Function powRational

src/compute-engine/interval/elementary.ts:261–286

(
  base: Interval | IntervalResult,
  p: number,
  q: number
)

 * bracket the range. A negative exponent has a pole at 0.
 */
export function powRational(
  base: Interval | IntervalResult,
  p: number,
  q: number
): IntervalResult {
  const unwrapped = unwrapOrPropagate(base);
  if (!Array.isArray(unwrapped)) return unwrapped;
  const [b] = unwrapped;
  const exp = p / q;

  // Even denominator: real only for a non-negative base — `pow` already models
  // the empty/partial/monotone cases correctly.
  if (q % 2 === 0) return pow(ok(b), exp);

  // Odd denominator, non-negative base: ordinary `pow` is exact.
  if (b.lo >= 0) return pow(ok(b), exp);

  const hasZero = b.lo <= 0 && b.hi >= 0;
  // Negative exponent has a pole at 0.
  if (p < 0 && hasZero) return { kind: 'singular', at: 0 };

  const candidates = [realRatPow(b.lo, p, exp), realRatPow(b.hi, p, exp)];
  // 0 is the interior minimum for an even numerator (positive exponent).
  if (hasZero && p > 0) candidates.push(0);
  return ok({ lo: Math.min(...candidates), hi: Math.max(...candidates) });
}

/**
 * Integer nth root on an interval (`Root(x, n)`).