Calculators

bc/dc

The Unix calculators. Offer arbitrary precision since 1970, and now you know what the 'bc' stands for in this blog's title! 😉 Neither of them supports factorials out of the box, but hey, these are programming languages, not plain calculators. Examples for scripts computing factorials can be found on Rosettacode and on Stackoverflow, but be aware that these examples are horribly inefficient — for fast algorithms see Peter Luschny's page. Here's the “script” for dc:

dc -e '?[q]sQ[d1=Qd1-lFx*]dsFxp'

After typing a number like 1000, we get an exact result. Overflows somewhat below 67,000!. bc does not, but it's too slow to be of much use for very much larger numbers.

wcalc

Approximate results with principally arbitrary precision defined by the command line parameter P (which accepts only integers and is thus useless for really large numbers).

wcalc -P 10 'fact(1000000)'

Takes 1.42 s, overflows somewhat below 44,500,000!

calc

My default calculator on PCs. Gives exact results.

calc 1000000!

Takes 330 s, and overflows somewhat below 2,200,000,000!

hypercalc

When firing up hypercalc, we are greeted with ”Go ahead -- just TRY to make me overflow!”. And indeed, that's not an easy task at first. Hypercalc gives approximate results only, but essentially instantaneous ones even for absolutely monstrous numbers. The factorials we have considered so far are kids play for this program. Instead of the factorial of a million, a billion, a trillion, why not ask for the factorial of a Googol! Hypercalc tells us this number amounts to 1e+(9.9565705518098e+101), and that agrees with the solution from Wolfram Alpha (see below), the only tool, which can at least partly follow hypercalc into the realm of big numbers.

But not when it comes to really big ones. Let's have a look, for example, at Pickover's superfactorial n$. What about, say, 10$? That's completely out of reach for any program I know, but not for hypercalc: 8pt8e+23804068 (PT stands for PowerTower). But even this is still a very very very small number: hypercalc overflows only at 1e+308pt1e+34 or, equivalently in Donald Knuth's up-arrow notation, 10↑↑1.7976e+308.

All of this is contained in a Perl script available for download (or for installation in the AUR for Archlinux users), and additionally in a Javascript powered web interface.

iPython

So far, we haven't been able to get exact results faster than with Mathematica. Let's see how python is doing in this regard. There are two possibilities, the first using plain python, the second scipy:

In [1]: import math
In [2]: %timeit math.factorial(1000000)
6.5 s ± 19.4 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)
In [3]: from scipy.special import factorial
In [4]: %timeit factorial(1000000, exact=True)
6.53 s ± 17.9 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)

It's disappointing that scipy doesn't outperform regular python, but instead seems to use exactly the same code. A special function should, IMHO, perform better than Mathematica (which needs only 1 s for the same task).

You can use python also directly from the terminal, and write the result to disk instead of displaying it directly:

echo 1000000 | python -c 'import sys; import math; print(math.factorial(int(sys.stdin.readline())))' > fac1M.dat

julia

Python turned out to be a disappointment, what about Julia, which is advertised to be suitable for high-performance numerical analysis and computational science?

julia> @time factorial(big(1000000));
0.162650 seconds (1.57 k allocations: 53.524 MiB, 2.00% gc time)
julia> @time factorial(big(100000000));
45.550577 seconds (432.60 k allocations: 11.586 GiB, 0.90% gc time)

Now we're talking!

You can write the results to disk in this way:

julia> using DelimitedFiles
julia> fac1M=factorial(big(1000000));
julia> writedlm("fac1M.dat", fac1M)

qalculate!

A great general purpose calculator packed with features. Includes an RPN mode and a plotting interface to gnuplot, as well as excellent conversion utilities that can be updated daily (important for currencies). We can get an approximate result for 1,000,000! in under 1 s, but 100,000,000! takes so long that I didn't wait. Overflows reportedly at 922,337,203,854,775,808!, which is impressive, but of little use because of the comparably poor performance.

speedcrunch

Easy to use and insanely fast. Even on slow hardware, the result is there as soon as you type it. Overflows at 72,306,961!

Casio

When I look at my colleagues desks at the office, I frequently see Casio calculators from the 1980s. Even I have one, although I have no idea when and where I've acquired it (and I also don't remember ever using it). In any case, true to their roots, Casio offers a quite capable online calculator. It gives 1,000,000! in about 2 s and overflows only at 1e+100,000,000, just below 14,845,000!

Wolfram Alpha

More than a calculator: a knowledge engine. You may ask what's the weather in Berlin today, and get the interesting bit of information that on April 22nd, it was -4°C in 1997 and 31°C in 1968. But foremost, Wolfram Alpha is an arbitrary precision calculator. It gives exact results where appropriate (for reasonably sized outputs) and approximate ones when the output seems to large. Regardless the task, the answers take a few second, whether you calculate 2+2 or 1e+(9^9^9)! [1e+1e+(1.58274e+369693108)]. Overflows at (2e+1573347107)! or 1e+(1e+9.196824545990035)! That's much higher than Mathematica, but still nothing compared to hypercalc.