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This collects a number of changes I've been testing over the
last month.
* Bug fix: The single-precision float formatter did not always
round the last digit even in cases where there were two
possible outputs that were otherwise equally good.
* Algorithm simplification: The condition for determining
whether to widen or narrow the interval was more complex than
necessary. I now simply widen the interval for all even
significands.
* Code simplification: The single-precision float formatter now uses fewer
64-bit features. This eliminated some 32-bit vs. 64-bit conditionals in
exchange for a minor loss of performance (~2%).
* Minor performance tweaks: Steve Canon pointed out a few places
where I could avoid some extraneous arithmetic.
I've also rewritten a lot of comments to try to make the exposition
clearer.
The earlier testing regime focused on testing from first
principles. For example, I verified accuracy by feeding the
result back into the C library `strtof`, `strtod`, etc. and
checking round-trip exactness. Unfortunately, this approach
requires many checks for each value, limiting test performance.
It's also difficult to validate last-digit rounding.
For this round of updates, I've instead compared the digit
decompositions to other popular algorithms:
* David M. Gay's gdtoa library is a robust and well-tested
implementation based on Dragon4. It supports all formats, but
is slow. (netlib.org/fp)
* Grisu3 supports Float and Double. It is fast but incomplete,
failing on about 1% of all inputs.
(github.com/google/double-conversion)
* Errol4 is fast and complete but only supports Double. The
repository includes an implementation of the enumeration
algorithm described in the Errol paper.
(github.com/marcandrysco/errol)
The exact tests varied by format:
* Float: SwiftDtoa now generates the exact same digits as gdtoa
for every single-precision Float.
* Double: Testing against Grisu3 (with fallback to Errol4 when
Grisu3 failed) greatly improved test performance. This
allowed me to test 100 trillion (10^14) randomly-selected
doubles in a reasonable amount of time. I also checked all
values generated by the Errol enumeration algorithm.
* Float80: I compared the Float80 output to the gdtoa library
because neither Grisu3 nor Errol4 yet supports 80-bit extended
precision. All values generated by the Errol enumeration
algorithm have been checked, as well as several billion
randomly-selected values.
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