




UpperCAmelCase_		:					int          =  range(2, 20 + 1)
UpperCAmelCase_		:					Tuple          =  [10**k for k in range(ks[-1] + 1)]
UpperCAmelCase_		:					dict[int, dict[int, list[list[int]]]]          =  {}




def 		lowerCAmelCase_ (						lowerCamelCase						,  lowerCamelCase						,  lowerCamelCase						,  lowerCamelCase ):
				__magic_name__						:						Union[str, Any]										=sum(a_i[j] for j in range(lowerCamelCase						,  len(lowerCamelCase ) ) )
				__magic_name__						:						Any										=sum(a_i[j] * base[j] for j in range(min(len(lowerCamelCase )						,  lowerCamelCase ) ) )

				__magic_name__					,			__magic_name__						:						Tuple										=0, 0
				__magic_name__						:						Optional[Any]										=n - i

				__magic_name__						:						Union[str, Any]										=memo.get(lowerCamelCase )

				if sub_memo is not None:
								__magic_name__						:						int										=sub_memo.get(lowerCamelCase )

								if jumps is not None and len(lowerCamelCase ) > 0:
												# find and make the largest jump without going over
												__magic_name__						:						Dict										=-1
												for _k in range(len(lowerCamelCase ) - 1						,  -1						,  -1 ):
																if jumps[_k][2] <= k and jumps[_k][1] <= max_dn:
																				__magic_name__						:						Optional[Any]										=_k
																				break

												if max_jump >= 0:
																__magic_name__					,			__magic_name__					,			__magic_name__						:						Optional[int]										=jumps[max_jump]
																# since the difference between jumps is cached, add c
																__magic_name__						:						Tuple										=diff + c
																for j in range(min(lowerCamelCase						,  len(lowerCamelCase ) ) ):
																				__magic_name__					,			__magic_name__						:						Tuple										=divmod(lowerCamelCase						,  10 )
																if new_c > 0:
																				add(lowerCamelCase						,  lowerCamelCase						,  lowerCamelCase )

								else:
												__magic_name__						:						str										=[]
				else:
								__magic_name__						:						List[str]										={c: []}
								__magic_name__						:						List[str]										=sub_memo

				if dn >= max_dn or c + diff >= base[k]:
								return diff, dn

				if k > ks[0]:
								while True:
												# keep doing smaller jumps
												__magic_name__					,			__magic_name__						:						Union[str, Any]										=next_term(lowerCamelCase						,  k - 1						,  i + dn						,  lowerCamelCase )
												diff += _diff
												dn += terms_jumped

												if dn >= max_dn or c + diff >= base[k]:
																break
				else:
								# would be too small a jump, just compute sequential terms instead
								__magic_name__					,			__magic_name__						:						Optional[int]										=compute(lowerCamelCase						,  lowerCamelCase						,  i + dn						,  lowerCamelCase )
								diff += _diff
								dn += terms_jumped

				__magic_name__						:						Tuple										=sub_memo[c]

				# keep jumps sorted by # of terms skipped
				__magic_name__						:						List[Any]										=0
				while j < len(lowerCamelCase ):
								if jumps[j][1] > dn:
												break
								j += 1

				# cache the jump for this value digitsum(b) and c
				sub_memo[c].insert(lowerCamelCase						,  (diff, dn, k) )
				return (diff, dn)




def 		lowerCAmelCase_ (						lowerCamelCase						,  lowerCamelCase						,  lowerCamelCase						,  lowerCamelCase ):
				if i >= n:
								return 0, i
				if k > len(lowerCamelCase ):
								a_i.extend([0 for _ in range(k - len(lowerCamelCase ) )] )

				# note: a_i -> b * 10^k + c
				# ds_b -> digitsum(b)
				# ds_c -> digitsum(c)
				__magic_name__						:						Tuple										=i
				__magic_name__					,			__magic_name__					,			__magic_name__						:						Tuple										=0, 0, 0
				for j in range(len(lowerCamelCase ) ):
								if j >= k:
												ds_b += a_i[j]
								else:
												ds_c += a_i[j]

				while i < n:
								i += 1
								__magic_name__						:						Optional[Any]										=ds_c + ds_b
								diff += addend
								__magic_name__						:						str										=0
								for j in range(lowerCamelCase ):
												__magic_name__						:						int										=a_i[j] + addend
												__magic_name__					,			__magic_name__						:						Any										=divmod(lowerCamelCase						,  10 )

												ds_c += a_i[j]

								if addend > 0:
												break

				if addend > 0:
								add(lowerCamelCase						,  lowerCamelCase						,  lowerCamelCase )
				return diff, i - start_i




def 		lowerCAmelCase_ (						lowerCamelCase						,  lowerCamelCase						,  lowerCamelCase ):
				for j in range(lowerCamelCase						,  len(lowerCamelCase ) ):
								__magic_name__						:						Tuple										=digits[j] + addend
								if s >= 10:
												__magic_name__					,			__magic_name__						:						int										=divmod(lowerCamelCase						,  10 )
												__magic_name__						:						int										=addend // 10 + quotient
								else:
												__magic_name__						:						Dict										=s
												__magic_name__						:						Any										=addend // 10

								if addend == 0:
												break

				while addend > 0:
								__magic_name__					,			__magic_name__						:						Union[str, Any]										=divmod(lowerCamelCase						,  10 )
								digits.append(lowerCamelCase )




def 		lowerCAmelCase_ (						lowerCamelCase = 10**15 ):
				__magic_name__						:						List[str]										=[1]
				__magic_name__						:						str										=1
				__magic_name__						:						str										=0
				while True:
								__magic_name__					,			__magic_name__						:						List[str]										=next_term(lowerCamelCase						,  20						,  i + dn						,  lowerCamelCase )
								dn += terms_jumped
								if dn == n - i:
												break

				__magic_name__						:						int										=0
				for j in range(len(lowerCamelCase ) ):
								a_n += digits[j] * 10**j
				return a_n


if __name__ == "__main__":
							print(F"""{solution() = }""")










# DISCLAIMER: This file is strongly influenced by https://github.com/yang-song/score_sde_pytorch

import math
from typing import Union

import torch

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




class 		__A   (      UpperCamelCase__       ,    UpperCamelCase__      ):
						UpperCamelCase      =       1

						@register_to_config
						def __init__(		self :Any     ,			__snake_case :Tuple=20_00     ,			__snake_case :Optional[Any]=0.1     ,			__snake_case :Any=20     ,			__snake_case :Optional[int]=1E-3			):
										'''simple docstring'''

										__magic_name__						:						Dict										=None
										__magic_name__						:						List[str]										=None
										__magic_name__						:						str										=None

						def      A__    (		self :Dict     ,			__snake_case :Optional[int]     ,			__snake_case :Union[str, torch.device] = None			):
										'''simple docstring'''

										__magic_name__						:						Union[str, Any]										=torch.linspace(1     ,			self.config.sampling_eps     ,			__snake_case     ,			device=__snake_case			)

						def      A__    (		self :List[str]     ,			__snake_case :List[str]     ,			__snake_case :int     ,			__snake_case :int     ,			__snake_case :List[str]=None			):
										'''simple docstring'''

										if self.timesteps is None:
														raise ValueError(
														    """`self.timesteps` is not set, you need to run 'set_timesteps' after creating the scheduler"""			)

										# TODO(Patrick) better comments + non-PyTorch
										# postprocess model score
										__magic_name__						:						int										=(
										    -0.25 * t**2 * (self.config.beta_max - self.config.beta_min) - 0.5 * t * self.config.beta_min
										)
										__magic_name__						:						Optional[int]										=torch.sqrt(1.0 - torch.exp(2.0 * log_mean_coeff			)			)
										__magic_name__						:						str										=std.flatten()
										while len(std.shape			) < len(score.shape			):
														__magic_name__						:						List[str]										=std.unsqueeze(-1			)
										__magic_name__						:						Union[str, Any]										=-score / std

										# compute
										__magic_name__						:						Tuple										=-1.0 / len(self.timesteps			)

										__magic_name__						:						int										=self.config.beta_min + t * (self.config.beta_max - self.config.beta_min)
										__magic_name__						:						Dict										=beta_t.flatten()
										while len(beta_t.shape			) < len(x.shape			):
														__magic_name__						:						Any										=beta_t.unsqueeze(-1			)
										__magic_name__						:						Dict										=-0.5 * beta_t * x

										__magic_name__						:						Optional[int]										=torch.sqrt(__snake_case			)
										__magic_name__						:						int										=drift - diffusion**2 * score
										__magic_name__						:						List[str]										=x + drift * dt

										# add noise
										__magic_name__						:						Optional[int]										=randn_tensor(x.shape     ,			layout=x.layout     ,			generator=__snake_case     ,			device=x.device     ,			dtype=x.dtype			)
										__magic_name__						:						Optional[Any]										=x_mean + diffusion * math.sqrt(-dt			) * noise

										return x, x_mean


						def __len__(		self :List[Any]			):
										'''simple docstring'''

										return self.config.num_train_timesteps




