// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.

package math

/*
	Hypot -- sqrt(p*p + q*q), but overflows only if the result does.
*/

// Hypot returns [Sqrt](p*p + q*q), taking care to avoid
// unnecessary overflow and underflow.
//
// Special cases are:
//
//	Hypot(±Inf, q) = +Inf
//	Hypot(p, ±Inf) = +Inf
//	Hypot(NaN, q) = NaN

      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);

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

	// 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);

// 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

   *     sqrt(x)} is not an integer
   */
  @GwtIncompatible // TODO
  @SuppressWarnings("fallthrough")
  public static BigInteger sqrt(BigInteger x, RoundingMode mode) {
    checkNonNegative("x", x);
    if (fitsInLong(x)) {
      return BigInteger.valueOf(LongMath.sqrt(x.longValue(), mode));
    }
    BigInteger sqrtFloor = sqrtFloor(x);

with an architecture tag, followed by a colon and a quoted Go-style
regexp to be matched. For example, the following test:

  func Sqrt(x float64) float64 {
  	   // amd64:"SQRTSD"
  	   // arm64:"FSQRTD"
  	   return math.Sqrt(x)
  }

verifies that math.Sqrt calls are intrinsified to a SQRTSD instruction
on amd64, and to a FSQRTD instruction on arm64.

        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);

   *     sqrt(x)} is not an integer
   */
  @GwtIncompatible // TODO
  @SuppressWarnings("fallthrough")
  public static BigInteger sqrt(BigInteger x, RoundingMode mode) {
    checkNonNegative("x", x);
    if (fitsInLong(x)) {
      return BigInteger.valueOf(LongMath.sqrt(x.longValue(), mode));
    }
    BigInteger sqrtFloor = sqrtFloor(x);