273 lines
12 KiB
Markdown
273 lines
12 KiB
Markdown
<!-- Copyright © 2018–2019 Trevor Spiteri -->
|
||
|
||
<!-- Copying and distribution of this file, with or without
|
||
modification, are permitted in any medium without royalty provided the
|
||
copyright notice and this notice are preserved. This file is offered
|
||
as-is, without any warranty. -->
|
||
|
||
# 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` ≤ <i>n</i> and <i>n</i> is the total number of bits. When
|
||
`Frac` = 0, the fixed-point number behaves like an <i>n</i>-bit
|
||
integer. When `Frac` = <i>n</i>, the value <i>x</i> lies in the range
|
||
−0.5 ≤ <i>x</i> < 0.5 for signed numbers, and in the range
|
||
0 ≤ <i>x</i> < 1 for unsigned numbers.
|
||
|
||
Currently the [*typenum* crate] is used for the fractional bit count
|
||
`Frac`; it is planned to move to [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`.
|
||
|
||
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 0.5.0 news (2019-12-06)
|
||
|
||
* The *fixed* crate now requires rustc version 1.39.0 or later.
|
||
* The following methods were added to all fixed-point types and to
|
||
the [`Fixed`] trait:
|
||
* [`from_be_bytes`], [`from_le_bytes`], [`from_ne_bytes`]
|
||
* [`to_be_bytes`], [`to_le_bytes`], [`to_ne_bytes`]
|
||
* [`div_euclid_int`], [`rem_euclid_int`]
|
||
* [`checked_div_euclid_int`], [`checked_rem_euclid_int`]
|
||
* [`wrapping_div_euclid_int`], [`wrapping_rem_euclid_int`]
|
||
* [`overflowing_div_euclid_int`], [`overflowing_rem_euclid_int`]
|
||
|
||
[`Fixed`]: https://docs.rs/fixed/0.5.0/fixed/traits/trait.Fixed.html
|
||
[`checked_div_euclid_int`]: https://docs.rs/fixed/0.5.0/fixed/struct.FixedI32.html#method.checked_div_euclid_int
|
||
[`checked_rem_euclid_int`]: https://docs.rs/fixed/0.5.0/fixed/struct.FixedI32.html#method.checked_rem_euclid_int
|
||
[`div_euclid_int`]: https://docs.rs/fixed/0.5.0/fixed/struct.FixedI32.html#method.div_euclid_int
|
||
[`from_be_bytes`]: https://docs.rs/fixed/0.5.0/fixed/struct.FixedI32.html#method.from_be_bytes
|
||
[`from_le_bytes`]: https://docs.rs/fixed/0.5.0/fixed/struct.FixedI32.html#method.from_le_bytes
|
||
[`from_ne_bytes`]: https://docs.rs/fixed/0.5.0/fixed/struct.FixedI32.html#method.from_ne_bytes
|
||
[`overflowing_div_euclid_int`]: https://docs.rs/fixed/0.5.0/fixed/struct.FixedI32.html#method.overflowing_div_euclid_int
|
||
[`overflowing_rem_euclid_int`]: https://docs.rs/fixed/0.5.0/fixed/struct.FixedI32.html#method.overflowing_rem_euclid_int
|
||
[`rem_euclid_int`]: https://docs.rs/fixed/0.5.0/fixed/struct.FixedI32.html#method.rem_euclid_int
|
||
[`to_be_bytes`]: https://docs.rs/fixed/0.5.0/fixed/struct.FixedI32.html#method.to_be_bytes
|
||
[`to_le_bytes`]: https://docs.rs/fixed/0.5.0/fixed/struct.FixedI32.html#method.to_le_bytes
|
||
[`to_ne_bytes`]: https://docs.rs/fixed/0.5.0/fixed/struct.FixedI32.html#method.to_ne_bytes
|
||
[`wrapping_div_euclid_int`]: https://docs.rs/fixed/0.5.0/fixed/struct.FixedI32.html#method.wrapping_div_euclid_int
|
||
[`wrapping_rem_euclid_int`]: https://docs.rs/fixed/0.5.0/fixed/struct.FixedI32.html#method.wrapping_rem_euclid_int
|
||
|
||
#### Incompatible changes
|
||
|
||
* Deprecated methods and modules were removed.
|
||
|
||
### Version 0.4.6 news (2019-10-16)
|
||
|
||
* Conversions to/from [`bf16`] are now provided when the `f16`
|
||
option is enabled.
|
||
* The following methods are now `const` functions:
|
||
[`saturating_neg`], [`saturating_add`], [`saturating_sub`],
|
||
[`saturating_mul_int`], [`saturating_abs`]
|
||
* Support for casts using the [*az* crate] was added.
|
||
|
||
[`saturating_abs`]: https://docs.rs/fixed/0.5.0/fixed/struct.FixedI32.html#method.saturating_abs
|
||
[`saturating_add`]: https://docs.rs/fixed/0.5.0/fixed/struct.FixedI32.html#method.saturating_add
|
||
[`saturating_mul_int`]: https://docs.rs/fixed/0.5.0/fixed/struct.FixedI32.html#method.saturating_mul_int
|
||
[`saturating_sub`]: https://docs.rs/fixed/0.5.0/fixed/struct.FixedI32.html#method.saturating_sub
|
||
[`saturating_neg`]: https://docs.rs/fixed/0.5.0/fixed/struct.FixedI32.html#method.saturating_neg
|
||
|
||
### 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::<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
|
||
<code>[FixedI32][`FixedI32`]<[U12][`U12`]></code>. 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 = "0.5.0"
|
||
```
|
||
|
||
The *fixed* crate requires rustc version 1.39.0 or later.
|
||
|
||
## Optional features
|
||
|
||
The *fixed* crate has three 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].
|
||
|
||
To enable features, you can add the dependency like this to
|
||
[*Cargo.toml*]:
|
||
|
||
```toml
|
||
[dependencies.fixed]
|
||
version = "0.5.0"
|
||
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
|
||
[*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/std/fmt/trait.Binary.html
|
||
[`Display`]: https://doc.rust-lang.org/nightly/std/fmt/trait.Display.html
|
||
[`FixedI128`]: https://docs.rs/fixed/0.5.0/fixed/struct.FixedI128.html
|
||
[`FixedI16`]: https://docs.rs/fixed/0.5.0/fixed/struct.FixedI16.html
|
||
[`FixedI32`]: https://docs.rs/fixed/0.5.0/fixed/struct.FixedI32.html
|
||
[`FixedI64`]: https://docs.rs/fixed/0.5.0/fixed/struct.FixedI64.html
|
||
[`FixedI8`]: https://docs.rs/fixed/0.5.0/fixed/struct.FixedI8.html
|
||
[`FixedU128`]: https://docs.rs/fixed/0.5.0/fixed/struct.FixedU128.html
|
||
[`FixedU16`]: https://docs.rs/fixed/0.5.0/fixed/struct.FixedU16.html
|
||
[`FixedU32`]: https://docs.rs/fixed/0.5.0/fixed/struct.FixedU32.html
|
||
[`FixedU64`]: https://docs.rs/fixed/0.5.0/fixed/struct.FixedU64.html
|
||
[`FixedU8`]: https://docs.rs/fixed/0.5.0/fixed/struct.FixedU8.html
|
||
[`FromFixed`]: https://docs.rs/fixed/0.5.0/fixed/traits/trait.FromFixed.html
|
||
[`FromStr`]: https://doc.rust-lang.org/nightly/std/str/trait.FromStr.html
|
||
[`From`]: https://doc.rust-lang.org/nightly/std/convert/trait.From.html
|
||
[`I20F12`]: https://docs.rs/fixed/0.5.0/fixed/types/type.I20F12.html
|
||
[`I4F12`]: https://docs.rs/fixed/0.5.0/fixed/types/type.I4F12.html
|
||
[`I4F4`]: https://docs.rs/fixed/0.5.0/fixed/types/type.I4F4.html
|
||
[`Into`]: https://doc.rust-lang.org/nightly/std/convert/trait.Into.html
|
||
[`LossyFrom`]: https://docs.rs/fixed/0.5.0/fixed/traits/trait.LossyFrom.html
|
||
[`LossyInto`]: https://docs.rs/fixed/0.5.0/fixed/traits/trait.LossyInto.html
|
||
[`LowerHex`]: https://doc.rust-lang.org/nightly/std/fmt/trait.LowerHex.html
|
||
[`Octal`]: https://doc.rust-lang.org/nightly/std/fmt/trait.Octal.html
|
||
[`ToFixed`]: https://docs.rs/fixed/0.5.0/fixed/traits/trait.ToFixed.html
|
||
[`U12`]: https://docs.rs/fixed/0.5.0/fixed/types/extra/type.U12.html
|
||
[`U20F12`]: https://docs.rs/fixed/0.5.0/fixed/types/type.U20F12.html
|
||
[`UpperHex`]: https://doc.rust-lang.org/nightly/std/fmt/trait.UpperHex.html
|
||
[`bf16`]: https://docs.rs/half/^1/half/struct.bf16.html
|
||
[`checked_from_num`]: https://docs.rs/fixed/0.5.0/fixed/struct.FixedI32.html#method.checked_from_num
|
||
[`f16`]: https://docs.rs/half/^1/half/struct.f16.html
|
||
[`from_num`]: https://docs.rs/fixed/0.5.0/fixed/struct.FixedI32.html#method.from_num
|
||
[`from_str_binary`]: https://docs.rs/fixed/0.5.0/fixed/struct.FixedI32.html#method.from_str_binary
|
||
[`from_str_hex`]: https://docs.rs/fixed/0.5.0/fixed/struct.FixedI32.html#method.from_str_hex
|
||
[`from_str_octal`]: https://docs.rs/fixed/0.5.0/fixed/struct.FixedI32.html#method.from_str_octal
|
||
[`to_num`]: https://docs.rs/fixed/0.5.0/fixed/struct.FixedI32.html#method.to_num
|
||
[const generics]: https://github.com/rust-lang/rust/issues/44580
|