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);

 */

const blockSize = 64 // A DCT block is 8x8.

type block [blockSize]int32

const (
	w1 = 2841 // 2048*sqrt(2)*cos(1*pi/16)
	w2 = 2676 // 2048*sqrt(2)*cos(2*pi/16)
	w3 = 2408 // 2048*sqrt(2)*cos(3*pi/16)
	w5 = 1609 // 2048*sqrt(2)*cos(5*pi/16)
	w6 = 1108 // 2048*sqrt(2)*cos(6*pi/16)
	w7 = 565  // 2048*sqrt(2)*cos(7*pi/16)

	w1pw7 = w1 + w7
	w1mw7 = w1 - w7
	w2pw6 = w2 + w6
	w2mw6 = w2 - w6
	w3pw5 = w3 + w5

	// standard deviation is sqrt( average( (x - xbar)^2 ) )
	// = sqrt( Integral( x^2 + xbar^2 -2*x*xbar dt ) / Duration )
	// = sqrt( ( Integral( x^2 dt ) + Duration * xbar^2 - 2*xbar*Integral(x dt) ) / Duration)
	// = sqrt( Integral(x^2 dt)/Duration - xbar^2 )
	variance := igr.IntegralXX/igr.ElapsedSeconds - avg*avg
	if variance >= 0 {
		return avg, math.Sqrt(variance)
	}
	return avg, math.NaN()

        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);

		// Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x)
		if x >= Two302 { // x > 2**302

			// (x >> n**2)
			//          Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
			//          Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
			//          Let s=sin(x), c=cos(x),
			//              xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
			//
			//                 n    sin(xn)*sqt2    cos(xn)*sqt2
			//              ----------------------------------

	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);

		{name: "NEGF", argLength: 1, reg: fp11, asm: "NEGF"},   // -arg0, float32
		{name: "NEGD", argLength: 1, reg: fp11, asm: "NEGD"},   // -arg0, float64
		{name: "SQRTD", argLength: 1, reg: fp11, asm: "SQRTD"}, // sqrt(arg0), float64
		{name: "SQRTF", argLength: 1, reg: fp11, asm: "SQRTF"}, // sqrt(arg0), float32
		{name: "ABSD", argLength: 1, reg: fp11, asm: "ABSD"},   // abs(arg0), float64

          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);
      }
    }