formula.md

$$\begin {aligned} \sum _{n=0}^{\infty }\frac {((2n)!)^3}{(n!)^6}\frac {42n+5}{2^{12n}}&=\frac {16}\pi \\\\\ \end {aligned}$$

Ramanujan.scala

import scala.compiletime.ops.int
import scala.compiletime.ops.long
import scala.compiletime.ops.double

type Factorial[A <: Int] =
  Factorial0[A, 1L]

type Factorial0[A <: Int, B <: Long] <: Long =
  int.<[A, 1] match
    case true =>
      B
    case _ =>
      Factorial0[
        int.-[A, 1],
        long.*[
          int.ToLong[A],
          B
        ]
      ]

type Exponential[A <: Long, N <: Int] =
  Exponential0[A, 1L, N]

type Exponential0[A <: Long, B <: Long, N <: Int] <: Long =
  N match
    case 0 =>
      B
    case _ =>
      Exponential0[A, long.*[A, B], int.-[N, 1]]

type F1[N <: Int] =
  double.*[
    double./[
      long.ToDouble[
        Exponential[
          Factorial[
            int.*[
              2,
              N
            ]
          ],
          3
        ]
      ],
      long.ToDouble[
        Exponential[
          Factorial[N],
          6
        ]
      ]
    ],
    double./[
      long.ToDouble[
        long.+[
          long.*[
            42L,
            int.ToLong[N]
          ],
          5L
        ]
      ],
      long.ToDouble[
        Exponential[
          2L,
          int.*[12, N]
        ]
      ]
    ]
  ]

type F2[N <: Int, A <: Double] <: Double =
  int.<[N, 0] match {
    case true =>
      A
    case _ =>
      F2[int.-[N, 1], double.+[F1[N], A]]
  }

type F[N <: Int] =
  double./[
    16.0,
    F2[N, 0.0]
  ]

def f = {
  summon[Exponential[2L, 3] =:= 8L]
  summon[Exponential[4L, 5] =:= 1024L]

  summon[Factorial[4] =:= 24L]
  summon[Factorial[5] =:= 120L]
  summon[Factorial[6] =:= 720L]

  summon[F[5] =:= 3.141592655748593]
}