Fixed Math

This library implements analytic/trigonometric functions for fixed point numbers.

Implemented functions:

• sqrt from trait Sqrt
  • Sqrt is not implemented for FixedI* types with 0 integer bits since that would overflow.
• sin_cos, sin, cos, tan from trait SinCos
  • SinCos is not implemented for fixed numbers with less than 7 integer bits because there are table values that would overflow on those types.
  • All calculations are made in degrees
    • except that there is a sin_cos_rad function which is very imprecise; check source code to see why, feel free to fix it (it is not important for me)

Optional Features

• std : std feature of fixed, there is no proper no-std support yet...
• right_angles : by using an additional match, sin_cos gives exact result from an angle that is a multiple of 90 degrees; could be useful on a grid.

Examples

There are traits and standalone functions, see examples on how to use them.

Errors

Check the examples to see about how much error this implementation produces.

Usually sqrt has an error of around 1-2 Delta.
(Delta = the distance to the next representable number)

sin_cos may produce bigger errors, around 1-2 decimal places.

Benchmarks

You can check or run the benchmarks in benches.

Here are some conclusions I've got to:

SinCos

2022-09-28

Calculation time for sin_cos varies with the fixed number's byte size.

• I10F6: ~ 8ns
• I16F16: ~ 9ns
• I32F32: ~ 18ns
• I32F96: ~ 210ns

Notes:

• there are many different int bit / frac bit combinations; I did not test them
  (int bits must be >= 10 (but maybe I can do something to relax that further))
• these are all calculations in degrees
• code was compiled with native cpu features
• go for FixedI32 instead of FixedI16 unless you are limited by memory much
• I did a benchmark in the same style on cordic's sin_cos on FixedI64
  • keep in mind that cordic works with radians, I used the same angle values
  • so they can take sin_cos of a lot bigger angle on the same number representation size
  • this crate was about 1.5-2 times faster on same angle sizes

2023-07-30

System info:

                   -`                    
                  .o+`                   --------
                 `ooo/                   OS: Arch Linux x86_64
                `+oooo:                  Host: X570 AORUS ELITE -CF
               `+oooooo:                 Kernel: 6.4.7-arch1-1
               -+oooooo+:                
             `/:-:++oooo+:               
            `/++++/+++++++:              Shell: fish 3.6.1
           `/++++++++++++++:             Resolution: 3840x2160
          `/+++ooooooooooooo/`           DE: Hyprland
         ./ooosssso++osssssso+`          
        .oossssso-````/ossssss+`         
       -osssssso.      :ssssssso.        Terminal: WezTerm
      :osssssss/        osssso+++.       CPU: AMD Ryzen 7 5800X (16) @ 3.800GHz
     /ossssssss/        +ssssooo/-       GPU: AMD ATI Radeon RX 7900 XT/7900 XTX
   `/ossssso+/:-        -:/+osssso+-     Memory: 32014MiB
  `+sso+:-`                 `.-/+oso:
 `++:.                           `-/+/
 .`                                 `/

Performance is different now and without native cpu features:

• I10F6: ~ 430ps
• I16F16: ~ 15ns
• I32F32: ~ 25ns
• I32F96: ~ 255ns

With native cpu features : RUSTFLAGS="-C target-cpu=native" cargo bench:

• I10F6: ~ 420ps
• I16F16: ~ 8ns
• I32F32: ~ 22ns
• I32F96: ~ 222ns

Notes:

• performance regressed at I32F32 and I32F96
• performance improved a lot at I10F6, so we might consider using 16bit fixed point values where we do not need much precision
• cordic benchmark not checked...

License

All code in this repository is dual-licensed under either:

• MIT License (LICENSE-MIT or http://opensource.org/licenses/MIT)
• Apache License, Version 2.0 (LICENSE-APACHE or http://www.apache.org/licenses/LICENSE-2.0)

at your option.

Contributing

Unless you explicitly state otherwise, any contribution intentionally submitted for inclusion in the work by you, as defined in the Apache-2.0 license, shall be dual licensed as above, without any additional terms or conditions.

Attribution

We use code modified from cordic, licensed as BSD-3-Clause:

• LICENSE_CORDIC.