Chop - MATLAB code for rounding matrix elements to lower precision

About

chop is a MATLAB function for rounding the elements of a matrix to a lower precision arithmetic with one of several forms of rounding. Its intended use is for simulating arithmetic of different precisions (less than double) with various rounding modes. The input to chop should be single precision or double precision and the output will have the same type: the lower precision numbers are stored within a higher precision type.

The arithmetic formats supported are

• 'b', 'bfloat16' - bfloat16,
• 'h', 'half', 'fp16' - IEEE half precision (the default),
• 's', 'single', 'fp32' - IEEE single precision,
• 'd', 'double', 'fp64' - IEEE double precision,
• 'c', 'custom' - custom format.

Subnormal numbers can be supported or not, and in the latter case they are flushed to zero.

Several rounding modes are supported:

• Round to nearest using round to even last bit to break ties (the default).
• Round towards plus infinity (round up).
• Round towards minus infinity (round down).
• Round towards zero.
• Stochastic rounding - round to the next larger or next smaller floating-point number with probability proportional to the distance to those floating-point numbers.
• Stochastic rounding - round to the next larger or next smaller floating-point number with equal probability.

Optionally, each element of the rounded result has, with a specified probability defaulting to 0.5, a randomly chosen bit in its significand flipped. This option is useful for simulating soft errors

A further option causes the exponent limit for the specified arithmetic to be ignored, so overflow, underflow, or subnormal numbers will be produced only if necessary for the data type of the input. This option is useful for exploring low precisions indepdent of range limitations.

Demonstration function:

• demo-harmonic computes the harmonic series in several arithmetic formats using all the supported rounding modes.

Other M-file:

• roundit is a function for rounding a matrix to have integer entries. It is used by chop and is not intended to be called directly.

Test functions:

• test_chop is a test function for chop.
• test_roundit is a test function for roundit.

Each test function should print "All tests successful!".

The function chop is a successor to a function of the same name in the The Matrix Computation Toolbox (also available on File Exchange).

The test function test_chop needs the function float_paramsfrom the repository float_params. That function is included in this repository for convenience, but may not be the latest version.

Usage

There are two main usages of chop. First, one can pass options with every call:

options.format = 's'; options.round = 5; options.subnormal = 1; 
...
A(i,j) = chop(A(i,j) - chop(A(i,k) * A(k,j),options),options);

Here, options.format = 's' specifies that the precision is single, options.round = 5 specifies stochastic rounding, mode 1 and options.subnormal = 1 specifies that subnormal numbers are not flushed to zero. For full details of the options see the help lines in chop.m.

The above usage is rather tedious and produces cluttered code. Instead we can set up the arithmetic parameters on a call of the form chop([],options) and exploit the fact that subsequent calls with just one input argument will reuse the previously specified options:

options.format = 's'; options.round = 5; options.subnormal = 1; 
chop([],options)
...
A(i,j) = chop(A(i,j) - chop(A(i,k)*A(k,j))); 

The current value of options is stored inside the function (in a persistent variable, whose value is retained during the session until the function is cleared with clear chop and can be obtained with

[~,options] = chop

Requirements

The code was developed in MATLAB R2018b to R2020a and works with versions at least back to R2016a.

Reference

Nicholas J. Higham and Srikara Pranesh, Simulating Low Precision Floating-Point Arithmetic, SIAM J. Sci. Comput., 41(4):A2536-A2551, 2019.

Acknowledgements

The code was written by Nick Higham and Sri Pranesh. Max Fasi and Mantas Mikaitis have contributed improvements.

License

See license.txt for licensing information.