import math
from numbers import Number

import torch
from torch.distributions import constraints
from torch.distributions.distribution import Distribution
from torch.distributions.utils import broadcast_all


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::

        >>> m = Cauchy(torch.tensor([0.0]), torch.tensor([1.0]))
        >>> m.sample()  # sample from a Cauchy distribution with loc=0 and scale=1
         2.3214
        [torch.FloatTensor of size 1]

    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(Cauchy, self).__init__(batch_shape, validate_args=validate_args)

    @property
    def mean(self):
        return self.loc.new_tensor(float('nan')).expand(self._extended_shape())

    @property
    def variance(self):
        return self.loc.new_tensor(float('inf')).expand(self._extended_shape())

    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() - (1 + ((value - self.loc) / self.scale)**2).log()

    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):
        if self._validate_args:
            self._validate_sample(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()
