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);
      }
    }

      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
      assertTrue(x4.compareTo(plusHalfSquared) <= 0);
      BigInteger minusHalfSquared = result.pow(2).subtract(result).shiftLeft(2).add(ONE);

        .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));

      int j = i & ARRAY_MASK;
      tmp += BigIntegerMath.log10(positive[j], mode);
    }
    return tmp;
  }

  @Benchmark
  int sqrt(int reps) {
    int tmp = 0;
    for (int i = 0; i < reps; i++) {
      int j = i & ARRAY_MASK;
      tmp += BigIntegerMath.sqrt(positive[j], mode).intValue();
    }
    return tmp;
  }

  @Benchmark
  int divide(int reps) {
    int tmp = 0;

        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);

   *     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(LongMath.sqrt(x, mode)));
      }
    }
  }

  /* Relies on the correctness of sqrt(long, FLOOR). */
  @GwtIncompatible // TODO
  public void testSqrtExactMatchesFloorOrThrows() {
    for (long x : POSITIVE_LONG_CANDIDATES) {
      long sqrtFloor = LongMath.sqrt(x, FLOOR);

      int j = i & ARRAY_MASK;
      tmp += IntMath.log10(positive[j], mode);
    }
    return tmp;
  }

  @Benchmark
  int sqrt(int reps) {
    int tmp = 0;
    for (int i = 0; i < reps; i++) {
      int j = i & ARRAY_MASK;
      tmp += IntMath.sqrt(positive[j], mode);
    }
    return tmp;
  }

  @Benchmark
  int divide(int reps) {
    int tmp = 0;
    for (int i = 0; i < reps; i++) {

      int j = i & ARRAY_MASK;
      tmp += BigIntegerMath.log10(positive[j], mode);
    }
    return tmp;
  }

  @Benchmark
  int sqrt(int reps) {
    int tmp = 0;
    for (int i = 0; i < reps; i++) {
      int j = i & ARRAY_MASK;
      tmp += BigIntegerMath.sqrt(positive[j], mode).intValue();
    }
    return tmp;
  }

  @Benchmark
  int divide(int reps) {
    int tmp = 0;

     * Math.sqrt(k * k) <= Math.sqrt(x) <= Math.sqrt((k + 1) * (k + 1))
     *          since Math.sqrt is monotonic.
     * (long) Math.sqrt(k * k) <= (long) Math.sqrt(x) <= (long) Math.sqrt((k + 1) * (k + 1))
     *          since casting to long is monotonic
     * k <= (long) Math.sqrt(x) <= k + 1
     *          since (long) Math.sqrt(k * k) == k, as checked exhaustively in