# 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. These types can have `Frac` fractional bits, where 0 ≤ `Frac` ≤ n and n is the total number of bits. When `Frac` = 0, the fixed-point number behaves like an n-bit integer. When `Frac` = n, the value x lies in the range −0.5 ≤ x < 0.5 for signed numbers, and in the range 0 ≤ x < 1 for unsigned numbers. 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 they are supported by the Rust compiler. The main features are * Representation of 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 general analytic functions. * No algebraic functions are provided, for example no `sqrt` or `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 [*fixed-sqrt* crate] provides the square root operation. 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 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`]. * 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`] and [`UpperHex`]. The output is rounded to the nearest, with ties rounded to even. ## What’s new ### Version 1.0.0 news (unreleased) * The crate now requires rustc version 1.43.0 or later. * All deprecated items were removed. * The [`PHI`][phi-1-0] and [`FRAC_1_PHI`][f1phi-1-0] constants were added to the [`consts`][cons-1-0] module and as [associated constants][f-phi-1-0] for fixed-point types. [cons-1-0]: https://tspiteri.gitlab.io/fixed/dev/fixed/consts/index.html [f-phi-1-0]: https://tspiteri.gitlab.io/fixed/dev/fixed/struct.FixedI32.html#associatedconstant.PHI [f1phi-1-0]: https://tspiteri.gitlab.io/fixed/dev/fixed/consts/constant.FRAC_1_PHI.html [phi-1-0]: https://tspiteri.gitlab.io/fixed/dev/fixed/consts/constant.PHI.html ### Version 0.5.6 news (2020-05-01) Other news in this release: * The following method were added to signed fixed-point types and to the [`FixedSigned`][tfs-0-5-6] trait: * [`checked_signum`][f-csig-0-5-6], [`saturating_signum`][f-ssig-0-5-6], [`wrapping_signum`][f-wsig-0-5-6], [`overflowing_signum`][f-osig-0-5-6] * The [`LossyFrom`][lf-0-5-6] and [`LossyInto`][li-0-5-6] traits were added to [the prelude][pre-0-5-6]. * Casts deprecated in version 0.3.1 of the *az* crate were marked as deprecated. [f-csig-0-5-6]: https://docs.rs/fixed/0.5.6/fixed/struct.FixedI32.html#method.checked_signum [f-osig-0-5-6]: https://docs.rs/fixed/0.5.6/fixed/struct.FixedI32.html#method.overflowing_signum [f-ssig-0-5-6]: https://docs.rs/fixed/0.5.6/fixed/struct.FixedI32.html#method.saturating_signum [f-wsig-0-5-6]: https://docs.rs/fixed/0.5.6/fixed/struct.FixedI32.html#method.wrapping_signum [lf-0-5-6]: https://docs.rs/fixed/0.5.6/fixed/traits/trait.LossyFrom.html [li-0-5-6]: https://docs.rs/fixed/0.5.6/fixed/traits/trait.LossyInto.html [pre-0-5-6]: https://docs.rs/fixed/0.5.6/fixed/prelude/index.html [tfs-0-5-6]: https://docs.rs/fixed/0.5.6/fixed/traits/trait.FixedSigned.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][`FixedI32`]<[U12][`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. ## 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 = "0.5.6" ``` The *fixed* crate requires rustc version 1.43.0 or later. ## Optional features The *fixed* crate has four optional feature: 1. `az`, disabled by default. This implements the cast traits provided by the [*az* crate]. 2. `f16`, disabled by default. This provides conversion to/from [`f16`] and [`bf16`]. This features requires the [*half* 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`]. To enable features, you can add the dependency like this to [*Cargo.toml*]: ```toml [dependencies.fixed] version = "0.5.6" features = ["f16", "serde"] ``` ## 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 [*az* crate]: https://crates.io/crates/az [*fixed* crate]: https://crates.io/crates/fixed [*fixed-sqrt* crate]: https://crates.io/crates/fixed-sqrt [*half* crate]: https://crates.io/crates/half [*serde* crate]: https://crates.io/crates/serde [*typenum* crate]: https://crates.io/crates/typenum [LICENSE-APACHE]: https://www.apache.org/licenses/LICENSE-2.0 [LICENSE-MIT]: https://opensource.org/licenses/MIT [`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/0.5.6/fixed/struct.FixedI128.html [`FixedI16`]: https://docs.rs/fixed/0.5.6/fixed/struct.FixedI16.html [`FixedI32`]: https://docs.rs/fixed/0.5.6/fixed/struct.FixedI32.html [`FixedI64`]: https://docs.rs/fixed/0.5.6/fixed/struct.FixedI64.html [`FixedI8`]: https://docs.rs/fixed/0.5.6/fixed/struct.FixedI8.html [`FixedU128`]: https://docs.rs/fixed/0.5.6/fixed/struct.FixedU128.html [`FixedU16`]: https://docs.rs/fixed/0.5.6/fixed/struct.FixedU16.html [`FixedU32`]: https://docs.rs/fixed/0.5.6/fixed/struct.FixedU32.html [`FixedU64`]: https://docs.rs/fixed/0.5.6/fixed/struct.FixedU64.html [`FixedU8`]: https://docs.rs/fixed/0.5.6/fixed/struct.FixedU8.html [`FromFixed`]: https://docs.rs/fixed/0.5.6/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/0.5.6/fixed/types/type.I20F12.html [`I4F12`]: https://docs.rs/fixed/0.5.6/fixed/types/type.I4F12.html [`I4F4`]: https://docs.rs/fixed/0.5.6/fixed/types/type.I4F4.html [`Into`]: https://doc.rust-lang.org/nightly/core/convert/trait.Into.html [`LossyFrom`]: https://docs.rs/fixed/0.5.6/fixed/traits/trait.LossyFrom.html [`LossyInto`]: https://docs.rs/fixed/0.5.6/fixed/traits/trait.LossyInto.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/0.5.6/fixed/struct.ParseFixedError.html [`ToFixed`]: https://docs.rs/fixed/0.5.6/fixed/traits/trait.ToFixed.html [`U12`]: https://docs.rs/fixed/0.5.6/fixed/types/extra/type.U12.html [`U20F12`]: https://docs.rs/fixed/0.5.6/fixed/types/type.U20F12.html [`UpperHex`]: https://doc.rust-lang.org/nightly/core/fmt/trait.UpperHex.html [`bf16`]: https://docs.rs/half/^1/half/struct.bf16.html [`checked_from_num`]: https://docs.rs/fixed/0.5.6/fixed/struct.FixedI32.html#method.checked_from_num [`f16`]: https://docs.rs/half/^1/half/struct.f16.html [`from_num`]: https://docs.rs/fixed/0.5.6/fixed/struct.FixedI32.html#method.from_num [`from_str_binary`]: https://docs.rs/fixed/0.5.6/fixed/struct.FixedI32.html#method.from_str_binary [`from_str_hex`]: https://docs.rs/fixed/0.5.6/fixed/struct.FixedI32.html#method.from_str_hex [`from_str_octal`]: https://docs.rs/fixed/0.5.6/fixed/struct.FixedI32.html#method.from_str_octal [`to_num`]: https://docs.rs/fixed/0.5.6/fixed/struct.FixedI32.html#method.to_num [const generics]: https://github.com/rust-lang/rust/issues/44580