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);

	MOVDW	F4, F5			// 85081b01
	NEGF	F4, F5			// 85141401
	NEGD	F4, F5			// 85181401
	ABSD	F4, F5			// 85081401
	TRUNCDW	F4, F5			// 85881a01
	TRUNCFW	F4, F5			// 85841a01
	SQRTF	F4, F5			// 85441401
	SQRTD	F4, F5			// 85481401

	DBAR	 			// 00007238
	NOOP	 			// 00004003

	MOVWR	R4, result+16(FP) 	// 6460402f
	MOVWR	R4, 1(R5) 		// a404402f
	MOVWR	y+8(FP), R4 		// 6440402e

// This decomposition is only correct inside XLA as it ignores use_locking
// attribute.
// alpha <- learning_rate * sqrt(1 - beta2^t) / (1 - beta1^t)
// m_t <- beta1 * m_{t-1} + (1 - beta1) * g_t
// v_t <- beta2 * v_{t-1} + (1 - beta2) * g_t * g_t
// variable <- variable - alpha * m_t / (sqrt(v_t) + epsilon)
def DecomposeResourceApplyAdamNonNesterov :
  Pattern<
    (TF_ResourceApplyAdamOp:$src_op

        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);

			if !testModSqrt(t, &elt, &mod, &sq, &sqrt) {
				t.Errorf("#%d: failed (sqrt(%d,%d) = %s)", x, &elt, &mod, &sqrt)
			}
			isSquare[sq.Uint64()] = true
		}

		// test all non-squares
		for x := 1; x < n; x++ {
			sq.SetInt64(int64(x))
			z := sqrt.ModSqrt(&sq, &mod)
			if !isSquare[x] && z != nil {
				t.Errorf("#%d: failed (sqrt(%d,%d) = nil)", x, &sqrt, &mod)
			}
		}
	}
}

        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

    if (!erf_user_user) return failure();

    // `mul 1/sqrt(2)`
    auto* erf_input = GetInputOpWithOneUse(op, 0);
    if (!erf_input) return failure();

    // `mul 0.5`
    auto* erf_user_user_input = GetInputOpWithOneUse(erf_user_user, 0);
    if (!erf_user_user_input) return failure();

    // Check `mul 0.5` and `mul 1/sqrt(2)` refer to the same input.

     * 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