# Copyright 2018 The JAX Authors.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
#     https://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.


import numpy as np

from jax import lax
from jax._src.lax.lax import _const as _lax_const
from jax._src.numpy.util import promote_args_inexact
from jax.numpy import arctan
from jax._src.typing import Array, ArrayLike


def logpdf(x: ArrayLike, loc: ArrayLike = 0, scale: ArrayLike = 1) -> Array:
  r"""Cauchy log probability distribution function.

  JAX implementation of :obj:`scipy.stats.cauchy` ``logpdf``.

  The Cauchy probability distribution function is

  .. math::

     f(x) = \frac{1}{\pi(1 + x^2)}

  Args:
    x: arraylike, value at which to evaluate the PDF
    loc: arraylike, distribution offset parameter
    scale: arraylike, distribution scale parameter

  Returns:
    array of logpdf values

  See Also:
    - :func:`jax.scipy.stats.cauchy.cdf`
    - :func:`jax.scipy.stats.cauchy.pdf`
    - :func:`jax.scipy.stats.cauchy.sf`
    - :func:`jax.scipy.stats.cauchy.logcdf`
    - :func:`jax.scipy.stats.cauchy.logsf`
    - :func:`jax.scipy.stats.cauchy.isf`
    - :func:`jax.scipy.stats.cauchy.ppf`
  """
  x, loc, scale = promote_args_inexact("cauchy.logpdf", x, loc, scale)
  pi = _lax_const(x, np.pi)
  scaled_x = lax.div(lax.sub(x, loc), scale)
  normalize_term = lax.log(lax.mul(pi, scale))
  return lax.neg(lax.add(normalize_term, lax.log1p(lax.mul(scaled_x, scaled_x))))


def pdf(x: ArrayLike, loc: ArrayLike = 0, scale: ArrayLike = 1) -> Array:
  r"""Cauchy probability distribution function.

  JAX implementation of :obj:`scipy.stats.cauchy` ``pdf``.

  The Cauchy probability distribution function is

  .. math::

     f(x) = \frac{1}{\pi(1 + x^2)}

  Args:
    x: arraylike, value at which to evaluate the PDF
    loc: arraylike, distribution offset parameter
    scale: arraylike, distribution scale parameter

  Returns:
    array of pdf values

  See Also:
    - :func:`jax.scipy.stats.cauchy.cdf`
    - :func:`jax.scipy.stats.cauchy.sf`
    - :func:`jax.scipy.stats.cauchy.logcdf`
    - :func:`jax.scipy.stats.cauchy.logpdf`
    - :func:`jax.scipy.stats.cauchy.logsf`
    - :func:`jax.scipy.stats.cauchy.isf`
    - :func:`jax.scipy.stats.cauchy.ppf`
  """
  return lax.exp(logpdf(x, loc, scale))


def cdf(x: ArrayLike, loc: ArrayLike = 0, scale: ArrayLike = 1) -> Array:
  r"""Cauchy cumulative distribution function.

  JAX implementation of :obj:`scipy.stats.cauchy` ``cdf``.

  The cdf is defined as

  .. math::

     f_{cdf} = \int_{-\infty}^x f_{pdf}(y) \mathrm{d}y

  where here :math:`f_{pdf}` is the Cauchy probability distribution function,
  :func:`jax.scipy.stats.cauchy.pdf`.

  Args:
    x: arraylike, value at which to evaluate the CDF
    loc: arraylike, distribution offset parameter
    scale: arraylike, distribution scale parameter

  Returns:
    array of cdf values.

  See Also:
    - :func:`jax.scipy.stats.cauchy.pdf`
    - :func:`jax.scipy.stats.cauchy.sf`
    - :func:`jax.scipy.stats.cauchy.logcdf`
    - :func:`jax.scipy.stats.cauchy.logpdf`
    - :func:`jax.scipy.stats.cauchy.logsf`
    - :func:`jax.scipy.stats.cauchy.isf`
    - :func:`jax.scipy.stats.cauchy.ppf`
  """
  x, loc, scale = promote_args_inexact("cauchy.cdf", x, loc, scale)
  pi = _lax_const(x, np.pi)
  scaled_x = lax.div(lax.sub(x, loc), scale)
  return lax.add(_lax_const(x, 0.5), lax.mul(lax.div(_lax_const(x, 1.), pi), arctan(scaled_x)))


def logcdf(x: ArrayLike, loc: ArrayLike = 0, scale: ArrayLike = 1) -> Array:
  r"""Cauchy log cumulative distribution function.

  JAX implementation of :obj:`scipy.stats.cauchy` ``logcdf``

  The cdf is defined as

  .. math::

     f_{cdf} = \int_{-\infty}^x f_{pdf}(y) \mathrm{d}y

  where here :math:`f_{pdf}` is the Cauchy probability distribution function,
  :func:`jax.scipy.stats.cauchy.pdf`.

  Args:
    x: arraylike, value at which to evaluate the CDF
    loc: arraylike, distribution offset parameter
    scale: arraylike, distribution scale parameter

  Returns:
    array of logcdf values.

  See Also:
    - :func:`jax.scipy.stats.cauchy.cdf`
    - :func:`jax.scipy.stats.cauchy.pdf`
    - :func:`jax.scipy.stats.cauchy.sf`
    - :func:`jax.scipy.stats.cauchy.logpdf`
    - :func:`jax.scipy.stats.cauchy.logsf`
    - :func:`jax.scipy.stats.cauchy.isf`
    - :func:`jax.scipy.stats.cauchy.ppf`
  """
  return lax.log(cdf(x, loc, scale))


def sf(x: ArrayLike, loc: ArrayLike = 0, scale: ArrayLike = 1) -> Array:
  r"""Cauchy distribution log survival function.

  JAX implementation of :obj:`scipy.stats.cauchy` ``sf``.

  The survival function is defined as

  .. math::

     f_{sf}(x) = 1 - f_{cdf}(x)

  where :math:`f_{cdf}(x)` is the cumulative distribution function,
  :func:`jax.scipy.stats.cauchy.cdf`.

  Args:
    x: arraylike, value at which to evaluate the SF
    loc: arraylike, distribution offset parameter
    scale: arraylike, distribution scale parameter

  Returns:
    array of sf values

  See Also:
    - :func:`jax.scipy.stats.cauchy.cdf`
    - :func:`jax.scipy.stats.cauchy.pdf`
    - :func:`jax.scipy.stats.cauchy.logcdf`
    - :func:`jax.scipy.stats.cauchy.logpdf`
    - :func:`jax.scipy.stats.cauchy.logsf`
    - :func:`jax.scipy.stats.cauchy.isf`
    - :func:`jax.scipy.stats.cauchy.ppf`
  """
  x, loc, scale = promote_args_inexact("cauchy.sf", x, loc, scale)
  return cdf(-x, -loc, scale)


def logsf(x: ArrayLike, loc: ArrayLike = 0, scale: ArrayLike = 1) -> Array:
  r"""Cauchy distribution log survival function.

  JAX implementation of :obj:`scipy.stats.cauchy` ``logsf``

  The survival function is defined as

  .. math::

     f_{sf}(x) = 1 - f_{cdf}(x)

  where :math:`f_{cdf}(x)` is the cumulative distribution function,
  :func:`jax.scipy.stats.cauchy.cdf`.

  Args:
    x: arraylike, value at which to evaluate the SF
    loc: arraylike, distribution offset parameter
    scale: arraylike, distribution scale parameter

  Returns:
    array of logsf values.

  See Also:
    - :func:`jax.scipy.stats.cauchy.cdf`
    - :func:`jax.scipy.stats.cauchy.pdf`
    - :func:`jax.scipy.stats.cauchy.sf`
    - :func:`jax.scipy.stats.cauchy.logcdf`
    - :func:`jax.scipy.stats.cauchy.logpdf`
    - :func:`jax.scipy.stats.cauchy.isf`
    - :func:`jax.scipy.stats.cauchy.ppf`
  """
  x, loc, scale = promote_args_inexact("cauchy.logsf", x, loc, scale)
  return logcdf(-x, -loc, scale)


def isf(q: ArrayLike, loc: ArrayLike = 0, scale: ArrayLike = 1) -> Array:
  r"""Cauchy distribution inverse survival function.

  JAX implementation of :obj:`scipy.stats.cauchy` ``isf``.

  Returns the inverse of the survival function,
  :func:`jax.scipy.stats.cauchy.sf`.

  Args:
    q: arraylike, value at which to evaluate the ISF
    loc: arraylike, distribution offset parameter
    scale: arraylike, distribution scale parameter

  Returns:
    array of isf values.

  See Also:
    - :func:`jax.scipy.stats.cauchy.cdf`
    - :func:`jax.scipy.stats.cauchy.pdf`
    - :func:`jax.scipy.stats.cauchy.sf`
    - :func:`jax.scipy.stats.cauchy.logcdf`
    - :func:`jax.scipy.stats.cauchy.logpdf`
    - :func:`jax.scipy.stats.cauchy.logsf`
    - :func:`jax.scipy.stats.cauchy.ppf`
  """
  q, loc, scale = promote_args_inexact("cauchy.isf", q, loc, scale)
  pi = _lax_const(q, np.pi)
  half_pi = _lax_const(q, np.pi / 2)
  unscaled = lax.tan(lax.sub(half_pi, lax.mul(pi, q)))
  return lax.add(lax.mul(unscaled, scale), loc)


def ppf(q: ArrayLike, loc: ArrayLike = 0, scale: ArrayLike = 1) -> Array:
  r"""Cauchy distribution percent point function.

  JAX implementation of :obj:`scipy.stats.cauchy` ``ppf``.

  The percent point function is defined as the inverse of the
  cumulative distribution function, :func:`jax.scipy.stats.cauchy.cdf`.

  Args:
    q: arraylike, value at which to evaluate the PPF
    loc: arraylike, distribution offset parameter
    scale: arraylike, distribution scale parameter

  Returns:
    array of ppf values.

  See Also:
    - :func:`jax.scipy.stats.cauchy.cdf`
    - :func:`jax.scipy.stats.cauchy.pdf`
    - :func:`jax.scipy.stats.cauchy.sf`
    - :func:`jax.scipy.stats.cauchy.logcdf`
    - :func:`jax.scipy.stats.cauchy.logpdf`
    - :func:`jax.scipy.stats.cauchy.logsf`
    - :func:`jax.scipy.stats.cauchy.isf`
  """
  q, loc, scale = promote_args_inexact("cauchy.ppf", q, loc, scale)
  pi = _lax_const(q, np.pi)
  half_pi = _lax_const(q, np.pi / 2)
  unscaled = lax.tan(lax.sub(lax.mul(pi, q), half_pi))
  return lax.add(lax.mul(unscaled, scale), loc)