

"""simple docstring"""




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

import numpy as np
import torch

from ..configuration_utils import ConfigMixin, register_to_config
from ..utils import BaseOutput, randn_tensor
from .scheduling_utils import SchedulerMixin





@dataclass
class 				lowerCamelCase  (A__     ):
			lowerCamelCase__				:						torch.FloatTensor
			lowerCamelCase__				:						torch.FloatTensor

def 		A    (       snake_case__						,			snake_case__=0.9_99						,			snake_case__="cosine"						,			):




			'''simple docstring'''


			if alpha_transform_type == "cosine":

						def alpha_bar_fn(snake_case__      ):
									return math.cos((t + 0.0_08) / 1.0_08 * math.pi / 2      ) ** 2

			elif alpha_transform_type == "exp":

						def alpha_bar_fn(snake_case__      ):
									return math.exp(t * -12.0      )

			else:
						raise ValueError(f"""Unsupported alpha_tranform_type: {alpha_transform_type}"""      )

			SCREAMING_SNAKE_CASE__									= []
			for i in range(snake_case__      ):
						SCREAMING_SNAKE_CASE__									= i / num_diffusion_timesteps
						SCREAMING_SNAKE_CASE__									= (i + 1) / num_diffusion_timesteps
						betas.append(min(1 - alpha_bar_fn(snake_case__      ) / alpha_bar_fn(snake_case__      )						,			snake_case__      )      )
			return torch.tensor(snake_case__						,			dtype=torch.floataa      )





class 				lowerCamelCase  (A__     ,A__     ):
			lowerCamelCase__				:						Dict   						=							1

			@register_to_config
			def __init__(   self			:  Union[str, Any]      ,			__UpperCAmelCase			:  int = 1_0_0_0      ,			__UpperCAmelCase			:  float = 0.0_001      ,			__UpperCAmelCase			:  float = 0.02      ,			__UpperCAmelCase			:  str = "linear"      ,			__UpperCAmelCase			:  float = 0.0      ,			__UpperCAmelCase			:  Optional[np.ndarray] = None      ,			__UpperCAmelCase			:  bool = True      ,			)				->      Union[str, Any]:
						if trained_betas is not None:
									SCREAMING_SNAKE_CASE__									= torch.from_numpy(__UpperCAmelCase    )
						elif beta_schedule == "linear":
									SCREAMING_SNAKE_CASE__									= torch.linspace(__UpperCAmelCase      ,			__UpperCAmelCase      ,			__UpperCAmelCase      ,			dtype=torch.floataa    )
						elif beta_schedule == "scaled_linear":
									# this schedule is very specific to the latent diffusion model.
									SCREAMING_SNAKE_CASE__									= (
									    torch.linspace(beta_start**0.5      ,			beta_end**0.5      ,			__UpperCAmelCase      ,			dtype=torch.floataa    ) ** 2
									)
						elif beta_schedule == "squaredcos_cap_v2":
									# Glide cosine schedule
									SCREAMING_SNAKE_CASE__									= betas_for_alpha_bar(__UpperCAmelCase    )
						elif beta_schedule == "sigmoid":
									# GeoDiff sigmoid schedule
									SCREAMING_SNAKE_CASE__									= torch.linspace(-6      ,			6      ,			__UpperCAmelCase    )
									SCREAMING_SNAKE_CASE__									= torch.sigmoid(__UpperCAmelCase    ) * (beta_end - beta_start) + beta_start
						else:
									raise NotImplementedError(F"""{beta_schedule} does is not implemented for {self.__class__}"""    )

						SCREAMING_SNAKE_CASE__									= 1.0 - self.betas
						SCREAMING_SNAKE_CASE__									= torch.cumprod(self.alphas      ,			dim=0    )
						SCREAMING_SNAKE_CASE__									= torch.tensor(1.0    )

						SCREAMING_SNAKE_CASE__									= torch.tensor(1.0    )

						# standard deviation of the initial noise distribution
						SCREAMING_SNAKE_CASE__									= 1.0

						# setable values
						SCREAMING_SNAKE_CASE__									= None
						SCREAMING_SNAKE_CASE__									= torch.from_numpy(np.arange(0      ,			__UpperCAmelCase    )[::-1].copy()    )

						SCREAMING_SNAKE_CASE__									= eta

			def    SCREAMING_SNAKE_CASE			(   self			:  List[str]      ,			__UpperCAmelCase			:  torch.FloatTensor      ,			__UpperCAmelCase			:  Optional[int] = None    )				->      torch.FloatTensor:
						return sample

			def    SCREAMING_SNAKE_CASE			(   self			:  Optional[Any]      ,			__UpperCAmelCase			:  int      ,			__UpperCAmelCase			:  int = 1_0      ,			__UpperCAmelCase			:  int = 1_0      ,			__UpperCAmelCase			:  Union[str, torch.device] = None      ,			)				->      List[str]:
						SCREAMING_SNAKE_CASE__									= min(self.config.num_train_timesteps      ,			__UpperCAmelCase    )
						SCREAMING_SNAKE_CASE__									= num_inference_steps

						SCREAMING_SNAKE_CASE__									= []

						SCREAMING_SNAKE_CASE__									= {}
						for j in range(0      ,			num_inference_steps - jump_length      ,			__UpperCAmelCase    ):
									SCREAMING_SNAKE_CASE__									= jump_n_sample - 1

						SCREAMING_SNAKE_CASE__									= num_inference_steps
						while t >= 1:
									SCREAMING_SNAKE_CASE__									= t - 1
									timesteps.append(__UpperCAmelCase    )

									if jumps.get(__UpperCAmelCase      ,			0    ) > 0:
												SCREAMING_SNAKE_CASE__									= jumps[t] - 1
												for _ in range(__UpperCAmelCase    ):
															SCREAMING_SNAKE_CASE__									= t + 1
															timesteps.append(__UpperCAmelCase    )

						SCREAMING_SNAKE_CASE__									= np.array(__UpperCAmelCase    ) * (self.config.num_train_timesteps // self.num_inference_steps)
						SCREAMING_SNAKE_CASE__									= torch.from_numpy(__UpperCAmelCase    ).to(__UpperCAmelCase    )

			def    SCREAMING_SNAKE_CASE			(   self			:  List[Any]      ,			__UpperCAmelCase			:  Optional[int]    )				->      int:
						SCREAMING_SNAKE_CASE__									= t - self.config.num_train_timesteps // self.num_inference_steps

						SCREAMING_SNAKE_CASE__									= self.alphas_cumprod[t]
						SCREAMING_SNAKE_CASE__									= self.alphas_cumprod[prev_timestep] if prev_timestep >= 0 else self.final_alpha_cumprod
						SCREAMING_SNAKE_CASE__									= 1 - alpha_prod_t
						SCREAMING_SNAKE_CASE__									= 1 - alpha_prod_t_prev

						# For t > 0, compute predicted variance βt (see formula (6) and (7) from
						# https://arxiv.org/pdf/2006.11239.pdf) and sample from it to get
						# previous sample x_{t-1} ~ N(pred_prev_sample, variance) == add
						# variance to pred_sample
						# Is equivalent to formula (16) in https://arxiv.org/pdf/2010.02502.pdf
						# without eta.
						# variance = (1 - alpha_prod_t_prev) / (1 - alpha_prod_t) * self.betas[t]
						SCREAMING_SNAKE_CASE__									= (beta_prod_t_prev / beta_prod_t) * (1 - alpha_prod_t / alpha_prod_t_prev)

						return variance

			def    SCREAMING_SNAKE_CASE			(   self			:  Optional[int]      ,			__UpperCAmelCase			:  torch.FloatTensor      ,			__UpperCAmelCase			:  int      ,			__UpperCAmelCase			:  torch.FloatTensor      ,			__UpperCAmelCase			:  torch.FloatTensor      ,			__UpperCAmelCase			:  torch.FloatTensor      ,			__UpperCAmelCase			:  Optional[torch.Generator] = None      ,			__UpperCAmelCase			:  bool = True      ,			)				->      Union[RePaintSchedulerOutput, Tuple]:
						SCREAMING_SNAKE_CASE__									= timestep
						SCREAMING_SNAKE_CASE__									= timestep - self.config.num_train_timesteps // self.num_inference_steps

						# 1. compute alphas, betas
						SCREAMING_SNAKE_CASE__									= self.alphas_cumprod[t]
						SCREAMING_SNAKE_CASE__									= self.alphas_cumprod[prev_timestep] if prev_timestep >= 0 else self.final_alpha_cumprod
						SCREAMING_SNAKE_CASE__									= 1 - alpha_prod_t

						# 2. compute predicted original sample from predicted noise also called
						# "predicted x_0" of formula (15) from https://arxiv.org/pdf/2006.11239.pdf
						SCREAMING_SNAKE_CASE__									= (sample - beta_prod_t**0.5 * model_output) / alpha_prod_t**0.5

						# 3. Clip "predicted x_0"
						if self.config.clip_sample:
									SCREAMING_SNAKE_CASE__									= torch.clamp(__UpperCAmelCase      ,			-1      ,			1    )

						# We choose to follow RePaint Algorithm 1 to get x_{t-1}, however we
						# substitute formula (7) in the algorithm coming from DDPM paper
						# (formula (4) Algorithm 2 - Sampling) with formula (12) from DDIM paper.
						# DDIM schedule gives the same results as DDPM with eta = 1.0
						# Noise is being reused in 7. and 8., but no impact on quality has
						# been observed.

						# 5. Add noise
						SCREAMING_SNAKE_CASE__									= model_output.device
						SCREAMING_SNAKE_CASE__									= randn_tensor(model_output.shape      ,			generator=__UpperCAmelCase      ,			device=__UpperCAmelCase      ,			dtype=model_output.dtype    )
						SCREAMING_SNAKE_CASE__									= self.eta * self._get_variance(__UpperCAmelCase    ) ** 0.5

						SCREAMING_SNAKE_CASE__									= 0
						if t > 0 and self.eta > 0:
									SCREAMING_SNAKE_CASE__									= std_dev_t * noise

						# 6. compute "direction pointing to x_t" of formula (12)
						# from https://arxiv.org/pdf/2010.02502.pdf
						SCREAMING_SNAKE_CASE__									= (1 - alpha_prod_t_prev - std_dev_t**2) ** 0.5 * model_output

						# 7. compute x_{t-1} of formula (12) from https://arxiv.org/pdf/2010.02502.pdf
						SCREAMING_SNAKE_CASE__									= alpha_prod_t_prev**0.5 * pred_original_sample + pred_sample_direction + variance

						# 8. Algorithm 1 Line 5 https://arxiv.org/pdf/2201.09865.pdf
						SCREAMING_SNAKE_CASE__									= (alpha_prod_t_prev**0.5) * original_image + ((1 - alpha_prod_t_prev) ** 0.5) * noise

						# 9. Algorithm 1 Line 8 https://arxiv.org/pdf/2201.09865.pdf
						SCREAMING_SNAKE_CASE__									= mask * prev_known_part + (1.0 - mask) * prev_unknown_part

						if not return_dict:
									return (
									    pred_prev_sample,
									    pred_original_sample,
									)

						return RePaintSchedulerOutput(prev_sample=__UpperCAmelCase      ,			pred_original_sample=__UpperCAmelCase    )

			def    SCREAMING_SNAKE_CASE			(   self			:  Optional[Any]      ,			__UpperCAmelCase			:  str      ,			__UpperCAmelCase			:  List[str]      ,			__UpperCAmelCase			:  Tuple=None    )				->      List[str]:
						SCREAMING_SNAKE_CASE__									= self.config.num_train_timesteps // self.num_inference_steps

						for i in range(__UpperCAmelCase    ):
									SCREAMING_SNAKE_CASE__									= self.betas[timestep + i]
									if sample.device.type == "mps":
												# randn does not work reproducibly on mps
												SCREAMING_SNAKE_CASE__									= randn_tensor(sample.shape      ,			dtype=sample.dtype      ,			generator=__UpperCAmelCase    )
												SCREAMING_SNAKE_CASE__									= noise.to(sample.device    )
									else:
												SCREAMING_SNAKE_CASE__									= randn_tensor(sample.shape      ,			generator=__UpperCAmelCase      ,			device=sample.device      ,			dtype=sample.dtype    )

									# 10. Algorithm 1 Line 10 https://arxiv.org/pdf/2201.09865.pdf
									SCREAMING_SNAKE_CASE__									= (1 - beta) ** 0.5 * sample + beta**0.5 * noise

						return sample

			def    SCREAMING_SNAKE_CASE			(   self			:  List[Any]      ,			__UpperCAmelCase			:  torch.FloatTensor      ,			__UpperCAmelCase			:  torch.FloatTensor      ,			__UpperCAmelCase			:  torch.IntTensor      ,			)				->      torch.FloatTensor:
						raise NotImplementedError("""Use `DDPMScheduler.add_noise()` to train for sampling with RePaint."""    )





			def __len__(   self			:  Optional[Any]    )				->      Optional[int]:
						return self.config.num_train_timesteps





"""simple docstring"""




import math

def 		A    (       snake_case__      ):




			'''simple docstring'''


			if 1 < number < 4:
						# 2 and 3 are primes
						return True
			elif number < 2 or number % 2 == 0 or number % 3 == 0:
						# Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
						return False

			# All primes number are in format of 6k +/- 1
			for i in range(5						,			int(math.sqrt(snake_case__      ) + 1      )						,			6      ):
						if number % i == 0 or number % (i + 2) == 0:
									return False
			return True

def 		A    (       snake_case__ = 10_00						,			snake_case__ = 10_00      ):




			'''simple docstring'''


			SCREAMING_SNAKE_CASE__									= [0, 0, 0]  # length, a, b
			for a in range((a_limit * -1) + 1						,			snake_case__      ):
						for b in range(2						,			snake_case__      ):
									if is_prime(snake_case__      ):
												SCREAMING_SNAKE_CASE__									= 0
												SCREAMING_SNAKE_CASE__									= 0
												while is_prime((n**2) + (a * n) + b      ):
															count += 1
															n += 1
												if count > longest[0]:
															SCREAMING_SNAKE_CASE__									= [count, a, b]
			SCREAMING_SNAKE_CASE__									= longest[1] * longest[2]
			return ans


if __name__ == "__main__":
						print(solution(1_000, 1_000))


