# 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. [`generic_const_exprs` feature]: https://github.com/rust-lang/rust/issues/76560 [alpha version]: https://docs.rs/fixed/2.0.0-alpha/fixed/ 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][`checked_from_num`]. * 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`][tf-1-28] trait: * [`unchecked_add`][f-ua-1-28], [`unchecked_sub`][f-us-1-28], [`unchecked_mul_int`][f-umi-1-28] * For all fixed-point numbers and the [`Fixed`][tf-1-28] trait, the following methods were renamed. The old method names are deprecated. * [`round_ties_to_even`][f-rtte-1-28] renamed to [`round_ties_even`][f-rte-1-28] * [`checked_round_ties_to_even`][f-crtte-1-28] renamed to [`checked_round_ties_even`][f-crte-1-28] * [`saturating_round_ties_to_even`][f-srtte-1-28] renamed to [`saturating_round_ties_even`][f-srte-1-28] * [`wrapping_round_ties_to_even`][f-wrtte-1-28] renamed to [`wrapping_round_ties_even`][f-wrte-1-28] * [`unwrapped_round_ties_to_even`][f-urtte-1-28] renamed to [`unwrapped_round_ties_even`][f-urte-1-28] * [`overflowing_round_ties_to_even`][f-ortte-1-28] renamed to [`overflowing_round_ties_even`][f-orte-1-28] * The following constants were added to the [`consts`][c-1-28] module and as associated constants to fixed-point types: * [`SQRT_2PI`][c-r2p-1-28], [`FRAC_1_SQRT_2PI`][c-1r2p-1-28] * The experimental feature [`nightly-float`][feat-exp-1-28] was added. * For the experimental feature [`num-traits`][feat-exp-1-28], the following traits were implemented for all fixed-point numbers: * [`ConstZero`][nt-0-2-cz], [`ConstOne`][nt-0-2-co] * [`ToBytes`][nt-0-2-tb], [`FromBytes`][nt-0-2-fb] [c-1-28]: https://docs.rs/fixed/~1.28/fixed/consts/index.html [c-1r2p-1-28]: https://docs.rs/fixed/~1.28/fixed/consts/constant.FRAC_1_SQRT_2PI.html [c-r2p-1-28]: https://docs.rs/fixed/~1.28/fixed/consts/constant.SQRT_2PI.html [f-crte-1-28]: https://docs.rs/fixed/~1.28/fixed/struct.FixedI32.html#method.checked_round_ties_even [f-crtte-1-28]: https://docs.rs/fixed/~1.28/fixed/struct.FixedI32.html#method.checked_round_ties_to_even [f-orte-1-28]: https://docs.rs/fixed/~1.28/fixed/struct.FixedI32.html#method.overflowing_round_ties_even [f-ortte-1-28]: https://docs.rs/fixed/~1.28/fixed/struct.FixedI32.html#method.overflowing_round_ties_to_even [f-rte-1-28]: https://docs.rs/fixed/~1.28/fixed/struct.FixedI32.html#method.round_ties_even [f-rtte-1-28]: https://docs.rs/fixed/~1.28/fixed/struct.FixedI32.html#method.round_ties_to_even [f-srte-1-28]: https://docs.rs/fixed/~1.28/fixed/struct.FixedI32.html#method.saturating_round_ties_even [f-srtte-1-28]: https://docs.rs/fixed/~1.28/fixed/struct.FixedI32.html#method.saturating_round_ties_to_even [f-ua-1-28]: https://docs.rs/fixed/~1.28/fixed/struct.FixedI32.html#method.unchecked_add [f-umi-1-28]: https://docs.rs/fixed/~1.28/fixed/struct.FixedI32.html#method.unchecked_mul_int [f-urte-1-28]: https://docs.rs/fixed/~1.28/fixed/struct.FixedI32.html#method.unwrapped_round_ties_even [f-urtte-1-28]: https://docs.rs/fixed/~1.28/fixed/struct.FixedI32.html#method.unwrapped_round_ties_to_even [f-us-1-28]: https://docs.rs/fixed/~1.28/fixed/struct.FixedI32.html#method.unchecked_sub [f-wrte-1-28]: https://docs.rs/fixed/~1.28/fixed/struct.FixedI32.html#method.wrapping_round_ties_even [f-wrtte-1-28]: https://docs.rs/fixed/~1.28/fixed/struct.FixedI32.html#method.wrapping_round_ties_to_even [feat-exp-1-28]: https://docs.rs/fixed/~1.28/fixed/index.html#experimental-optional-features [nt-0-2-co]: https://docs.rs/num-traits/^0.2/num_traits/identities/trait.ConstOne.html [nt-0-2-cz]: https://docs.rs/num-traits/^0.2/num_traits/identities/trait.ConstZero.html [nt-0-2-fb]: https://docs.rs/num-traits/^0.2/num_traits/ops/bytes/trait.FromBytes.html [nt-0-2-tb]: https://docs.rs/num-traits/^0.2/num_traits/ops/bytes/trait.ToBytes.html [tf-1-28]: https://docs.rs/fixed/~1.28/fixed/traits/trait.Fixed.html ### Other releases Details on other releases can be found in [*RELEASES.md*]. [*RELEASES.md*]: https://gitlab.com/tspiteri/fixed/blob/master/RELEASES.md ## Quick examples ```rust 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::(), 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. ```rust 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. ```rust 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][*fixed* crate]. To use it in your crate, add it as a dependency inside [*Cargo.toml*]: ```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*]: ```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.) [`f128`]: https://doc.rust-lang.org/nightly/std/primitive.f128.html [`f16`]: https://doc.rust-lang.org/nightly/std/primitive.f16.html ## 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][half::f16] and [half]::[bf16][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][LICENSE-APACHE] or * the [MIT License][LICENSE-MIT] 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. [*Cargo.toml*]: https://doc.rust-lang.org/cargo/guide/dependencies.html [*arbitrary* crate]: https://crates.io/crates/arbitrary [*borsh* crate]: https://crates.io/crates/borsh [*cordic* crate]: https://crates.io/crates/cordic [*fixed* crate]: https://crates.io/crates/fixed [*half* crate]: https://crates.io/crates/half [*num-traits* crate]: https://crates.io/crates/num-traits [*serde* crate]: https://crates.io/crates/serde [*typenum* crate]: https://crates.io/crates/typenum [CORDIC]: https://en.wikipedia.org/wiki/CORDIC [FixedI32]: https://docs.rs/fixed/~1.28/fixed/struct.FixedI32.html [FixedU32]: https://docs.rs/fixed/~1.28/fixed/struct.FixedU32.html [LICENSE-APACHE]: https://www.apache.org/licenses/LICENSE-2.0 [LICENSE-MIT]: https://opensource.org/licenses/MIT [U0]: https://docs.rs/fixed/~1.28/fixed/types/extra/type.U0.html [U12]: https://docs.rs/fixed/~1.28/fixed/types/extra/type.U12.html [U24]: https://docs.rs/fixed/~1.28/fixed/types/extra/type.U24.html [U32]: https://docs.rs/fixed/~1.28/fixed/types/extra/type.U32.html [`Binary`]: https://doc.rust-lang.org/nightly/core/fmt/trait.Binary.html [`Display`]: https://doc.rust-lang.org/nightly/core/fmt/trait.Display.html [`Error`]: https://doc.rust-lang.org/nightly/std/error/trait.Error.html [`FixedI128`]: https://docs.rs/fixed/~1.28/fixed/struct.FixedI128.html [`FixedI16`]: https://docs.rs/fixed/~1.28/fixed/struct.FixedI16.html [`FixedI32`]: https://docs.rs/fixed/~1.28/fixed/struct.FixedI32.html [`FixedI64`]: https://docs.rs/fixed/~1.28/fixed/struct.FixedI64.html [`FixedI8`]: https://docs.rs/fixed/~1.28/fixed/struct.FixedI8.html [`FixedU128`]: https://docs.rs/fixed/~1.28/fixed/struct.FixedU128.html [`FixedU16`]: https://docs.rs/fixed/~1.28/fixed/struct.FixedU16.html [`FixedU32`]: https://docs.rs/fixed/~1.28/fixed/struct.FixedU32.html [`FixedU64`]: https://docs.rs/fixed/~1.28/fixed/struct.FixedU64.html [`FixedU8`]: https://docs.rs/fixed/~1.28/fixed/struct.FixedU8.html [`FromFixed`]: https://docs.rs/fixed/~1.28/fixed/traits/trait.FromFixed.html [`FromStr`]: https://doc.rust-lang.org/nightly/core/str/trait.FromStr.html [`From`]: https://doc.rust-lang.org/nightly/core/convert/trait.From.html [`I20F12`]: https://docs.rs/fixed/~1.28/fixed/types/type.I20F12.html [`I4F12`]: https://docs.rs/fixed/~1.28/fixed/types/type.I4F12.html [`I4F4`]: https://docs.rs/fixed/~1.28/fixed/types/type.I4F4.html [`Into`]: https://doc.rust-lang.org/nightly/core/convert/trait.Into.html [`LosslessTryFrom`]: https://docs.rs/fixed/~1.28/fixed/traits/trait.LosslessTryFrom.html [`LosslessTryInto`]: https://docs.rs/fixed/~1.28/fixed/traits/trait.LosslessTryInto.html [`LossyFrom`]: https://docs.rs/fixed/~1.28/fixed/traits/trait.LossyFrom.html [`LossyInto`]: https://docs.rs/fixed/~1.28/fixed/traits/trait.LossyInto.html [`LowerExp`]: https://doc.rust-lang.org/nightly/core/fmt/trait.LowerExp.html [`LowerHex`]: https://doc.rust-lang.org/nightly/core/fmt/trait.LowerHex.html [`Octal`]: https://doc.rust-lang.org/nightly/core/fmt/trait.Octal.html [`ParseFixedError`]: https://docs.rs/fixed/~1.28/fixed/struct.ParseFixedError.html [`ToFixed`]: https://docs.rs/fixed/~1.28/fixed/traits/trait.ToFixed.html [`U20F12`]: https://docs.rs/fixed/~1.28/fixed/types/type.U20F12.html [`UpperExp`]: https://doc.rust-lang.org/nightly/core/fmt/trait.UpperExp.html [`UpperHex`]: https://doc.rust-lang.org/nightly/core/fmt/trait.UpperHex.html [`az`]: https://docs.rs/az/^1/az/index.html [`bytemuck`]: https://docs.rs/bytemuck/^1/bytemuck/index.html [`checked_from_num`]: https://docs.rs/fixed/~1.28/fixed/struct.FixedI32.html#method.checked_from_num [`from_num`]: https://docs.rs/fixed/~1.28/fixed/struct.FixedI32.html#method.from_num [`from_str_binary`]: https://docs.rs/fixed/~1.28/fixed/struct.FixedI32.html#method.from_str_binary [`from_str_hex`]: https://docs.rs/fixed/~1.28/fixed/struct.FixedI32.html#method.from_str_hex [`from_str_octal`]: https://docs.rs/fixed/~1.28/fixed/struct.FixedI32.html#method.from_str_octal [`i32`]: https://doc.rust-lang.org/nightly/core/primitive.i32.html [`lit`]: https://docs.rs/fixed/~1.28/fixed/struct.FixedI32.html#method.lit [`to_num`]: https://docs.rs/fixed/~1.28/fixed/struct.FixedI32.html#method.to_num [`u32`]: https://doc.rust-lang.org/nightly/core/primitive.u32.html [half::bf16]: https://docs.rs/half/^2/half/struct.bf16.html [half::f16]: https://docs.rs/half/^2/half/struct.f16.html [half]: https://docs.rs/half/^2/half/index.html