User blog:Nayuta Ito/Mashimo Function (translation)

$$Mashimo function$$, or $$M(x)$$, is a function designed to make various sizes of large numbers. Its domain is real number and range is positive real number. It is a monotonically increasing function. There is an index of large numbers called Mashimo scale. Mashimo is a character in Sushi Kokuuhen.

If $$x\geq1$$,

$$ M(x)=max(10^{10x}, ^{x/5}e, H(^{x/20}2,2), H(^{H(x-70,2)}3,3), $$ $$V(x-72), V(3^{x-80}), BHO(x-83), O(x-85), OF(x-86), $$ $$L(x-90), D(5(x-94),10), FC(x-80), RC(x-120)) $$

If $$0\leq x < 1$$, $$M(x)=10^{10x}$$

If $$x<0$$, $$M(x)=M(-x)^{-1}$$

Max function is a function which returns the biggest number in the argument. Sub functions H,V,BHO,O,OF,L,D,FC,RC are defined below. Generalization of tetration is like this:

$$^x a=log_a(^{x+1} a) \,\, for x\leq -1$$

$$^x a=x+1 \,\,\,\,\, for -1 < x \leq 0$$

$$^x a= a^{(^{x-1} a)} \,\, for 0<x$$