Arbitrary-precision numbers

Rug provides integers and floating-point numbers with arbitrary precision and correct rounding. Its main features are

• bignum [integers][Integer] with arbitrary precision,
• bignum [rational numbers][Rational] with arbitrary precision,
• multi-precision [floating-point numbers][Float] with correct rounding, and
• multi-precision [complex numbers][Complex] with correct rounding.

Rug is a high-level interface to the following [GNU] libraries:

• [GMP] for integers and rational numbers,
• [MPFR] for floating-point numbers, and
• [MPC] for complex numbers.

Rug is free software: you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. See the full text of the [GNU LGPL] and [GNU GPL] for details.

What’s new

Version 1.1.0 news

• Support for [i128] and [u128] conversions and comparisons was added, conditional on compiler support.
• Conditional on compiler support, [TryFrom] conversions were implemented for conversions
  • from [Integer] values to integer primitives,
  • from floating-point primitives to [Rational] numbers, and
  • from [Float] values to [Rational] numbers.
• A new [Float::get_significand] method was added.
• New methods [Float::u_pow_u] and [Float::i_pow_u] were added.
• New methods [from_digits], [to_digits], [assign_digits], [write_digits] and [significant_digits] were added to [Integer], providing reading from and writing to slices of unsigned integer primitives.

Other releases

Details on other releases can be found in [RELEASES.md].

Quick example

```rust use rug::{Assign, Integer}; let mut int = Integer::new(); asserteq!(int, 0); int.assign(14); asserteq!(int, 14);

let decimal = "98765432109876543210"; int.assign(Integer::parse(decimal).unwrap()); assert!(int > 100000000);

let hex160 = "ffff0000ffff0000ffff0000ffff0000ffff0000"; int.assign(Integer::parseradix(hex160, 16).unwrap()); asserteq!(int.significantbits(), 160); int = (int >> 128) - 1; asserteq!(int, 0xfffeffffu32); ```

• [Integer::new()][Integer::new] creates a new [Integer] intialized to zero.
• To assign values to Rug types, we use the [Assign] trait and its method [assign][Assign::assign]. We do not use the assignment operator = as that would drop the left-hand-side operand and replace it with a right-hand-side operand of the same type, which is not what we want here.
• Arbitrary precision numbers can hold numbers that are too large to fit in a primitive type. To assign such a number to the large types, we use strings rather than primitives; in the example this is done using [Integer::parse] and [Integer::parse_radix].
• We can compare Rug types to primitive types or to other Rug types using the normal comparison operators, for example int > 100_000_000.
• Most arithmetic operations are supported with Rug types and primitive types on either side of the operator, for example int >> 128.

Using with primitive types

With Rust primitive types, arithmetic operators usually operate on two values of the same type, for example 12i32 + 5i32. Unlike primitive types, conversion to and from Rug types can be expensive, so the arithmetic operators are overloaded to work on many combinations of Rug types and primitives. The following are provided:

1. Where they make sense, all arithmetic operators are overloaded to work with Rug types and the primitives [i32], [u32], [f32] and [f64].
2. Where they make sense, conversions using the [From] trait and assignments using the [Assign] trait are supported for all the primitives in 1 above as well as the other primitives [i8], [i16], [i64], [isize], [u8], [u16], [u64] and [usize]. This also applies to [i128] and [u128] if they are supported by the compiler.
3. Comparisons between Rug types and all the primitives listed in 1 and 2 above are supported.
4. For [Rational] numbers, conversions and comparisons are also supported for tuples containing two integer primitives: the first is the numerator and the second is the denominator which must not be zero. The two primitives do not need to be of the same type.
5. For [Complex] numbers, conversions and comparisons are also supported for tuples containing two primitives: the first is the real part and the second is the imaginary part. The two primitives do not need to be of the same type.

Operators

Operators are overloaded to work on Rug types alone or on a combination of Rug types and Rust primitives. When at least one operand is an owned value of a Rug type, the operation will consume that value and return a value of the Rug type. For example

rust use rug::Integer; let a = Integer::from(10); let b = 5 - a; assert_eq!(b, 5 - 10);

Here a is consumed by the subtraction, and b is an owned [Integer].

If on the other hand there are no owned Rug types and there are references instead, the returned value is not the final value, but an incomplete-computation value. For example

rust use rug::Integer; let (a, b) = (Integer::from(10), Integer::from(20)); let incomplete = &a - &b; // This would fail to compile: assert_eq!(incomplete, -10); let sub = Integer::from(incomplete); assert_eq!(sub, -10);

Here a and b are not consumed, and incomplete is not the final value. It still needs to be converted or assigned into an [Integer]. This is covered in more detail in the [Incomplete-computation values] section.

Shifting operations

The left shift << and right shift >> operators support shifting by negative values, for example a << 5 is equivalent to a >> -5.

The shifting operators are also supported for the [Float] and [Complex] number types, where they are equivalent to multiplication or division by a power of two. Only the exponent of the value is affected; the mantissa is unchanged.

Exponentiation

Exponentiation (raising to a power) does not have a dedicated operator in Rust. In order to perform exponentiation of Rug types, the [Pow] trait has to be brought into scope, for example

rust use rug::Integer; use rug::ops::Pow; let base = Integer::from(10); let power = base.pow(5); assert_eq!(power, 100_000);

Compound assignments to right-hand-side operands

Traits are provided for compound assignment to right-hand-side operands. This can be useful for non-commutative operations like subtraction. The names of the traits and their methods are similar to Rust compound assignment traits, with the suffix “Assign” replaced with “From”. For example the counterpart to [SubAssign] is [SubFrom]:

rust use rug::Integer; use rug::ops::SubFrom; let mut rhs = Integer::from(10); // set rhs = 100 - rhs rhs.sub_from(100); assert_eq!(rhs, 90);

Incomplete-computation values

There are two main reasons why operations like &a - &b do not perform a complete computation and return a Rug type:

1. Sometimes we need to assign the result to an object that already exists. Since Rug types require memory allocations, this can help reduce the number of allocations. (While the allocations might not affect performance noticeably for computationally intensive functions, they can have a much more significant effect on faster functions like addition.)
2. For the [Float] and [Complex] number types, we need to know the precision when we create a value, and the operation itself does not convey information about what precision is desired for the result.

There are two things that can be done with incomplete-computation values:

1. Assign them to an existing object without unnecessary allocations. This is usually achieved using the [Assign] trait or a similar method, for example [int.assign(incomplete)][Assign::assign] and [float.assign_round(incomplete, Round::Up)][assign_round].
2. Convert them to the final value using the [From] trait or a similar method, for example [Integer::from(incomplete)][From::from] and [Float::with_val(53, incomplete)][Float::with_val].

Let us consider a couple of examples.

rust use rug::{Assign, Integer}; let mut buffer = Integer::new(); // ... buffer can be used and reused ... let (a, b) = (Integer::from(10), Integer::from(20)); let incomplete = &a - &b; buffer.assign(incomplete); assert_eq!(buffer, -10);

Here the assignment from incomplete into buffer does not require an allocation unless the result does not fit in the current capacity of buffer. If &a - &b returned an [Integer] instead, then an allocation would take place even if it is not necessary.

rust use rug::Float; use rug::float::Constant; // x has a precision of 10 bits let x = Float::with_val(10, 180); // y has a precision of 50 bits let y = Float::with_val(50, Constant::Pi); let incomplete = &x / &y; // z has a precision of 45 bits let z = Float::with_val(45, incomplete); assert!(57.295 < z && z < 57.296);

The precision to use for the result depends on the requirements of the algorithm being implemented. Here z is created with a precision of 45.

Many operations can return incomplete-computation values:

• unary operators applied to references, for example -&int;
• binary operators applied to two references, for example &int1 + &int2;
• binary operators applied to a primitive and a reference, for example &int * 10;
• methods that take a reference, for example [int.abs_ref()][Integer::abs_ref];
• methods that take two references, for example [int1.gcd_ref(&int2)][Integer::gcd_ref];
• string parsing, for example [Integer::parse("12")][Integer::parse];
• and more.

These operations return objects that can be stored in temporary variables like incomplete in the last few examples. However, the names of the types are not public, and consequently, the incomplete-computation values cannot be for example stored in a struct. If you need to store the value in a struct, convert it to its final type and value.

Using Rug

Rug is available on crates.io. To use Rug in your crate, add it as a dependency inside [Cargo.toml]:

toml [dependencies] rug = "1.0"

You also need to declare it by adding this to your crate root (usually lib.rs or main.rs):

rust extern crate rug;

Rug requires rustc version 1.18.0 or later.

Rug also depends on the [GMP], [MPFR] and [MPC] libraries through the low-level FFI bindings in the gmp-mpfr-sys crate, which needs some setup to build; the gmp-mpfr-sys documentation has some details on usage under GNU/Linux, macOS and Windows.

Optional features

The Rug crate has six optional features:

1. integer, enabled by default. Required for the [Integer] type and its supporting features.
2. rational, enabled by default. Required for the [Rational] number type and its supporting features. This feature requires the integer feature.
3. float, enabled by default. Required for the [Float] type and its supporting features.
4. complex, enabled by default. Required for the [Complex] number type and its supporting features. This feature requires the float feature.
5. rand, enabled by default. Required for the [RandState] type and its supporting features. This feature requires the integer feature.
6. serde, disabled by default. This provides serialization support for the [Integer], [Rational], [Float] and [Complex] number types, providing that they are enabled. This feature requires the [serde crate].

The first five optional features are enabled by default; to use features selectively, you can add the dependency like this to [Cargo.toml]:

toml [dependencies.rug] version = "1.0" default-features = false features = ["integer", "float", "rand"]

Here only the integer, float and rand features are enabled. If none of the features are selected, the gmp-mpfr-sys crate is not required and thus not enabled. In that case, only the [Assign] trait and the traits that are in the [ops] module are provided by the crate.