// It's definitely safe to multiply into numerator and denominator.
              numerator *= n;
              denominator *= i;
              numeratorBits += nBits;
            } else {
              // It might not be safe to multiply into numerator and denominator,
              // so multiply (numerator / denominator) into result.
              result = multiplyFraction(result, numerator, denominator);
              numerator = n;

}

// ceilFrac takes a numerator and denominator representing a fraction
// and returns its ceiling. If denominator is 0, it returns 0 instead
// of crashing.
func ceilFrac(numerator, denominator int64) (ceil int64) {
	if denominator == 0 {
		// do nothing on invalid input
		return ceil
	}
	// Make denominator positive
	if denominator < 0 {
		numerator = -numerator
		denominator = -denominator
	}

      // a rounded ratio and a remainder which can be expressed as ints, without risk of overflow:
      int quotient = (int) LongMath.divide(numerator, scale, RoundingMode.DOWN);
      int remainder = (int) (numerator - (long) quotient * scale);
      selectInPlace(quotient, dataset, 0, dataset.length - 1);
      if (remainder == 0) {
        return dataset[quotient];
      } else {

   * quantities.
   *
   * <p><b>Java 8+ users:</b> use {@link Integer#divideUnsigned(int, int)} instead.
   *
   * @param dividend the dividend (numerator)
   * @param divisor the divisor (denominator)
   * @throws ArithmeticException if divisor is 0
   */
  public static int divide(int dividend, int divisor) {
    return (int) (toLong(dividend) / toLong(divisor));

    for (int i = 1; i < k; i++) {
      int p = n - i;
      int q = i + 1;

      // log2(p) >= bits - 1, because p >= n/2

      if (numeratorBits + bits >= Long.SIZE - 1) {
        // The numerator is as big as it can get without risking overflow.
        // Multiply numeratorAccum / denominatorAccum into accum.
        accum =
            accum
                .multiply(BigInteger.valueOf(numeratorAccum))