floating point equality in Python and in general

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As has been shown comparing two floats (or doubles etc) can be problematic. Generally, instead of comparing for exact equality they should be checked against an error bound. If they are within the error bound, they are considered equal.

That is much easier said than done. The nature of floating point make a fixed error bound worthless. A small error bound (like 2*float_epsilon) works well when the values are near 0.0, but will fail if the value are near 1000. An error bound for values as large as 1,000,000.0 will be far too lax for values near 0.0.

The best solution is know the domain of your math and pick an approriate err bound on a case by case basis.

When this is impractical or you’re being lazy, Units in the Last Place (ULPs) is a very novel and robust solution. The full details are quite involved, you can read more here.

The basic idea is this, a floating point number has two pieces, mantissa and exponent. Generally rounding errors only change the mantissa by a few steps. When the value is near 0.0 those steps are exactly float_epsilon. When the floating point value is nearer to 1,000,000, the steps will be nearly as large as 1.

Google test uses ULP to compare floating point numbers. They chose a default of 4 ULPs for a two floating point numbers to be compared equal. You could also use their code as reference to build your own ULP style float comparator.

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