Why 0.1 + 0.2 != 0.3, and how to compare floats

Question

A reconciliation job flags a break because a computed fill price "doesn't equal" the exchange's drop-copy price:

>>> 0.1 + 0.2 == 0.3
False
>>> 0.1 + 0.2
0.30000000000000004

Explain what's happening, and write an approx_equal(a, b) that a production system could actually use. Why does it need both a relative and an absolute tolerance?

Accepted Answer

Binary floating point represents numbers as m 	imes 2^e. Just as 1/3 has no finite decimal expansion, 0.1 has no finite binary expansion, the stored value is the nearest representable double, off by about 10^{-17}. Adding two such approximations and rounding again lands on a double that differs from the (also approximated) 0.3 in the last bit. Nothing is "broken"; exact equality was never the right test for computed floats. A robust comparison needs two tolerances: - Relative, doubles have about 15–16 significant decimal digits, so the natural error scale of a value near 10^9 is around 10^{-…

Why 0.1 + 0.2 != 0.3, and how to compare floats

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