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);

$$\text{lr}_t := \mathrm{lr} \cdot \frac{\sqrt{1 - \beta_2^t}}{1 - \beta_1^t}$$
$$m_t := \beta_1 \cdot m_{t-1} + (1 - \beta_1) \cdot g$$
$$v_t := \beta_2 \cdot v_{t-1} + (1 - \beta_2) \cdot g^2$$
$$\text{var} := \begin{cases} \text{var} - (m_t \beta_1 + g \cdot (1 - \beta_1))\cdot\text{lr}_t/(\sqrt{v_t} + \epsilon), &\text{if use_nesterov}\\\\  \text{var} - m_t \cdot \text{lr}_t /(\sqrt{v_t} + \epsilon), &\text{otherwise} \end{cases}$$
  }];

// 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);

    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.

// MODEL-DEBUG: %[[f_sqrt1:.*]] = "tfl.sqrt"(%[[f_conv]]
// MODEL-DEBUG: %[[q_sqrt1:.*]] = "tfl.sqrt"(%[[dq0]]
// MODEL-DEBUG: %[[q1:.*]] = "tfl.quantize"(%[[q_sqrt1]])
// MODEL-DEBUG: %[[dq1:.*]] = "tfl.dequantize"(%[[q1]])
// MODEL-DEBUG: %[[f_sqrt2:.*]] = "tfl.sqrt"(%[[f_sqrt1]])
// MODEL-DEBUG-NOT: debug_
// MODEL-DEBUG-SAME: (tensor<?x1x1x3xf32>)
// MODEL-DEBUG: %[[q_sqrt2:.*]] = "tfl.sqrt"(%[[dq1]]
// MODEL-DEBUG-NOT: debug_

    if (!fp_type) {
      return failure();
    }

    // result = (x - mean) * scale / sqrt(variance + epsilon) + offset
    // Let multiplier = scale / sqrt(variance + epsilon), to compute
    // (x - mean) * scale / sqrt(variance + epsilon) + offset,
    // is then to compute (x * multiplier) + (offset - mean * multiplier).

    auto epsilon = materializeEpsilon(

// operations. Specifically, performs the following calculation:
//
//   (x - mean) * scale / sqrt(variance + epsilon) + offset
//
// Let multiplier = scale / sqrt(variance + epsilon),
// to compute
//   (x - mean) * scale / sqrt(variance + epsilon) + offset,
// is then to compute
//   (x * multiplier) + (offset - mean * multiplier).
//