> No, I think all customers care about penny precision when you are talking about money they have or owe.

That wasn't my personal experience working on a fixed income desk. We regularly interacted with CFOs and corporate treasurers and the 'bills' were in the millions. And why would they? As they say, 'penny wise, pound foolish'.

> No, you're generally not throwing out precision when you round in this case. You're throwing out error.

That's not always true from a numerical analysis standpoint. By the way, I highly recommend this old but still useful doc which goes through the math carefully:

https://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.h...

Rounding can introduce up to 0.5 ulp (unit in the last place) of error, where ulp here is the precision that you round to. This is pretty easy to show:

let intSum = 0;
let floatSum = 0;
let roundedFloatSum = 0;
for (let i = 0; i < 100; i++) {
// Generate numbers up to 1000, with up to 3 decimals
const randInt = Math.floor(Math.random() * 100 * 1000);
const randFloat = randInt / 1000;

intSum += randInt; // Sum in exact precision, scaled by 1000
floatSum += randFloat; // Sum using full FP precision

// Sum while rounding to 2 decimals along the way
roundedFloatSum += randFloat;

roundedFloatSum = Math.round(roundedFloatSum * 100) / 100;
}
intSum /= 1000;

console.log(intSum, floatSum, roundedFloatSum);
console.log(Math.abs(intSum - floatSum), Math.abs(roundedFloatSum - intSum))

Here we're generating some 3-fig decimal numbers and summing them up, once in exact precision, once with floats, and once with intermediate rounding to the 2nd decimal place.

On the last run, this outputs:

5567.347 5567.347000000001 5567.39
9.094947017729282e-13 0.0430000000005748

'Rounding as you go' for your intermediate results here introduced an unnecessary 0.043 of absolute error.

Every guide to numerical analysis I've ever seen recommends keeping all calculations in the same format (whether fixnums or floats) all the way through, then only doing rounding at the very end, for this very reason.

In fact, we can prove that on reasonable accounting inputs and simple calculations, doing everything using doubles and then rounding at the very end gives you the _exact_ result.

Let's assume the numbers you're summing, multiplying, etc. remain bounded under <$1 mm, and you're manipulating at most 1 million of those numbers.

1. Doubles have 52-bits, or about 15 decimal digits of precision

2. Basic arithmetic operations (addition, subtraction, multiplication, division) introduce at most 0.5 ulp of error per operation. Using our assumptions, each decimal number is accurate to at least the billionth (9th digit) place, whereas we only need accuracy to the hundredths.

3. The cumulative error of a chain of a million operations, each with an error in the 9th digit place, can at most only affect the 3rd decimal digit. The 2nd decimal will always be correct

> No, you're generally not throwing out precision when you round in this case. You're throwing out error.

Floating point calculations round to the available precision after every operation. If introducing extraneous rounding to a much lower precision magically 'fixed' errors, then how could long floating point calculations themselves be inaccurate? In fact, there are algorithms that lower overall error by carefully shepherding the low-order digits, e.g. https://en.wikipedia.org/wiki/Kahan_summation_algorithm

Rounding is often appropriate if the rounded value is the actual source of truth, e.g. if, after some long sequence of calculations, you've told the customer that they have $10.15 in the bank, then you should try to store that value rather than the raw float result. Even that's pretty subtle though -- e.g. if their account balance is a result of interest payments, one can show that you will introduce more error in the total interest paid over time if you discard lower order digits rather than reusing them for the next interest calculation.

One last thing: When you hand your exact precision results to an accountant, auditor, or customer, Excel's a pretty common tool that they use for basically everything right? It must be sporting some fancy arbitrary precision or decimal machinery under the hood, right?

Nope, just floats all the way down! https://en.wikipedia.org/wiki/Numeric_precision_in_Microsoft...

I think you should be strongly questioning your assumption that floats aren't 'good enough' if the very first thing every customer you interact with does is cast your results into a floats to do their own calcs.