The Kahan summation algorithm is a "compensated summation" that "*significantly reduces the numerical error in the total obtained by adding a sequence of finite precision floating point numbers, compared to the obvious approach*".

Matt Adereth wrote a blog post a few days ago demonstrated some advantages of using Kahan, which I show below using Factor.

## The Problem

To demonstrate the problem, let's consider the harmonic series (e.g., `1 + 1/2 + 1/3 + 1/4 + ...`

) as a series of rational numbers:

: harmonic-ratios ( n -- seq ) [1,b] [ recip ] map ;

It gives the first `n`

of the harmonic series:

IN: scratchpad 6 harmonic-ratios . { 1 1/2 1/3 1/4 1/5 1/6 }

We define a "simple sum" as just adding the numbers in order:

: simple-sum ( seq -- n ) 0 [ + ] reduce ;

The simple sum of the first 10,000 harmonic ratios as a floating point number is:

IN: scratchpad 10,000 harmonic-ratios simple-sum >float . 9.787606036044382

But, if we use floating points instead of ratios to represent the harmonic numbers (e.g., `1.0 + 0.5 + 0.33333333 + 0.25 + ...`

):

: harmonic-floats ( n -- seq ) harmonic-ratios [ >float ] map! ;

You can see that an error has been introduced (ending in "`48`

" instead of "`82`

"):

IN: scratchpad 10,000 harmonic-floats simple-sum . 9.787606036044348

If we reverse the sequence and add them from smallest to largest, there is a slightly different error (ending in "`86`

"):

IN: scratchpad 10,000 harmonic-floats reverse simple-sum . 9.787606036044386

## The Solution

The pseudocode for the Kahan algorithm can be seen on the Wikipedia page, using a running compensation for lost low-order bits:

function KahanSum(input) var sum = 0.0 var c = 0.0 for i = 1 to input.length do var y = input[i] - c var t = sum + y c = (t - sum) - y sum = t return sum

We could translate this directly using local variables to hold state, but instead I tried to make it a bit more concatenative (and possibly harder to read in this case):

: kahan+ ( c sum elt -- c' sum' ) rot - 2dup + [ -rot [ - ] bi@ ] keep ; : kahan-sum ( seq -- n ) [ 0.0 0.0 ] dip [ kahan+ ] each nip ;

You can see both forward and backward errors no longer exist:

IN: scratchpad 10,000 harmonic-floats kahan-sum . 9.787606036044382 IN: scratchpad 10,000 harmonic-floats reverse kahan-sum . 9.787606036044382

Nifty! Thanks, Matt!

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