Finding Factorials with Gamma

Question

Introduction

We know that the factorial notation is valid for all natural numbers. However, Euler had extended it for all positive real numbers, as well as for complex numbers by defining a function, which is known as the Gamma Function. It is represented by Γ.

Challenge

You will be given a non-negative floating point number, say 'n' (with a maximum of 2 decimal places in it), and you need to output the factorial of that number correct to at least 3 decimal places. Assume all inputs to be valid, and n <= 7.

Now talking about Gamma function, there is an integral which can be used to find the value of any factorial. You may choose to use it. You can find more information here. The Gamma function is recursive in nature. One important property, which would probably be useful to you is:

Γ(n) = (n-1)! , which means that, n! = Γ(n) . n

Also note that we really do not need to know Γ(n) for all values of n. For instance, look at the number 4.5 - we can write its factorial as:

4.5! = 4.5 * 3.5 * 2.5 * 1.5 * 0.5! = 59.0625 * 0.5!

So here we need only the value of 0.5! , which means the value of Γ(1.5). Also, similarly, we can find the factorial of any decimal if we know the values of factorials of numbers between 0 and 1, i.e, values of gamma function of numbers between 1 and 2. You need it, so that is why there is a table below for your help:

Using this table, we can see that 0.5!, i.e, the value of Γ(1.5) = 0.88623, which gives our result: (4.5)! = 59.0625 * 0.88623 = 52.3429594 , which is indeed a good result. Similarly,

(5.7)! = (5.7)*(4.7)*(3.7)*(2.7)*(1.7)*(0.7)! = 454.97457 * 0.90864 = 413.408093

Also, if you do not wish to use the table for calculating the gamma function, here is a simple integral to help you out:

Remember that you can provide your answers only up to 3 decimal places if you want to. But the value should have minimum error. Like, +- 0.003 is acceptable.

You are free to decide in what way you want to compute the factorial - whether by the integral, or by the table, or using any other method.

Examples

Please note that here the answer is correct to 3 decimal places.

2.4 -> 2.981
3.9 -> 20.667
2.59 -> 3.675
1.7 -> 1.545
0.5 -> 0.886

Scoring

This is code-golf, so the shortest code wins!

Answer 1

Community wiki for built-in solutions

This answer is intended to collect answers that are simply built-ins. It's a community wiki, so anybody can edit in without approval. Go ahead and include any built-in only solution here, following xnor's proposal and according to our consensus.

Jelly, 1 byte

!

Try it online!

J , 1 byte

!

Try it online!

2sable, 1 byte

!

Try it online!

Pyth, 2 bytes

.!

Try it online!

R, 9 bytes

factorial

Try it online!

Stax, 2 bytes

|F

Try and debug online!

Wolfram Language (Mathematica), 3 bytes

#!&

Try it online!

Answer 2

Perl 5, -p 40 bytes

Very slow. One run takes about 13 minutes for me.

#!/usr/bin/perl -p
$\=5e8**++$_/$_;$\/=1+$'/$_ for//..5e8}{

Try it online!

The TIO link uses 1e7 instead of 5e8 to avoid timeouts and is therefore not accurate enough

Old version, -p 60 bytes, not accurate enough

Uses a Stirling approximation. But even with the first correction term that is pretty inaccurate for small n. So instead I approximate (n+2)!/(n+2)/(n+1).

Notice that I don't bother trying to get the first 3 digits right, but the relative error seems to properly be < 0.001 in the range [0, 7]. E.g. 0! outputs 0.999481433067179 which would be 1.000 if rounded to 3 digits.

#!/usr/bin/perl -p
$_=(($_+=2)/exp 1)**$_*(2+1/6/$_)/$_/($_-1)*sqrt$_*atan2 1,0

Try it online!

Answer 3

APL (Dyalog Unicode), 1 byte

!

Try it online!

In APL, the factorial primitive has always included the entire domain of the gamma function.