*
 * @author Louis Wasserman
 * @since 11.0
 */
@GwtCompatible(emulated = true)
@ElementTypesAreNonnullByDefault
public final class LongMath {
  // NOTE: Whenever both tests are cheap and functional, it's faster to use &, | instead of &&, ||

  @VisibleForTesting static final long MAX_SIGNED_POWER_OF_TWO = 1L << (Long.SIZE - 2);

  /**
   * Returns the smallest power of two greater than or equal to {@code x}. This is equivalent to

 * }</pre>
 *
 * <h3>Alternatives</h3>
 *
 * <h4>Frameworks</h4>
 *
 * <p>When chaining together a graph of asynchronous operations, you will often find it easier to
 * use a framework. Frameworks automate the process, often adding features like monitoring,
 * debugging, and cancellation. Examples of frameworks include:
 *
 * <ul>

 *       <i>removing</i> that key from the multimap.
 *   <li>The total entry count is available as {@link #size}.
 *   <li>Many complex operations become easier; for example, {@code
 *       Collections.min(multimap.values())} finds the smallest value across all keys.
 * </ul>
 *
 * <h3>Implementations</h3>
 *
 * <ul>
 *   <li>{@link ImmutableListMultimap}

   * equivalently, if this inverse view contains a key that is equal to {@code value}).
   *
   * <p>Due to the property that values in a BiMap are unique, this will tend to execute in
   * faster-than-linear time.
   *
   * @param value the object to search for in the values of this BiMap
   * @return true if a mapping exists from a key to the specified value
   */
  @Override

                return String.valueOf(QUALIFIERS.indexOf(""));
            }

            int i = QUALIFIERS.indexOf(qualifier);

            // Just returning an Integer with the index here is faster, but requires a lot of if/then/else to check for
            // -1
            //  or QUALIFIERS.size and then resort to lexical ordering. Most comparisons are decided by the first
            // character,

   * equal, ignoring the case of any ASCII alphabetic characters between {@code 'a'} and {@code 'z'}
   * or {@code 'A'} and {@code 'Z'} inclusive.
   *
   * <p>This method is significantly faster than {@link String#equalsIgnoreCase} and should be used
   * in preference if at least one of the parameters is known to contain only ASCII characters.
   *

    return CollectSpliterators.indexed(size(), SPLITERATOR_CHARACTERISTICS, this::get);
  }

  @Override
  int copyIntoArray(@Nullable Object[] dst, int offset) {
    // this loop is faster for RandomAccess instances, which ImmutableLists are
    int size = size();
    for (int i = 0; i < size; i++) {
      dst[offset + i] = get(i);
    }
    return offset + size;
  }

  /**

   * equivalently, if this inverse view contains a key that is equal to {@code value}).
   *
   * <p>Due to the property that values in a BiMap are unique, this will tend to execute in
   * faster-than-linear time.
   *
   * @param value the object to search for in the values of this BiMap
   * @return true if a mapping exists from a key to the specified value
   */
  @Override

 * worst case time complexity of O(N^2). You are extremely unlikely to hit this quadratic case on
 * randomly ordered data (the probability decreases faster than exponentially in N), but if you are
 * passing in unsanitized user data then a malicious user could force it. A light shuffle of the
 * data using an unpredictable seed should normally be enough to thwart this attack.
 *

   * <p>Many of the filtered multiset's methods, such as {@code size()}, iterate across every
   * element in the underlying multiset and determine which elements satisfy the filter. When a live
   * view is <i>not</i> needed, it may be faster to copy the returned multiset and use the copy.
   *
   * <p><b>Warning:</b> {@code predicate} must be <i>consistent with equals</i>, as documented at
   * {@link Predicate#apply}. Do not provide a predicate such as {@code