def msum(iterable):
    "Full precision summation using multiple floats for intermediate values"
    # Rounded x+y stored in hi with the round-off stored in lo.  Together
    # hi+lo are exactly equal to x+y.  The inner loop applies hi/lo summation
    # to each partial so that the list of partial sums remains exact.
    # Depends on IEEE-754 arithmetic guarantees.  See proof of correctness at:
    # www-2.cs.cmu.edu/afs/cs/project/quake/public/papers/robust-arithmetic.ps

    partials = []               # sorted, non-overlapping partial sums
    for x in iterable:
        i = 0
        for y in partials:
            if abs(x) < abs(y):
                x, y = y, x
            hi = x + y
            lo = y - (hi - x)
            if lo:
                partials[i] = lo
                i += 1
            x = hi
        partials[i:] = [x]
    return sum(partials, 0.0)
Full precision summation using multiple floats for intermediate values
recipes/Python/393090_Binary_floating_point_summatiaccurate_full/recipe-393090.py