*/
      sqrt0 = sqrtApproxWithDoubles(x.shiftRight(shift)).shiftLeft(shift >> 1);
    }
    BigInteger sqrt1 = sqrt0.add(x.divide(sqrt0)).shiftRight(1);
    if (sqrt0.equals(sqrt1)) {
      return sqrt0;
    }
    do {
      sqrt0 = sqrt1;
      sqrt1 = sqrt0.add(x.divide(sqrt0)).shiftRight(1);
    } while (sqrt1.compareTo(sqrt0) < 0);
    return sqrt0;
  }

  @GwtIncompatible // TODO

        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);

        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);

      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);

      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));

          }
          k >>= 1;
          if (k > 0) {
            if (-FLOOR_SQRT_MAX_INT > b | b > FLOOR_SQRT_MAX_INT) {
              return limit;
            }
            b *= b;
          }
      }
    }
  }

  @VisibleForTesting static final int FLOOR_SQRT_MAX_INT = 46340;

  /**

        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);

    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())

     * 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