-- 1-15-Haskell.hs
--
-- Haskell is a functional programming language.
-- Variables are immutable, and repetition is commonly
-- expressed using recursion rather than loops that
-- repeatedly modify variables.
--
-- A function definition also often has a separate type
-- signature. For example,
--
--     mySin :: Double -> Double
--
-- says that mySin takes a Double and returns a Double.

main :: IO ()
main = do
    let x = 3.2

    putStrLn $ "mySin x: " ++ show (mySin x)
    putStrLn $ "sin x:   " ++ show (sin x)
    putStrLn $ "reference: "
        ++ show (-0.058374143427579909137)

    putStrLn $ "mySin (1.0/0.0): "
        ++ show (mySin (1.0/0.0))

-- mySin :: Double -> Double
--
-- Approximate sin(x) using the Taylor series
--
--                 x^3   x^5   x^7
--     sin(x) = x - --- + --- - --- + ...
--                  3!    5!    7!
--
-- Rather than calculating each power and factorial
-- separately, calculate each term from the previous
-- term:
--
--                    x^2
--     next = -term -----------,
--                  n(n + 1)
--
-- where n takes the values 2, 4, 6, ...
--
-- Haskell does not modify result, term, and n.
-- Instead, each recursive call supplies new values
-- for them.
mySin :: Double -> Double
mySin x
    | isNaN x || isInfinite x = 0.0/0.0
    | otherwise               = next x x 2.0
  where
    next result term n =
        let nextTerm = -term*(x/n)*(x/(n + 1.0))
        in
            if abs nextTerm < 1e-16
            then result
            else next (result + nextTerm)
                      nextTerm
                      (n + 2.0)