ErrorfreeArithmetic.jl

Error-free transformations are used to get results with extra accuracy.

Copyright © 2016-2020 by Jeffrey Sarnoff. Released under the MIT License.

exports

The number that begins a function name always matches the number of values returned.
- the values returned are of descending magnitude and non-overlapping when added.
The number that begins a function name often matches the number of arguments expected.
- two_inv and two_sqrt are single argument functions returning two values

These are error-free transformations.

two_sum, two_diff, two_prod
two_square, two_cube
three_sum, three_diff, three_prod
two_fma, three_fma
four_sum, four_diff

These are error-free transformations with magnitude sorted arguments.

two_hilo_sum, two_lohi_sum
two_hilo_diff, two_lohi_diff
three_hilo_sum, three_lohi_sum
three_hilo_diff, three_lohi_diff
four_hilo_sum, four_lohi_sum
four_hilo_diff, four_lohi_diff

These are least-error transformations, as close to error-free as possible.

two_inv, two_sqrt
two_div

naming

The routines named with the prefix two_ return a two-tuple holding (high_order_part, low_order_part).

Those named with the prefix three_ return a three-tuple holding (high_part, mid_part, low_part).

introduction

Error-free transformations return a tuple of the nominal result and the residual from the result (the left-over part).

Error-free addition sums two floating point values (a, b) and returns two floating point values (hi, lo) such that:

(+)(a, b) == hi
|hi| > |lo| and (+)(hi, lo) == hi abs(hi) and abs(lo) do not share significant bits

Here is how it is done:

function add_errorfree(a::T, b::T) where T<:Union{Float64, Float32}
    a_plus_b_hipart = a + b
    b_asthe_summand = a_plus_b_hipart - a
    a_plus_b_lopart = (a - (a_plus_b_hipart - b_asthe_summand)) + (b - b_asthe_summand)
    return a_plus_b_hipart, a_plus_b_lopart
end

a = Float32(1/golden^2)                           #   0.3819_6602f0
b = Float32(pi^3)                                 #  31.0062_7700f0
a_plus_b = a + b                                  #  31.3882_4300f0

hi, lo = add_errorfree(a,b)                       # (31.3882_4300f0, 3.8743_0270f-7)

a_plus_b == hi                                    # true
abs(hi) > abs(lo) && hi + lo == hi                # true

The lo part is a portion of the accurate value, it is (most of) the residuum that the hi part could not represent.
The hi part runs out of significant bits before the all of the accurate value is represented. We can see this:

a = Float32(1/golden^2)                           #   0.3819_6602f0
b = Float32(pi^3)                                 #  31.0062_7700f0

hi, lo = add_errorfree(a,b)                       # (31.3882_4300f0, 3.8743_0270f-7)

a_plus_b_accurate = BigFloat(a) + BigFloat(b)
lo_accurate  = Float32(a_plus_b_accurate - hi)

lo == lo_accurate                                 # true

use

This package is intended to be used in the support of other work.
All routines expect Float64 or Float32 or Float16 values.

references

Takeshi Ogita, Siegfried M. Rump, and Shin'ichi Oishi
Accurate Sum and Dot Product
SIAM J. Sci. Comput., 26(6), 1955–1988.
Published online: 25 July 2006
DOI: 10.1137/030601818

Stef Graillat, Valérie Ménissier-Morain
Error-Free Transformations in Real and Complex Floating Point Arithmetic
International Symposium on Nonlinear Theory and its Applications (NOLTA'07), Sep 2007, Vancouver, Canada.
Proceedings of International Symposium on Nonlinear Theory and its Applications (NOLTA'07), pp.341-344.
https://hal.archives-ouvertes.fr/hal-01306229

Sylvie Boldo, Stef Graillat, and Jean-Michel Muller
On the robustness of the 2Sum and Fast2Sum algorithms
ACM Transactions on Mathematical Software, Association for Computing Machinery, 2017
https://hal.inria.fr/ensl-01310023