/**
   * Returns x rounded to either the greatest double less than or equal to the precise value of x,
   * or the least double greater than or equal to the precise value of x.
   */
  abstract double roundToDoubleArbitrarily(X x);

  /** Returns the sign of x: either -1, 0, or 1. */
  abstract int sign(X x);

  /** Returns d's value as an X, rounded with the specified mode. */

      int x = 1 << exponent;
      intValues.add(x, x + 1, x - 1);
    }
    intValues.add(9999).add(10000).add(10001).add(1000000); // near powers of 10
    intValues.add(5792).add(5793); // sqrt(2^25) rounded up and down
    POSITIVE_INTEGER_CANDIDATES = intValues.build();
    NEGATIVE_INTEGER_CANDIDATES =
        ImmutableList.copyOf(
            Iterables.concat(

    // to make sure bit twiddling works as expected.
    return ~~(x - y) >>> (Integer.SIZE - 1);
  }

  /**
   * Returns the base-2 logarithm of {@code x}, rounded according to the specified rounding mode.
   *
   * @throws IllegalArgumentException if {@code x <= 0}
   * @throws ArithmeticException if {@code mode} is {@link RoundingMode#UNNECESSARY} and {@code x}

      int result = BigIntegerMath.log2(x, HALF_UP);
      BigInteger x2 = x.pow(2);
      // x^2 < 2^(2 * result + 1), or else we would have rounded up
      assertThat(ZERO.setBit(2 * result + 1)).isGreaterThan(x2);
      // x^2 >= 2^(2 * result - 1), or else we would have rounded down
      assertTrue(result == 0 || ZERO.setBit(2 * result - 1).compareTo(x2) <= 0);
    }
  }

  public void testLog2HalfDown() {

    // to make sure bit twiddling works as expected.
    return (int) (~~(x - y) >>> (Long.SIZE - 1));
  }

  /**
   * Returns the base-2 logarithm of {@code x}, rounded according to the specified rounding mode.
   *
   * @throws IllegalArgumentException if {@code x <= 0}
   * @throws ArithmeticException if {@code mode} is {@link RoundingMode#UNNECESSARY} and {@code x}

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										${f:h(fe:label(countEntry.key))}
										<span class="badge rounded-pill text-bg-secondary float-end">${f:h(countEntry.value)}</span>
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						</c:if>
					</c:forEach>
				</ul>
			</c:if>
		</c:forEach>

      // Since scale is a positive int, index is in [0, scale], and (dataset.length - 1) is a
      // non-negative int, we can do long-arithmetic on index * (dataset.length - 1) / scale to get
      // 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);

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										${f:h(fe:label(countEntry.key))}
										<span class="badge rounded-pill text-bg-secondary float-end">${f:h(countEntry.value)}</span>
								</la:link></li>
						</c:if>
					</c:forEach>
				</ul>
			</c:if>
		</c:forEach>

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