code: from typing import Dict, Optional, Tuple
  
  import torch
  
  def      SCREAMING_SNAKE_CASE_  (		snake_case__							)  ->			torch.Tensor:
  							lowerCAmelCase				=	boundaries[1] - boundaries[0]
  							lowerCAmelCase				=	boundaries + step / 2
  							lowerCAmelCase				=	torch.cat([bin_centers, (bin_centers[-1] + step).unsqueeze(-1							)]     ,						dim=0							)
  							return bin_centers
  
  def      SCREAMING_SNAKE_CASE_  (		snake_case__     ,						snake_case__     ,						)  ->			Tuple[torch.Tensor, torch.Tensor]:
  							lowerCAmelCase				=	_calculate_bin_centers(snake_case__							)
  							return (
  							    torch.sum(aligned_distance_error_probs * bin_centers     ,						dim=-1							),
  							    bin_centers[-1],
  							)
  
  def      SCREAMING_SNAKE_CASE_  (		snake_case__     ,						snake_case__ = 3_1     ,						snake_case__ = 6_4     ,						**snake_case__     ,						)  ->			Dict[str, torch.Tensor]:
  							lowerCAmelCase				=	torch.linspace(0     ,						snake_case__     ,						steps=(no_bins - 1)     ,						device=logits.device							)
  
  							lowerCAmelCase				=	torch.nn.functional.softmax(snake_case__     ,						dim=-1							)
  							lowerCAmelCase	,   lowerCAmelCase				=	_calculate_expected_aligned_error(
  							    alignment_confidence_breaks=snake_case__     ,						aligned_distance_error_probs=snake_case__     ,						)
  
  							return {
  							    "aligned_confidence_probs": aligned_confidence_probs,
  							    "predicted_aligned_error": predicted_aligned_error,
  							    "max_predicted_aligned_error": max_predicted_aligned_error,
  							}
  
  
  
  
  
  
  def      SCREAMING_SNAKE_CASE_  (		snake_case__     ,						snake_case__ = None     ,						snake_case__ = 3_1     ,						snake_case__ = 6_4     ,						snake_case__ = 1E-8     ,						**snake_case__     ,						)  ->			torch.Tensor:
  							if residue_weights is None:
  														lowerCAmelCase				=	logits.new_ones(logits.shape[-2]							)
  
  							lowerCAmelCase				=	torch.linspace(0     ,						snake_case__     ,						steps=(no_bins - 1)     ,						device=logits.device							)
  
  							lowerCAmelCase				=	_calculate_bin_centers(snake_case__							)
  							torch.sum(snake_case__							)
  							lowerCAmelCase				=	logits.shape[-2]
  							lowerCAmelCase				=	max(snake_case__     ,						1_9							)
  
  							lowerCAmelCase				=	1.24 * (clipped_n - 1_5) ** (1.0 / 3) - 1.8
  
  							lowerCAmelCase				=	torch.nn.functional.softmax(snake_case__     ,						dim=-1							)
  
  							lowerCAmelCase				=	1.0 / (1 + (bin_centers**2) / (da**2))
  							lowerCAmelCase				=	torch.sum(probs * tm_per_bin     ,						dim=-1							)
  
  							lowerCAmelCase				=	residue_weights / (eps + residue_weights.sum())
  							lowerCAmelCase				=	torch.sum(predicted_tm_term * normed_residue_mask     ,						dim=-1							)
  
  							lowerCAmelCase				=	per_alignment * residue_weights
  
  							lowerCAmelCase				=	(weighted == torch.max(snake_case__							)).nonzero()[0]
  							return per_alignment[tuple(snake_case__							)]
  
  
  
  
style_context: import math
  
  def      SCREAMING_SNAKE_CASE_  (		snake_case__							)  ->			float:
  							lowerCAmelCase				=	math.log(2     ,						1_0							) * number
  							lowerCAmelCase				=	round((log_number - int(snake_case__							))     ,						1_5							)
  
  							return difference
  
  def      SCREAMING_SNAKE_CASE_  (		snake_case__ = 6_7_8_9_1_0							)  ->			int:
  							lowerCAmelCase				=	9_0
  							lowerCAmelCase				=	0
  
  							lowerCAmelCase				=	math.log(1.23     ,						1_0							)
  							lowerCAmelCase				=	math.log(1.24     ,						1_0							)
  							lowerCAmelCase				=	0
  
  							while position < number:
  														lowerCAmelCase				=	log_difference(snake_case__							)
  
  														if difference >= upper_limit:
  																					power_iterator += 9_3
  
  														elif difference < lower_limit:
  																					power_iterator += 1_9_6
  
  														else:
  																					lowerCAmelCase				=	power_iterator
  																					power_iterator += 1_9_6
  																					position += 1
  
  							return previous_power
  
  
  if __name__ == "__main__":
  	import doctest
  
  	doctest.testmod()
  
  	print(f'{solution() = }')
  
  
  
  