[VA] SRC #016 - Pi Day 2024 Special

[C.Ret]

      
Bonjour à toutes et à tous.

Happy \(\pi\) Day 2024 !

This is my humble participation to SRC #16 - Pi Day 2024 Special

1. First appearance

I am trying hard to solve this equation with my HP-15C, hoping that numeric integration, solver and hyperbolic trigonometric are of any use.

My obviously perfectible solution is a three step algorithm:

1. Grab yourHP-15C and power it up.
   From far the easiest and fastest part of the proposed solution.
   Make sure you wake up your HP-15C in radian trigonometric mode and properly format its display.
   ON
   g RAD
   f FIX 8
   g P/R
   f CLEAR PRGM
2. Compute the left side integral
   The left side of the equation is a constant, I need a short program to compute the integral: \( R0 = \int_{0}^{1}\frac{{tan^{-1}(tanh^{-1}(x))}}{x}\: \mathrm{d}x \)
   
   After entering this short code:
   001-42,21, 0   f LBL 0
   002-43,22,25   g HYP-¹ TAN
   003- 43 25   g TAN-¹
   004-   34     x⇋y
   005-   10      ÷
   006- 43 32   g RTN
   
   The value of this left side integral may be compute and store in register R0.
   0 ENTER 1 ∫yx 0
        ... Running ... 
        1.02576051
   STO 0
3. Solve for a value of x that balance the equation
   \( R1=x\cdot\left( ln(\Gamma(\frac{1}{x}))-ln(\Gamma(\frac{1}{2}+\frac{1}{x}))-\frac{1}{2}\cdot ln\,x\right)=x\cdot ln\left(\frac{\Gamma(1/x)}{\Gamma(1/x+0.5)\cdot\sqrt{x}}\right) \)
   The right side of the equation may be code in memory as the following program :
   
   007-42,21, 1   f LBL 1
   008-    15    1/x
   009-    1     1
   010-    30    -
   011-  42 0   f x!
   012-  43 36  g LST x
   013-    48    .
   014-     5    5
   015-    40    +
   016-  42 0   f x!
   017-    10    ÷
   018-    34    x⇋y
   019-    11    √x
   020-    10    ÷
   021-  43 12  g LN   
   022-    20    ×
   023-45,30, 0    RCL-0
   024-  43 32  g RTN
   
   The expected result may be display using the solver and store in register R1.
   1 ENTER 6 SOLVE 1
        ... Running ...
         3.14159265
   STO 1

Currently looking for the calculator I may used for the next appearance.

Thanks to Valentin for this nice PI-day special !