Continued Fraction Approximations

I wrote this Python script to find increasingly accurate rational approximations to any number using convergents and continued fraction representations. Below is the output for the Golden ratio φ ≈ 1.618, π ≈  3.14159, √2 ≈ 1.4142, and Euler's constant e ≈ 2.718.

    >>> phi = (math.sqrt(5)+1)/2

    >>> convergents2floats(convergents(contfrac(phi)), phi)

       Fraction       Approximation    Difference

          1/1         1.00000000000  -0.61803398875

          2/1         2.00000000000   0.38196601125

          3/2         1.50000000000  -0.11803398875

          5/3         1.66666666667   0.04863267792

          8/5         1.60000000000  -0.01803398875

         13/8         1.62500000000   0.00696601125

         21/13        1.61538461538  -0.00264937337

            Target:   1.61803398875

    >>> convergents2floats(convergents(contfrac(math.pi)), math.pi)

       Fraction       Approximation    Difference

          3/1         3.00000000000  -0.14159265359

         22/7         3.14285714286   0.00126448927

        333/106       3.14150943396  -0.00008321963

        355/113       3.14159292035   0.00000026676

     103993/33102     3.14159265301  -0.00000000058

     104348/33215     3.14159265392   0.00000000033

     208341/66317     3.14159265347  -0.00000000012

            Target:   3.14159265359

    >>> convergents2floats(convergents(contfrac(math.sqrt(2))), math.sqrt(2))

       Fraction       Approximation    Difference

          1/1         1.00000000000  -0.41421356237

          3/2         1.50000000000   0.08578643763

          7/5         1.40000000000  -0.01421356237

         17/12        1.41666666667   0.00245310429

         41/29        1.41379310345  -0.00042045892

         99/70        1.41428571429   0.00007215191

        239/169       1.41420118343  -0.00001237894

            Target:   1.41421356237

    >>> convergents2floats(convergents(contfrac(math.e)), math.e)

       Fraction       Approximation    Difference

          2/1         2.00000000000  -0.71828182846

          3/1         3.00000000000   0.28171817154

          8/3         2.66666666667  -0.05161516179

         11/4         2.75000000000   0.03171817154

         19/7         2.71428571429  -0.00399611417

         87/32        2.71875000000   0.00046817154

        106/39        2.71794871795  -0.00033311051

            Target:   2.71828182846