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# Source code for torch.distributions.cauchy

    import math
    from numbers import Number

    import torch
    from torch import inf, nan
    from torch.distributions import constraints
    from torch.distributions.distribution import Distribution
    from torch.distributions.utils import broadcast_all

    __all__ = ["Cauchy"]

    class Cauchy(Distribution):
        r"""
        Samples from a Cauchy (Lorentz) distribution. The distribution of the ratio of
        independent normally distributed random variables with means `0` follows a
        Cauchy distribution.

        Example::

            >>> # xdoctest: +IGNORE_WANT("non-deterinistic")
            >>> m = Cauchy(torch.tensor([0.0]), torch.tensor([1.0]))
            >>> m.sample()  # sample from a Cauchy distribution with loc=0 and scale=1
            tensor([ 2.3214])

        Args:
            loc (float or Tensor): mode or median of the distribution.
            scale (float or Tensor): half width at half maximum.
        """
        arg_constraints = {"loc": constraints.real, "scale": constraints.positive}
        support = constraints.real
        has_rsample = True

        def __init__(self, loc, scale, validate_args=None):
            self.loc, self.scale = broadcast_all(loc, scale)
            if isinstance(loc, Number) and isinstance(scale, Number):
                batch_shape = torch.Size()
            else:
                batch_shape = self.loc.size()
            super().__init__(batch_shape, validate_args=validate_args)

        def expand(self, batch_shape, _instance=None):
            new = self._get_checked_instance(Cauchy, _instance)
            batch_shape = torch.Size(batch_shape)
            new.loc = self.loc.expand(batch_shape)
            new.scale = self.scale.expand(batch_shape)
            super(Cauchy, new).__init__(batch_shape, validate_args=False)
            new._validate_args = self._validate_args
            return new

        @property
        def mean(self):
            return torch.full(
                self._extended_shape(), nan, dtype=self.loc.dtype, device=self.loc.device
            )

        @property
        def mode(self):
            return self.loc

        @property
        def variance(self):
            return torch.full(
                self._extended_shape(), inf, dtype=self.loc.dtype, device=self.loc.device
            )

        def rsample(self, sample_shape=torch.Size()):
            shape = self._extended_shape(sample_shape)
            eps = self.loc.new(shape).cauchy_()
            return self.loc + eps * self.scale

        def log_prob(self, value):
            if self._validate_args:
                self._validate_sample(value)
            return (
                -math.log(math.pi)
                - self.scale.log()
                - (((value - self.loc) / self.scale) ** 2).log1p()
            )

        def cdf(self, value):
            if self._validate_args:
                self._validate_sample(value)
            return torch.atan((value - self.loc) / self.scale) / math.pi + 0.5

        def icdf(self, value):
            return torch.tan(math.pi * (value - 0.5)) * self.scale + self.loc

        def entropy(self):
            return math.log(4 * math.pi) + self.scale.log()

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