Fixed-point numbers

Alpha: This is an alpha release of the new major version 2.0.0 that makes use of const generics instead of the typenum crate. This version requires the nightly compiler with the generic_const_exprs feature enabled. The stable version 2.0.0 itself will not be released before the generic_const_exprs feature is stabilized. See the documentation for porting from version 1 to version 2.

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, and n − f integer bits. For example, FixedI32<24> is a 32-bit signed fixed-point number with n = 32 total bits, f = 24 fractional bits, and n − f = 8 integer bits. FixedI32<0> behaves like i32, and FixedU32<0> 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<0>, 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<32>, and in the range 0 ≤ x < 1 for unsigned numbers like FixedU32<32>.

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 2.0.0-alpha.28.0 news (2024-07-25)

• The crate now requires the nightly compiler with the generic_const_exprs feature enabled.
• The crate now uses generic constant expressions to specify the number of fractional bits.
• The Fixed trait constraints have been relaxed. Methods that required the number of fractional bits to be bounded have been moved to the new subtrait FixedBoundFrac.
• The INT_NBITS and FRAC_NBITS associated constants were replaced with INT_BITS and FRAC_BITS which can be negative.
• The Unwrapped methods from_str_binary, from_str_octal and from_str_hex return the value directly instead of a Result.
• The deprecated F128Bits and F128 structs have been removed. They were replaced by the f128 primitive.
• The deprecated const_fixed_from_int macro has been removed. It was replaced by the const_from_int method in version 1.20.0.
• The deprecated optional features az and f16 were removed. These features had no effect, as the functionality they enabled is now always enabled.
• All deprecated methods have been removed.
• The new generic associated type Fixed was added to the FixedBits trait.
• The following methods of the Fixed trait and of the Saturating, Wrapping and Unwrapped wrappers now have some parameters and return types that can be generic:
  • mul_add, add_prod, mul_acc
• The following methods of the Fixed trait now have some parameters and return types that can be generic:
  • checked_mul_add, checked_add_prod, checked_mul_acc
  • saturating_mul_add, saturating_add_prod, saturating_mul_acc
  • wrapping_mul_add, wrapping_add_prod, wrapping_mul_acc
  • unwrapped_mul_add, unwrapped_add_prod, unwrapped_mul_acc
  • overflowing_mul_add, overflowing_add_prod, overflowing_mul_acc

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

#![feature(generic_const_exprs)]

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<12>. 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.

#![feature(generic_const_exprs)]

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.

#![feature(generic_const_exprs)]

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 = "2.0.0-alpha.28.0"

This alpha version of the fixed crate requires the nightly compiler with the generic_const_exprs feature enabled.

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. With this feature, serialization is only supported for fixed-point numbers where the number of fractional bits is from zero to the total number of bits. 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 = "2.0.0-alpha.28.0"

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.)

Porting from version 1 to version 2

To port from version 1 to version 2, the following is required:

• Temporary change required until the generic_const_exprs feature are stabilized: use the nightly compiler and enable the generic_const_exprs feature using

  #![feature(generic_const_exprs)]
  

• Use integer literals instead of typenum integer constants, for example FixedI32<8> instead of FixedI32<U8>.

• The Fixed trait constraints have been relaxed. Methods that required the number of fractional bits to be bounded have been moved to the new subtrait FixedBoundFrac. For code that uses these trait methods, Fixed should be replaced by FixedBoundFrac.

• The FRAC_NBITS and INT_NBITS associated constants of type u32 were replaced by FRAC_BITS and INT_BITS of type i32.

• For the Unwrapped wrapper, the methods from_str_binary, from_str_octal and from_str_hex return the value directly instead of a Result.

• The deprecated F128Bits and F128 structs have been removed. They were replaced by the f128 primitive.

• The deprecated const_fixed_from_int macro has been removed. It was replaced by the const_from_int method in version 1.20.0.

• The deprecated optional features az and f16 were removed. These features had no effect, as their functionality has been unconditionally enabled since version 1.7.0.

• All deprecated methods have been removed.

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.