# Algorithm: turining a fraction into a decimal expansion string

So I’m basicly implementing a number type that can represent all fractions and was working on an algorithm to compute the decimal expansion for said fractions.

Let’s say we have the reduced fraction $$\frac{n}{m}$$. For converting is into it’s decimal expansion I have now two algorithms.

The first algorithm is simply sing long division to caluclate the decimal expansion up to a given number of decimal places.

The second is:

Let be $$a \in \{1,2,\ldots\}$$ a specifier for accuracy.

Calculate: \begin{align} p &= \lceil \log_{10}(m) \rceil + a \\ f &= \lfloor \frac{10^p}{m} \rfloor \\ v &= n \cdot f \end{align} Then in $$v$$ insert the decimal comma at the correct place or add 0. with leading zeros.

Which works good but it is hard to control the accuarcy with $$a$$. For example if I have the fraction $$\dfrac{884279719003555}{281474976710656} \approx \pi$$ then I get:

 a | dec. exp. ---|--------------------------------      v acc 0  1 | 3.0949790165124425        v acc 1  2 | 3.13919300246262025         v acc 1  3 | 3.14096156190062736              v acc 7  8 | 3.14159264580768862709685               v acc 8  9 | 3.14159265288192637912529                   v acc 12 10 | 3.141592653589350154328134                   v acc 12 11 | 3.141592653589350154328134                   v acc 12 12 | 3.141592653589350154328134                      v acc 15 f  = 3.1415926535897931159979634...  pi = 3.1415926535897932384626433... 

So its seams I can controll with $$a$$ that at least $$a-1$$ decimal places are correct.

But I’m not sure if this will alwasy be the case.

Also I benchmarked both algorithms, and the second is more than 5 times faster. So I really want it to be controllable.

| Method |       Mean |    Error |   StdDev | |--------|-----------:|---------:|---------:| |  first | 4,929.2 ns | 24.34 ns | 20.33 ns | | second |   848.8 ns |  4.00 ns |  3.54 ns | 

So my question basicly is does anybody have suggestion on improving the algorithm or is maybe another algorithm that does the job even better (a.i. fast)?