Cut the Cards
08-29-2020, 04:02 PM
Post: #18
 Albert Chan Senior Member Posts: 1,890 Joined: Jul 2018
RE: Cut the Cards
(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))])

$$\qquad\left(\begin{array}{cc} 1 & 0.999999666188 \\ 2 & 2.00000000607 \\ 3 & 3.00000000018 \\ 4 & 4.0 \\ 5 & 5.0 \end{array}\right)$$
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 Messages In This Thread Cut the Cards - David Hayden - 07-30-2020, 08:00 PM RE: Cut the Cards - Albert Chan - 07-30-2020, 08:58 PM RE: Cut the Cards - Albert Chan - 08-21-2020, 11:00 PM RE: Cut the Cards - Jim Horn - 07-30-2020, 09:49 PM RE: Cut the Cards - John Keith - 07-31-2020, 12:24 AM RE: Cut the Cards - Gerson W. Barbosa - 08-24-2020, 01:57 PM RE: Cut the Cards - Albert Chan - 08-25-2020, 06:14 PM RE: Cut the Cards - Albert Chan - 07-30-2020, 10:21 PM RE: Cut the Cards - pinkman - 08-24-2020, 09:49 PM RE: Cut the Cards - Gerson W. Barbosa - 08-25-2020, 11:41 PM RE: Cut the Cards - Albert Chan - 08-26-2020, 03:06 AM RE: Cut the Cards - Gerson W. Barbosa - 08-26-2020, 08:23 AM RE: Cut the Cards - Albert Chan - 08-26-2020, 02:13 PM RE: Cut the Cards - Gerson W. Barbosa - 08-26-2020, 06:13 PM RE: Cut the Cards - Gerson W. Barbosa - 08-27-2020, 10:07 PM RE: Cut the Cards - Albert Chan - 08-28-2020, 09:26 PM RE: Cut the Cards - Albert Chan - 08-29-2020 04:02 PM RE: Cut the Cards - Gerson W. Barbosa - 08-28-2020, 11:39 PM RE: Cut the Cards - Albert Chan - 06-23-2021, 12:08 AM

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