Maintain an accumulated array of denominators and an exact remaining fraction. The usual greedy algorithm can require a denominator larger than Int.MaxValue, even for an input with Int components. The following implementation uses arbitrary-precision integers internally. If that representational limit is reached, it returns the always-valid alternative consisting of num copies of . Repeated denominators are permitted by the given Egyptian fraction representation.
def fromRat(num: Int, den: Int): Array[Int] = {
  require(0 <= num && num < den)
  val start = Reduce(num, den)
  var n = BigInt(start._1)
  var d = BigInt(start._2)
  var e = Array.empty[Int]
  while (n > 0) {
    val q = (d + n - 1) / n
    if (!q.isValidInt) return Array.fill(num)(den)
    e = e :+ q.toInt
    val nextN = n * q - d
    val nextD = d * q
    val g = nextN.gcd(nextD)
    n = nextN / g
    d = nextD / g
  }
  e
}
The loop invariant is , , all entries of e are positive, and
(1)
Appending and replacing the residual by preserves this equality; division by the greatest common divisor preserves the residual's value. The loop variant is , by the proof in part (b). If nextN is zero, its greatest common divisor with nextD is nextD, giving residual . On normal exit the residual is zero and the returned array has exactly the required value. On the early-return branch, num copies of have that same value directly; the partial greedy Egyptian fraction expansion is discarded. num=0 returns an empty array.
The fallback is a correctness provision for the fixed result type, not an efficiency claim: a large repeated-term array can be impractical. An array of arbitrary-precision integers would retain the greedy Egyptian fraction expansion without this fallback. Allocation, as with any output-producing program, requires enough available memory. The BigInt remainder, greatest common divisor and checked conversion operations are documented in [ref].

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