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)
			}
		}
	}
}

    // CHECK: [[LR_MULTIPLY:%.*]] = "tf.Mul"([[LR]], [[GRAD]]) : (tensor<f32>, tensor<f32>) -> tensor<f32>
    // CHECK: [[SQRT:%.*]] = "tf.Sqrt"([[NEW_ACC]]) : (tensor<*xf32>) -> tensor<*xf32>
    // CHECK: [[DIVISOR:%.*]] = "tf.AddV2"([[SQRT]], [[EPSILON]]) : (tensor<*xf32>, tensor<f32>) -> tensor<*xf32>

                const std::vector<Output>& grad_inputs,
                std::vector<Output>* grad_outputs) {
  // y = asin(x)
  // dy/dx = 1 / sqrt(1 - x^2)
  auto x2 = Square(scope, op.input(0));
  auto one = Cast(scope, Const(scope, 1.0), op.input(0).type());
  auto dydx = Reciprocal(scope, Sqrt(scope, Sub(scope, one, x2)));
  // grad(x) = grad(y) * conj(dy/dx)
  auto dx = Mul(scope, grad_inputs[0], ConjugateHelper(scope, dydx));

 *
 * <pre>{@code
 * public static double sqrt(double value) {
 *   if (value < 0) {
 *     throw new IllegalArgumentException("input is negative: " + value);
 *   }
 *   // calculate square root
 * }
 * }</pre>
 *
 * <p>to be replaced with the more compact
 *
 * <pre>{@code
 * public static double sqrt(double value) {
 *   checkArgument(value >= 0, "input is negative: %s", value);

 *
 * <pre>{@code
 * public static double sqrt(double value) {
 *   if (value < 0) {
 *     throw new IllegalArgumentException("input is negative: " + value);
 *   }
 *   // calculate square root
 * }
 * }</pre>
 *
 * <p>to be replaced with the more compact
 *
 * <pre>{@code
 * public static double sqrt(double value) {
 *   checkArgument(value >= 0, "input is negative: %s", value);

//
// The pattern lower FusedBatchNormV3 to arithmetic ops.
// 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).
//
// def : Pattern<

   *      = ImmutableListMultimap.of("a", 4, "a", 16, "b", 9);
   * Function<Integer, Double> sqrt =
   *     new Function<Integer, Double>() {
   *       public Double apply(Integer in) {
   *         return Math.sqrt((int) in);
   *       }
   *     };
   * ListMultimap<String, Double> transformed = Multimaps.transformValues(map,
   *     sqrt);
   * System.out.println(transformed);
   * }</pre>
   *

   *      = ImmutableListMultimap.of("a", 4, "a", 16, "b", 9);
   * Function<Integer, Double> sqrt =
   *     new Function<Integer, Double>() {
   *       public Double apply(Integer in) {
   *         return Math.sqrt((int) in);
   *       }
   *     };
   * ListMultimap<String, Double> transformed = Multimaps.transformValues(map,
   *     sqrt);
   * System.out.println(transformed);
   * }</pre>
   *

		{name: "FSQRTD", argLength: 1, reg: fp11, asm: "FSQRTD"},                              // sqrt(arg0), float64
		{name: "FSQRTS", argLength: 1, reg: fp11, asm: "FSQRTS"},                              // sqrt(arg0), float32
		{name: "FMIND", argLength: 2, reg: fp21, asm: "FMIND"},                                // min(arg0, arg1)

(Or(32|16|8) ...) => (OR ...)
(Xor(32|16|8) ...) => (XOR ...)

// unary ops
(Neg(32|16|8) x) => (RSBconst [0] x)
(Neg(32|64)F ...) => (NEG(F|D) ...)

(Com(32|16|8) ...) => (MVN ...)

(Sqrt ...) => (SQRTD ...)
(Sqrt32 ...) => (SQRTF ...)
(Abs ...) => (ABSD ...)

// TODO: optimize this for ARMv5 and ARMv6
(Ctz32NonZero ...) => (Ctz32 ...)
(Ctz16NonZero ...) => (Ctz32 ...)
(Ctz8NonZero ...) => (Ctz32 ...)