
def  lowerCamelCase__ (					_a							,			_a):

	# Check if the input is valid
	if not len(_lowerCamelCase) == len(_lowerCamelCase) == 3:
		raise ValueError("Please enter a valid equation.")
	if equationa[0] == equationa[1] == equationa[0] == equationa[1] == 0:
		raise ValueError("Both a & b of two equations can\'t be zero.")

	# Extract the coefficients
	SCREAMING_SNAKE_CASE     ,SCREAMING_SNAKE_CASE     ,SCREAMING_SNAKE_CASE   :      Tuple				     =       equationa
	SCREAMING_SNAKE_CASE     ,SCREAMING_SNAKE_CASE     ,SCREAMING_SNAKE_CASE   :      List[str]				     =       equationa

	# Calculate the determinants of the matrices
	SCREAMING_SNAKE_CASE   :      List[str]				     =       aa * ba - aa * ba
	SCREAMING_SNAKE_CASE   :      Optional[int]				     =       ca * ba - ca * ba
	SCREAMING_SNAKE_CASE   :      Any				     =       aa * ca - aa * ca

	# Check if the system of linear equations has a solution (using Cramer's rule)
	if determinant == 0:
		if determinant_x == determinant_y == 0:
			raise ValueError("Infinite solutions. (Consistent system)")
		else:
			raise ValueError("No solution. (Inconsistent system)")
	else:
		if determinant_x == determinant_y == 0:
			# Trivial solution (Inconsistent system)
			return (0.0, 0.0)
		else:
			SCREAMING_SNAKE_CASE   :      List[str]				     =       determinant_x / determinant
			SCREAMING_SNAKE_CASE   :      Any				     =       determinant_y / determinant
			# Non-Trivial Solution (Consistent system)
			return (x, y)


from dataclasses import dataclass
from typing import Optional, Tuple, Union

import flax
import jax.numpy as jnp
from jax import random

from ..configuration_utils import ConfigMixin, register_to_config
from ..utils import BaseOutput
from .scheduling_utils_flax import FlaxSchedulerMixin






@flax.struct.dataclass
class _UpperCamelCase							:
			'''simple docstring'''







			lowerCamelCase__           =None
			lowerCamelCase__           =None
			lowerCamelCase__           =None  # sigma(t_i)
			@classmethod
			def 							__UpperCamelCase (			cls							: Tuple	)							->       int:

				"""simple docstring"""

				return cls()






@dataclass
class _UpperCamelCase							(							__A   ):
			'''simple docstring'''







			lowerCamelCase__           =42
			lowerCamelCase__           =42
			lowerCamelCase__           =42






class _UpperCamelCase							(							__A							,	__A   ):
			'''simple docstring'''







			@property
			def 							__UpperCamelCase (			self							: List[str]	)							->       Optional[int]:

				"""simple docstring"""

				return True
			@register_to_config
			def __init__(			self							: List[str]      ,      a							: float = 0.02      ,      a							: float = 100      ,      a							: float = 1.007      ,      a							: float = 80      ,      a							: float = 0.05      ,      a							: float = 50      ,      )							->       Optional[Any]:

				"""simple docstring"""

				pass
			def 							__UpperCamelCase (			self							: Dict	)							->       str:

				"""simple docstring"""

				return KarrasVeSchedulerState.create()
			def 							__UpperCamelCase (			self							: Tuple      ,      a							: KarrasVeSchedulerState      ,      a							: int      ,      a							: Tuple = ()	)							->       KarrasVeSchedulerState:

				"""simple docstring"""

				SCREAMING_SNAKE_CASE   :      Any				     =       jnp.arange(0      ,      a	)[::-1].copy()
				SCREAMING_SNAKE_CASE   :      int				     =       [
				    (
				        self.config.sigma_max**2
				        * (self.config.sigma_min**2 / self.config.sigma_max**2) ** (i / (num_inference_steps - 1))
				    )
				    for i in timesteps
				]

				return state.replace(
				    num_inference_steps=a      ,      schedule=jnp.array(a      ,      dtype=jnp.floataa	)      ,      timesteps=a      ,      )
			def 							__UpperCamelCase (			self							: str      ,      a							: KarrasVeSchedulerState      ,      a							: jnp.ndarray      ,      a							: float      ,      a							: random.KeyArray      ,      )							->       Tuple[jnp.ndarray, float]:

				"""simple docstring"""

				if self.config.s_min <= sigma <= self.config.s_max:
					SCREAMING_SNAKE_CASE   :      Dict				     =       min(self.config.s_churn / state.num_inference_steps      ,      2**0.5 - 1	)
				else:
					SCREAMING_SNAKE_CASE   :      Any				     =       0

				# sample eps ~ N(0, S_noise^2 * I)
				SCREAMING_SNAKE_CASE   :      Any				     =       random.split(a      ,      num=1	)
				SCREAMING_SNAKE_CASE   :      List[str]				     =       self.config.s_noise * random.normal(key=a      ,      shape=sample.shape	)
				SCREAMING_SNAKE_CASE   :      Optional[int]				     =       sigma + gamma * sigma
				SCREAMING_SNAKE_CASE   :      str				     =       sample + ((sigma_hat**2 - sigma**2) ** 0.5 * eps)

				return sample_hat, sigma_hat
			def 							__UpperCamelCase (			self							: Union[str, Any]      ,      a							: KarrasVeSchedulerState      ,      a							: jnp.ndarray      ,      a							: float      ,      a							: float      ,      a							: jnp.ndarray      ,      a							: bool = True      ,      )							->       Union[FlaxKarrasVeOutput, Tuple]:

				"""simple docstring"""

				SCREAMING_SNAKE_CASE   :      int				     =       sample_hat + sigma_hat * model_output
				SCREAMING_SNAKE_CASE   :      Optional[Any]				     =       (sample_hat - pred_original_sample) / sigma_hat
				SCREAMING_SNAKE_CASE   :      str				     =       sample_hat + (sigma_prev - sigma_hat) * derivative

				if not return_dict:
					return (sample_prev, derivative, state)

				return FlaxKarrasVeOutput(prev_sample=a      ,      derivative=a      ,      state=a	)
			def 							__UpperCamelCase (			self							: Optional[int]      ,      a							: KarrasVeSchedulerState      ,      a							: jnp.ndarray      ,      a							: float      ,      a							: float      ,      a							: jnp.ndarray      ,      a							: jnp.ndarray      ,      a							: jnp.ndarray      ,      a							: bool = True      ,      )							->       Union[FlaxKarrasVeOutput, Tuple]:

				"""simple docstring"""

				SCREAMING_SNAKE_CASE   :      Union[str, Any]				     =       sample_prev + sigma_prev * model_output
				SCREAMING_SNAKE_CASE   :      Union[str, Any]				     =       (sample_prev - pred_original_sample) / sigma_prev
				SCREAMING_SNAKE_CASE   :      List[Any]				     =       sample_hat + (sigma_prev - sigma_hat) * (0.5 * derivative + 0.5 * derivative_corr)

				if not return_dict:
					return (sample_prev, derivative, state)

				return FlaxKarrasVeOutput(prev_sample=a      ,      derivative=a      ,      state=a	)






			def 							__UpperCamelCase (			self							: int      ,      a							: KarrasVeSchedulerState      ,      a							: Optional[Any]      ,      a							: int      ,      a							: Dict	)							->       List[str]:

				"""simple docstring"""

				raise NotImplementedError()