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);
      }
    }

       */
      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

      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);

        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(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);

     * 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

    for (int exp : asList(-1022, -50, -1, 0, 1, 2, 3, 4, 100, 1022, 1023)) {
      for (RoundingMode mode : asList(HALF_EVEN, HALF_UP, HALF_DOWN)) {
        double x = Math.scalb(Math.sqrt(2) + 0.001, exp);
        double y = Math.scalb(Math.sqrt(2) - 0.001, exp);
        assertEquals(exp + 1, DoubleMath.log2(x, mode));
        assertEquals(exp, DoubleMath.log2(y, mode));
      }
    }
  }

import static com.google.common.base.Preconditions.checkNotNull;
import static com.google.common.collect.CollectPreconditions.checkNonnegative;
import static com.google.common.math.IntMath.sqrt;
import static java.lang.Math.max;
import static java.util.Objects.requireNonNull;

import com.google.common.annotations.GwtCompatible;
import com.google.common.annotations.GwtIncompatible;

	XVMODW		X3, X2, X1	// 410ce374
	XVMODV		X3, X2, X1	// 418ce374
	XVMODBU		X3, X2, X1	// 410ce674
	XVMODHU		X3, X2, X1	// 418ce674
	XVMODWU		X3, X2, X1	// 410ce774
	XVMODVU		X3, X2, X1	// 418ce774

	// [X]VF{SQRT/RECIP/RSQRT}{F/D} instructions
	VFSQRTF		V1, V2		// 22e49c72
	VFSQRTD		V1, V2		// 22e89c72
	VFRECIPF	V1, V2		// 22f49c72
	VFRECIPD	V1, V2		// 22f89c72
	VFRSQRTF	V1, V2		// 22049d72
	VFRSQRTD	V1, V2		// 22089d72

   * {@snippet :
   * ListMultimap<String, Integer> multimap =
   *      ImmutableListMultimap.of("a", 4, "a", 16, "b", 9);
   * Function<Integer, Double> sqrt = (Integer in) -> Math.sqrt((int) in);
   * ListMultimap<String, Double> transformed = Multimaps.transformValues(map,
   *     sqrt);
   * System.out.println(transformed);
   * }
   *
   * ... prints {@code {a=[2.0, 4.0], b=[3.0]}}.
   *