






from typing import Callable, Dict, Optional, Tuple

import torch
from torch import nn
from torch.distributions import (
    AffineTransform,
    Distribution,
    Independent,
    NegativeBinomial,
    Normal,
    StudentT,
    TransformedDistribution,
)







class 	_UpperCAmelCase  (  _A		):




      """simple docstring"""



      def __init__(   self    ,	_lowerCAmelCase    ,	_lowerCAmelCase=None    ,	_lowerCAmelCase=None    ,	_lowerCAmelCase=0		):



          '''simple docstring'''
          lowerCAmelCase__   :Optional[Any]      							=				1.0 if scale is None else scale
          lowerCAmelCase__   :Any      							=				0.0 if loc is None else loc

          super().__init__(_lowerCAmelCase    ,	[AffineTransform(loc=self.loc    ,	scale=self.scale    ,	event_dim=_lowerCAmelCase		)]		)
      @property
      def 			snake_case_      (   self		):



          '''simple docstring'''
          return self.base_dist.mean * self.scale + self.loc
      @property
      def 			snake_case_      (   self		):



          '''simple docstring'''
          return self.base_dist.variance * self.scale**2



      @property
      def 			snake_case_      (   self		):



          '''simple docstring'''
          return self.variance.sqrt()







class 	_UpperCAmelCase  (  nn.Module		):




      """simple docstring"""



      def __init__(   self    ,	_lowerCAmelCase    ,	_lowerCAmelCase    ,	_lowerCAmelCase    ,	**_lowerCAmelCase		):



          '''simple docstring'''
          super().__init__(**_lowerCAmelCase		)
          lowerCAmelCase__   :Any      							=				args_dim
          lowerCAmelCase__   :Optional[int]      							=				nn.ModuleList([nn.Linear(_lowerCAmelCase    ,	_lowerCAmelCase		) for dim in args_dim.values()]		)
          lowerCAmelCase__   :Any      							=				domain_map



      def 			snake_case_      (   self    ,	_lowerCAmelCase		):



          '''simple docstring'''
          lowerCAmelCase__   :str      							=				[proj(_lowerCAmelCase		) for proj in self.proj]

          return self.domain_map(*_lowerCAmelCase		)







class 	_UpperCAmelCase  (  nn.Module		):




      """simple docstring"""



      def __init__(   self    ,	_lowerCAmelCase		):



          '''simple docstring'''
          super().__init__()
          lowerCAmelCase__   :Tuple      							=				function



      def 			snake_case_      (   self    ,	_lowerCAmelCase    ,	*_lowerCAmelCase		):



          '''simple docstring'''
          return self.function(_lowerCAmelCase    ,	*_lowerCAmelCase		)







class 	_UpperCAmelCase  :




      """simple docstring"""



      A          =    42
      A          =    42
      A          =    42
      def __init__(   self    ,	_lowerCAmelCase = 1		):



          '''simple docstring'''
          lowerCAmelCase__   :Optional[Any]      							=				dim
          lowerCAmelCase__   :Optional[int]      							=				{k: dim * self.args_dim[k] for k in self.args_dim}
      def 			snake_case_      (   self    ,	_lowerCAmelCase		):



          '''simple docstring'''
          if self.dim == 1:
              return self.distribution_class(*_lowerCAmelCase		)
          else:
              return Independent(self.distribution_class(*_lowerCAmelCase		)    ,	1		)
      def 			snake_case_      (   self    ,	_lowerCAmelCase    ,	_lowerCAmelCase = None    ,	_lowerCAmelCase = None    ,	):



          '''simple docstring'''
          lowerCAmelCase__   :Optional[Any]      							=				self._base_distribution(_lowerCAmelCase		)
          if loc is None and scale is None:
              return distr
          else:
              return AffineTransformed(_lowerCAmelCase    ,	loc=_lowerCAmelCase    ,	scale=_lowerCAmelCase    ,	event_dim=self.event_dim		)
      @property
      def 			snake_case_      (   self		):



          '''simple docstring'''
          return () if self.dim == 1 else (self.dim,)
      @property
      def 			snake_case_      (   self		):



          '''simple docstring'''
          return len(self.event_shape		)
      @property
      def 			snake_case_      (   self		):



          '''simple docstring'''
          return 0.0
      def 			snake_case_      (   self    ,	_lowerCAmelCase		):



          '''simple docstring'''
          return ParameterProjection(
              in_features=_lowerCAmelCase    ,	args_dim=self.args_dim    ,	domain_map=LambdaLayer(self.domain_map		)    ,	)
      def 			snake_case_      (   self    ,	*_lowerCAmelCase		):



          '''simple docstring'''
          raise NotImplementedError()



      @staticmethod
      def 			snake_case_      (   _lowerCAmelCase		):



          '''simple docstring'''
          return (x + torch.sqrt(torch.square(_lowerCAmelCase		) + 4.0		)) / 2.0







class 	_UpperCAmelCase  (  _A		):




      """simple docstring"""



      A          =    {"df": 1, "loc": 1, "scale": 1}
      A          =    StudentT
      @classmethod
      def 			snake_case_      (   cls    ,	_lowerCAmelCase    ,	_lowerCAmelCase    ,	_lowerCAmelCase		):



          '''simple docstring'''
          lowerCAmelCase__   :str      							=				cls.squareplus(_lowerCAmelCase		).clamp_min(torch.finfo(scale.dtype		).eps		)
          lowerCAmelCase__   :Union[str, Any]      							=				2.0 + cls.squareplus(_lowerCAmelCase		)
          return df.squeeze(-1		), loc.squeeze(-1		), scale.squeeze(-1		)







class 	_UpperCAmelCase  (  _A		):




      """simple docstring"""



      A          =    {"loc": 1, "scale": 1}
      A          =    Normal
      @classmethod
      def 			snake_case_      (   cls    ,	_lowerCAmelCase    ,	_lowerCAmelCase		):



          '''simple docstring'''
          lowerCAmelCase__   :str      							=				cls.squareplus(_lowerCAmelCase		).clamp_min(torch.finfo(scale.dtype		).eps		)
          return loc.squeeze(-1		), scale.squeeze(-1		)







class 	_UpperCAmelCase  (  _A		):




      """simple docstring"""



      A          =    {"total_count": 1, "logits": 1}
      A          =    NegativeBinomial
      @classmethod
      def 			snake_case_      (   cls    ,	_lowerCAmelCase    ,	_lowerCAmelCase		):



          '''simple docstring'''
          lowerCAmelCase__   :List[str]      							=				cls.squareplus(_lowerCAmelCase		)
          return total_count.squeeze(-1		), logits.squeeze(-1		)
      def 			snake_case_      (   self    ,	_lowerCAmelCase		):



          '''simple docstring'''
          lowerCAmelCase__							,lowerCAmelCase__   :int      							=				distr_args
          if self.dim == 1:
              return self.distribution_class(total_count=_lowerCAmelCase    ,	logits=_lowerCAmelCase		)
          else:
              return Independent(self.distribution_class(total_count=_lowerCAmelCase    ,	logits=_lowerCAmelCase		)    ,	1		)



      def 			snake_case_      (   self    ,	_lowerCAmelCase    ,	_lowerCAmelCase = None    ,	_lowerCAmelCase = None		):



          '''simple docstring'''
          lowerCAmelCase__							,lowerCAmelCase__   :Any      							=				distr_args

          if scale is not None:
              # See scaling property of Gamma.
              logits += scale.log()

          return self._base_distribution((total_count, logits)		)











from __future__ import annotations

import math
from collections.abc import Callable


def 				snake_case__							(							UpperCAmelCase							:  Callable[[int | float], int | float]    ,			UpperCAmelCase							:  int | float    ,			UpperCAmelCase							:  int | float    ,			UpperCAmelCase							:  int = 1_0_0    ,			):
    lowerCAmelCase__   :int      							=				x_start
    lowerCAmelCase__   :int      							=				fnc(UpperCAmelCase       )
    lowerCAmelCase__   :List[Any]      							=				0.0

    for _ in range(UpperCAmelCase       ):
        # Approximates curve as a sequence of linear lines and sums their length
        lowerCAmelCase__   :Dict      							=				(x_end - x_start) / steps + xa
        lowerCAmelCase__   :Optional[Any]      							=				fnc(UpperCAmelCase       )
        length += math.hypot(xa - xa    ,			fxa - fxa       )

        # Increment step
        lowerCAmelCase__   :Union[str, Any]      							=				xa
        lowerCAmelCase__   :str      							=				fxa

    return length


if __name__ == "__main__":


 def 				snake_case__							(							UpperCAmelCase							:  int       ):
     return math.sin(1_0 * x       )

 print("""f(x) = sin(10 * x)""")
 print("""The length of the curve from x = -10 to x = 10 is:""")
 _a    :   Any        =			10
 while i <= 10_0000:
  print(f"""With {i} steps: {line_length(f, -10, 10, i)}""")
  i *= 10



