analysis.egi

--
-- Originally auto-generated by egison-translator; modernized for the
-- typed CAS (declare symbol; expectations follow the current normal
-- forms: the Pythagorean auto rule rewrites cos^2 to 1 - sin^2,
-- casRewriteExp merges exp products and powers, and function-symbol
-- derivative marks are positional and sorted -- Schwarz canonical
-- form, so the mixed second-order Taylor terms merge into one).
--

declare symbol x, y, z, a

assertEqual "d/d - case 1" (d/d (x ^ 2) x) (2 * x)

assertEqual "d/d - case 2" (d/d (a ^ (x ^ 2)) x) (2 * a ^ (x ^ 2) * log a * x)

assertEqual "d/d - case 3" (d/d (cos x * sin x) x) (1 - 2 * sin x ^ 2)

assertEqual
  "d/d - case 4"
  (d/d (sigmoid z) z)
  (exp (- z) / (1 + 2 * exp (- z) + exp (-2 * z)))

assertEqual "d/d - case 5" (d/d (d/d (log x) x) x) ((-1) / x ^ 2)

assertEqual
  "taylor-expansion - case 1"
  (take 4 (taylorExpansion (e ^ (i * x)) x 0))
  [1, i * x, -1 * x ^ 2 / 2, -1 * i * x ^ 3 / 6]

def f := function (x, y)

assertEqual
  "multivariate-tailor-expansion - case 1"
  (take 3 (multivariateTaylorExpansion (f x y) [|x, y|] [|0, 0|]))
  [ f 0 0
  , ((userRefs f [1]) 0 0) * x + ((userRefs f [2]) 0 0) * y
  , (1 / 2) * ((userRefs f [1, 1]) 0 0) * x ^ 2
      + ((userRefs f [1, 2]) 0 0) * x * y
      + (1 / 2) * ((userRefs f [2, 2]) 0 0) * y ^ 2 ]

assertEqual
  "function expr"
  (let g := function (x, y)
    in d/d g y)
  (let g := function (x, y)
    in userRefs g [y])

assertEqual
  "analytic derivative keeps FunctionData and registered chain rules"
  (∂/∂ (sin (f ^ 2)) x)
  (2 * (userRefs f [1]) * f * cos (f ^ 2))

assertEqual
  "analytic derivative maps coordinate tensors"
  (∂/∂ f [|x, y|])
  [|userRefs f [1], userRefs f [2]|]

def derivativeVector := generateTensor (\[i] -> function (x, y)) [2]

def functionDataHead (value: MathValue) : MathValue :=
  match value as mathValue with
    | func $head _ -> head
    | _ -> 0

assertEqual
  "symbolIndices preserves component and derivative index kinds"
  (symbolIndices
    (functionDataHead
      (partialDiffMV
        (partialDiffMV derivativeVector_2 y) x)))
  [SubIndex 2, UserIndex 1, UserIndex 2]