Fixed-point numbers

The fixed crate provides fixed-point numbers.

• FixedI8 and FixedU8 are eight-bit fixed-point numbers.
• FixedI16 and FixedU16 are 16-bit fixed-point numbers.
• FixedI32 and FixedU32 are 32-bit fixed-point numbers.
• FixedI64 and FixedU64 are 64-bit fixed-point numbers.
• FixedI128 and FixedU128 are 128-bit fixed-point numbers.

An n-bit fixed-point number has f = Frac fractional bits where 0 ≤ f ≤ n, and n − f integer bits. For example, FixedI32<U24> is a 32-bit signed fixed-point number with n = 32 total bits, f = 24 fractional bits, and n − f = 8 integer bits. FixedI32<U0> behaves like i32, and FixedU32<U0> behaves like u32.

The difference between any two successive representable numbers is constant throughout the possible range for a fixed-point number: Δ = 1/2^f. When f = 0, like in FixedI32<U0>, Δ = 1 because representable numbers are integers, and the difference between two successive integers is 1. When f = n, Δ = 1/2ⁿ and the value lies in the range −0.5 ≤ x < 0.5 for signed numbers like FixedI32<U32>, and in the range 0 ≤ x < 1 for unsigned numbers like FixedU32<U32>.

In version 1 the typenum crate is used for the fractional bit count Frac; the plan is to to have a major version 2 with const generics when the Rust compiler’s generic_const_exprs feature is ready and stabilized. An alpha version is already available.

The main features are

• Representation of binary fixed-point numbers up to 128 bits wide.
• Conversions between fixed-point numbers and numeric primitives.
• Comparisons between fixed-point numbers and numeric primitives.
• Parsing from strings in decimal, binary, octal and hexadecimal.
• Display as decimal, binary, octal and hexadecimal.
• Arithmetic and logic operations.

This crate does not provide decimal fixed-point numbers. For example 0.001 cannot be represented exactly, as it is 1/10³. It is binary fractions like 1/2⁴ (0.0625) that can be represented exactly, provided there are enough fractional bits.

This crate does not provide general analytic functions.

• No algebraic functions are provided, for example no pow.
• No trigonometric functions are provided, for example no sin or cos.
• No other transcendental functions are provided, for example no log or exp.

These functions are not provided because different implementations can have different trade-offs, for example trading some correctness for speed. Implementations can be provided in other crates.

• The cordic crate provides various functions implemented using the CORDIC algorithm.

The conversions supported cover the following cases.

• Infallible lossless conversions between fixed-point numbers and numeric primitives are provided using From and Into. These never fail (infallible) and do not lose any bits (lossless).
• Infallible lossy conversions between fixed-point numbers and numeric primitives are provided using the LossyFrom and LossyInto traits. The source can have more fractional bits than the destination.
• Checked lossless conversions between fixed-point numbers and numeric primitives are provided using the LosslessTryFrom and LosslessTryInto traits. The source cannot have more fractional bits than the destination.
• Checked conversions between fixed-point numbers and numeric primitives are provided using the FromFixed and ToFixed traits, or using the from_num and to_num methods and their checked versions.
• Additionally, az casts are implemented for conversion between fixed-point numbers and numeric primitives.
• Fixed-point numbers can be parsed from decimal strings using FromStr, and from binary, octal and hexadecimal strings using the from_str_binary, from_str_octal and from_str_hex methods. The result is rounded to the nearest, with ties rounded to even.
• Fixed-point numbers can be converted to strings using Display, Binary, Octal, LowerHex, UpperHex, LowerExp and UpperExp. The output is rounded to the nearest, with ties rounded to even.
• All fixed-point numbers are plain old data, so bytemuck bit casting conversions can be used.

What’s new

Version 1.28.0 news (2024-07-25)

• The crate now requires rustc version 1.79.0 or later.
• The following methods were added to all fixed-point numbers and to the Fixed trait:
  • unchecked_add, unchecked_sub, unchecked_mul_int
• For all fixed-point numbers and the Fixed trait, the following methods were renamed. The old method names are deprecated.
  • round_ties_to_even renamed to round_ties_even
  • checked_round_ties_to_even renamed to checked_round_ties_even
  • saturating_round_ties_to_even renamed to saturating_round_ties_even
  • wrapping_round_ties_to_even renamed to wrapping_round_ties_even
  • unwrapped_round_ties_to_even renamed to unwrapped_round_ties_even
  • overflowing_round_ties_to_even renamed to overflowing_round_ties_even
• The following constants were added to the consts module and as associated constants to fixed-point types:
  • SQRT_2PI, FRAC_1_SQRT_2PI
• The experimental feature nightly-float was added.
• For the experimental feature num-traits, the following traits were implemented for all fixed-point numbers:
  • ConstZero, ConstOne
  • ToBytes, FromBytes

Other releases

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

Quick examples

use fixed::types::I20F12;

// 19/3 = 6 1/3
let six_and_third = I20F12::from_num(19) / 3;
// four decimal digits for 12 binary digits
assert_eq!(six_and_third.to_string(), "6.3333");
// find the ceil and convert to i32
assert_eq!(six_and_third.ceil().to_num::<i32>(), 7);
// we can also compare directly to integers
assert_eq!(six_and_third.ceil(), 7);

The type I20F12 is a 32-bit fixed-point signed number with 20 integer bits and 12 fractional bits. It is an alias to FixedI32<U12>. The unsigned counterpart would be U20F12. Aliases are provided for all combinations of integer and fractional bits adding up to a total of eight, 16, 32, 64 or 128 bits.

use fixed::types::{I4F4, I4F12};

// -8 ≤ I4F4 < 8 with steps of 1/16 (~0.06)
let a = I4F4::from_num(1);
// multiplication and division by integers are possible
let ans1 = a / 5 * 17;
// 1 / 5 × 17 = 3 2/5 (3.4), but we get 3 3/16 (~3.2)
assert_eq!(ans1, I4F4::from_bits((3 << 4) + 3));
assert_eq!(ans1.to_string(), "3.2");

// -8 ≤ I4F12 < 8 with steps of 1/4096 (~0.0002)
let wider_a = I4F12::from(a);
let wider_ans = wider_a / 5 * 17;
let ans2 = I4F4::from_num(wider_ans);
// now the answer is the much closer 3 6/16 (~3.4)
assert_eq!(ans2, I4F4::from_bits((3 << 4) + 6));
assert_eq!(ans2.to_string(), "3.4");

The second example shows some precision and conversion issues. The low precision of a means that a / 5 is 3⁄16 instead of 1⁄5, leading to an inaccurate result ans1 = 3 3⁄16 (~3.2). With a higher precision, we get wider_a / 5 equal to 819⁄4096, leading to a more accurate intermediate result wider_ans = 3 1635⁄4096. When we convert back to four fractional bits, we get ans2 = 3 6⁄16 (~3.4).

Note that we can convert from I4F4 to I4F12 using From, as the target type has the same number of integer bits and a larger number of fractional bits. Converting from I4F12 to I4F4 cannot use From as we have less fractional bits, so we use from_num instead.

Writing fixed-point constants and values literally

The lit method, which is available as a const function, can be used to parse literals. It supports

• underscores as separators;
• prefixes “0b”, “0o” and “0x” for binary, octal and hexadecimal numbers;
• an optional decimal exponent with separator “e” or “E” for decimal, binary and octal numbers, or with separator “@” for all supported radices including hexadecimal.

use fixed::types::I16F16;

// 0.1275e2 is 12.75
const TWELVE_POINT_75: I16F16 = I16F16::lit("0.127_5e2");
// 1.8 hexadecimal is 1.5 decimal, and 18@-1 is 1.8
const ONE_POINT_5: I16F16 = I16F16::lit("0x_18@-1");
// 12.75 + 1.5 = 14.25
let sum = TWELVE_POINT_75 + ONE_POINT_5;
assert_eq!(sum, 14.25);

Using the fixed crate

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

[dependencies]
fixed = "1.28"

The fixed crate requires rustc version 1.79.0 or later.

Optional features

The fixed crate has these optional feature:

1. arbitrary, disabled by default. This provides the generation of arbitrary fixed-point numbers from raw, unstructured data. This feature requires the arbitrary crate.
2. borsh, disabled by default. This implements serialization and deserialization using the borsh crate.
3. serde, disabled by default. This provides serialization support for the fixed-point types. This feature requires the serde crate.
4. std, disabled by default. This is for features that are not possible under no_std: currently the implementation of the Error trait for ParseFixedError.
5. serde-str, disabled by default. Fixed-point numbers are serialized as strings showing the value when using human-readable formats. This feature requires the serde and the std optional features. Warning: numbers serialized when this feature is enabled cannot be deserialized when this feature is disabled, and vice versa.

To enable features, you can add the dependency like this to Cargo.toml:

[dependencies.fixed]
features = ["serde"]
version = "1.28"

Experimental optional features

It is not considered a breaking change if the following experimental features are removed. The removal of experimental features would however require a minor version bump. Similarly, on a minor version bump, optional dependencies can be updated to an incompatible newer version.

1. num-traits, disabled by default. This implements some traits from the num-traits crate. (The plan is to promote this to an optional feature once the num-traits crate reaches version 1.0.0.)
2. nightly-float, disabled by default. This requires the nightly compiler, and implements conversions and comparisons with the experimental f16 and f128 primitives. (The plan is to always implement the conversions and comparisons and remove this experimental feature once the primitives are stabilized.)

Deprecated optional features

The following optional features are deprecated and will be removed in the next major version of the crate.

1. az, has no effect. Previously required for the az cast traits. Now these cast traits are always provided.
2. f16, has no effect. Previously required for conversion to/from half::f16 and half::bf16. Now these conversions are always provided.

License

This crate is free software: you can redistribute it and/or modify it under the terms of either

• the Apache License, Version 2.0 or
• the MIT License

at your option.

Contribution

Unless you explicitly state otherwise, any contribution intentionally submitted for inclusion in the work by you, as defined in the Apache License, Version 2.0, shall be dual licensed as above, without any additional terms or conditions.