Rounding floats so that they sum to precisely 1

Question

I have a rather gnarly bit of code that must more-or-less randomly generate a bunch of percentages, stored as decimal floats. That is, it decides that material one makes up

Answer 1

If you mean to find two values that add up to 1.0

I understand that you want to pick two floating-point numbers between 0.0 and 1.0 such that they add to 1.0.

Do this:

• pick the largest L of the two. It has to be between 0.5 and 1.0.
• define the smallest number S as 1.0 - L.

Then in floating-point, S + L is exactly 1.0.

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If for some reason you obtain the smallest number S first in your algorithm, compute L = 1.0 - S and then S0 = 1.0 - L. Then L and S0 add up exactly to 1.0. Consider S0 the “rounded” version of S.

If you mean several values X₁, X₂, …, X_N

Here is an alternative solution if you are adding N numbers, each between 0.0 and 1.0, and expect the operations X₁ + X₂ + … and 1.0 - X₁ … to behave like they do in math.

Each time you obtain a new number X_i, do: X_i ← 1.0 - (1.0 - X_i). Only use this new value of X_i from that point onwards. This assignment will slightly round X_i so that it behaves well in all sums whose intermediate results are between 0.0 and 1.0.

EDIT: after doing the above for values X₁, …, X_N-1, compute X_N as 1 - X₁ - … - X_N-1. This floating-point computation will be exact (despite involving floating-point), so that you will have X₁ + … + X_N = 1 exactly.