Why summing 0.1 ten times misses 1.0, and how to fix it

Question

A risk report totals thousands of small position values. A junior notices a sanity check fails:

>>> sum([0.1] * 10)
0.9999999999999999
>>> sum([0.1] * 10) == 1.0
False

Explain why a plain running sum drifts, and write a compensated_sum(values) that recovers almost all of the lost precision. What is its cost?

Accepted Answer

Each 0.1 is already the nearest double to one tenth, off by about 10^{-17}. That alone is tiny. The real damage is accumulation: every time you add a small number to a large running total, the small number's low-order bits fall off the end of the 53-bit mantissa and are silently rounded away. Add n numbers and you can lose up to about n roundings' worth of precision. Over millions of terms this becomes visible dollars. The fix is Kahan (compensated) summation: carry a second variable c that tracks the low-order bits that were dropped last time, and feed them back on the next add. Algebraicall…

Why summing 0.1 ten times misses 1.0, and how to fix it

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