# Fixed-point numbers

The fixed crate provides 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 instead when the Rust compiler support for them is powerful enough.

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 conversions supported cover the following cases.

## 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 fixed-macro crate provides a convenient macro to write down fixed-point constants literally in the code.

```use fixed::types::I16F16;
use fixed_macro::fixed;

const NUM1: I16F16 = fixed!(12.75: I16F16);
let num2 = NUM1 + fixed!(13.125: I16F16);
assert_eq!(num2, 25.875);```

## 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 = "1.6"
``````

The fixed crate requires rustc version 1.47.0 or later.

## Optional features

The fixed crate has these 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`.
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]
version = "1.6"
features = ["f16", "serde"]
``````

## Experimental optional features

It is not considered a breaking change if 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.

There is one experimental feature:

1. `num-traits`, disabled by default. This implements some traits from the num-traits crate. (The plan is to upgrade this to an optional feature once the num-traits crate reaches version 1.0.0.)

## License

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

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.

## Modules

 consts Mathematical constants. prelude A prelude to import useful traits. traits Traits for conversions and for generic use of fixed-point numbers. types Type aliases for all supported fixed-point numbers.

## Macros

 const_fixed_from_int Defines constant fixed-point numbers from integer expressions.

## Structs

 FixedI8 An eight-bit fixed-point signed number with `Frac` fractional bits. FixedI16 A 16-bit fixed-point signed number with `Frac` fractional bits. FixedI32 A 32-bit fixed-point signed number with `Frac` fractional bits. FixedI64 A 64-bit fixed-point signed number with `Frac` fractional bits. FixedI128 A 128-bit fixed-point signed number with `Frac` fractional bits. FixedU8 An eight-bit fixed-point unsigned number with `Frac` fractional bits. FixedU16 A 16-bit fixed-point unsigned number with `Frac` fractional bits. FixedU32 A 32-bit fixed-point unsigned number with `Frac` fractional bits. FixedU64 A 64-bit fixed-point unsigned number with `Frac` fractional bits. FixedU128 A 128-bit fixed-point unsigned number with `Frac` fractional bits. ParseFixedError An error which can be returned when parsing a fixed-point number. Unwrapped Provides arithmetic operations that panic on overflow even when debug assertions are disabled. Wrapping Provides intentionally wrapped arithmetic on fixed-point numbers.

## Enums

 RadixParseFixedError An error which can be returned when parsing a fixed-point number with a given radix.