

import math
from decimal import Decimal, getcontext
def   _lowerCAmelCase(						__A = 4000000      ):

  try:
    UpperCAmelCase          =	int(__A      )
  except (TypeError, ValueError):
    raise TypeError("Parameter n must be int or castable to int."      )
  if n <= 0:
    raise ValueError("Parameter n must be greater than or equal to one."      )
  UpperCAmelCase          =	100
  UpperCAmelCase          =	(Decimal(5      ) ** Decimal(0.5      ) + 1) / Decimal(2      )

  UpperCAmelCase          =	(math.floor(math.log(n * (phi + 2) ,							__A      ) - 1      ) // 3) * 3 + 2
  UpperCAmelCase          =	Decimal(round(phi ** Decimal(index + 1      )      )      ) / (phi + 2)
  UpperCAmelCase          =	num // 2
  return int(__A      )


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






from typing import Dict, Optional, Tuple

import torch
def   _lowerCAmelCase(						__A      ):
  UpperCAmelCase          =	boundaries[1] - boundaries[0]
  UpperCAmelCase          =	boundaries + step / 2
  UpperCAmelCase          =	torch.cat([bin_centers, (bin_centers[-1] + step).unsqueeze(-1      )] ,							dim=0      )
  return bin_centers
def   _lowerCAmelCase(						__A ,							__A ,							):
  UpperCAmelCase          =	_calculate_bin_centers(__A      )
  return (
      torch.sum(aligned_distance_error_probs * bin_centers ,							dim=-1      ),
      bin_centers[-1],
  )
def   _lowerCAmelCase(						__A ,							__A = 31 ,							__A = 64 ,							**__A ,							):
  UpperCAmelCase          =	torch.linspace(0 ,							__A ,							steps=(no_bins - 1) ,							device=logits.device      )

  UpperCAmelCase          =	torch.nn.functional.softmax(__A ,							dim=-1      )
  UpperCAmelCase   ,						UpperCAmelCase          =	_calculate_expected_aligned_error(
      alignment_confidence_breaks=__A ,							aligned_distance_error_probs=__A ,							)

  return {
      "aligned_confidence_probs": aligned_confidence_probs,
      "predicted_aligned_error": predicted_aligned_error,
      "max_predicted_aligned_error": max_predicted_aligned_error,
  }





def   _lowerCAmelCase(						__A ,							__A = None ,							__A = 31 ,							__A = 64 ,							__A = 1E-8 ,							**__A ,							):
  if residue_weights is None:
    UpperCAmelCase          =	logits.new_ones(logits.shape[-2]      )

  UpperCAmelCase          =	torch.linspace(0 ,							__A ,							steps=(no_bins - 1) ,							device=logits.device      )

  UpperCAmelCase          =	_calculate_bin_centers(__A      )
  torch.sum(__A      )
  UpperCAmelCase          =	logits.shape[-2]
  UpperCAmelCase          =	max(__A ,							19      )

  UpperCAmelCase          =	1.24 * (clipped_n - 15) ** (1.0 / 3) - 1.8

  UpperCAmelCase          =	torch.nn.functional.softmax(__A ,							dim=-1      )

  UpperCAmelCase          =	1.0 / (1 + (bin_centers**2) / (da**2))
  UpperCAmelCase          =	torch.sum(probs * tm_per_bin ,							dim=-1      )

  UpperCAmelCase          =	residue_weights / (eps + residue_weights.sum())
  UpperCAmelCase          =	torch.sum(predicted_tm_term * normed_residue_mask ,							dim=-1      )

  UpperCAmelCase          =	per_alignment * residue_weights

  UpperCAmelCase          =	(weighted == torch.max(__A      )).nonzero()[0]
  return per_alignment[tuple(__A      )]



