TwoFloat

An implementation of DoubleWord Arithmetic as a simple C++ templated class.

It implements algorithms from

1. Mioara Maria Joldes, Jean-Michel Muller, Valentina Popescu. | Tight and rigorous error bounds for basic building blocks of double-word arithmetic." ACM Transactions on Mathematical Software, 2017, 44 (2), pp.1 - 27. .10.1145/3121432.. https://hal.science/hal-01351529v3

2. Vincent Lefèvre, Nicolas Louvet, Jean-Michel Muller, Joris Picot, Laurence Rideau. "Accurate Calculation of Euclidean Norms using Double-Word Arithmetic." ACM Transactions on Mathematical Software, 2023, 49.  https://hal.science/hal-03482567

3. S. M. Rump and M. Lange. "Faithfully Rounded Floating-Point Computation." ACM Transactions on Mathematical Software, 46(3):1–20, jul 2020. https://www.tuhh.de/ti3/paper/rump/LaRu2017b.pdf

The code is contained in just one file TwoFloat.h. The file consists in four sections:

1. Basic Block of DoubleWord Arithmetic imported from a previous project in C.
2. The TwoFloat class itself
3. A set of operators implementing the 4 arithmetic operations and comparisons
4. A small set of elementary functions plus some utilities

"numeric_limits" specializations are defined in the file TwoFloatLimits.h included by TwoFloat.h.

Usage

The TwoFloat class is designed to be a seamless drop-in replacement for float-types in any code, included templated libraries such as Eigen.

Implicit conversions to and from the single-word type (the template argument) are supported. That will also allow the implicit promotion to double following the C++ language rule often producing not the result the coder expected.

Implicit promotion to double can be easily detected using compiler option "-Wdouble-promotion".

Conversion can be forced to be explicit defining the macro "TWOFLOAT_EXPLICIT"

Choice of algorithms

By default Sum is implemented using Algo 5 from Ref 1 (aka SloppySum), Multiplication by Algo 11 and Division by Algo 17.

More precise algoritms can be selected by defining macro "TWOFLOAT_PRECISE_SUM" (Algo 6 for Sum), "TWOFLOAT_PRECISE_MULT" (Algo 12 for Multiplication) and "TWOFLOAT_PRECISE_DIV" (Algo 18 for Division)

Please refer to table 1 in Ref 1 for details about precision and "latency" of the algorithms.

Debugging aids

Sometimes the results are not those expected, in particular when using TwoFloat in an external software package. This is usually due to implicit assumptions that the type is a C++ floating point (with a IEEE754 format). Missing specializations are of course a major issue.

Compiling with "-DTWOFLOAT_EXPLICIT" and "-Wdouble-promotion" should already help in detecting all cases of conversions and promotions some of which maybe problematic.

TwoFloat declares also two functions: void "trapTwoFloat(float h, float l)" and similar for double that are called by all constructors and assign operators if the macro "TWOFLOAT_TRAP" is defined. The functions can be defined in the user application to detect problematic occurrencies (such as NaN) and used to set a breakpoint in gdb to identify thier origin.

Examples and tests

Examples and tests can be found in the "tests" directory for both CPU and GPU