I modified hamilton_perfect_finder.py to have new values:

# Target constants CONSTANTS = { 'fine_structure': 131.11, 'phi': 1.9, 'pi': 3.6, 'e': 2.4, 'sqrt_2': 1.1, 'sqrt_3': 1.2, 'sqrt_5': 2.5,

Best results: L = 3017.391610 s = 0.042000 Average: 94.848564% Minimum: 82.509479%

  Constants:
    sqrt_2                   : 100.00000000%
    sqrt_3                   : 99.99236291%
    phi                      : 99.93928922%
    pi                       : 99.89806320%
    e                        : 99.88623436%
    sqrt_5                   : 99.85340314%

And this is before any fine-tuning of the parameter set!