## Executive Summary

The problem facing the AI industry today is that many of the functions at the heart of machine learning processes were written over forty years ago when computer hardware and compiler technology were very different.

Our library, Doctor Syn, addresses this problem by using the three technologies of SIMD, multithreading and autovectorisation. We achieve 30x or more speedups over traditional libraries in C, C++, Rust and Fortran, without making the code platform or language specific.

Doctor Syn’s primary focus at present is to generate accurate polynomial approximations to key functions important to the execution of many programs. You are probably familiar with many of the functions we are targeting:

Rust Function calculates
f32/f64::sin $$\sin{x}$$
f32/f64::cos $$\cos{x}$$
f32/f64::atan2 $$\arctan{y/x}$$
f32/f64::exp $$e^x$$
f32/f64::ln $$\log{x}$$

While improving these functions has a lot of value, we are currently focusing most of our effort on statistical functions such as:

R Function distribution role calculates
dnorm normal pdf $$\frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x - \mu}{\sigma}\right)^2}$$
pnorm normal cdf $$\frac{1}{2}\left[1 + \operatorname{erf}\left( \frac{x-\mu}{\sigma\sqrt{2}}\right)\right]$$
qnorm normal quantile $$\mu+\sigma\sqrt{2} \operatorname{erf}^{-1}(2p-1)$$
rnorm normal random $$\operatorname{qnorm}(\operatorname{runif}(i))$$

These functions are used extensively in finance and bioinformatics to perform statistical inference, stochastic modelling, AI and Machine learning. For example, rnorm is a key part of many MCMC algorithms and variational techniques, as well as a key part of monte carlo simulations, such as those that may be used to solve stochastic differential equations.

Using this library, combined with parallel iterators, we generate more efficient versions of

• Numpy
• R
• GNU Octave

and many others.

We have also targeted new architectures like Arm SVE which do not fit the X86 model. We are working with the Isambard A64FX cluster to attempt to improve existing algorithms.

This approach to function generation should fit perfectly with the A64FX’s SVE architecture as SVE has a variable length SIMD architecture which will run the same binary on many different word-length machines. SVE requires Autovectorisation to work effectively.

## The AI sound barrier

While great effort has been expended in key function optimization, current techniques are unable to efficiently utilize modern compiler technology such as auto-vectorisation and thread based parallelism. Research in function approximation has focused on squeezing out the last half bit of precision of functions at the expense of making functions every more complex and very much slower.

In practice, machine learning algorithms can tolerate a large amount of error and giving users the ability to choose the level of accuracy that a function delivers as well as the domain of inputs can make those algorithms orders of magnitude faster.

For example, using 32 bit floating point instead of 64 bit often has a 4:1 performance advantage in compute and a 2:1 advantage in memory performance. With modern computers, the memory bandwidth is very often the limiting factor and finding smarter ways to represent data becomes the key to fast algorithms. If we know that a vector of numbers does not contain NaN values, then we can skip NaN checks on every calculation.

But primarily, we need to make our functions simple enough to be vectorisable - once we have achieved this, we get remarkable performance improvements.

## The challenge of vectorisation.

Existing functions will not vectorise primarily because:

• They are in shared or static libraries.
• They contain branches and look-up tables.

In the 1970’s when many of these functions were written, this was the state of the art. Today however, this is a problem: with the advances of modern vectorising processors, these implementations are substantially less efficient than they could be.

There is no short and simple fix to this problem either, these problems are fundamental ones that preclude any efficient vectorisation. The only viable approach to this problem is with substantial and novel changes such as the one that Doctor Syn proposes.

## Autovectorisation

In the past we have implemented fast functions in assembler or even written assembler to write functions in machine code. These days however, we often try to take a more “civilized” approach where possible. Assembler functions are fast, but difficult to read, difficult to improve, and difficult to generalize. Despite these problems, many such functions end up hanging around like a bad smell, and more are being produced by chip vendors for special architectures. Even excellent libraries like Sleef are done this way with machine specific intrinsics.

To solve the problems with these architecture specific assembly implementations, we try and write portable code; we wish our code to be able to run on both x86 architectures with SIMD as well as the new ARM SVE with variable sized registers. The way we achieve this is by trying to write code in such a way as that it will be automatically vectorised by modern compilers.

use std::arch::x86_64::*;

pub fn inc_doubles_simd(x: &mut [f64]) {
unsafe {
for x in x.chunks_exact_mut(4) {
let a = _mm256_loadu_pd(&x[0] as *const f64);
_mm256_storeu_pd(&mut x[0] as *mut f64, b);
}
for x in x.chunks_exact_mut(4).into_remainder() {
*x += 1.0;
}
}
}


we simply write:

fn inc_doubles_scalar(x: &mut [f64]) {
for x in x {
*x += 1;
}
}


This is much easier to read, works on all known hardware without modifications and does not specify a vector size, which might be variable.

Vectorisers are fickle beasts however. If the wind blows in the wrong direction, the compiler will often fail to vectorise or worse - vectorise in the IR and then convert the vector operations to a long series of library calls.

For example, the following rather innocent function, which absolutely should be vectorisable, converts itself into a series of function calls:

#include <math.h>

void vector_sin(double *d, int len) {
while (len--) {
*d = sin(*d);
++d;
}
}


Clang gives:

.LBB0_7:                                # =>This Inner Loop Header: Depth=1
vmovsd  xmm0, qword ptr [rbx + 8*rbp]   # xmm0 = mem[0],zero
call    sin
vmovsd  qword ptr [rbx + 8*rbp], xmm0
vmovsd  xmm0, qword ptr [rbx + 8*rbp + 8] # xmm0 = mem[0],zero
call    sin
vmovsd  qword ptr [rbx + 8*rbp + 8], xmm0
vmovsd  xmm0, qword ptr [rbx + 8*rbp + 16] # xmm0 = mem[0],zero
call    sin
vmovsd  qword ptr [rbx + 8*rbp + 16], xmm0
vmovsd  xmm0, qword ptr [rbx + 8*rbp + 24] # xmm0 = mem[0],zero
call    sin
vmovsd  qword ptr [rbx + 8*rbp + 24], xmm0
cmp     r14d, ebp
jne     .LBB0_7


Each call will take several hundred cycles.

## Making library functions that vectorise

Library functions make things bad for themselves by introducing branching. So even if we can inline the function, the functions will not vectorise. To get better accuracy, they divide the domain of a function - for example $$[-\pi, \pi]$$ for $$\sin(x)$$ into many small parts. This is often done using a switch statement which will not vectorise. Alternatives include using a lookup table of coefficients but many CPUs have not yet implemented an efficient gather operation which can do table lookups in reasonable time. The exception to this is GPUs, which do commonly have efficient gather but these are likely to hurt the cache performance unless you use non-temporal loads and stores.

Doctor Syn generates functions that are free of complex control flow which would inhibit vectorisation. The functions are all available as source code allowing inlining and as a result the chance of them inlining is much increased.

## Example - sampling from the normal distribution.

We tested some of our generated functions against one of the best stats distribution libraries in the Rust world - rand_distr.

Combined with rayon the parallel execution library this would have been the best choice for monte carlo experiments.

We started with a uniform random number generator, based on a xorshift hash and tested this against rust’s ThreadRng.

By using a hash of an integer index instead of a sequence, we are able to parallelise random number generation.

pub fn runif(index: usize) -> f64 {
let mut z = (index + 1) as u64 * 0x9e3779b97f4a7c15;
z = (z ^ (z >> 30)) * 0xbf58476d1ce4e5b9;
z = (z ^ (z >> 27)) * 0x94d049bb133111eb;
z = z ^ (z >> 31);
from_bits((z >> 2) | 0x3ff0000000000000_u64) - 1.0
}


We tested both single and multi-threaded versions of these functions - easy in Rust as it is a naturally multi-threaded language - on a four core X86 laptop.

Library Function ns per iteration (smaller is better)
Doctor Syn runif 0.8
Doctor Syn parallel runif 0.6
rand ThreadRnd::gen() 5.1
rand parallel ThreadRnd::gen() 2.1
R runif 35.0
Numpy numpy.random.uniform 35.0
C rand() * (1.0/RAND_MAX) -O3 6.0
C++ uniform_real_distribution -O3 13.6

So clearly, we do well against even the best Rust version and much better (over 30 times better) than R and Numpy.

Moving to normal random number generation, we use the quantile (or probit) function to shape the random variable - this is a very simple version good to about six decimal digits, more accurate versions using log and sqrt are also available.

fn qnorm(arg: fty) -> fty {
let scaled: fty = arg - 0.5;
let x = scaled;
let recip: fty = 1.0 / (x * x - 0.5 * 0.5);
let y: fty = (177186111.131545818686411653000483 as fty)
.mul_add(x * x, -219058235.58919835 as fty)
.mul_add(x * x, 117054121.857504129646289572504640 as fty)
.mul_add(x * x, -35345955.68660036 as fty)
.mul_add(x * x, 6623473.609141078534685775398250 as fty)
.mul_add(x * x, -796318.1973069897 as fty)
.mul_add(x * x, 61391.409088151006196662227193 as fty)
.mul_add(x * x, -2938.7971360761 as fty)
.mul_add(x * x, 83.911295471202339471921364 as fty)
.mul_add(x * x, 0.012702493639562371692090 as fty)
.mul_add(x * x, 1.856861340488065073103038 as fty)
.mul_add(x * x, -0.626662948075053 as fty)
* x;
y * recip
}

/// Use qnorm to shape the uniform random number.
pub fn rnorm(index: usize) -> f64 {
qnorm(runif(index) * 0.999 + 0.0005)
}

/// Parallel version in Rust.
#[target_feature(enable = "avx2,fma")]
unsafe fn test_par_rnorm(d: &mut [f64]) {
do_par(d, |d| *d = rnorm(ref_to_usize(d)));
}

Library Function ns per iteration (smaller is better)
Doctor Syn rnorm 2.4
Doctor Syn parallel rnorm 0.9
rand_distr Normal::sample() 6.9
rand_distr parallel Normal::sample() 1.7
R rnorm 65.0
Numpy numpy.random.uniform 60.4
C++ normal_distribution<double> -O3 31.0

So more than 60x speedup over the R and python versions on a four core laptop and about 30x for C++.

## Future work

With an implementation of just two Doctor Syn functions, we have shown a significant performance boost over even the best-in-class Rust distribution system. This excellent result on a toy example shows excellent promise for what the generalised Doctor Syn system is capable of.

Work has started on support for ARM SVE, The Doctor Syn method also provides flexibility in the accuracy of the solution it provides, and we are exploring super-accurate versions of our functions by using larger sizes, table lookups or fixed point integer arithmetic. Finally, we are working on the substantial task of full verification as well as generation of R, python and Octave libraries.

Atomic Increment is developing this technology in partnership with Embecosm. If you want to use this technology, get in touch. We are keen to develop industry partnerships with companies who require extra performance in their machine learning and computational processes.

andy@atomicincrement.com jeremy.bennett@embecosm.com