// 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;

func TestCeilFrac(t *testing.T) {
	cases := []struct {
		numerator, denominator, ceiling int64
	}{
		{0, 1, 0},
		{-1, 2, 0},
		{1, 2, 1},
		{1, 1, 1},
		{3, 2, 2},
		{54, 11, 5},
		{45, 11, 5},
		{-4, 3, -1},
		{4, -3, -1},
		{-4, -3, 2},
		{3, 0, 0},
	}
	for i, testCase := range cases {
		ceiling := ceilFrac(testCase.numerator, testCase.denominator)
		if ceiling != testCase.ceiling {

}

// 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
	}
	// 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 Long#divideUnsigned(long, long)} instead.
   *
   * @param dividend the dividend (numerator)
   * @param divisor the divisor (denominator)
   * @throws ArithmeticException if divisor is 0
   */
  public static long divide(long dividend, long divisor) {
    if (divisor < 0) { // i.e., divisor >= 2^63:

   * 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))

215E          ; mapped                 ; 0037 2044 0038 #1.1  VULGAR FRACTION SEVEN EIGHTHS
215F          ; mapped                 ; 0031 2044     # 1.1  FRACTION NUMERATOR ONE
2160          ; mapped                 ; 0069          # 1.1  ROMAN NUMERAL ONE
2161          ; mapped                 ; 0069 0069     # 1.1  ROMAN NUMERAL TWO