assertThat(INTEGER_MANY_VALUES_STATS_ITERABLE.populationStandardDeviation())
        .isWithin(ALLOWED_ERROR * sqrt(INTEGER_MANY_VALUES_SUM_OF_SQUARES_OF_DELTAS))
        .of(sqrt(INTEGER_MANY_VALUES_SUM_OF_SQUARES_OF_DELTAS / INTEGER_MANY_VALUES_COUNT));
    assertThat(LONG_MANY_VALUES_STATS_ITERATOR.populationStandardDeviation())

        .isWithin(ALLOWED_ERROR * sqrt(LONG_MANY_VALUES_SUM_OF_SQUARES_OF_DELTAS))
        .of(sqrt(LONG_MANY_VALUES_SUM_OF_SQUARES_OF_DELTAS / LONG_MANY_VALUES_COUNT));
    assertThat(longManyValuesAccumulatorByAddAllVarargs.populationStandardDeviation())
        .isWithin(ALLOWED_ERROR * sqrt(LONG_MANY_VALUES_SUM_OF_SQUARES_OF_DELTAS))
        .of(sqrt(LONG_MANY_VALUES_SUM_OF_SQUARES_OF_DELTAS / LONG_MANY_VALUES_COUNT));

        .isWithin(ALLOWED_ERROR * sqrt(LONG_MANY_VALUES_SUM_OF_SQUARES_OF_DELTAS))
        .of(sqrt(LONG_MANY_VALUES_SUM_OF_SQUARES_OF_DELTAS / LONG_MANY_VALUES_COUNT));
    assertThat(longManyValuesAccumulatorByAddAllVarargs.populationStandardDeviation())
        .isWithin(ALLOWED_ERROR * sqrt(LONG_MANY_VALUES_SUM_OF_SQUARES_OF_DELTAS))
        .of(sqrt(LONG_MANY_VALUES_SUM_OF_SQUARES_OF_DELTAS / LONG_MANY_VALUES_COUNT));

      BigInteger plusHalfSquared = result.pow(2).add(result).shiftLeft(2).add(ONE);
      BigInteger x4 = x.shiftLeft(2);
      // sqrt(x) <= result + 0.5, so 4 * x <= (result + 0.5)^2 * 4
      // (result + 0.5)^2 * 4 = (result^2 + result)*4 + 1
      assertThat(plusHalfSquared).isAtLeast(x4);
      BigInteger minusHalfSquared = result.pow(2).subtract(result).shiftLeft(2).add(ONE);

          fractionalBuilder.add(x);
        }
      }
    }
    INTEGRAL_DOUBLE_CANDIDATES = integralBuilder.build();
    fractionalBuilder.add(1.414).add(1.415).add(Math.sqrt(2));
    fractionalBuilder.add(5.656).add(5.657).add(4 * Math.sqrt(2));
    for (double d : INTEGRAL_DOUBLE_CANDIDATES) {
      double x = 1 / d;
      if (x != Math.rint(x)) {
        fractionalBuilder.add(x);
      }
    }

   *     sqrt(x)} is not an integer
   */
  @GwtIncompatible // TODO
  @SuppressWarnings("fallthrough")
  public static BigInteger sqrt(BigInteger x, RoundingMode mode) {
    checkNonNegative("x", x);
    if (fitsInLong(x)) {
      return BigInteger.valueOf(LongMath.sqrt(x.longValue(), mode));
    }
    BigInteger sqrtFloor = sqrtFloor(x);

        assertEquals(BigIntegerMath.sqrt(valueOf(x), mode), valueOf(IntMath.sqrt(x, mode)));
      }
    }
  }

  /* Relies on the correctness of sqrt(int, FLOOR). */
  @GwtIncompatible // sqrt
  public void testSqrtExactMatchesFloorOrThrows() {
    for (int x : POSITIVE_INTEGER_CANDIDATES) {
      int floor = IntMath.sqrt(x, FLOOR);

        assertEquals(BigIntegerMath.sqrt(valueOf(x), mode), valueOf(IntMath.sqrt(x, mode)));
      }
    }
  }

  /* Relies on the correctness of sqrt(int, FLOOR). */
  @GwtIncompatible // sqrt
  public void testSqrtExactMatchesFloorOrThrows() {
    for (int x : POSITIVE_INTEGER_CANDIDATES) {
      int floor = IntMath.sqrt(x, FLOOR);

   * @throws ArithmeticException if {@code mode} is {@link RoundingMode#UNNECESSARY} and {@code
   *     sqrt(x)} is not an integer
   */
  @GwtIncompatible // need BigIntegerMath to adequately test
  @SuppressWarnings("fallthrough")
  public static int sqrt(int x, RoundingMode mode) {
    checkNonNegative("x", x);
    int sqrtFloor = sqrtFloor(x);
    switch (mode) {
      case UNNECESSARY:

   * R*R)} of the population standard deviation of {@code y}, where {@code R} is the Pearson's
   * correlation coefficient (as given by {@link #pearsonsCorrelationCoefficient()}).
   *
   * <p>The corresponding root-mean-square error in {@code x} as a function of {@code y} is a
   * fraction {@code sqrt(1/(R*R) - 1)} of the population standard deviation of {@code x}. This fit