Loader's number

Loader's number is the output of loader.c, a program by Ralph Loader that came in first place for the Bignum Bakeoff contest, whose objective was to write a C program (in 512 characters or less) that generates the largest possible output on a theoretical machine with infinite memory. It is among the largest computable numbers ever devised.

The program diagonalizes over the Huet-Coquand calculus of constructions. Its output, affectionately nicknamed Loader's number, is defined as \(D^5(99)=D(D(D(D(D(99)))))\), where \(D(k)\) is the sum of all possible bit strings described by the first k expressions of the calculus of constructions (encoding everything as binary numbers).

David Moews has shown that \(D(99)\) is larger than \(2↑↑30,419\) (where ↑↑ is tetration), and that even \(D^2(99)\) would be much larger than \(f_{\varepsilon_0+\omega^3}(1,000,000)\) in the fast-growing hierarchy, using Cantor's definition of fundamental sequences. \(D^2(99)\) thus is obviously much larger than the output of Marxen.c, which is upper bounded at the aforementioned \(f_{\varepsilon_0+\omega^3}(1,000,000)\).

The final output of \(D^5(99)\) is much larger than TREE(3), SCG(13), and, say, BH(100). It is probably overpowered by finite promise games and greedy clique sequences. Loader's function is computable, so \(\Sigma(n) > D^5(99)\) for relatively small n, say, n = 100.

Code

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 * 2) define P P (
 * 3) define L L (
 * 4) define T S (v, y, c,
 * 5) define C ),
 * 6) define X x)
 * 7) define F );}

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 * 1) define U = S(4,13,-4,
 * 1) define B (x /= 2) % 2 && (