Introducing glam and mathbench

glam is a simple and fast Rust linear algebra library for games and graphics.

mathbench is a set of unit tests and benchmarks comparing the performance of glam with the popular Rust linear algebra libraries cgmath and nalgebra .

The following is a table of benchmarks produced by mathbench comparing glam performance to cgmath and nalgebra on f32 data.

benchmark glam cgmath nalgebra euler 2d 9.063 us 8.977 us 27.94 us euler 3d 15.78 us 29.52 us 195.4 us mat2 determinant 1.3140 ns 1.0528 ns 1.0825 ns mat2 inverse 2.0524 ns 2.6565 ns 2.7056 ns mat2 mul mat2 2.1091 ns 3.0495 ns 3.6305 ns mat2 transform vec2 2.1865 ns 2.4035 ns 6.8877 ns mat2 transpose 0.7117 ns 1.3231 ns 1.8297 ns mat3 determinant 2.2130 ns 2.5348 ns 2.5766 ns mat3 inverse 7.7296 ns 7.9519 ns 9.3100 ns mat3 mul mat3 4.9371 ns 9.4619 ns 8.3076 ns mat3 transform vec3 2.2893 ns 4.0822 ns 8.0838 ns mat3 transpose 2.0035 ns 3.5175 ns 8.9548 ns mat4 determinant 7.9524 ns 12.2581 ns 54.2954 ns mat4 inverse 21.0455 ns 43.6983 ns 55.8788 ns mat4 mul mat4 6.7517 ns 9.2621 ns 16.2498 ns mat4 transform vec4 2.5298 ns 3.5734 ns 4.1222 ns mat4 transpose 2.7026 ns 8.0926 ns 11.5411 ns quat conjugate 0.8866 ns 1.8263 ns 1.7650 ns quat mul quat 2.8647 ns 5.6678 ns 5.4807 ns quat transform vec3 4.2107 ns 6.5798 ns 7.0398 ns vec3 cross 2.0677 ns 2.8890 ns 2.8725 ns vec3 dot 1.3789 ns 1.6669 ns 1.6666 ns vec3 length 2.0412 ns 2.0508 ns 88.1277 ns vec3 normalize 4.0433 ns 4.1260 ns 87.2455 ns

These benchmarks were performed on my Intel i7-4710HQ CPU on Linux. They were compiled with the stable 1.36 Rust compiler. Lower (better) numbers are highlighted. I hope it’s clear that glam is outperforming cgmath and nalgebra in most of these benchmarks.

Generally glam functionality is the same as the libraries for the functions tested here. One exception is cgmath and nalgebra matrix inverse functions return an Option type which is None if the matrix wasn’t invertible, glam assumes the input was invertible and returns a Mat4 . There may be other differences I’m not aware of.

Like glam , cgmath also targets games and graphics. nalgebra is a general purpose linear algebra library which has a broader audience than games and graphics programmers.

There are some strange outliers in the mathbench results, primarily nalgebra vec3 length and vec3 normalize . I haven’t looked into why they’re slow, if it’s a problem with the bench or nalgebra but given dot is perfectly fast it seems odd, the only difference between dot and length should be a square root.

The reason glam is faster is primarily due to it using SIMD internally for all types with the exception of Vec2 .

You can see a similar table at the bottom of this post with glam SIMD support disabled. TL;DR it’s similar performance to cgmath which is expected.

See the full mathbench report for more detailed results.

Why write another math library?

My goal writing glam and the trade offs I made are primarily being focused on good f32 performance by utilising SIMD and a simple API. This is at the expense of genericity, there is no Vector3<T> type, just Vec3 which is f32 based. It would be possible to support f64 or generic types in the future but it’s not a high priority for me right now, in my experience most games only use f32 .

glam also avoids baking mathematical correctness into the type system. There are no Point3 or UnitQuaternion types for example, it is up to the programmer if they want their Vector3 to behave as a point or a vector or if their quaternion is normalised. This decision sacrifices enforced runtime correctness via the type system for performance and a simple API.

SIMD

I noticed when optimising a path tracer was that I got some good performance wins from implementing a Vector3 with SSE2 (see SIMD path tracing). I wanted to take that further and build a full library that utilised SIMD for vector, matrix and quaternion types. I’d seen Rust math libraries that use SIMD for some functions but I hadn’t see one that uses SIMD vectors for storage.

SIMD primer

SIMD stands for Single Instruction Multiple Data. The multiple data part varies depending on CPU. Recent Intel CPUs have SIMD instruction sets that can operate on 128-bit (SSE), 256-bit (AVX) and 512-bit data sizes. That is 4, 8 and 16 f32 values processed with a single instruction.

Using a simple summation as an example, the difference between the scalar and SIMD operations is illustrated below.

With conventional scalar operations, four add instructions must be executed one after another to obtain the sums. Meanwhile, SIMD uses only one add instruction to achieve the same result. Requiring fewer instructions to process a given mass of data, SIMD operations yield higher efficiency than scalar operations.

SIMD values are typically stored in a special type. In the case of SSE2, the __m128 type is used to store 4 floats. One significant difference between this type and say [f32; 4] is __m128 is 16 byte aligned. While you can load unaligned data into SSE2 registers, this is slower than loading aligned data.

There are some down sides to using SIMD for a vector math library. One thing you may notice with the above example is there are 4 values, while a Vec3 only contains 3 values. The 4th value is effectively ignored and is just wasted space.

Another down side is while SIMD is great for adding two vectors together, “horizontal operations” are awkward and not so fast. An example of a horizontal operation would be to sum the values A0 , A1 , A2 and A3 , or more commonly in a vector math library, performing a dot product is a horizontal operation.

If your CPU supported wider instruction sets such as AVX2, a Vec3 can’t take advantage of it, after all it only has 3 floats even if your instruction set can operate on 8.

The most effective way to take advantage of SIMD is to structure your data as Structure of Arrays (SoA) and feed that through the widest SIMD vectors you have on your CPU. What I mean by that is instead of an Array of Structs (AoS):

struct Vec3AoS { xyz : Vec < ( f32 , f32 , f32 ) > , }

You store each component of the vector in it’s own array:

struct Vec3SoA { x : Vec < f32 > , y : Vec < f32 > , z : Vec < f32 > , }

Then load 4 x , y and z values at a time and add them. glam does not help you with that.

For more information on doing SIMD the “right way” I recommend reading through this GDC2015 presentation on SIMD at Insomniac Games.

You should always be able to out perform glam by rewriting your data structures to be SoA and using SIMD instructions directly, I still think it’s valuable to have a math library that performs well if you haven’t done that work.

Although 25% of glam::Vec3 is wasted space, it still performs better than the equivalent scalar code since we are still operating on 3 float with one instructions a lot of the time. And even though horizontal operations such as dot products are awkward, they still have slightly better performance than the scalar equivalent as you can see in the mathbench results above for Vec3 dot .

API design

Built around SIMD

Many glam types use __m128 for data storage internally to get the best SSE2 performance. This has a few implications for API design.

The __m128 is opaque. You do not have direct access to the underlying f32 components. All component data must be read and written via accessors. For example this is how the get and set of the y component of a glam::Vec3 are implemented:

impl Vec3 { #[inline] pub fn y ( self ) -> f32 { unsafe { _ mm_cvtss_f32 ( _ mm_shuffle_ps ( self . 0 , self . 0 , 0b01_01_01_01 )) } } #[inline] pub fn set_y ( & mut self , y : f32 ) { unsafe { let mut t = _ mm_move_ss ( self . 0 , _ mm_set_ss ( y )); t = _ mm_shuffle_ps ( t , t , 0b11_10_00_00 ); self . 0 = _ mm_move_ss ( t , self . 0 ); } } }

These getter and setter methods are using SSE2 intrinsics to load and store scalar values to and from the y lane of the __m128 vector.

Avoiding complexity

I wanted to come up with an API which is very low friction for developers to use. Something that covers the common needs for someone working in games and graphics and doesn’t require much effort learn. I also wanted something to was easy for me to write.

From the outset I wanted to avoid using traits and generics. Traits and generics are wonderful language features, but I didn’t see a great reason to use them in glam .

Using SIMD types for storage would complicate generics, as I think 2 generic types would be required, a scalar type (e.g. f32 ) and a generic for the storage type (e.g. __m128 if available), that already sounds complicated!

That’s not to say that glam will never contain vectors of integer or generic types, but for the sake of simplicity I wanted to avoid them for a while until the API feels stable and there’s a compelling reason to use them.

Feeling Rusty

I also wanted to make glam “Rusty”. One thing I’ve noticed about the Rust standard library is everything is a method, for example sin is a method on f32 and f64 . It feels a bit weird coming from other languages. It turns out there are reasons for this, from the Rust API guidelines:

Methods have numerous advantages over functions: They do not need to be imported or qualified to be used: all you need is a value of the appropriate type.

Their invocation performs auto-borrowing (including mutable borrows).

They make it easy to answer the question “what can I do with a value of type T” (especially when using rustdoc).

They provide self notation, which is more concise and often more clearly conveys ownership distinctions.

Things like a.lerp(b, 0.5) feel a bit odd, but I could always add functions for this kind of thing in addition to the methods.

I’ve tried to follow the Rust API Guidelines in general.

Mathematical conventions

glam interprets vectors as column matrices (also known as column vectors) meaning when transforming a vector with a matrix the matrix goes on the left, e.g. v' = Mv . DirectX uses row vectors, OpenGL uses column vectors. There are pros and cons to both.

Matrices are stored in column major format. Each column vector is stored in contiguous memory.

Rotations follow the left-hand rule.

Some libraries support both column and row vectors my preference with glam is to pick one convention and stick with it (of course I didn’t do that, I started with row vectors and switched to column vectors).

Test coverage

I wanted to aim for 100% test coverage, especially because there are multiple implementations of many types depending on whether SIMD is available or not.

To determine if I actually did have 100% test coverage I’ve been using a cargo plugin called tarpaulin . It only supports x86_64 processors running Linux and does give some false negatives, but it’s been pretty good for my needs. I have it integrated into my travis-ci build and posting results to coveralls.io .

According to coveralls.io glam has 87% test coverage, I think the real figure is probably a bit higher due to tarpaulin reporting some lines being untested when they actually are.

Performance

If you want to say your library is fast, you better be measuring performance!

Micro-benchmarking with Criterion.rs

I’ve primarily been monitoring the performance of glam using the criterion crate. criterion fills a similar niche to Rust’s built-in bench but unlike bench it works on stable. It also has a bunch of other nice features.

Micro-benchmarking is a useful indication of performance, but it’s not necessarily going to tell you the full story of what code will perform like under less artificial conditions (benchmarks tend to be quite limited in scope). All the same, they’re definitely better than no metrics and I’ve found them to be a pretty reliable source of performance information when optimising some code.

Comparing with other libraries

glam has a bunch of criterion benchmarks that are useful for optimising glam but they don’t tell me how glam performs relative to other math libraries.

I created mathbench for this purpose. mathbench is a collection of unit tests and benchmarks. The unit tests are to check that glam is producing the same output as other libraries (with some floating point tolerance) and the benchmarks compare performance. The table at the top of this post is produced from these benchmarks.

mathbench could also be of use to other library authors to see how their library is performing and for performing optimisations.

One of the nice features of criterion is you can set up one benchmark that runs multiple functions on the same data and produces a report comparing the results. Here’s an example from mathbench :

fn bench_mat4_mul_mat4 ( c : & mut Criterion ) { use criterion :: Benchmark ; use std :: ops :: Mul ; c .bench ( "mat4 mul mat4" , Benchmark :: new ( "glam" , | b | { use glam :: Mat4 ; bench_binop! ( b , op => mul_mat4 , ty1 => Mat4 , ty2 => Mat4 ) }) .with_function ( "cgmath" , | b | { use cgmath :: Matrix4 ; bench_binop! ( b , op => mul , ty1 => Matrix4 < f32 > , ty2 => Matrix4 < f32 > ) }) .with_function ( "nalgebra" , | b | { use nalgebra :: Matrix4 ; bench_binop! ( b , op => mul , ty1 => Matrix4 < f32 > , ty2 => Matrix4 < f32 > ) }), ); }

If you have gnuplot installed criterion will produce reports with charts like the violin plot below:

Viewing assembly

While mathbench is a library, most of it’s utility is from the tests and benchmarks. One use I have had for the library though is creating public functions for inspecting assembly with cargo-asm . This is primarily because it was the easiest way to get cargo-asm to show me something.

For example, in mathbench I’ve defined a simple wrapper function for glam::Mat4 * glam::Vec4 :

pub fn glam_mat4_mul_vec4 ( lhs : & glam :: Mat4 , rhs : & glam :: Vec4 ) -> glam :: Vec4 { * lhs * * rhs }

And then run cargo asm mathbench::glam_mat4_mul_vec4 to get the following output:

mov rax, rdi movaps xmm0, xmmword, ptr, [rdx] movaps xmm1, xmm0 shufps xmm1, xmm0, 0 mulps xmm1, xmmword, ptr, [rsi] movaps xmm2, xmm0 shufps xmm2, xmm0, 85 mulps xmm2, xmmword, ptr, [rsi, +, 16] addps xmm2, xmm1 movaps xmm1, xmm0 shufps xmm1, xmm0, 170 mulps xmm1, xmmword, ptr, [rsi, +, 32] addps xmm1, xmm2 shufps xmm0, xmm0, 255 mulps xmm0, xmmword, ptr, [rsi, +, 48] addps xmm0, xmm1 movaps xmmword, ptr, [rdi], xmm0 ret

It’s a pretty convenient way of viewing assembly, with the catch that you can’t see #[inline] blocks.

Profiling benchmarks

You can run individual benchmarks through a profiler to get a better understanding of their performance. When you run cargo bench it prints the path to each benchmark executable that it is running. You can pass the same parameters to this executable that you pass to cargo bench .

For example, if I wanted to profile the glam Mat4 inverse method I could run this command via my preferred profiler:

target/release/deps/mat4bench-e528128e2a9ccbc9 "mat4 inverse/glam"

There will be a bit of noise from the benchmarking code but it’s usually not too hard to work out what is what. If your code is being inlined then it will probably appear inside criterion::Bencher::iter .

What next for glam

I consider glam to be a minimum viable math library right now, so I’ll slowly add more functionality over time. That will depend a bit on if other people adopt it and if so what features they are missing.

The main outstanding thing for me right now is documenting what is there.

There are of course more optimisations that could be done.

I’m interested in using the packed_simd crate instead of using SSE2 directly as packed_simd supports multiple architectures out of the box. However packed_simd currently requires nightly and I’m not sure what the development status of this crate is right now.

Inspirations

There were many inspirations for the interface and internals of glam from the Rust and C++ worlds. In no particular order:

The Rust ecosystem is awesome

Writing Rust is purely recreational for me. My day job for the last 14 years has been writing C++ game engine and gameplay code.

It boggles my mind how much great tooling I have at my fingertips with Rust that either comes with the Rust installer or is easily accessible. Things like:

A common build system via cargo build

A common packaging system via crates.io and cargo update

Built in code formatting via cargo fmt

Built in linting via cargo clippy

Built in testing framework via cargo test

High quality third party micro-benchmarking - criterion and cargo bench

and Code coverage - tarpaulin and coveralls.io

and Continuous integration - travis-ci

Setting up and maintaining this kind of infrastructure for a C++ project is a huge amount of work, especially for something I’m just doing in my evenings.

Rust is pretty new and there are certainly a lot of gaps in the tooling compared to established languages like C++, but the out of the box experience is far far superior.

glam without SSE2

glam ’s performance is around on par with cgmath if SSE2 is disabled using the scalar-math feature. Some glam functions got faster without SIMD.