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# std.bigint

Arbitrary-precision ('bignum') arithmetic.
Performance is optimized for numbers below ~1000 decimal digits. For X86 machines, highly optimised assembly routines are used.
The following algorithms are currently implemented:
• Karatsuba multiplication
• Squaring is optimized independently of multiplication
• Divide-and-conquer division
• Binary exponentiation
For very large numbers, consider using the GMP library instead.
Authors:
Don Clugston

Source: std/bigint.d

struct `BigInt`;
A struct representing an arbitrary precision integer.
All arithmetic operations are supported, except unsigned shift right (>>>). Bitwise operations (|, &, ^, ~) are supported, and behave as if `BigInt` was an infinite length 2's complement number.
`BigInt` implements value semantics using copy-on-write. This means that assignment is cheap, but operations such as x++ will cause heap allocation. (But note that for most bigint operations, heap allocation is inevitable anyway.)
Examples:
```BigInt a = "9588669891916142";
BigInt b = "7452469135154800";
auto c = a * b;
assert(c == BigInt("71459266416693160362545788781600"));
auto d = b * a;
assert(d == BigInt("71459266416693160362545788781600"));
assert(d == c);
d = c * BigInt("794628672112");
assert(d == BigInt("56783581982794522489042432639320434378739200"));
auto e = c + d;
assert(e == BigInt("56783581982865981755459125799682980167520800"));
auto f = d + c;
assert(f == e);
auto g = f - c;
assert(g == d);
g = f - d;
assert(g == c);
e = 12345678;
g = c + e;
auto h = g / b;
auto i = g % b;
assert(h == a);
assert(i == e);
BigInt j = "-0x9A56_57f4_7B83_AB78";
j ^^= 11;
```
this(Range)(Range `s`)
if (isBidirectionalRange!Range && isSomeChar!(ElementType!Range) && !isInfinite!Range);
Construct a BigInt from a decimal or hexadecimal string. The number must be in the form of a decimal or hex literal. It may have a leading + or - sign, followed by 0x or 0X if hexadecimal. Underscores are permitted in any location after the 0x and/or the sign of the number.
Parameters:
 Range `s` a finite bidirectional range of any character type
Throws:
ConvException if the string doesn't represent a valid number
pure nothrow this(T)(T `x`)
if (isIntegral!T);
Construct a BigInt from a built-in integral type.
Examples:
```// @system due to failure in FreeBSD32
ulong data = 1_000_000_000_000;
auto bigData = BigInt(data);
assert(data == BigInt("1_000_000_000_000"));
```
pure nothrow this(T)(T `x`)
if (is(Unqual!T == BigInt));
Construct a BigInt from another BigInt.
Examples:
```const(BigInt) b1 = BigInt("1_234_567_890");
BigInt b2 = BigInt(b1);
assert(b2 == BigInt("1_234_567_890"));
```
pure nothrow BigInt `opAssign`(T)(T `x`)
if (isIntegral!T);
Assignment from built-in integer types.
Examples:
```auto b = BigInt("123");
b = 456;
assert(b == BigInt("456"));
```
pure @nogc BigInt `opAssign`(T : BigInt)(T `x`);
Assignment from another BigInt.
Examples:
```auto b1 = BigInt("123");
auto b2 = BigInt("456");
b2 = b1;
assert(b2 == BigInt("123"));
```
pure nothrow BigInt `opOpAssign`(string op, T)(T `y`)
if ((op == "+" || op == "-" || op == "*" || op == "/" || op == "%" || op == ">>" || op == "<<" || op == "^^" || op == "|" || op == "&" || op == "^") && isIntegral!T);
Implements assignment operators from built-in integers of the form BigInt op= integer.
Examples:
```//@system because opOpAssign is @system
auto b = BigInt("1_000_000_000");

b += 12345;
assert(b == BigInt("1_000_012_345"));

b /= 5;
assert(b == BigInt("200_002_469"));
```
pure nothrow BigInt `opOpAssign`(string op, T)(T `y`)
if ((op == "+" || op == "-" || op == "*" || op == "|" || op == "&" || op == "^" || op == "/" || op == "%") && is(T : BigInt));
Implements assignment operators of the form BigInt op= BigInt.
Examples:
```// @system because opOpAssign is @system
auto x = BigInt("123");
auto y = BigInt("321");
x += y;
assert(x == BigInt("444"));
```
const pure nothrow BigInt `opBinary`(string op, T)(T `y`)
if ((op == "+" || op == "*" || op == "-" || op == "|" || op == "&" || op == "^" || op == "/" || op == "%") && is(T : BigInt));
Implements binary operators between BigInts.
Examples:
```auto x = BigInt("123");
auto y = BigInt("456");
BigInt z = x * y;
assert(z == BigInt("56088"));
```
const pure nothrow BigInt `opBinary`(string op, T)(T `y`)
if ((op == "+" || op == "*" || op == "-" || op == "/" || op == "|" || op == "&" || op == "^" || op == ">>" || op == "<<" || op == "^^") && isIntegral!T);
Implements binary operators between BigInt's and built-in integers.
Examples:
```auto x = BigInt("123");
x *= 300;
assert(x == BigInt("36900"));
```
const pure nothrow auto `opBinary`(string op, T)(T `y`)
if (op == "%" && isIntegral!T);
Implements a narrowing remainder operation with built-in integer types.
This binary operator returns a narrower, built-in integer type where applicable, according to the following table.
 BigInt % long → long BigInt % ulong → BigInt BigInt % other type → int
Examples:
```auto  x  = BigInt("1_000_000_500");
long  l  = 1_000_000L;
ulong ul = 2_000_000UL;
int   i  = 500_000;
short s  = 30_000;

assert(is(typeof(x % l)  == long)   && x % l  == 500L);
assert(is(typeof(x % ul) == BigInt) && x % ul == BigInt(500));
assert(is(typeof(x % i)  == int)    && x % i  == 500);
assert(is(typeof(x % s)  == int)    && x % s  == 10500);
```
const pure nothrow BigInt `opBinaryRight`(string op, T)(T `y`)
if ((op == "+" || op == "*" || op == "|" || op == "&" || op == "^") && isIntegral!T);

const pure nothrow BigInt `opBinaryRight`(string op, T)(T `y`)
if (op == "-" && isIntegral!T);

const pure nothrow T `opBinaryRight`(string op, T)(T `x`)
if ((op == "%" || op == "/") && isIntegral!T);
Implements operators with built-in integers on the left-hand side and BigInt on the right-hand side.
Examples:
```auto x = BigInt("100");
BigInt y = 123 + x;
assert(y == BigInt("223"));

BigInt z = 123 - x;
assert(z == BigInt("23"));

// Dividing a built-in integer type by BigInt always results in
// something that fits in a built-in type, so the built-in type is
// returned, not BigInt.
assert(is(typeof(1000 / x) == int));
assert(1000 / x == 10);
```
const pure nothrow BigInt `opUnary`(string op)()
if (op == "+" || op == "-" || op == "~");

pure nothrow BigInt `opUnary`(string op)()
if (op == "++" || op == "--");
Implements BigInt unary operators.
Examples:
```auto x = BigInt("1234");
assert(-x == BigInt("-1234"));

++x;
assert(x == BigInt("1235"));
```
const pure @nogc bool `opEquals`()(auto ref const BigInt `y`);

const pure nothrow @nogc bool `opEquals`(T)(T `y`)
if (isIntegral!T);
Implements BigInt equality test with other BigInt's and built-in integer types.
Examples:
```auto x = BigInt("12345");
auto y = BigInt("12340");
int z = 12345;
int w = 54321;

assert(x == x);
assert(x != y);
assert(x == y + 5);
assert(x == z);
assert(x != w);
```
const pure nothrow @nogc T `opCast`(T : bool)();
Implements casting to bool.
Examples:
```// Non-zero values are regarded as true
auto x = BigInt("1");
auto y = BigInt("10");
assert(x);
assert(y);

// Zero value is regarded as false
auto z = BigInt("0");
assert(!z);
```
const T `opCast`(T : ulong)();
Implements casting to integer types.
Throws:
std.conv.ConvOverflowException if the number exceeds the target type's range.
Examples:
```import std.conv : to, ConvOverflowException;
import std.exception : assertThrown;

assert(BigInt("0").to!int == 0);

assert(BigInt("0").to!ubyte == 0);
assert(BigInt("255").to!ubyte == 255);
assertThrown!ConvOverflowException(BigInt("256").to!ubyte);
assertThrown!ConvOverflowException(BigInt("-1").to!ubyte);
```
const pure nothrow @nogc T `opCast`(T)()
if (is(Unqual!T == BigInt));
Implements casting to/from qualified BigInt's.

Warning: Casting to/from const or immutable may break type system guarantees. Use with care.

Examples:
```const(BigInt) x = BigInt("123");
BigInt y = cast() x;    // cast away const
assert(y == x);
```
const pure nothrow @nogc int `opCmp`(ref const BigInt `y`);

const pure nothrow @nogc int `opCmp`(T)(T `y`)
if (isIntegral!T);

const pure nothrow @nogc int `opCmp`(T : BigInt)(const T `y`);
Implements 3-way comparisons of BigInt with BigInt or BigInt with built-in integers.
Examples:
```auto x = BigInt("100");
auto y = BigInt("10");
int z = 50;
const int w = 200;

assert(y < x);
assert(x > z);
assert(z > y);
assert(x < w);
```
const pure nothrow @nogc @safe long `toLong`();
Returns:
The value of this BigInt as a long, or +/- long.max if outside the representable range.
Examples:
```auto b = BigInt("12345");
long l = b.toLong();
assert(l == 12345);
```
const pure nothrow @nogc @safe int `toInt`();
Returns:
The value of this BigInt as an int, or +/- int.max if outside the representable range.
Examples:
```auto big = BigInt("5_000_000");
auto i = big.toInt();
assert(i == 5_000_000);

// Numbers that are too big to fit into an int will be clamped to int.max.
auto tooBig = BigInt("5_000_000_000");
i = tooBig.toInt();
assert(i == int.max);
```
const pure nothrow @nogc @property @safe size_t `uintLength`();
Number of significant uints which are used in storing this number.
The absolute value of this BigInt is always < 232*`uintLength`
const pure nothrow @nogc @property @safe size_t `ulongLength`();
Number of significant ulongs which are used in storing this number.
The absolute value of this BigInt is always < 264*`ulongLength`
const void `toString`(scope void delegate(const(char)[]) `sink`, string `formatString`);

const void `toString`(scope void delegate(const(char)[]) `sink`, ref FormatSpec!char `f`);
Convert the BigInt to string, passing it to the given `sink`.
Parameters:
void delegate(const(char)[]) `sink` A delegate for accepting possibly piecewise segments of the formatted string.
string `formatString` A format string specifying the output format.
 "d" Decimal "o" Octal "x" Hexadecimal, lower case "X" Hexadecimal, upper case "s" Default formatting (same as "d") `null` Default formatting (same as "d")
Examples:
`toString` is rarely directly invoked; the usual way of using it is via std.format.format:
```import std.format : format;

auto x = BigInt("1_000_000");
x *= 12345;

assert(format("%d", x) == "12345000000");
assert(format("%x", x) == "2_dfd1c040");
assert(format("%X", x) == "2_DFD1C040");
assert(format("%o", x) == "133764340100");
```
const nothrow @safe size_t `toHash`();
Returns:
A unique hash of the BigInt's value suitable for use in a hash table.
Examples:
`toHash` is rarely directly invoked; it is implicitly used when BigInt is used as the key of an associative array.
```string[BigInt] aa;
aa[BigInt(123)] = "abc";
aa[BigInt(456)] = "def";

assert(aa[BigInt(123)] == "abc");
assert(aa[BigInt(456)] == "def");
```
string `toDecimalString`(const(BigInt) `x`);
Parameters:
 const(BigInt) `x` The BigInt to convert to a decimal string.
Returns:
A string that represents the BigInt as a decimal number.
Examples:
```auto x = BigInt("123");
x *= 1000;
x += 456;

auto xstr = x.toDecimalString();
assert(xstr == "123456");
```
string `toHex`(const(BigInt) `x`);
Parameters:
 const(BigInt) `x` The BigInt to convert to a hexadecimal string.
Returns:
A string that represents the BigInt as a hexadecimal (base 16) number in upper case.
Examples:
```auto x = BigInt("123");
x *= 1000;
x += 456;

auto xstr = x.toHex();
assert(xstr == "1E240");
```
Unsigned!T `absUnsign`(T)(T `x`)
if (isIntegral!T);
Returns the absolute value of `x` converted to the corresponding unsigned type.
Parameters:
 T `x` The integral value to return the absolute value of.
Returns:
The absolute value of `x`.