Recently, I came across an article, highlighting how quickly and dramatically floating point roundoff errors can cause things to fall apart, given the right conditions1, in our modern programming languages.

Note: Some assumptions might not be fully correct. Feel free to correct me if spot any mistakes!

A representation problem

Modern computer use binary representation to store decimals. That’s mean they can only use a combination of $2^n$.

1 0 1 . 1 0
4 2 1 . $\frac{1}{2}$ $\frac{1}{4}$
$2^{2}$ $2^{1}$ $2^{0}$   $2^{-1}$ $2^{-2}$

This is 5.5 in binary ($4 + 1 + \frac{1}{2}$).

We’ll try now to represent another number, 0.1, in binary.

We could go continue like this for ages, and still only approximate 0.1.

The problem lies in the fact our metric system uses base 10 fractions, decimals, but floating-point numbers are usually represented as base 2 fractions.

Unfortunately, most decimal fractions cannot be represented exactly as binary fractions. A consequence is that, in general, the decimal floating-point numbers you enter are only approximated by the binary floating-point numbers actually stored in the machine.

Let’s illustrate with a simple example, in Swift.

let a: Float = 0.1
let b: Double = 0.1

The value of a and b, if we print it, is 0.1.

However it’s easy to forget that the actual stored value is an approximation to the original decimal fraction. Printing more digits, we would see:

(a) 0.10000000149011611938
(b) 0.10000000000000000555

The value is not exactly what we hoped. The same kind of things is observable in all languages that support IEEE-754 floating-point arithmetic.

Muller’s Recurrence

We could easily argue that we usually don’t need that much precision, and that the representation produced is generally enough for us. This is true. But sometimes when the right conditions are met, floating-point representation might break and produces unexpected results.

Jean-Michel Muller is a French computer scientist specialized in finding ways to break computers using math. One of his best-known problems is his recurrence formula:

Really straightforward, nothing fancy or complex.

We’re going to compute $x_{n}$ for a large value of $n$ and try to infer the limit.

We begin with $n = 25$.

First, we define a generic Swift code that implements the formula:

func f<T>(y: T, z: T) -> T {
    return 108 - (815-1500/z)/y

func xvalues<T>(n: Int) -> [T] {
    var xi: [T] = [T(4), T(17)/T(4)]
    guard n > 2 else {
        return xi
    for i in 2..<n {
        xi.append(f(y: xi[i-1], z: xi[i-2]))
    return xi

And the main of our program:

let N: Int = 25
let xi = xvalues(n: N)

for i in 0..<N {
    print(String(format: "%i | %@", i, xi[i].description))

We now use this with the double floating point arithmetic implementation of Swift, Double.

$i$ $x_i$
0 4.0
1 4.25
2 4.470588235294116
3 4.6447368421052175
4 4.770538243625083
5 4.855700712568563
6 4.91084749866063
7 4.945537395530508
8 4.966962408040999
9 4.980042204293014
10 4.987909232795786
11 4.991362641314552
12 4.967455095552268
13 4.42969049830883
14 -7.817236578459315
15 168.93916767106458
16 102.03996315205927
17 100.0999475162497
18 100.00499204097244
19 100.0002495792373
20 100.00001247862016
21 100.00000062392161
22 100.0000000311958
23 100.00000000155978
24 100.00000000007799

With this precision, as $i$ increases, the result converges toward 100. Great, we found the limit!

Muller's Recurrence, using Double

Unfortunately, this recurrence actually converges to 5.

Why this gigantic round-off error?

There are two main reasons: the design of this formula and the way the computer store the floating point number.

Computer has a finite memory and we saw they represent numbers a combination of $2^n$. This limited memory requires the need to make trades-off, especially regarding the number of combinations. The number of combinations defines the “precision” of the number it represents. For example Float has about 6 digits precision whereas Double has about 15 digits precision.

Go back to the representation of 0.1 and count the numbers of zeros after the dot, you’ll retrieve these numbers.

We could have used Float80 or Decimal, but they only have a few more digits precision. We would have hit the same limit.

Muller designed his recurrence to produce numbers with an increasing number of digits. The first values of the recurrence can be represented with our classic floating-point types. At a given point, the number cannot be represented exactly, and has to be rounded.

The recurrence is also designed so that a slight rounding error produces a tremendous round-off error. This happens when we wit the $15^{th}$ iteration in our example.

If you need further explanations, this paper gives more details2.

Fixed-point representation

Another classical representation of number is fixed-point. A value of a fixed-point is an integer that is scaled by a factor. This factor is the same for all number. And so the scaling factor defines the range of numbers you can represent.

For example, 1 234 000 can be represented as 1234 by a factor of 1000.

But does fixed-point solves our rounding problem? The short answer is no.

Same as floating-point number, fixed-point is limited by memory.

Neither fixed point nor floating point are immune to Muller’s recurrence. Both will eventually produce the wrong answer. The question is when?

Fractional’s representation

Since storing decimals is hard, why not using a method, like fixed-point, that can store without producing an error?

While digging, I stumble upon Lisp and one of his dialect, Clojure.

Clojure choose to represent integer division as fractions.

So why not use the same technique to our little problem? Using fractions as inputs just treats everything as a fraction, avoiding the problem of decimals.

Same as before, we run the program but now using fractions of Int.

$i$ $x_i$ ~$x_i$
0 4 ~4
1 $\frac{17}{4}$ ~4.25
2 $\frac{76}{17}$ ~4.4705882352941176470588235294117647058
3 $\frac{353}{76}$ ~4.6447368421052631578947368421052631578
4 $\frac{1684}{353}$ ~4.7705382436260623229461756373937677053
5 $\frac{8177}{1684}$ ~4.8557007125890736342042755344418052256
6 $\frac{40156}{8177}$ ~4.9108474990827932004402592637886755533
7 $\frac{198593}{40156}$ ~4.9455374041239167247733838031676461798
8 $\frac{986404}{198593}$ ~4.9669625817627005987119384872578590383
9 $\frac{4912337}{986404}$ ~4.9800457013556311612686079942903718963
10 $\frac{24502636}{4912337}$ ~4.9879794484783922601401328939769400999
11 $\frac{122336033}{24502636}$ ~4.9927702880620680974895925483282696604
12 $\frac{611148724}{122336033}$ ~4.9956558915066340266240282615670560447
13 $\frac{3054149297}{611148724}$ ~4.9973912683813441128938690216425944791
14 $\frac{15265963516}{3054149297}$ ~4.9984339439448169190138971781902382881
15 $\frac{76315468673}{15265963516}$ ~4.9990600719708938678168396062869249162
16 $\frac{381534296644}{76315468673}$ ~4.9994359371468391479978508864997899689
17 $\frac{1907542343057}{381534296644}$ ~4.9996615241037675377868879857270921122
18 $\frac{9537324294796}{1907542343057}$ ~4.9997969007134179126629930746330014204
19 $\frac{47685459212513}{9537324294796}$ ~4.9998781354779312492317320300204045157
20 $\frac{238423809278164}{47685459212513}$ ~4.9999268795045999044664529810255297667
21 $\frac{1192108586037617}{238423809278164}$ ~4.9999561270611577381190152822944886054
22 $\frac{5960511549128476}{1192108586037617}$ ~4.9999736760057124445790151493567643737
23 $\frac{29802463602463553}{5960511549128476}$ ~4.9999842055202727079241797881426658531
24 $\frac{149012035582781284}{29802463602463553}$ ~4.9999905232822276594072074700221895687

Muller's Recurrence, using fractions of Int

Great! Looks like we’re on the right path! But what happens for $x_{25}$ and so on?

Unfortunately, we hit the Int limit. The numerator can’t be represented using Int and we got a wonderful runtime error.

In order to test our theory, we need an arbitrary-precision arithmetic.

Using this, we are now able to compute $x_{100}$ and a get an accurate result:

After some tests, I lost precision of my approximation at $x_{173}$. I completely lost my approximation at $x_{273}$. With a little more work, I’m sure we can easily do better.

Finally, I was able to reach $x_{1000}$ in matter of seconds.

Muller's Recurrence, with $n=1000$

Wrapping up

Not many languages handle fractions natively although it’s a good way of storing decimals efficiently. Sure, there are some drawbacks, time consuming for example, but if you seek extensive exactness, I encourage you to explore this perspective.

A complete sample project is available if you wish to try it out yourself: