This module presents a type for storing numbers in the log-domain. The main reason for doing this is to prevent underflow when multiplying many small probabilities as is done in Hidden Markov Models and other statistical models often used for natural language processing. The log-domain also helps prevent overflow when multiplying many large numbers. In rare cases it can speed up numerical computation (since addition is faster than multiplication, though logarithms are exceptionally slow), but the primary goal is to improve accuracy of results. A secondary goal has been to maximize efficiency since these computations are frequently done within a O(n^3) loop.
- module Data.Number.Transfinite
- module Data.Number.RealToFrac
- data LogFloat
- logFloat :: (Real a, RealToFrac a Double) => a -> LogFloat
- logToLogFloat :: (Real a, RealToFrac a Double) => a -> LogFloat
- fromLogFloat :: (Fractional a, Transfinite a, RealToFrac Double a) => LogFloat -> a
- logFromLogFloat :: (Fractional a, Transfinite a, RealToFrac Double a) => LogFloat -> a
Exceptional numeric values
LogFloat data type and conversion functions
LogFloat is just a
Double with a special interpretation.
logFloat function is presented instead of the constructor,
in order to ensure semantic conversion. At present the
instance will convert back to the normal-domain, and so will
underflow at that point. This behavior may change in the future.
Performing operations in the log-domain is cheap, prevents
underflow, and is otherwise very nice for dealing with miniscule
probabilities. However, crossing into and out of the log-domain
is expensive and should be avoided as much as possible. In
particular, if you're doing a series of multiplications as in
lp * logFloat q * logFloat r it's faster to do
lp * logFloat
(q * r) if you're reasonably sure the normal-domain multiplication
won't underflow, because that way you enter the log-domain only
once, instead of twice.
Even more particularly, you should avoid addition whenever
possible. Addition is provided because it's necessary at times
and the proper implementation is not immediately transparent.
However, between two
LogFloats addition requires crossing the
exp/log boundary twice; with a
LogFloat and a regular number
it's three times since the regular number needs to enter the
log-domain first. This makes addition incredibly slow. Again,
if you can parenthesize to do plain operations first, do it!
A constructor which does semantic conversion from normal-domain to log-domain.
Constructor which assumes the argument is already in the log-domain.
Return our log-domain value back into normal-domain. Beware of overflow/underflow.