Source code for cvxpy.atoms.elementwise.log_normcdf

"""
Copyright 2021 the CVXPY developers
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
    http://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
import scipy.sparse

from cvxpy.atoms.affine.reshape import reshape
from cvxpy.atoms.affine.sum import sum as sum_
from cvxpy.atoms.elementwise.maximum import maximum
from cvxpy.expressions.expression import Expression


# ruff: noqa: E501
def log_normcdf(x):
    """Elementwise log of the cumulative distribution function of a standard normal random variable.

    The implementation is a quadratic approximation with modest accuracy over [-4, 4].
    For details on the nature of the approximation, refer to
    `CVXPY GitHub PR #1224 <https://github.com/cvxpy/cvxpy/pull/1224#issue-793221374>`_.

    .. note::

        SciPy's analog of ``log_normcdf`` is called `log_ndtr <https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.log_ndtr.html>`_.
        We opted not to use that name because its meaning would not be obvious to the casual user.
    """
    A = scipy.sparse.diags(
        np.sqrt(
            [
                0.02301291,
                0.08070214,
                0.16411522,
                0.09003495,
                0.08200854,
                0.01371543,
                0.04641081,
            ]
        )
    )
    b = np.array([[3.0, 2.0, 1.0, 0.0, -1.0, -2.5, -3.5]]).reshape(-1, 1)

    x = Expression.cast_to_const(x)
    flat_x = reshape(x, (1, x.size))

    y = A @ (b @ np.ones(flat_x.shape) - np.ones(b.shape) @ flat_x)
    out = -sum_(maximum(y, 0) ** 2, axis=0)

    return reshape(out, x.shape)