Cut the Cards

[Albert Chan]

(08-27-2020 10:07 PM)Gerson W. Barbosa Wrote:  How about « 1. Psi + EXP DUP 1. + 4. * INV OVER 1. EXP / + INV OVER 24. * + INV - .5 - » ?

FYI, above H^-1 is equvialent to Ψ^-1 (previous post)
Since H(n) = Ψ(n+1) - Ψ(1), we flip the sign of -0.5 to +0.5

XCas> invPsi1(k) := k - 1/(24k + 1/(k/e + 1/(4k+4))) + 1/2

\(k \rightarrow k-\frac{1}{24\cdot k+
\large\frac{1}{\frac{k}{e}+\frac{1}{4\cdot k+4}}}+\frac{1}{2}\)

XCas> invPsi2(k) := k + 1/(2 + 1/(6k - 1/(2 + 1/(k/e - 1/(2 + 1/(k+1/2))))))

\(k \rightarrow k+\frac{1}{2+\frac{1}{6\cdot k-\frac{1}{2+\frac{1}{\frac{k}{e}-\frac{1}{2+\frac{1}{k+\frac{1}{2}}}}}}}\)

XCas> simplify(invPsi1(k) - invPsi2(k))       → 0

Let's try a few x, to see if it round-trip.

XCas> invPsi(x) := invPsi1(exp^float(x))             // Note: k = e^x
XCas> map(range(1,6), x->[x, Psi(invPsi(x))])

\(\qquad\left(\begin{array}{cc}
1 & 0.999999666188 \\
2 & 2.00000000607 \\
3 & 3.00000000018 \\
4 & 4.0 \\
5 & 5.0
\end{array}\right)\)