of non-existent _mulSub.
var nyDigits = _newDigits(yUsed + 2);
nyDigits[yUsed] = 1;
_absSub(nyDigits, yUsed + 1, yDigits, yUsed, nyDigits);
// nyDigits is read-only and has yUsed digits (possibly including several
// leading zeros) plus a leading zero for 64-bit processing.
// resultDigits is modified during iteration.
// resultDigits[0..yUsed-1] is the current remainder.
// resultDigits[yUsed..resultUsed-1] is the current quotient.
--i;
while (j > 0) {
var d0 = _estimateQuotientDigit(args, resultDigits, i);
j -= d0;
var d1 =
_mulAdd(args, _quotientDigit, nyDigits, 0, resultDigits, j, yUsed);
// _estimateQuotientDigit and _mulAdd must agree on the number of digits
// to process.
assert(d0 == d1);
if (d0 == 1) {
if (resultDigits[i] < args[_quotientDigit]) {
// Reusing the already existing tmpDigits array.
var tmpUsed = _dlShiftDigits(nyDigits, yUsed, j, tmpDigits);
_absSub(resultDigits, resultUsed, tmpDigits, tmpUsed, resultDigits);
while (resultDigits[i] < --args[_quotientDigit]) {
_absSub(resultDigits, resultUsed, tmpDigits, tmpUsed, resultDigits);
}
}
} else {
assert(d0 == 2);
assert(resultDigits[i] <= args[_quotientHighDigit]);
if (resultDigits[i] < args[_quotientHighDigit] ||
resultDigits[i - 1] < args[_quotientDigit]) {
// Reusing the already existing tmpDigits array.
var tmpUsed = _dlShiftDigits(nyDigits, yUsed, j, tmpDigits);
_absSub(resultDigits, resultUsed, tmpDigits, tmpUsed, resultDigits);
if (args[_quotientDigit] == 0) {
--args[_quotientHighDigit];
}
--args[_quotientDigit];
assert(resultDigits[i] <= args[_quotientHighDigit]);
while (resultDigits[i] < args[_quotientHighDigit] ||
resultDigits[i - 1] < args[_quotientDigit]) {
_absSub(resultDigits, resultUsed, tmpDigits, tmpUsed, resultDigits);
if (args[_quotientDigit] == 0) {
--args[_quotientHighDigit];
}
--args[_quotientDigit];
assert(resultDigits[i] <= args[_quotientHighDigit]);
}
}
}
i -= d0;
}
// Cache result.
_lastDividendDigits = _digits;
_lastDividendUsed = _used;
_lastDivisorDigits = other._digits;
_lastDivisorUsed = other._used;
_lastQuoRemDigits = resultDigits;
_lastQuoRemUsed = resultUsed;
_lastRemUsed = yUsed;
_lastRem_nsh = nsh;
}
// Customized version of _rem() minimizing allocations for use in reduction.
// Input:
// xDigits[0..xUsed-1]: positive dividend.
// yDigits[0..yUsed-1]: normalized positive divisor.
// nyDigits[0..yUsed-1]: negated yDigits.
// nsh: normalization shift amount.
// args: top y digit(s) and place holder for estimated quotient digit(s).
// tmpDigits: temp array of 2*yUsed digits.
// resultDigits: result digits array large enough to temporarily hold
// concatenated quotient and normalized remainder.
// Output:
// resultDigits[0..resultUsed-1]: positive remainder.
// Returns resultUsed.
static int _remDigits(
Uint32List xDigits,
int xUsed,
Uint32List yDigits,
int yUsed,
Uint32List nyDigits,
int nsh,
Uint32List args,
Uint32List tmpDigits,
Uint32List resultDigits) {
// Initialize resultDigits to normalized positive dividend.
var resultUsed = _lShiftDigits(xDigits, xUsed, nsh, resultDigits);
// For 64-bit processing, make sure yUsed, i, and j are even.
assert(yUsed.isEven);
var i = resultUsed + (resultUsed & 1);
var j = i - yUsed;
var tmpUsed = _dlShiftDigits(yDigits, yUsed, j, tmpDigits);
// Explicit first division step in case normalized dividend is larger or
// equal to shifted normalized divisor.
if (_compareDigits(resultDigits, resultUsed, tmpDigits, tmpUsed) >= 0) {
assert(i == resultUsed);
resultDigits[resultUsed++] = 1; // Quotient = 1.
// Subtract divisor from remainder.
_absSub(resultDigits, resultUsed, tmpDigits, tmpUsed, resultDigits);
} else {
// Account for possible carry in _mulAdd step.
resultDigits[resultUsed++] = 0;
}
if (resultUsed.isOdd) {
resultDigits[resultUsed] = 0; // Leading zero for 64-bit processing.
}
// Negated yDigits passed in nyDigits allow the use of _mulAdd instead of
// unimplemented _mulSub.
// nyDigits is read-only and has yUsed digits (possibly including several
// leading zeros) plus a leading zero for 64-bit processing.
// resultDigits is modified during iteration.
// resultDigits[0..yUsed-1] is the current remainder.
// resultDigits[yUsed..resultUsed-1] is the current quotient.
--i;
while (j > 0) {
var d0 = _estimateQuotientDigit(args, resultDigits, i);
j -= d0;
var d1 =
_mulAdd(args, _quotientDigit, nyDigits, 0, resultDigits, j, yUsed);
// _estimateQuotientDigit and _mulAdd must agree on the number of digits
// to process.
assert(d0 == d1);
if (d0 == 1) {
if (resultDigits[i] < args[_quotientDigit]) {
var tmpUsed = _dlShiftDigits(nyDigits, yUsed, j, tmpDigits);
_absSub(resultDigits, resultUsed, tmpDigits, tmpUsed, resultDigits);
while (resultDigits[i] < --args[_quotientDigit]) {
_absSub(resultDigits, resultUsed, tmpDigits, tmpUsed, resultDigits);
}
}
} else {
assert(d0 == 2);
assert(resultDigits[i] <= args[_quotientHighDigit]);
if ((resultDigits[i] < args[_quotientHighDigit]) ||
(resultDigits[i - 1] < args[_quotientDigit])) {
var tmpUsed = _dlShiftDigits(nyDigits, yUsed, j, tmpDigits);
_absSub(resultDigits, resultUsed, tmpDigits, tmpUsed, resultDigits);
if (args[_quotientDigit] == 0) {
--args[_quotientHighDigit];
}
--args[_quotientDigit];
assert(resultDigits[i] <= args[_quotientHighDigit]);
while ((resultDigits[i] < args[_quotientHighDigit]) ||
(resultDigits[i - 1] < args[_quotientDigit])) {
_absSub(resultDigits, resultUsed, tmpDigits, tmpUsed, resultDigits);
if (args[_quotientDigit] == 0) {
--args[_quotientHighDigit];
}
--args[_quotientDigit];
assert(resultDigits[i] <= args[_quotientHighDigit]);
}
}
}
i -= d0;
}
// Return remainder, i.e. denormalized resultDigits[0..yUsed-1].
resultUsed = yUsed;
if (nsh > 0) {
// Denormalize remainder.
resultUsed = _rShiftDigits(resultDigits, resultUsed, nsh, resultDigits);
}
return resultUsed;
}
int get hashCode {
// This is the [Jenkins hash function][1] but using masking to keep
// values in SMI range.
//
// [1]: http://en.wikipedia.org/wiki/Jenkins_hash_function
int combine(int hash, int value) {
hash = 0x1fffffff & (hash + value);
hash = 0x1fffffff & (hash + ((0x0007ffff & hash) << 10));
return hash ^ (hash >> 6);
}
int finish(int hash) {
hash = 0x1fffffff & (hash + ((0x03ffffff & hash) << 3));
hash = hash ^ (hash >> 11);
return 0x1fffffff & (hash + ((0x00003fff & hash) << 15));
}
if (_isZero) return 6707; // Just a random number.
var hash = _isNegative ? 83585 : 429689; // Also random.
for (int i = 0; i < _used; i++) {
hash = combine(hash, _digits[i]);
}
return finish(hash);
}
/**
* Test whether this value is numerically equal to `other`.
*
* If [other] is a [_BigIntImpl] returns whether the two operands have the
* same value.
*
* Returns false if `other` is not a [_BigIntImpl].
*/
bool operator ==(Object other) =>
other is _BigIntImpl && compareTo(other) == 0;
/**
* Returns the minimum number of bits required to store this big integer.
*
* The number of bits excludes the sign bit, which gives the natural length
* for non-negative (unsigned) values. Negative values are complemented to
* return the bit position of the first bit that differs from the sign bit.
*
* To find the number of bits needed to store the value as a signed value,
* add one, i.e. use `x.bitLength + 1`.
*
* ```
* x.bitLength == (-x-1).bitLength
*
* new BigInt.from(3).bitLength == 2; // 00000011
* new BigInt.from(2).bitLength == 2; // 00000010
* new BigInt.from(1).bitLength == 1; // 00000001
* new BigInt.from(0).bitLength == 0; // 00000000
* new BigInt.from(-1).bitLength == 0; // 11111111
* new BigInt.from(-2).bitLength == 1; // 11111110
* new BigInt.from(-3).bitLength == 2; // 11111101
* new BigInt.from(-4).bitLength == 2; // 11111100
* ```
*/
int get bitLength {
if (_used == 0) return 0;
if (_isNegative) return (~this).bitLength;
return _digitBits * (_used - 1) + _digits[_used - 1].bitLength;
}
/**
* Truncating division operator.
*
* Performs a truncating integer division, where the remainder is discarded.
*
* The remainder can be computed using the [remainder] method.
*
* Examples:
* ```
* var seven = new BigInt.from(7);
* var three = new BigInt.from(3);
* seven ~/ three; // => 2
* (-seven) ~/ three; // => -2
* seven ~/ -three; // => -2
* seven.remainder(three); // => 1
* (-seven).remainder(three); // => -1
* seven.remainder(-three); // => 1
* ```
*/
_BigIntImpl operator ~/(BigInt bigInt) {
_BigIntImpl other = bigInt;
if (other._used == 0) {
throw const IntegerDivisionByZeroException();
}
return _div(other);
}
/**
* Returns the remainder of the truncating division of `this` by [other].
*
* The result `r` of this operation satisfies:
* `this == (this ~/ other) * other + r`.
* As a consequence the remainder `r` has the same sign as the divider `this`.
*/
_BigIntImpl remainder(BigInt bigInt) {
_BigIntImpl other = bigInt;
if (other._used == 0) {
throw const IntegerDivisionByZeroException();
}
return _rem(other);
}
/// Division operator.
double operator /(BigInt other) => this.toDouble() / other.toDouble();
/** Relational less than operator. */
bool operator <(BigInt other) => compareTo(other) < 0;
/** Relational less than or equal operator. */
bool operator <=(BigInt other) => compareTo(other) <= 0;
/** Relational greater than operator. */
bool operator >(BigInt other) => compareTo(other) > 0;
/** Relational greater than or equal operator. */
bool operator >=(BigInt other) => compareTo(other) >= 0;
/**
* Euclidean modulo operator.
*
* Returns the remainder of the Euclidean division. The Euclidean division of
* two integers `a` and `b` yields two integers `q` and `r` such that
* `a == b * q + r` and `0 <= r < b.abs()`.
*
* The sign of the returned value `r` is always positive.
*
* See [remainder] for the remainder of the truncating division.
*/
_BigIntImpl operator %(BigInt bigInt) {
_BigIntImpl other = bigInt;
if (other._used == 0) {
throw const IntegerDivisionByZeroException();
}
var result = _rem(other);
if (result._isNegative) {
if (other._isNegative) {
result = result - other;
} else {
result = result + other;
}
}
return result;
}
/**
* Returns the sign of this big integer.
*
* Returns 0 for zero, -1 for values less than zero and
* +1 for values greater than zero.
*/
int get sign {
if (_used == 0) return 0;
return _isNegative ? -1 : 1;
}
/// Whether this big integer is even.
bool get isEven => _used == 0 || (_digits[0] & 1) == 0;
/// Whether this big integer is odd.
bool get isOdd => !isEven;
/// Whether this number is negative.
bool get isNegative => _isNegative;
_BigIntImpl pow(int exponent) {
if (exponent < 0) {
throw new ArgumentError("Exponent must not be negative: $exponent");
}
if (exponent == 0) return one;
// Exponentiation by squaring.
var result = one;
var base = this;
while (exponent != 0) {
if ((exponent & 1) == 1) {
result *= base;
}
exponent >>= 1;
// Skip unnecessary operation.
if (exponent != 0) {
base *= base;
}
}
return result;
}
/**
* Returns this integer to the power of [exponent] modulo [modulus].
*
* The [exponent] must be non-negative and [modulus] must be
* positive.
*/
_BigIntImpl modPow(BigInt bigExponent, BigInt bigModulus) {
_BigIntImpl exponent = bigExponent;
_BigIntImpl modulus = bigModulus;
if (exponent._isNegative) {
throw new ArgumentError("exponent must be positive: $exponent");
}
if (modulus <= zero) {
throw new ArgumentError("modulus must be strictly positive: $modulus");
}
if (exponent._isZero) return one;
final exponentBitlen = exponent.bitLength;
if (exponentBitlen <= 0) return one;
final bool cannotUseMontgomery = modulus.isEven || abs() >= modulus;
if (cannotUseMontgomery || exponentBitlen < 64) {
_BigIntReduction z = (cannotUseMontgomery || exponentBitlen < 8)
? new _BigIntClassicReduction(modulus)
: new _BigIntMontgomeryReduction(modulus);
var resultDigits = _newDigits(2 * z._normModulusUsed + 2);
var result2Digits = _newDigits(2 * z._normModulusUsed + 2);
var gDigits = _newDigits(z._normModulusUsed);
var gUsed = z._convert(this, gDigits);
// Initialize result with g.
// Copy leading zero if any.
for (int j = gUsed + (gUsed & 1) - 1; j >= 0; j--) {
resultDigits[j] = gDigits[j];
}
var resultUsed = gUsed;
var result2Used;
for (int i = exponentBitlen - 2; i >= 0; i--) {
result2Used = z._sqr(resultDigits, resultUsed, result2Digits);
if (exponent._digits[i ~/ _digitBits] & (1 << (i % _digitBits)) != 0) {
resultUsed =
z._mul(result2Digits, result2Used, gDigits, gUsed, resultDigits);
} else {
// Swap result and result2.
var tmpDigits = resultDigits;
var tmpUsed = resultUsed;
resultDigits = result2Digits;
resultUsed = result2Used;
result2Digits = tmpDigits;
result2Used = tmpUsed;
}
}
return z._revert(resultDigits, resultUsed);
}
var k;
if (exponentBitlen < 18)
k = 1;
else if (exponentBitlen < 48)
k = 3;
else if (exponentBitlen < 144)
k = 4;
else if (exponentBitlen < 768)
k = 5;
else
k = 6;
_BigIntReduction z = new _BigIntMontgomeryReduction(modulus);
var n = 3;
final k1 = k - 1;
final km = (1 << k) - 1;
List gDigits = new List(km + 1);
List gUsed = new List(km + 1);
gDigits[1] = _newDigits(z._normModulusUsed);
gUsed[1] = z._convert(this, gDigits[1]);
if (k > 1) {
var g2Digits = _newDigits(2 * z._normModulusUsed + 2);
var g2Used = z._sqr(gDigits[1], gUsed[1], g2Digits);
while (n <= km) {
gDigits[n] = _newDigits(2 * z._normModulusUsed + 2);
gUsed[n] =
z._mul(g2Digits, g2Used, gDigits[n - 2], gUsed[n - 2], gDigits[n]);
n += 2;
}
}
var w;
var isOne = true;
var resultDigits = one._digits;
var resultUsed = one._used;
var result2Digits = _newDigits(2 * z._normModulusUsed + 2);
var result2Used;
var exponentDigits = exponent._digits;
var j = exponent._used - 1;
var i = exponentDigits[j].bitLength - 1;
while (j >= 0) {
if (i >= k1) {
w = (exponentDigits[j] >> (i - k1)) & km;
} else {
w = (exponentDigits[j] & ((1 << (i + 1)) - 1)) << (k1 - i);
if (j > 0) {
w |= exponentDigits[j - 1] >> (_digitBits + i - k1);
}
}
n = k;
while ((w & 1) == 0) {
w >>= 1;
--n;
}
if ((i -= n) < 0) {
i += _digitBits;
--j;
}
if (isOne) {
// r == 1, don't bother squaring or multiplying it.
resultDigits = _newDigits(2 * z._normModulusUsed + 2);
resultUsed = gUsed[w];
var gwDigits = gDigits[w];
var ri = resultUsed + (resultUsed & 1); // Copy leading zero if any.
while (--ri >= 0) {
resultDigits[ri] = gwDigits[ri];
}
isOne = false;
} else {
while (n > 1) {
result2Used = z._sqr(resultDigits, resultUsed, result2Digits);
resultUsed = z._sqr(result2Digits, result2Used, resultDigits);
n -= 2;
}
if (n > 0) {
result2Used = z._sqr(resultDigits, resultUsed, result2Digits);
} else {
var swapDigits = resultDigits;
var swapUsed = resultUsed;
resultDigits = result2Digits;
resultUsed = result2Used;
result2Digits = swapDigits;
result2Used = swapUsed;
}
resultUsed = z._mul(
result2Digits, result2Used, gDigits[w], gUsed[w], resultDigits);
}
while (j >= 0 && (exponentDigits[j] & (1 << i)) == 0) {
result2Used = z._sqr(resultDigits, resultUsed, result2Digits);
var swapDigits = resultDigits;
var swapUsed = resultUsed;
resultDigits = result2Digits;
resultUsed = result2Used;
result2Digits = swapDigits;
result2Used = swapUsed;
if (--i < 0) {
i = _digitBits - 1;
--j;
}
}
}
assert(!isOne);
return z._revert(resultDigits, resultUsed);
}
// If inv is false, returns gcd(x, y).
// If inv is true and gcd(x, y) = 1, returns d, so that c*x + d*y = 1.
// If inv is true and gcd(x, y) != 1, throws Exception("Not coprime").
static _BigIntImpl _binaryGcd(_BigIntImpl x, _BigIntImpl y, bool inv) {
var xDigits = x._digits;
var yDigits = y._digits;
var xUsed = x._used;
var yUsed = y._used;
var maxUsed = _max(xUsed, yUsed);
final maxLen = maxUsed + (maxUsed & 1);
xDigits = _cloneDigits(xDigits, 0, xUsed, maxLen);
yDigits = _cloneDigits(yDigits, 0, yUsed, maxLen);
int shiftAmount = 0;
if (inv) {
if ((yUsed == 1) && (yDigits[0] == 1)) return one;
if ((yUsed == 0) || (yDigits[0].isEven && xDigits[0].isEven)) {
throw new Exception("Not coprime");
}
} else {
if (x._isZero) {
throw new ArgumentError.value(0, "this", "must not be zero");
}
if (y._isZero) {
throw new ArgumentError.value(0, "other", "must not be zero");
}
if (((xUsed == 1) && (xDigits[0] == 1)) ||
((yUsed == 1) && (yDigits[0] == 1))) return one;
while (((xDigits[0] & 1) == 0) && ((yDigits[0] & 1) == 0)) {
_rsh(xDigits, xUsed, 1, xDigits);
_rsh(yDigits, yUsed, 1, yDigits);
shiftAmount++;
}
if (shiftAmount >= _digitBits) {
var digitShiftAmount = shiftAmount ~/ _digitBits;
xUsed -= digitShiftAmount;
yUsed -= digitShiftAmount;
maxUsed -= digitShiftAmount;
}
if ((yDigits[0] & 1) == 1) {
// Swap x and y.
var tmpDigits = xDigits;
var tmpUsed = xUsed;
xDigits = yDigits;
xUsed = yUsed;
yDigits = tmpDigits;
yUsed = tmpUsed;
}
}
var uDigits = _cloneDigits(xDigits, 0, xUsed, maxLen);
var vDigits = _cloneDigits(yDigits, 0, yUsed, maxLen + 2); // +2 for lsh.
final bool ac = (xDigits[0] & 1) == 0;
// Variables a, b, c, and d require one more digit.
final abcdUsed = maxUsed + 1;
final abcdLen = abcdUsed + (abcdUsed & 1) + 2; // +2 to satisfy _absAdd.
var aDigits, bDigits, cDigits, dDigits;
bool aIsNegative, bIsNegative, cIsNegative, dIsNegative;
if (ac) {
aDigits = _newDigits(abcdLen);
aIsNegative = false;
aDigits[0] = 1;
cDigits = _newDigits(abcdLen);
cIsNegative = false;
}
bDigits = _newDigits(abcdLen);
bIsNegative = false;
dDigits = _newDigits(abcdLen);
dIsNegative = false;
dDigits[0] = 1;
while (true) {
while ((uDigits[0] & 1) == 0) {
_rsh(uDigits, maxUsed, 1, uDigits);
if (ac) {
if (((aDigits[0] & 1) == 1) || ((bDigits[0] & 1) == 1)) {
if (aIsNegative) {
if ((aDigits[maxUsed] != 0) ||
(_compareDigits(aDigits, maxUsed, yDigits, maxUsed)) > 0) {
_absSub(aDigits, abcdUsed, yDigits, maxUsed, aDigits);
} else {
_absSub(yDigits, maxUsed, aDigits, maxUsed, aDigits);
aIsNegative = false;
}
} else {
_absAdd(aDigits, abcdUsed, yDigits, maxUsed, aDigits);
}
if (bIsNegative) {
_absAdd(bDigits, abcdUsed, xDigits, maxUsed, bDigits);
} else if ((bDigits[maxUsed] != 0) ||
(_compareDigits(bDigits, maxUsed, xDigits, maxUsed) > 0)) {
_absSub(bDigits, abcdUsed, xDigits, maxUsed, bDigits);
} else {
_absSub(xDigits, maxUsed, bDigits, maxUsed, bDigits);
bIsNegative = true;
}
}
_rsh(aDigits, abcdUsed, 1, aDigits);
} else if ((bDigits[0] & 1) == 1) {
if (bIsNegative) {
_absAdd(bDigits, abcdUsed, xDigits, maxUsed, bDigits);
} else if ((bDigits[maxUsed] != 0) ||
(_compareDigits(bDigits, maxUsed, xDigits, maxUsed) > 0)) {
_absSub(bDigits, abcdUsed, xDigits, maxUsed, bDigits);
} else {
_absSub(xDigits, maxUsed, bDigits, maxUsed, bDigits);
bIsNegative = true;
}
}
_rsh(bDigits, abcdUsed, 1, bDigits);
}
while ((vDigits[0] & 1) == 0) {
_rsh(vDigits, maxUsed, 1, vDigits);
if (ac) {
if (((cDigits[0] & 1) == 1) || ((dDigits[0] & 1) == 1)) {
if (cIsNegative) {
if ((cDigits[maxUsed] != 0) ||
(_compareDigits(cDigits, maxUsed, yDigits, maxUsed) > 0)) {
_absSub(cDigits, abcdUsed, yDigits, maxUsed, cDigits);
} else {
_absSub(yDigits, maxUsed, cDigits, maxUsed, cDigits);
cIsNegative = false;
}
} else {
_absAdd(cDigits, abcdUsed, yDigits, maxUsed, cDigits);
}
if (dIsNegative) {
_absAdd(dDigits, abcdUsed, xDigits, maxUsed, dDigits);
} else if ((dDigits[maxUsed] != 0) ||
(_compareDigits(dDigits, maxUsed, xDigits, maxUsed) > 0)) {
_absSub(dDigits, abcdUsed, xDigits, maxUsed, dDigits);
} else {
_absSub(xDigits, maxUsed, dDigits, maxUsed, dDigits);
dIsNegative = true;
}
}
_rsh(cDigits, abcdUsed, 1, cDigits);
} else if ((dDigits[0] & 1) == 1) {
if (dIsNegative) {
_absAdd(dDigits, abcdUsed, xDigits, maxUsed, dDigits);
} else if ((dDigits[maxUsed] != 0) ||
(_compareDigits(dDigits, maxUsed, xDigits, maxUsed) > 0)) {
_absSub(dDigits, abcdUsed, xDigits, maxUsed, dDigits);
} else {
_absSub(xDigits, maxUsed, dDigits, maxUsed, dDigits);
dIsNegative = true;
}
}
_rsh(dDigits, abcdUsed, 1, dDigits);
}
if (_compareDigits(uDigits, maxUsed, vDigits, maxUsed) >= 0) {
_absSub(uDigits, maxUsed, vDigits, maxUsed, uDigits);
if (ac) {
if (aIsNegative == cIsNegative) {
var a_cmp_c = _compareDigits(aDigits, abcdUsed, cDigits, abcdUsed);
if (a_cmp_c > 0) {
_absSub(aDigits, abcdUsed, cDigits, abcdUsed, aDigits);
} else {
_absSub(cDigits, abcdUsed, aDigits, abcdUsed, aDigits);
aIsNegative = !aIsNegative && (a_cmp_c != 0);
}
} else {
_absAdd(aDigits, abcdUsed, cDigits, abcdUsed, aDigits);
}
}
if (bIsNegative == dIsNegative) {
var b_cmp_d = _compareDigits(bDigits, abcdUsed, dDigits, abcdUsed);
if (b_cmp_d > 0) {
_absSub(bDigits, abcdUsed, dDigits, abcdUsed, bDigits);
} else {
_absSub(dDigits, abcdUsed, bDigits, abcdUsed, bDigits);
bIsNegative = !bIsNegative && (b_cmp_d != 0);
}
} else {
_absAdd(bDigits, abcdUsed, dDigits, abcdUsed, bDigits);
}
} else {
_absSub(vDigits, maxUsed, uDigits, maxUsed, vDigits);
if (ac) {
if (cIsNegative == aIsNegative) {
var c_cmp_a = _compareDigits(cDigits, abcdUsed, aDigits, abcdUsed);
if (c_cmp_a > 0) {
_absSub(cDigits, abcdUsed, aDigits, abcdUsed, cDigits);
} else {
_absSub(aDigits, abcdUsed, cDigits, abcdUsed, cDigits);
cIsNegative = !cIsNegative && (c_cmp_a != 0);
}
} else {
_absAdd(cDigits, abcdUsed, aDigits, abcdUsed, cDigits);
}
}
if (dIsNegative == bIsNegative) {
var d_cmp_b = _compareDigits(dDigits, abcdUsed, bDigits, abcdUsed);
if (d_cmp_b > 0) {
_absSub(dDigits, abcdUsed, bDigits, abcdUsed, dDigits);
} else {
_absSub(bDigits, abcdUsed, dDigits, abcdUsed, dDigits);
dIsNegative = !dIsNegative && (d_cmp_b != 0);
}
} else {
_absAdd(dDigits, abcdUsed, bDigits, abcdUsed, dDigits);
}
}
// Exit loop if u == 0.
var i = maxUsed;
while ((i > 0) && (uDigits[i - 1] == 0)) --i;
if (i == 0) break;
}
if (!inv) {
if (shiftAmount > 0) {
maxUsed = _lShiftDigits(vDigits, maxUsed, shiftAmount, vDigits);
}
return new _BigIntImpl._(false, maxUsed, vDigits);
}
// No inverse if v != 1.
var i = maxUsed - 1;
while ((i > 0) && (vDigits[i] == 0)) --i;
if ((i != 0) || (vDigits[0] != 1)) {
throw new Exception("Not coprime");
}
if (dIsNegative) {
if ((dDigits[maxUsed] != 0) ||
(_compareDigits(dDigits, maxUsed, xDigits, maxUsed) > 0)) {
_absSub(dDigits, abcdUsed, xDigits, maxUsed, dDigits);
if ((dDigits[maxUsed] != 0) ||
(_compareDigits(dDigits, maxUsed, xDigits, maxUsed) > 0)) {
_absSub(dDigits, abcdUsed, xDigits, maxUsed, dDigits);
} else {
_absSub(xDigits, maxUsed, dDigits, maxUsed, dDigits);
dIsNegative = false;
}
} else {
_absSub(xDigits, maxUsed, dDigits, maxUsed, dDigits);
dIsNegative = false;
}
} else if ((dDigits[maxUsed] != 0) ||
(_compareDigits(dDigits, maxUsed, xDigits, maxUsed) > 0)) {
_absSub(dDigits, abcdUsed, xDigits, maxUsed, dDigits);
if ((dDigits[maxUsed] != 0) ||
(_compareDigits(dDigits, maxUsed, xDigits, maxUsed) > 0)) {
_absSub(dDigits, abcdUsed, xDigits, maxUsed, dDigits);
}
}
return new _BigIntImpl._(false, maxUsed, dDigits);
}
/**
* Returns the modular multiplicative inverse of this big integer
* modulo [modulus].
*
* The [modulus] must be positive.
*
* It is an error if no modular inverse exists.
*/
// Returns 1/this % modulus, with modulus > 0.
_BigIntImpl modInverse(BigInt bigInt) {
_BigIntImpl modulus = bigInt;
if (modulus <= zero) {
throw new ArgumentError("Modulus must be strictly positive: $modulus");
}
if (modulus == one) return zero;
var tmp = this;
if (tmp._isNegative || (tmp._absCompare(modulus) >= 0)) {
tmp %= modulus;
}
return _binaryGcd(modulus, tmp, true);
}
/**
* Returns the greatest common divisor of this big integer and [other].
*
* If either number is non-zero, the result is the numerically greatest
* integer dividing both `this` and `other`.
*
* The greatest common divisor is independent of the order,
* so `x.gcd(y)` is always the same as `y.gcd(x)`.
*
* For any integer `x`, `x.gcd(x)` is `x.abs()`.
*
* If both `this` and `other` is zero, the result is also zero.
*/
_BigIntImpl gcd(BigInt bigInt) {
_BigIntImpl other = bigInt;
if (_isZero) return other.abs();
if (other._isZero) return this.abs();
return _binaryGcd(this, other, false);
}
/**
* Returns the least significant [width] bits of this big integer as a
* non-negative number (i.e. unsigned representation). The returned value has
* zeros in all bit positions higher than [width].
*
* ```
* new BigInt.from(-1).toUnsigned(5) == 31 // 11111111 -> 00011111
* ```
*
* This operation can be used to simulate arithmetic from low level languages.
* For example, to increment an 8 bit quantity:
*
* ```
* q = (q + 1).toUnsigned(8);
* ```
*
* `q` will count from `0` up to `255` and then wrap around to `0`.
*
* If the input fits in [width] bits without truncation, the result is the
* same as the input. The minimum width needed to avoid truncation of `x` is
* given by `x.bitLength`, i.e.
*
* ```
* x == x.toUnsigned(x.bitLength);
* ```
*/
_BigIntImpl toUnsigned(int width) {
return this & ((one << width) - one);
}
/**
* Returns the least significant [width] bits of this integer, extending the
* highest retained bit to the sign. This is the same as truncating the value
* to fit in [width] bits using an signed 2-s complement representation. The
* returned value has the same bit value in all positions higher than [width].
*
* ```
* var big15 = new BigInt.from(15);
* var big16 = new BigInt.from(16);
* var big239 = new BigInt.from(239);
* V--sign bit-V
* big16.toSigned(5) == -big16 // 00010000 -> 11110000
* big239.toSigned(5) == big15 // 11101111 -> 00001111
* ^ ^
* ```
*
* This operation can be used to simulate arithmetic from low level languages.
* For example, to increment an 8 bit signed quantity:
*
* ```
* q = (q + 1).toSigned(8);
* ```
*
* `q` will count from `0` up to `127`, wrap to `-128` and count back up to
* `127`.
*
* If the input value fits in [width] bits without truncation, the result is
* the same as the input. The minimum width needed to avoid truncation of `x`
* is `x.bitLength + 1`, i.e.
*
* ```
* x == x.toSigned(x.bitLength + 1);
* ```
*/
_BigIntImpl toSigned(int width) {
// The value of binary number weights each bit by a power of two. The
// twos-complement value weights the sign bit negatively. We compute the
// value of the negative weighting by isolating the sign bit with the
// correct power of two weighting and subtracting it from the value of the
// lower bits.
var signMask = one << (width - 1);
return (this & (signMask - one)) - (this & signMask);
}
bool get isValidInt {
assert(_digitBits == 32);
return _used < 2 ||
(_used == 2 &&
(_digits[1] < 0x80000000 ||
(_isNegative && _digits[1] == 0x80000000 && _digits[0] == 0)));
}
int toInt() {
assert(_digitBits == 32);
if (_used == 0) return 0;
if (_used == 1) return _isNegative ? -_digits[0] : _digits[0];
if (_used == 2 && _digits[1] < 0x80000000) {
var result = (_digits[1] << _digitBits) | _digits[0];
return _isNegative ? -result : result;
}
return _isNegative ? _minInt : _maxInt;
}
/**
* Returns this [_BigIntImpl] as a [double].
*
* If the number is not representable as a [double], an
* approximation is returned. For numerically large integers, the
* approximation may be infinite.
*/
double toDouble() {
const int exponentBias = 1075;
// There are 11 bits for the exponent.
// 2047 (all bits set to 1) is reserved for infinity and NaN.
// When storing the exponent in the 11 bits, it is biased by exponentBias
// to support negative exponents.
const int maxDoubleExponent = 2046 - exponentBias;
if (_isZero) return 0.0;
// We fill the 53 bits little-endian.
var resultBits = new Uint8List(8);
var length = _digitBits * (_used - 1) + _digits[_used - 1].bitLength;
if (length - 53 > maxDoubleExponent) return double.INFINITY;
// The most significant bit is for the sign.
if (_isNegative) resultBits[7] = 0x80;
// Write the exponent into bits 1..12:
var biasedExponent = length - 53 + exponentBias;
resultBits[6] = (biasedExponent & 0xF) << 4;
resultBits[7] |= biasedExponent >> 4;
int cachedBits = 0;
int cachedBitsLength = 0;
int digitIndex = _used - 1;
int readBits(int n) {
// Ensure that we have enough bits in [cachedBits].
while (cachedBitsLength < n) {
int nextDigit;
int nextDigitLength = _digitBits; // May get updated.
if (digitIndex < 0) {
nextDigit = 0;
digitIndex--;
} else {
nextDigit = _digits[digitIndex];
if (digitIndex == _used - 1) nextDigitLength = nextDigit.bitLength;
digitIndex--;
}
cachedBits = (cachedBits << nextDigitLength) + nextDigit;
cachedBitsLength += nextDigitLength;
}
// Read the top [n] bits.
var result = cachedBits >> (cachedBitsLength - n);
// Remove the bits from the cache.
cachedBits -= result << (cachedBitsLength - n);
cachedBitsLength -= n;
return result;
}
// The first leading 1 bit is implicit in the double-representation and can
// be discarded.
var leadingBits = readBits(5) & 0xF;
resultBits[6] |= leadingBits;
for (int i = 5; i >= 0; i--) {
// Get the remaining 48 bits.
resultBits[i] = readBits(8);
}
void roundUp() {
// Simply consists of adding 1 to the whole 64 bit "number".
// It will update the exponent, if necessary.
// It might even round up to infinity (which is what we want).
var carry = 1;
for (int i = 0; i < 8; i++) {
if (carry == 0) break;
var sum = resultBits[i] + carry;
resultBits[i] = sum & 0xFF;
carry = sum >> 8;
}
}
if (readBits(1) == 1) {
if (resultBits[0].isOdd) {
// Rounds to even all the time.
roundUp();
} else {
// Round up, if there is at least one other digit that is not 0.
if (cachedBits != 0) {
// There is already one in the cachedBits.
roundUp();
} else {
for (int i = digitIndex; digitIndex >= 0; i--) {
if (_digits[i] != 0) {
roundUp();
break;
}
}
}
}
}
return resultBits.buffer.asByteData().getFloat64(0, Endian.little);
}
/**
* Returns a String-representation of this integer.
*
* The returned string is parsable by [parse].
* For any `_BigIntImpl` `i`, it is guaranteed that
* `i == _BigIntImpl.parse(i.toString())`.
*/
String toString() {
if (_used == 0) return "0";
if (_used == 1) {
if (_isNegative) return (-_digits[0]).toString();
return _digits[0].toString();
}
// Generate in chunks of 9 digits.
// The chunks are in reversed order.
var decimalDigitChunks = <String>[];
var rest = isNegative ? -this : this;
while (rest._used > 1) {
var digits9 = rest.remainder(_oneBillion).toString();
decimalDigitChunks.add(digits9);
var zeros = 9 - digits9.length;
if (zeros == 8) {
decimalDigitChunks.add("00000000");
} else {
if (zeros >= 4) {
zeros -= 4;
decimalDigitChunks.add("0000");
}
if (zeros >= 2) {
zeros -= 2;
decimalDigitChunks.add("00");
}
if (zeros >= 1) {
decimalDigitChunks.add("0");
}
}
rest = rest ~/ _oneBillion;
}
decimalDigitChunks.add(rest._digits[0].toString());
if (_isNegative) decimalDigitChunks.add("-");
return decimalDigitChunks.reversed.join();
}
int _toRadixCodeUnit(int digit) {
const int _0 = 48;
const int _a = 97;
if (digit < 10) return _0 + digit;
return _a + digit - 10;
}
/**
* Converts [this] to a string representation in the given [radix].
*
* In the string representation, lower-case letters are used for digits above
* '9', with 'a' being 10 an 'z' being 35.
*
* The [radix] argument must be an integer in the range 2 to 36.
*/
String toRadixString(int radix) {
if (radix > 36) throw new RangeError.range(radix, 2, 36);
if (_used == 0) return "0";
if (_used == 1) {
var digitString = _digits[0].toRadixString(radix);
if (_isNegative) return "-" + digitString;
return digitString;
}
if (radix == 16) return _toHexString();
var base = new _BigIntImpl._fromInt(radix);
var reversedDigitCodeUnits = <int>[];
var rest = this.abs();
while (!rest._isZero) {
var digit = rest.remainder(base).toInt();
rest = rest ~/ base;
reversedDigitCodeUnits.add(_toRadixCodeUnit(digit));
}
var digitString = new String.fromCharCodes(reversedDigitCodeUnits.reversed);
if (_isNegative) return "-" + digitString;
return digitString;
}
String _toHexString() {
var chars = <int>[];
for (int i = 0; i < _used - 1; i++) {
int chunk = _digits[i];
for (int j = 0; j < (_digitBits ~/ 4); j++) {
chars.add(_toRadixCodeUnit(chunk & 0xF));
chunk >>= 4;
}
}
var msbChunk = _digits[_used - 1];
while (msbChunk != 0) {
chars.add(_toRadixCodeUnit(msbChunk & 0xF));
msbChunk >>= 4;
}
if (_isNegative) {
const _dash = 45;
chars.add(_dash);
}
return new String.fromCharCodes(chars.reversed);
}
}
// Interface for modular reduction.
abstract class _BigIntReduction {
int get _normModulusUsed;
// Return the number of digits used by resultDigits.
int _convert(_BigIntImpl x, Uint32List resultDigits);
int _mul(Uint32List xDigits, int xUsed, Uint32List yDigits, int yUsed,
Uint32List resultDigits);
int _sqr(Uint32List xDigits, int xUsed, Uint32List resultDigits);
// Return x reverted to _BigIntImpl.
_BigIntImpl _revert(Uint32List xDigits, int xUsed);
}
// Montgomery reduction on _BigIntImpl.
class _BigIntMontgomeryReduction implements _BigIntReduction {
final _BigIntImpl _modulus;
int _normModulusUsed; // Even if processing 64-bit (digit pairs).
Uint32List _modulusDigits;
Uint32List _args;
int _digitsPerStep; // Number of digits processed in one step. 1 or 2.
static const int _xDigit = 0; // Index of digit of x.
static const int _xHighDigit = 1; // Index of high digit of x (64-bit only).
static const int _rhoDigit = 2; // Index of digit of rho.
static const int _rhoHighDigit = 3; // Index of high digit of rho (64-bit).
static const int _muDigit = 4; // Index of mu.
static const int _muHighDigit = 5; // Index of high 32-bits of mu (64-bit).
_BigIntMontgomeryReduction(this._modulus) {
_modulusDigits = _modulus._digits;
_args = _newDigits(6);
// Determine if we can process digit pairs by calling an intrinsic.
_digitsPerStep = _mulMod(_args, _args, 0);
_args[_xDigit] = _modulusDigits[0];
if (_digitsPerStep == 1) {
_normModulusUsed = _modulus._used;
_invDigit(_args);
} else {
assert(_digitsPerStep == 2);
_normModulusUsed = _modulus._used + (_modulus._used & 1);
_args[_xHighDigit] = _modulusDigits[1];
_invDigitPair(_args);
}
}
// Calculates -1/x % _digitBase, x is 32-bit digit.
// xy == 1 (mod m)
// xy = 1+km
// xy(2-xy) = (1+km)(1-km)
// x(y(2-xy)) = 1-k^2 m^2
// x(y(2-xy)) == 1 (mod m^2)
// if y is 1/x mod m, then y(2-xy) is 1/x mod m^2
// Should reduce x and y(2-xy) by m^2 at each step to keep size bounded.
//
// Operation:
// args[_rhoDigit] = 1/args[_xDigit] mod _digitBase.
static void _invDigit(Uint32List args) {
var x = args[_xDigit];
var y = x & 3; // y == 1/x mod 2^2
y = (y * (2 - (x & 0xf) * y)) & 0xf; // y == 1/x mod 2^4
y = (y * (2 - (x & 0xff) * y)) & 0xff; // y == 1/x mod 2^8
y = (y * (2 - (((x & 0xffff) * y) & 0xffff))) & 0xffff; // y == 1/x mod 2^16
y = (y * (2 - x * y % _BigIntImpl._digitBase)) % _BigIntImpl._digitBase;
// y == 1/x mod _digitBase
y = -y; // We really want the negative inverse.
args[_rhoDigit] = y & _BigIntImpl._digitMask;
assert(((x * y) & _BigIntImpl._digitMask) == _BigIntImpl._digitMask);
}
// Calculates -1/x % _digitBase^2, x is a pair of 32-bit digits.
// Operation:
// args[_rhoDigit.._rhoHighDigit] =
// 1/args[_xDigit.._xHighDigit] mod _digitBase^2.
static void _invDigitPair(Uint32List args) {
var two = _BigIntImpl.two;
var mask32 = _BigIntImpl._oneDigitMask;
var mask64 = _BigIntImpl._twoDigitMask;
var xl = args[_xDigit]; // Lower 32-bit digit of x.
var y = xl & 3; // y == 1/x mod 2^2
y = (y * (2 - (xl & 0xf) * y)) & 0xf; // y == 1/x mod 2^4
y = (y * (2 - (xl & 0xff) * y)) & 0xff; // y == 1/x mod 2^8
y = (y * (2 - (((xl & 0xffff) * y) & 0xffff))) & 0xffff;
// y == 1/x mod 2^16
y = (y * (2 - ((xl * y) & 0xffffffff))) & 0xffffffff; // y == 1/x mod 2^32
var x = (args[_xHighDigit] << _BigIntImpl._digitBits) | xl;
y *= 2 - x * y; // Masking with 2^64-1 is implied by 64-bit arithmetic.
// y == 1/x mod _digitBase^2
y = -y; // We really want the negative inverse.
args[_rhoDigit] = y & _BigIntImpl._digitMask;
args[_rhoHighDigit] =
(y >> _BigIntImpl._digitBits) & _BigIntImpl._digitMask;
assert(x * y == -1);
}
// Operation:
// args[_muDigit] = args[_rhoDigit]*digits[i] mod _digitBase.
// Returns 1.
// Note: Intrinsics on 64-bit platforms process digit pairs at even indices:
// args[_muDigit.._muHighDigit] =
// args[_rhoDigit.._rhoHighDigit] * digits[i..i+1] mod _digitBase^2.
// Returns 2.
static int _mulMod(Uint32List args, Uint32List digits, int i) {
var rhol = args[_rhoDigit] & _BigIntImpl._halfDigitMask;
var rhoh = args[_rhoDigit] >> _BigIntImpl._halfDigitBits;
var dh = digits[i] >> _BigIntImpl._halfDigitBits;
var dl = digits[i] & _BigIntImpl._halfDigitMask;
args[_muDigit] = (dl * rhol +
(((dl * rhoh + dh * rhol) & _BigIntImpl._halfDigitMask) <<
_BigIntImpl._halfDigitBits)) &
_BigIntImpl._digitMask;
return 1;
}
// result = x*R mod _modulus.
// Returns resultUsed.
int _convert(_BigIntImpl x, Uint32List resultDigits) {
// Montgomery reduction only works if abs(x) < _modulus.
assert(x.abs() < _modulus);
assert(_digitsPerStep == 1 || _normModulusUsed.isEven);
var result = x.abs()._dlShift(_normModulusUsed)._rem(_modulus);
if (x._isNegative && !result._isNegative && result._used > 0) {
result = _modulus - result;
}
var used = result._used;
var digits = result._digits;
var i = used + (used & 1);
while (--i >= 0) {
resultDigits[i] = digits[i];
}
return used;
}
_BigIntImpl _revert(Uint32List xDigits, int xUsed) {
var resultDigits = _newDigits(2 * _normModulusUsed);
var i = xUsed + (xUsed & 1);
while (--i >= 0) {
resultDigits[i] = xDigits[i];
}
var resultUsed = _reduce(resultDigits, xUsed);
return new _BigIntImpl._(false, resultUsed, resultDigits);
}
// x = x/R mod _modulus.
// Returns xUsed.
int _reduce(Uint32List xDigits, int xUsed) {
while (xUsed < 2 * _normModulusUsed) {
// Pad x so _mulAdd has enough room later.
xDigits[xUsed++] = 0;
}
var i = 0;
while (i < _normModulusUsed) {
var d = _mulMod(_args, xDigits, i);
assert(d == _digitsPerStep);
d = _BigIntImpl._mulAdd(
_args, _muDigit, _modulusDigits, 0, xDigits, i, _normModulusUsed);
assert(d == _digitsPerStep);
i += d;
}
// Clamp x.
while (xUsed > 0 && xDigits[xUsed - 1] == 0) {
--xUsed;
}
xUsed = _BigIntImpl._drShiftDigits(xDigits, xUsed, i, xDigits);
if (_BigIntImpl._compareDigits(
xDigits, xUsed, _modulusDigits, _normModulusUsed) >=
0) {
_BigIntImpl._absSub(
xDigits, xUsed, _modulusDigits, _normModulusUsed, xDigits);
}
// Clamp x.
while (xUsed > 0 && xDigits[xUsed - 1] == 0) {
--xUsed;
}
return xUsed;
}
int _sqr(Uint32List xDigits, int xUsed, Uint32List resultDigits) {
var resultUsed = _BigIntImpl._sqrDigits(xDigits, xUsed, resultDigits);
return _reduce(resultDigits, resultUsed);
}
int _mul(Uint32List xDigits, int xUsed, Uint32List yDigits, int yUsed,
Uint32List resultDigits) {
var resultUsed =
_BigIntImpl._mulDigits(xDigits, xUsed, yDigits, yUsed, resultDigits);
return _reduce(resultDigits, resultUsed);
}
}
// Modular reduction using "classic" algorithm.
class _BigIntClassicReduction implements _BigIntReduction {
final _BigIntImpl _modulus; // Modulus.
int _normModulusUsed;
_BigIntImpl _normModulus; // Normalized _modulus.
Uint32List _normModulusDigits;
Uint32List _negNormModulusDigits; // Negated _normModulus digits.
int _modulusNsh; // Normalization shift amount.
Uint32List _args; // Top _normModulus digit(s) and place holder for estimated
// quotient digit(s).
Uint32List _tmpDigits; // Temporary digits used during reduction.
_BigIntClassicReduction(this._modulus) {
// Preprocess arguments to _remDigits.
var nsh =
_BigIntImpl._digitBits - _modulus._digits[_modulus._used - 1].bitLength;
// For 64-bit processing, make sure _negNormModulusDigits has an even number
// of digits.
if (_modulus._used.isOdd) {
nsh += _BigIntImpl._digitBits;
}
_modulusNsh = nsh;
_normModulus = _modulus << nsh;
_normModulusUsed = _normModulus._used;
_normModulusDigits = _normModulus._digits;
assert(_normModulusUsed.isEven);
_args = _newDigits(4);
_args[_BigIntImpl._divisorLowTopDigit] =
_normModulusDigits[_normModulusUsed - 2];
_args[_BigIntImpl._divisorTopDigit] =
_normModulusDigits[_normModulusUsed - 1];
// Negate _normModulus so we can use _mulAdd instead of
// unimplemented _mulSub.
var negNormModulus =
_BigIntImpl.one._dlShift(_normModulusUsed) - _normModulus;
if (negNormModulus._used < _normModulusUsed) {
_negNormModulusDigits = _BigIntImpl._cloneDigits(
negNormModulus._digits, 0, _normModulusUsed, _normModulusUsed);
} else {
_negNormModulusDigits = negNormModulus._digits;
}
// _negNormModulusDigits is read-only and has _normModulusUsed digits (possibly
// including several leading zeros) plus a leading zero for 64-bit
// processing.
_tmpDigits = _newDigits(2 * _normModulusUsed);
}
int _convert(_BigIntImpl x, Uint32List resultDigits) {
var digits;
var used;
if (x._isNegative || x.compareTo(_modulus) >= 0) {
var remainder = x._rem(_modulus);
if (x._isNegative && !remainder._isNegative && remainder._used > 0) {
remainder = _modulus - remainder;
}
assert(!remainder._isNegative);
used = remainder._used;
digits = remainder._digits;
} else {
used = x._used;
digits = x._digits;
}
var i = used + (used & 1); // Copy leading zero if any.
while (--i >= 0) {
resultDigits[i] = digits[i];
}
return used;
}
_BigIntImpl _revert(Uint32List xDigits, int xUsed) {
return new _BigIntImpl._(false, xUsed, xDigits);
}
int _reduce(Uint32List xDigits, int xUsed) {
if (xUsed < _modulus._used) {
return xUsed;
}
// The function _BigIntImpl._remDigits(...) is optimized for reduction and
// equivalent to calling
// 'convert(revert(xDigits, xUsed)._rem(_normModulus), xDigits);'
return _BigIntImpl._remDigits(
xDigits,
xUsed,
_normModulusDigits,
_normModulusUsed,
_negNormModulusDigits,
_modulusNsh,
_args,
_tmpDigits,
xDigits);
}
int _sqr(Uint32List xDigits, int xUsed, Uint32List resultDigits) {
var resultUsed = _BigIntImpl._sqrDigits(xDigits, xUsed, resultDigits);
return _reduce(resultDigits, resultUsed);
}
int _mul(Uint32List xDigits, int xUsed, Uint32List yDigits, int yUsed,
Uint32List resultDigits) {
var resultUsed =
_BigIntImpl._mulDigits(xDigits, xUsed, yDigits, yUsed, resultDigits);
return _reduce(resultDigits, resultUsed);
}
}