elementwise_functions.py

__all__ = [
    "abs",
    "acos",
    "acosh",
    "add",
    "asin",
    "asinh",
    "atan",
    "atan2",
    "atanh",
    "bitwise_and",
    "bitwise_left_shift",
    "bitwise_invert",
    "bitwise_or",
    "bitwise_right_shift",
    "bitwise_xor",
    "ceil",
    "clip",
    "conj",
    "copysign",
    "cos",
    "cosh",
    "divide",
    "equal",
    "exp",
    "expm1",
    "floor",
    "floor_divide",
    "greater",
    "greater_equal",
    "hypot",
    "imag",
    "isfinite",
    "isinf",
    "isnan",
    "less",
    "less_equal",
    "log",
    "log1p",
    "log2",
    "log10",
    "logaddexp",
    "logical_and",
    "logical_not",
    "logical_or",
    "logical_xor",
    "maximum",
    "minimum",
    "multiply",
    "negative",
    "nextafter",
    "not_equal",
    "positive",
    "pow",
    "real",
    "reciprocal",
    "remainder",
    "round",
    "sign",
    "signbit",
    "sin",
    "sinh",
    "square",
    "sqrt",
    "subtract",
    "tan",
    "tanh",
    "trunc",
]


from ._types import Optional, Union, array


def abs(x: array, /) -> array:
    r"""
    Calculates the absolute value for each element ``x_i`` of the input array ``x``.

    For real-valued input arrays, the element-wise result has the same magnitude as the respective element in ``x`` but has positive sign.

    .. note::
       For signed integer data types, the absolute value of the minimum representable integer is implementation-dependent.

    .. note::
       For complex floating-point operands, the complex absolute value is known as the norm, modulus, or magnitude and, for a complex number :math:`z = a + bj` is computed as

       .. math::
          \operatorname{abs}(z) = \sqrt{a^2 + b^2}

    .. note::
       For complex floating-point operands, conforming implementations should take care to avoid undue overflow or underflow during intermediate stages of computation.

    ..
       TODO: once ``hypot`` is added to the specification, remove the special cases for complex floating-point operands and the note concerning guarding against undue overflow/underflow, and state that special cases must be handled as if implemented as ``hypot(real(x), imag(x))``.

    Parameters
    ----------
    x: array
        input array. Should have a numeric data type.

    Returns
    -------
    out: array
        an array containing the absolute value of each element in ``x``. If ``x`` has a real-valued data type, the returned array must have the same data type as ``x``. If ``x`` has a complex floating-point data type, the returned array must have a real-valued floating-point data type whose precision matches the precision of ``x`` (e.g., if ``x`` is ``complex128``, then the returned array must have a ``float64`` data type).

    Notes
    -----

    **Special Cases**

    For real-valued floating-point operands,

    - If ``x_i`` is ``NaN``, the result is ``NaN``.
    - If ``x_i`` is ``-0``, the result is ``+0``.
    - If ``x_i`` is ``-infinity``, the result is ``+infinity``.

    For complex floating-point operands, let ``a = real(x_i)``, ``b = imag(x_i)``, and

    - If ``a`` is either ``+infinity`` or ``-infinity`` and ``b`` is any value (including ``NaN``), the result is ``+infinity``.
    - If ``a`` is any value (including ``NaN``) and ``b`` is either ``+infinity`` or ``-infinity``, the result is ``+infinity``.
    - If ``a`` is either ``+0`` or ``-0``, the result is equal to ``abs(b)``.
    - If ``b`` is either ``+0`` or ``-0``, the result is equal to ``abs(a)``.
    - If ``a`` is ``NaN`` and ``b`` is a finite number, the result is ``NaN``.
    - If ``a`` is a finite number and ``b`` is ``NaN``, the result is ``NaN``.
    - If ``a`` is ``NaN`` and ``b`` is ``NaN``, the result is ``NaN``.

    .. versionchanged:: 2022.12
       Added complex data type support.
    """


def acos(x: array, /) -> array:
    r"""
    Calculates an implementation-dependent approximation of the principal value of the inverse cosine for each element ``x_i`` of the input array ``x``.

    Each element-wise result is expressed in radians.

    .. note::
       The principal value of the arc cosine of a complex number :math:`z` is

       .. math::
          \operatorname{acos}(z) = \frac{1}{2}\pi + j\ \ln(zj + \sqrt{1-z^2})

       For any :math:`z`,

       .. math::
          \operatorname{acos}(z) = \pi - \operatorname{acos}(-z)

    .. note::
       For complex floating-point operands, ``acos(conj(x))`` must equal ``conj(acos(x))``.

    .. note::
       The inverse cosine (or arc cosine) is a multi-valued function and requires a branch cut on the complex plane. By convention, a branch cut is placed at the line segments :math:`(-\infty, -1)` and :math:`(1, \infty)` of the real axis.

       Accordingly, for complex arguments, the function returns the inverse cosine in the range of a strip unbounded along the imaginary axis and in the interval :math:`[0, \pi]` along the real axis.

       *Note: branch cuts follow C99 and have provisional status* (see :ref:`branch-cuts`).

    Parameters
    ----------
    x: array
        input array. Should have a floating-point data type.

    Returns
    -------
    out: array
        an array containing the inverse cosine of each element in ``x``. The returned array must have a floating-point data type determined by :ref:`type-promotion`.

    Notes
    -----

    **Special cases**

    For real-valued floating-point operands,

    - If ``x_i`` is ``NaN``, the result is ``NaN``.
    - If ``x_i`` is greater than ``1``, the result is ``NaN``.
    - If ``x_i`` is less than ``-1``, the result is ``NaN``.
    - If ``x_i`` is ``1``, the result is ``+0``.

    For complex floating-point operands, let ``a = real(x_i)``, ``b = imag(x_i)``, and

    - If ``a`` is either ``+0`` or ``-0`` and ``b`` is ``+0``, the result is ``π/2 - 0j``.
    - If ``a`` is either ``+0`` or ``-0`` and ``b`` is ``NaN``, the result is ``π/2 + NaN j``.
    - If ``a`` is a finite number and ``b`` is ``+infinity``, the result is ``π/2 - infinity j``.
    - If ``a`` is a nonzero finite number and ``b`` is ``NaN``, the result is ``NaN + NaN j``.
    - If ``a`` is ``-infinity`` and ``b`` is a positive (i.e., greater than ``0``) finite number, the result is ``π - infinity j``.
    - If ``a`` is ``+infinity`` and ``b`` is a positive (i.e., greater than ``0``) finite number, the result is ``+0 - infinity j``.
    - If ``a`` is ``-infinity`` and ``b`` is ``+infinity``, the result is ``3π/4 - infinity j``.
    - If ``a`` is ``+infinity`` and ``b`` is ``+infinity``, the result is ``π/4 - infinity j``.
    - If ``a`` is either ``+infinity`` or ``-infinity`` and ``b`` is ``NaN``, the result is ``NaN ± infinity j`` (sign of the imaginary component is unspecified).
    - If ``a`` is ``NaN`` and ``b`` is a finite number, the result is ``NaN + NaN j``.
    - If ``a`` is ``NaN`` and ``b`` is ``+infinity``, the result is ``NaN - infinity j``.
    - If ``a`` is ``NaN`` and ``b`` is ``NaN``, the result is ``NaN + NaN j``.

    .. versionchanged:: 2022.12
       Added complex data type support.
    """


def acosh(x: array, /) -> array:
    r"""
    Calculates an implementation-dependent approximation to the inverse hyperbolic cosine for each element ``x_i`` of the input array ``x``.

    .. note::
       The principal value of the inverse hyperbolic cosine of a complex number :math:`z` is

       .. math::
          \operatorname{acosh}(z) = \ln(z + \sqrt{z+1}\sqrt{z-1})

       For any :math:`z`,

       .. math::
          \operatorname{acosh}(z) = \frac{\sqrt{z-1}}{\sqrt{1-z}}\operatorname{acos}(z)

       or simply

       .. math::
          \operatorname{acosh}(z) = j\ \operatorname{acos}(z)

       in the upper half of the complex plane.

    .. note::
       For complex floating-point operands, ``acosh(conj(x))`` must equal ``conj(acosh(x))``.

    .. note::
       The inverse hyperbolic cosine is a multi-valued function and requires a branch cut on the complex plane. By convention, a branch cut is placed at the line segment :math:`(-\infty, 1)` of the real axis.

       Accordingly, for complex arguments, the function returns the inverse hyperbolic cosine in the interval :math:`[0, \infty)` along the real axis and in the interval :math:`[-\pi j, +\pi j]` along the imaginary axis.

       *Note: branch cuts follow C99 and have provisional status* (see :ref:`branch-cuts`).

    Parameters
    ----------
    x: array
        input array whose elements each represent the area of a hyperbolic sector. Should have a floating-point data type.

    Returns
    -------
    out: array
        an array containing the inverse hyperbolic cosine of each element in ``x``. The returned array must have a floating-point data type determined by :ref:`type-promotion`.

    Notes
    -----

    **Special cases**

    For real-valued floating-point operands,

    - If ``x_i`` is ``NaN``, the result is ``NaN``.
    - If ``x_i`` is less than ``1``, the result is ``NaN``.
    - If ``x_i`` is ``1``, the result is ``+0``.
    - If ``x_i`` is ``+infinity``, the result is ``+infinity``.

    For complex floating-point operands, let ``a = real(x_i)``, ``b = imag(x_i)``, and

    - If ``a`` is either ``+0`` or ``-0`` and ``b`` is ``+0``, the result is ``+0 + πj/2``.
    - If ``a`` is a finite number and ``b`` is ``+infinity``, the result is ``+infinity + πj/2``.
    - If ``a`` is a nonzero finite number and ``b`` is ``NaN``, the result is ``NaN + NaN j``.
    - If ``a`` is ``+0`` and ``b`` is ``NaN``, the result is ``NaN ± πj/2`` (sign of imaginary component is unspecified).
    - If ``a`` is ``-infinity`` and ``b`` is a positive (i.e., greater than ``0``) finite number, the result is ``+infinity + πj``.
    - If ``a`` is ``+infinity`` and ``b`` is a positive (i.e., greater than ``0``) finite number, the result is ``+infinity + 0j``.
    - If ``a`` is ``-infinity`` and ``b`` is ``+infinity``, the result is ``+infinity + 3πj/4``.
    - If ``a`` is ``+infinity`` and ``b`` is ``+infinity``, the result is ``+infinity + πj/4``.
    - If ``a`` is either ``+infinity`` or ``-infinity`` and ``b`` is ``NaN``, the result is ``+infinity + NaN j``.
    - If ``a`` is ``NaN`` and ``b`` is a finite number, the result is ``NaN + NaN j``.
    - If ``a`` is ``NaN`` and ``b`` is ``+infinity``, the result is ``+infinity + NaN j``.
    - If ``a`` is ``NaN`` and ``b`` is ``NaN``, the result is ``NaN + NaN j``.

    .. versionchanged:: 2022.12
       Added complex data type support.
    """


def add(
    x1: Union[array, int, float, complex], x2: Union[array, int, float, complex], /
) -> array:
    """
    Calculates the sum for each element ``x1_i`` of the input array ``x1`` with the respective element ``x2_i`` of the input array ``x2``.

    Parameters
    ----------
    x1: Union[array, int, float, complex]
        first input array. Should have a numeric data type.
    x2: Union[array, int, float, complex]
        second input array. Must be compatible with ``x1`` (see :ref:`broadcasting`). Should have a numeric data type.

    Returns
    -------
    out: array
        an array containing the element-wise sums. The returned array must have a data type determined by :ref:`type-promotion`.

    Notes
    -----

    -   At least one of ``x1`` or ``x2`` must be an array.

    **Special cases**

    For real-valued floating-point operands,

    - If either ``x1_i`` or ``x2_i`` is ``NaN``, the result is ``NaN``.
    - If ``x1_i`` is ``+infinity`` and ``x2_i`` is ``-infinity``, the result is ``NaN``.
    - If ``x1_i`` is ``-infinity`` and ``x2_i`` is ``+infinity``, the result is ``NaN``.
    - If ``x1_i`` is ``+infinity`` and ``x2_i`` is ``+infinity``, the result is ``+infinity``.
    - If ``x1_i`` is ``-infinity`` and ``x2_i`` is ``-infinity``, the result is ``-infinity``.
    - If ``x1_i`` is ``+infinity`` and ``x2_i`` is a finite number, the result is ``+infinity``.
    - If ``x1_i`` is ``-infinity`` and ``x2_i`` is a finite number, the result is ``-infinity``.
    - If ``x1_i`` is a finite number and ``x2_i`` is ``+infinity``, the result is ``+infinity``.
    - If ``x1_i`` is a finite number and ``x2_i`` is ``-infinity``, the result is ``-infinity``.
    - If ``x1_i`` is ``-0`` and ``x2_i`` is ``-0``, the result is ``-0``.
    - If ``x1_i`` is ``-0`` and ``x2_i`` is ``+0``, the result is ``+0``.
    - If ``x1_i`` is ``+0`` and ``x2_i`` is ``-0``, the result is ``+0``.
    - If ``x1_i`` is ``+0`` and ``x2_i`` is ``+0``, the result is ``+0``.
    - If ``x1_i`` is either ``+0`` or ``-0`` and ``x2_i`` is a nonzero finite number, the result is ``x2_i``.
    - If ``x1_i`` is a nonzero finite number and ``x2_i`` is either ``+0`` or ``-0``, the result is ``x1_i``.
    - If ``x1_i`` is a nonzero finite number and ``x2_i`` is ``-x1_i``, the result is ``+0``.
    - In the remaining cases, when neither ``infinity``, ``+0``, ``-0``, nor a ``NaN`` is involved, and the operands have the same mathematical sign or have different magnitudes, the sum must be computed and rounded to the nearest representable value according to IEEE 754-2019 and a supported round mode. If the magnitude is too large to represent, the operation overflows and the result is an `infinity` of appropriate mathematical sign.

    .. note::
       Floating-point addition is a commutative operation, but not always associative.

    For complex floating-point operands, addition is defined according to the following table. For real components ``a`` and ``c`` and imaginary components ``b`` and ``d``,

    +------------+------------+------------+----------------+
    |            | c          | dj         | c + dj         |
    +============+============+============+================+
    | **a**      | a + c      | a + dj     | (a+c) + dj     |
    +------------+------------+------------+----------------+
    | **bj**     | c + bj     | (b+d)j     | c + (b+d)j     |
    +------------+------------+------------+----------------+
    | **a + bj** | (a+c) + bj | a + (b+d)j | (a+c) + (b+d)j |
    +------------+------------+------------+----------------+

    For complex floating-point operands, real-valued floating-point special cases must independently apply to the real and imaginary component operations involving real numbers as described in the above table. For example, let ``a = real(x1_i)``, ``b = imag(x1_i)``, ``c = real(x2_i)``, ``d = imag(x2_i)``, and

    - If ``a`` is ``-0`` and ``c`` is ``-0``, the real component of the result is ``-0``.
    - Similarly, if ``b`` is ``+0`` and ``d`` is ``-0``, the imaginary component of the result is ``+0``.

    Hence, if ``z1 = a + bj = -0 + 0j`` and ``z2 = c + dj = -0 - 0j``, the result of ``z1 + z2`` is ``-0 + 0j``.

    .. versionchanged:: 2022.12
       Added complex data type support.

    .. versionchanged:: 2024.12
       Added scalar argument support.
    """


def asin(x: array, /) -> array:
    r"""
    Calculates an implementation-dependent approximation of the principal value of the inverse sine for each element ``x_i`` of the input array ``x``.

    Each element-wise result is expressed in radians.

    .. note::
       The principal value of the arc sine of a complex number :math:`z` is

       .. math::
          \operatorname{asin}(z) = -j\ \ln(zj + \sqrt{1-z^2})

       For any :math:`z`,

       .. math::
          \operatorname{asin}(z) = \operatorname{acos}(-z) - \frac{\pi}{2}

    .. note::
       For complex floating-point operands, ``asin(conj(x))`` must equal ``conj(asin(x))``.

    .. note::
       The inverse sine (or arc sine) is a multi-valued function and requires a branch cut on the complex plane. By convention, a branch cut is placed at the line segments :math:`(-\infty, -1)` and :math:`(1, \infty)` of the real axis.

       Accordingly, for complex arguments, the function returns the inverse sine in the range of a strip unbounded along the imaginary axis and in the interval :math:`[-\pi/2, +\pi/2]` along the real axis.

       *Note: branch cuts follow C99 and have provisional status* (see :ref:`branch-cuts`).

    Parameters
    ----------
    x: array
        input array. Should have a floating-point data type.

    Returns
    -------
    out: array
        an array containing the inverse sine of each element in ``x``. The returned array must have a floating-point data type determined by :ref:`type-promotion`.

    Notes
    -----

    **Special cases**

    For real-valued floating-point operands,

    - If ``x_i`` is ``NaN``, the result is ``NaN``.
    - If ``x_i`` is greater than ``1``, the result is ``NaN``.
    - If ``x_i`` is less than ``-1``, the result is ``NaN``.
    - If ``x_i`` is ``+0``, the result is ``+0``.
    - If ``x_i`` is ``-0``, the result is ``-0``.

    For complex floating-point operands, special cases must be handled as if the operation is implemented as ``-1j * asinh(x*1j)``.

    .. versionchanged:: 2022.12
       Added complex data type support.
    """


def asinh(x: array, /) -> array:
    r"""
    Calculates an implementation-dependent approximation to the inverse hyperbolic sine for each element ``x_i`` in the input array ``x``.

    .. note::
       The principal value of the inverse hyperbolic sine of a complex number :math:`z` is

       .. math::
          \operatorname{asinh}(z) = \ln(z + \sqrt{1+z^2})

       For any :math:`z`,

       .. math::
          \operatorname{asinh}(z) = \frac{\operatorname{asin}(zj)}{j}

    .. note::
       For complex floating-point operands, ``asinh(conj(x))`` must equal ``conj(asinh(x))`` and ``asinh(-z)`` must equal ``-asinh(z)``.

    .. note::
       The inverse hyperbolic sine is a multi-valued function and requires a branch cut on the complex plane. By convention, a branch cut is placed at the line segments :math:`(-\infty j, -j)` and :math:`(j, \infty j)` of the imaginary axis.

       Accordingly, for complex arguments, the function returns the inverse hyperbolic sine in the range of a strip unbounded along the real axis and in the interval :math:`[-\pi j/2, +\pi j/2]` along the imaginary axis.

       *Note: branch cuts follow C99 and have provisional status* (see :ref:`branch-cuts`).

    Parameters
    ----------
    x: array
        input array whose elements each represent the area of a hyperbolic sector. Should have a floating-point data type.

    Returns
    -------
    out: array
        an array containing the inverse hyperbolic sine of each element in ``x``. The returned array must have a floating-point data type determined by :ref:`type-promotion`.

    Notes
    -----

    **Special cases**

    For real-valued floating-point operands,

    - If ``x_i`` is ``NaN``, the result is ``NaN``.
    - If ``x_i`` is ``+0``, the result is ``+0``.
    - If ``x_i`` is ``-0``, the result is ``-0``.
    - If ``x_i`` is ``+infinity``, the result is ``+infinity``.
    - If ``x_i`` is ``-infinity``, the result is ``-infinity``.

    For complex floating-point operands, let ``a = real(x_i)``, ``b = imag(x_i)``, and

    - If ``a`` is ``+0`` and ``b`` is ``+0``, the result is ``+0 + 0j``.
    - If ``a`` is a positive (i.e., greater than ``0``) finite number and ``b`` is ``+infinity``, the result is ``+infinity + πj/2``.
    - If ``a`` is a finite number and ``b`` is ``NaN``, the result is ``NaN + NaN j``.
    - If ``a`` is ``+infinity`` and ``b`` is a positive (i.e., greater than ``0``) finite number, the result is ``+infinity + 0j``.
    - If ``a`` is ``+infinity`` and ``b`` is ``+infinity``, the result is ``+infinity + πj/4``.
    - If ``a`` is ``NaN`` and ``b`` is ``+0``, the result is ``NaN + 0j``.
    - If ``a`` is ``NaN`` and ``b`` is a nonzero finite number, the result is ``NaN + NaN j``.
    - If ``a`` is ``NaN`` and ``b`` is ``+infinity``, the result is ``±infinity + NaN j`` (sign of the real component is unspecified).
    - If ``a`` is ``NaN`` and ``b`` is ``NaN``, the result is ``NaN + NaN j``.

    .. versionchanged:: 2022.12
       Added complex data type support.
    """


def atan(x: array, /) -> array:
    r"""
    Calculates an implementation-dependent approximation of the principal value of the inverse tangent for each element ``x_i`` of the input array ``x``.

    Each element-wise result is expressed in radians.

    .. note::
       The principal value of the inverse tangent of a complex number :math:`z` is

       .. math::
          \operatorname{atan}(z) = -\frac{\ln(1 - zj) - \ln(1 + zj)}{2}j

    .. note::
       For complex floating-point operands, ``atan(conj(x))`` must equal ``conj(atan(x))``.

    .. note::
       The inverse tangent (or arc tangent) is a multi-valued function and requires a branch on the complex plane. By convention, a branch cut is placed at the line segments :math:`(-\infty j, -j)` and :math:`(+j, \infty j)` of the imaginary axis.

       Accordingly, for complex arguments, the function returns the inverse tangent in the range of a strip unbounded along the imaginary axis and in the interval :math:`[-\pi/2, +\pi/2]` along the real axis.

       *Note: branch cuts follow C99 and have provisional status* (see :ref:`branch-cuts`).

    Parameters
    ----------
    x: array
        input array. Should have a floating-point data type.

    Returns
    -------
    out: array
        an array containing the inverse tangent of each element in ``x``. The returned array must have a floating-point data type determined by :ref:`type-promotion`.

    Notes
    -----

    **Special cases**

    For real-valued floating-point operands,

    - If ``x_i`` is ``NaN``, the result is ``NaN``.
    - If ``x_i`` is ``+0``, the result is ``+0``.
    - If ``x_i`` is ``-0``, the result is ``-0``.
    - If ``x_i`` is ``+infinity``, the result is an implementation-dependent approximation to ``+π/2``.
    - If ``x_i`` is ``-infinity``, the result is an implementation-dependent approximation to ``-π/2``.

    For complex floating-point operands, special cases must be handled as if the operation is implemented as ``-1j * atanh(x*1j)``.

    .. versionchanged:: 2022.12
       Added complex data type support.
    """


def atan2(x1: Union[array, int, float], x2: Union[array, int, float], /) -> array:
    """
    Calculates an implementation-dependent approximation of the inverse tangent of the quotient ``x1/x2``, having domain ``[-infinity, +infinity] x [-infinity, +infinity]`` (where the ``x`` notation denotes the set of ordered pairs of elements ``(x1_i, x2_i)``) and codomain ``[-π, +π]``, for each pair of elements ``(x1_i, x2_i)`` of the input arrays ``x1`` and ``x2``, respectively. Each element-wise result is expressed in radians.

    The mathematical signs of ``x1_i`` and ``x2_i`` determine the quadrant of each element-wise result. The quadrant (i.e., branch) is chosen such that each element-wise result is the signed angle in radians between the ray ending at the origin and passing through the point ``(1,0)`` and the ray ending at the origin and passing through the point ``(x2_i, x1_i)``.

    .. note::
       Note the role reversal: the "y-coordinate" is the first function parameter; the "x-coordinate" is the second function parameter. The parameter order is intentional and traditional for the two-argument inverse tangent function where the y-coordinate argument is first and the x-coordinate argument is second.

    By IEEE 754 convention, the inverse tangent of the quotient ``x1/x2`` is defined for ``x2_i`` equal to positive or negative zero and for either or both of ``x1_i`` and ``x2_i`` equal to positive or negative ``infinity``.

    Parameters
    ----------
    x1: Union[array, int, float]
        input array corresponding to the y-coordinates. Should have a real-valued floating-point data type.
    x2: Union[array, int, float]
        input array corresponding to the x-coordinates. Must be compatible with ``x1`` (see :ref:`broadcasting`). Should have a real-valued floating-point data type.

    Returns
    -------
    out: array
        an array containing the inverse tangent of the quotient ``x1/x2``. The returned array must have a real-valued floating-point data type determined by :ref:`type-promotion`.

    Notes
    -----

    -   At least one of ``x1`` or ``x2`` must be an array.

    **Special cases**

    For floating-point operands,

    - If either ``x1_i`` or ``x2_i`` is ``NaN``, the result is ``NaN``.
    - If ``x1_i`` is greater than ``0`` and ``x2_i`` is ``+0``, the result is an implementation-dependent approximation to ``+π/2``.
    - If ``x1_i`` is greater than ``0`` and ``x2_i`` is ``-0``, the result is an implementation-dependent approximation to ``+π/2``.
    - If ``x1_i`` is ``+0`` and ``x2_i`` is greater than ``0``, the result is ``+0``.
    - If ``x1_i`` is ``+0`` and ``x2_i`` is ``+0``, the result is ``+0``.
    - If ``x1_i`` is ``+0`` and ``x2_i`` is ``-0``, the result is an implementation-dependent approximation to ``+π``.
    - If ``x1_i`` is ``+0`` and ``x2_i`` is less than ``0``, the result is an implementation-dependent approximation to ``+π``.
    - If ``x1_i`` is ``-0`` and ``x2_i`` is greater than ``0``, the result is ``-0``.
    - If ``x1_i`` is ``-0`` and ``x2_i`` is ``+0``, the result is ``-0``.
    - If ``x1_i`` is ``-0`` and ``x2_i`` is ``-0``, the result is an implementation-dependent approximation to ``-π``.
    - If ``x1_i`` is ``-0`` and ``x2_i`` is less than ``0``, the result is an implementation-dependent approximation to ``-π``.
    - If ``x1_i`` is less than ``0`` and ``x2_i`` is ``+0``, the result is an implementation-dependent approximation to ``-π/2``.
    - If ``x1_i`` is less than ``0`` and ``x2_i`` is ``-0``, the result is an implementation-dependent approximation to ``-π/2``.
    - If ``x1_i`` is greater than ``0``, ``x1_i`` is a finite number, and ``x2_i`` is ``+infinity``, the result is ``+0``.
    - If ``x1_i`` is greater than ``0``, ``x1_i`` is a finite number, and ``x2_i`` is ``-infinity``, the result is an implementation-dependent approximation to ``+π``.
    - If ``x1_i`` is less than ``0``, ``x1_i`` is a finite number, and ``x2_i`` is ``+infinity``, the result is ``-0``.
    - If ``x1_i`` is less than ``0``, ``x1_i`` is a finite number, and ``x2_i`` is ``-infinity``, the result is an implementation-dependent approximation to ``-π``.
    - If ``x1_i`` is ``+infinity`` and ``x2_i`` is a finite number, the result is an implementation-dependent approximation to ``+π/2``.
    - If ``x1_i`` is ``-infinity`` and ``x2_i`` is a finite number, the result is an implementation-dependent approximation to ``-π/2``.
    - If ``x1_i`` is ``+infinity`` and ``x2_i`` is ``+infinity``, the result is an implementation-dependent approximation to ``+π/4``.
    - If ``x1_i`` is ``+infinity`` and ``x2_i`` is ``-infinity``, the result is an implementation-dependent approximation to ``+3π/4``.
    - If ``x1_i`` is ``-infinity`` and ``x2_i`` is ``+infinity``, the result is an implementation-dependent approximation to ``-π/4``.
    - If ``x1_i`` is ``-infinity`` and ``x2_i`` is ``-infinity``, the result is an implementation-dependent approximation to ``-3π/4``.

    .. versionchanged:: 2024.12
       Added scalar argument support.
    """


def atanh(x: array, /) -> array:
    r"""
    Calculates an implementation-dependent approximation to the inverse hyperbolic tangent for each element ``x_i`` of the input array ``x``.

    .. note::
       The principal value of the inverse hyperbolic tangent of a complex number :math:`z` is

       .. math::
          \operatorname{atanh}(z) = \frac{\ln(1+z)-\ln(z-1)}{2}

       For any :math:`z`,

       .. math::
          \operatorname{atanh}(z) = \frac{\operatorname{atan}(zj)}{j}

    .. note::
       For complex floating-point operands, ``atanh(conj(x))`` must equal ``conj(atanh(x))`` and ``atanh(-x)`` must equal ``-atanh(x)``.

    .. note::
       The inverse hyperbolic tangent is a multi-valued function and requires a branch cut on the complex plane. By convention, a branch cut is placed at the line segments :math:`(-\infty, 1]` and :math:`[1, \infty)` of the real axis.

       Accordingly, for complex arguments, the function returns the inverse hyperbolic tangent in the range of a half-strip unbounded along the real axis and in the interval :math:`[-\pi j/2, +\pi j/2]` along the imaginary axis.

       *Note: branch cuts follow C99 and have provisional status* (see :ref:`branch-cuts`).

    Parameters
    ----------
    x: array
        input array whose elements each represent the area of a hyperbolic sector. Should have a floating-point data type.

    Returns
    -------
    out: array
        an array containing the inverse hyperbolic tangent of each element in ``x``. The returned array must have a floating-point data type determined by :ref:`type-promotion`.

    Notes
    -----

    **Special cases**

    For real-valued floating-point operands,

    - If ``x_i`` is ``NaN``, the result is ``NaN``.
    - If ``x_i`` is less than ``-1``, the result is ``NaN``.
    - If ``x_i`` is greater than ``1``, the result is ``NaN``.
    - If ``x_i`` is ``-1``, the result is ``-infinity``.
    - If ``x_i`` is ``+1``, the result is ``+infinity``.
    - If ``x_i`` is ``+0``, the result is ``+0``.
    - If ``x_i`` is ``-0``, the result is ``-0``.

    For complex floating-point operands, let ``a = real(x_i)``, ``b = imag(x_i)``, and

    - If ``a`` is ``+0`` and ``b`` is ``+0``, the result is ``+0 + 0j``.
    - If ``a`` is ``+0`` and ``b`` is ``NaN``, the result is ``+0 + NaN j``.
    - If ``a`` is ``1`` and ``b`` is ``+0``, the result is ``+infinity + 0j``.
    - If ``a`` is a positive (i.e., greater than ``0``) finite number and ``b`` is ``+infinity``, the result is ``+0 + πj/2``.
    - If ``a`` is a nonzero finite number and ``b`` is ``NaN``, the result is ``NaN + NaN j``.
    - If ``a`` is ``+infinity`` and ``b`` is a positive (i.e., greater than ``0``) finite number, the result is ``+0 + πj/2``.
    - If ``a`` is ``+infinity`` and ``b`` is ``+infinity``, the result is ``+0 + πj/2``.
    - If ``a`` is ``+infinity`` and ``b`` is ``NaN``, the result is ``+0 + NaN j``.
    - If ``a`` is ``NaN`` and ``b`` is a finite number, the result is ``NaN + NaN j``.
    - If ``a`` is ``NaN`` and ``b`` is ``+infinity``, the result is ``±0 + πj/2`` (sign of the real component is unspecified).
    - If ``a`` is ``NaN`` and ``b`` is ``NaN``, the result is ``NaN + NaN j``.

    .. versionchanged:: 2022.12
       Added complex data type support.
    """


def bitwise_and(x1: Union[array, int, bool], x2: Union[array, int, bool], /) -> array:
    """
    Computes the bitwise AND of the underlying binary representation of each element ``x1_i`` of the input array ``x1`` with the respective element ``x2_i`` of the input array ``x2``.

    Parameters
    ----------
    x1: Union[array, int, bool]
        first input array. Should have an integer or boolean data type.
    x2: Union[array, int, bool]
        second input array. Must be compatible with ``x1`` (see :ref:`broadcasting`). Should have an integer or boolean data type.

    Returns
    -------
    out: array
        an array containing the element-wise results. The returned array must have a data type determined by :ref:`type-promotion`.

    Notes
    -----

    -   At least one of ``x1`` or ``x2`` must be an array.

    .. versionchanged:: 2024.12
       Added scalar argument support.
    """


def bitwise_left_shift(x1: Union[array, int], x2: Union[array, int], /) -> array:
    """
    Shifts the bits of each element ``x1_i`` of the input array ``x1`` to the left by appending ``x2_i`` (i.e., the respective element in the input array ``x2``) zeros to the right of ``x1_i``.

    Parameters
    ----------
    x1: Union[array, int]
        first input array. Should have an integer data type.
    x2: Union[array, int]
        second input array. Must be compatible with ``x1`` (see :ref:`broadcasting`). Should have an integer data type. Each element must be greater than or equal to ``0``.

    Returns
    -------
    out: array
        an array containing the element-wise results. The returned array must have a data type determined by :ref:`type-promotion`.

    Notes
    -----

    -   At least one of ``x1`` or ``x2`` must be an array.

    .. versionchanged:: 2024.12
       Added scalar argument support.
    """


def bitwise_invert(x: array, /) -> array:
    """
    Inverts (flips) each bit for each element ``x_i`` of the input array ``x``.

    Parameters
    ----------
    x: array
        input array. Should have an integer or boolean data type.

    Returns
    -------
    out: array
        an array containing the element-wise results. The returned array must have the same data type as ``x``.
    """


def bitwise_or(x1: Union[array, int, bool], x2: Union[array, int, bool], /) -> array:
    """
    Computes the bitwise OR of the underlying binary representation of each element ``x1_i`` of the input array ``x1`` with the respective element ``x2_i`` of the input array ``x2``.

    Parameters
    ----------
    x1: Union[array, int, bool]
        first input array. Should have an integer or boolean data type.
    x2: Union[array, int, bool]
        second input array. Must be compatible with ``x1`` (see :ref:`broadcasting`). Should have an integer or boolean data type.

    Returns
    -------
    out: array
        an array containing the element-wise results. The returned array must have a data type determined by :ref:`type-promotion`.

    Notes
    -----

    -   At least one of ``x1`` or ``x2`` must be an array.

    .. versionchanged:: 2024.12
       Added scalar argument support.
    """


def bitwise_right_shift(x1: Union[array, int], x2: Union[array, int], /) -> array:
    """
    Shifts the bits of each element ``x1_i`` of the input array ``x1`` to the right according to the respective element ``x2_i`` of the input array ``x2``.

    .. note::
       This operation must be an arithmetic shift (i.e., sign-propagating) and thus equivalent to floor division by a power of two.

    Parameters
    ----------
    x1: Union[array, int]
        first input array. Should have an integer data type.
    x2: Union[array, int]
        second input array. Must be compatible with ``x1`` (see :ref:`broadcasting`). Should have an integer data type. Each element must be greater than or equal to ``0``.

    Returns
    -------
    out: array
        an array containing the element-wise results. The returned array must have a data type determined by :ref:`type-promotion`.

    Notes
    -----

    -   At least one of ``x1`` or ``x2`` must be an array.

    .. versionchanged:: 2024.12
       Added scalar argument support.
    """


def bitwise_xor(x1: Union[array, int, bool], x2: Union[array, int, bool], /) -> array:
    """
    Computes the bitwise XOR of the underlying binary representation of each element ``x1_i`` of the input array ``x1`` with the respective element ``x2_i`` of the input array ``x2``.

    Parameters
    ----------
    x1: Union[array, int, bool]
        first input array. Should have an integer or boolean data type.
    x2: Union[array, int, bool]
        second input array. Must be compatible with ``x1`` (see :ref:`broadcasting`). Should have an integer or boolean data type.

    Returns
    -------
    out: array
        an array containing the element-wise results. The returned array must have a data type determined by :ref:`type-promotion`.

    Notes
    -----

    -   At least one of ``x1`` or ``x2`` must be an array.

    .. versionchanged:: 2024.12
       Added scalar argument support.
    """


def ceil(x: array, /) -> array:
    """
    Rounds each element ``x_i`` of the input array ``x`` to the smallest (i.e., closest to ``-infinity``) integer-valued number that is not less than ``x_i``.

    Parameters
    ----------
    x: array
        input array. Should have a real-valued data type.

    Returns
    -------
    out: array
        an array containing the rounded result for each element in ``x``. The returned array must have the same data type as ``x``.

    Notes
    -----

    **Special cases**

    - If ``x_i`` is already integer-valued, the result is ``x_i``.

    For floating-point operands,

    - If ``x_i`` is ``+infinity``, the result is ``+infinity``.
    - If ``x_i`` is ``-infinity``, the result is ``-infinity``.
    - If ``x_i`` is ``+0``, the result is ``+0``.
    - If ``x_i`` is ``-0``, the result is ``-0``.
    - If ``x_i`` is ``NaN``, the result is ``NaN``.
    """


def clip(
    x: array,
    /,
    min: Optional[Union[int, float, array]] = None,
    max: Optional[Union[int, float, array]] = None,
) -> array:
    r"""
    Clamps each element ``x_i`` of the input array ``x`` to the range ``[min, max]``.

    Parameters
    ----------
    x: array
      input array. Should have a real-valued data type.
    min: Optional[Union[int, float, array]]
      lower-bound of the range to which to clamp. If ``None``, no lower bound must be applied. Must be compatible with ``x`` and ``max`` (see :ref:`broadcasting`). Should have the same data type as ``x``. Default: ``None``.
    max: Optional[Union[int, float, array]]
      upper-bound of the range to which to clamp. If ``None``, no upper bound must be applied. Must be compatible with ``x`` and ``min`` (see :ref:`broadcasting`). Should have the same data type as ``x``. Default: ``None``.

    Returns
    -------
    out: array
      an array containing element-wise results. The returned array should have the same data type as ``x``.

    Notes
    -----

    - This function is conceptually equivalent to ``maximum(minimum(x, max), min)`` when ``x``, ``min``, and ``max`` have the same data type.
    - If both ``min`` and ``max`` are ``None``, the elements of the returned array must equal the respective elements in ``x``.
    - If a broadcasted element in ``min`` is greater than a corresponding broadcasted element in ``max``, behavior is unspecified and thus implementation-dependent.
    - For scalar ``min`` and/or ``max``, the scalar values should follow type promotion rules for operations involving arrays and scalar operands (see :ref:`type-promotion`).
    - If ``x`` has an integral data type and a broadcasted element in ``min`` or ``max`` is outside the bounds of the data type of ``x``, behavior is unspecified and thus implementation-dependent.
    - If either ``min`` or ``max`` is an array having a different data type than ``x``, behavior is unspecified and thus implementation-dependent.

    **Special cases**

    - If ``x_i`` is ``NaN``, the result is ``NaN``.
    - If ``min_i`` is ``NaN``, the result is ``NaN``.
    - If ``max_i`` is ``NaN``, the result is ``NaN``.

    .. versionadded:: 2023.12

    .. versionchanged:: 2024.12
       Added special case behavior when one of the operands is ``NaN``.

    .. versionchanged:: 2024.12
       Clarified that behavior is only defined when ``x``, ``min``, and ``max`` resolve to arrays having the same data type.

    .. versionchanged:: 2024.12
       Clarified that behavior is only defined when elements of ``min`` and ``max`` are inside the bounds of the input array data type.
    """


def conj(x: array, /) -> array:
    """
    Returns the complex conjugate for each element ``x_i`` of the input array ``x``.

    For complex numbers of the form

    .. math::
       a + bj

    the complex conjugate is defined as

    .. math::
       a - bj

    Hence, the returned complex conjugates must be computed by negating the imaginary component of each element ``x_i``.

    Parameters
    ----------
    x: array
        input array. Must have a numeric data type.

    Returns
    -------
    out: array
        an array containing the element-wise results. The returned array must have the same data type as ``x``.

    Notes
    -----

    -   Whether the returned array and the input array share the same underlying memory is unspecified and thus implementation-defined.

    .. versionadded:: 2022.12

    .. versionchanged:: 2024.12
       Added support for real-valued arrays.
    """


def copysign(x1: Union[array, int, float], x2: Union[array, int, float], /) -> array:
    r"""
    Composes a floating-point value with the magnitude of ``x1_i`` and the sign of ``x2_i`` for each element of the input array ``x1``.

    Parameters
    ----------
    x1: Union[array, int, float]
       input array containing magnitudes. Should have a real-valued floating-point data type.
    x2: Union[array, int, float]
       input array whose sign bits are applied to the magnitudes of ``x1``. Must be compatible with ``x1`` (see :ref:`broadcasting`). Should have a real-valued floating-point data type.

    Returns
    -------
    out: array
       an array containing the element-wise results. The returned array must have a floating-point data type determined by :ref:`type-promotion`.

    Notes
    -----

    -   At least one of ``x1`` or ``x2`` must be an array.

    **Special cases**

    For real-valued floating-point operands, let ``|x|`` be the absolute value, and if ``x1_i`` is not ``NaN``,

    - If ``x2_i`` is less than ``0``, the result is ``-|x1_i|``.
    - If ``x2_i`` is ``-0``, the result is ``-|x1_i|``.
    - If ``x2_i`` is ``+0``, the result is ``|x1_i|``.
    - If ``x2_i`` is greater than ``0``, the result is ``|x1_i|``.
    - If ``x2_i`` is ``NaN`` and the sign bit of ``x2_i`` is ``1``, the result is ``-|x1_i|``.
    - If ``x2_i`` is ``NaN`` and the sign bit of ``x2_i`` is ``0``, the result is ``|x1_i|``.

    - If ``x1_i`` is ``NaN`` and ``x2_i`` is less than ``0``, the result is ``NaN`` with a sign bit of ``1``.
    - If ``x1_i`` is ``NaN`` and ``x2_i`` is ``-0``, the result is ``NaN`` with a sign bit of ``1``.
    - If ``x1_i`` is ``NaN`` and ``x2_i`` is ``+0``, the result is ``NaN`` with a sign bit of ``0``.
    - If ``x1_i`` is ``NaN`` and ``x2_i`` is greater than ``0``, the result is ``NaN`` with a sign bit of ``0``.
    - If ``x1_i`` is ``NaN`` and ``x2_i`` is ``NaN`` and the sign bit of ``x2_i`` is ``1``, the result is ``NaN`` with a sign bit of ``1``.
    - If ``x1_i`` is ``NaN`` and ``x2_i`` is ``NaN`` and the sign bit of ``x2_i`` is ``0``, the result is ``NaN`` with a sign bit of ``0``.

    .. versionadded:: 2023.12

    .. versionchanged:: 2024.12
       Added scalar argument support.
    """


def cos(x: array, /) -> array:
    r"""
    Calculates an implementation-dependent approximation to the cosine for each element ``x_i`` of the input array ``x``.

    Each element ``x_i`` is assumed to be expressed in radians.

    .. note::
       The cosine is an entire function on the complex plane and has no branch cuts.

    .. note::
       For complex arguments, the mathematical definition of cosine is

       .. math::
          \begin{align} \operatorname{cos}(x) &= \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!} x^{2n} \\ &= \frac{e^{jx} + e^{-jx}}{2} \\ &= \operatorname{cosh}(jx) \end{align}

       where :math:`\operatorname{cosh}` is the hyperbolic cosine.

    Parameters
    ----------
    x: array
        input array whose elements are each expressed in radians. Should have a floating-point data type.

    Returns
    -------
    out: array
        an array containing the cosine of each element in ``x``. The returned array must have a floating-point data type determined by :ref:`type-promotion`.

    Notes
    -----

    **Special cases**

    For real-valued floating-point operands,

    - If ``x_i`` is ``NaN``, the result is ``NaN``.
    - If ``x_i`` is ``+0``, the result is ``1``.
    - If ``x_i`` is ``-0``, the result is ``1``.
    - If ``x_i`` is ``+infinity``, the result is ``NaN``.
    - If ``x_i`` is ``-infinity``, the result is ``NaN``.

    For complex floating-point operands, special cases must be handled as if the operation is implemented as ``cosh(x*1j)``.

    .. versionchanged:: 2022.12
       Added complex data type support.
    """


def cosh(x: array, /) -> array:
    r"""
    Calculates an implementation-dependent approximation to the hyperbolic cosine for each element ``x_i`` in the input array ``x``.

    The mathematical definition of the hyperbolic cosine is

    .. math::
       \operatorname{cosh}(x) = \frac{e^x + e^{-x}}{2}

    .. note::
       The hyperbolic cosine is an entire function in the complex plane and has no branch cuts. The function is periodic, with period :math:`2\pi j`, with respect to the imaginary component.

    Parameters
    ----------
    x: array
        input array whose elements each represent a hyperbolic angle. Should have a floating-point data type.

    Returns
    -------
    out: array
        an array containing the hyperbolic cosine of each element in ``x``. The returned array must have a floating-point data type determined by :ref:`type-promotion`.

    Notes
    -----

    **Special cases**

    .. note::
       For all operands, ``cosh(x)`` must equal ``cosh(-x)``.

    For real-valued floating-point operands,

    - If ``x_i`` is ``NaN``, the result is ``NaN``.
    - If ``x_i`` is ``+0``, the result is ``1``.
    - If ``x_i`` is ``-0``, the result is ``1``.
    - If ``x_i`` is ``+infinity``, the result is ``+infinity``.
    - If ``x_i`` is ``-infinity``, the result is ``+infinity``.

    For complex floating-point operands, let ``a = real(x_i)``, ``b = imag(x_i)``, and

    .. note::
       For complex floating-point operands, ``cosh(conj(x))`` must equal ``conj(cosh(x))``.

    - If ``a`` is ``+0`` and ``b`` is ``+0``, the result is ``1 + 0j``.
    - If ``a`` is ``+0`` and ``b`` is ``+infinity``, the result is ``NaN + 0j`` (sign of the imaginary component is unspecified).
    - If ``a`` is ``+0`` and ``b`` is ``NaN``, the result is ``NaN + 0j`` (sign of the imaginary component is unspecified).
    - If ``a`` is a nonzero finite number and ``b`` is ``+infinity``, the result is ``NaN + NaN j``.
    - If ``a`` is a nonzero finite number and ``b`` is ``NaN``, the result is ``NaN + NaN j``.
    - If ``a`` is ``+infinity`` and ``b`` is ``+0``, the result is ``+infinity + 0j``.
    - If ``a`` is ``+infinity`` and ``b`` is a nonzero finite number, the result is ``+infinity * cis(b)``.
    - If ``a`` is ``+infinity`` and ``b`` is ``+infinity``, the result is ``+infinity + NaN j`` (sign of the real component is unspecified).
    - If ``a`` is ``+infinity`` and ``b`` is ``NaN``, the result is ``+infinity + NaN j``.
    - If ``a`` is ``NaN`` and ``b`` is either ``+0`` or ``-0``, the result is ``NaN + 0j`` (sign of the imaginary component is unspecified).
    - If ``a`` is ``NaN`` and ``b`` is a nonzero finite number, the result is ``NaN + NaN j``.
    - If ``a`` is ``NaN`` and ``b`` is ``NaN``, the result is ``NaN + NaN j``.

    where ``cis(v)`` is ``cos(v) + sin(v)*1j``.

    .. versionchanged:: 2022.12
       Added complex data type support.
    """


def divide(
    x1: Union[array, int, float, complex], x2: Union[array, int, float, complex], /
) -> array:
    r"""
    Calculates the division of each element ``x1_i`` of the input array ``x1`` with the respective element ``x2_i`` of the input array ``x2``.

    Parameters
    ----------
    x1: Union[array, int, float, complex]
        dividend input array. Should have a numeric data type.
    x2: Union[array, int, float, complex]
        divisor input array. Must be compatible with ``x1`` (see :ref:`broadcasting`). Should have a numeric data type.

    Returns
    -------
    out: array
        an array containing the element-wise results. The returned array must have a floating-point data type determined by :ref:`type-promotion`.

    Notes
    -----

    -   At least one of ``x1`` or ``x2`` must be an array.

    -   If one or both of the input arrays have integer data types, the result is implementation-dependent, as type promotion between data type "kinds" (e.g., integer versus floating-point) is unspecified.

        Specification-compliant libraries may choose to raise an error or return an array containing the element-wise results. If an array is returned, the array must have a real-valued floating-point data type.

    **Special cases**

    For real-valued floating-point operands,

    - If either ``x1_i`` or ``x2_i`` is ``NaN``, the result is ``NaN``.
    - If ``x1_i`` is either ``+infinity`` or ``-infinity`` and ``x2_i`` is either ``+infinity`` or ``-infinity``, the result is ``NaN``.
    - If ``x1_i`` is either ``+0`` or ``-0`` and ``x2_i`` is either ``+0`` or ``-0``, the result is ``NaN``.
    - If ``x1_i`` is ``+0`` and ``x2_i`` is greater than ``0``, the result is ``+0``.
    - If ``x1_i`` is ``-0`` and ``x2_i`` is greater than ``0``, the result is ``-0``.
    - If ``x1_i`` is ``+0`` and ``x2_i`` is less than ``0``, the result is ``-0``.
    - If ``x1_i`` is ``-0`` and ``x2_i`` is less than ``0``, the result is ``+0``.
    - If ``x1_i`` is greater than ``0`` and ``x2_i`` is ``+0``, the result is ``+infinity``.
    - If ``x1_i`` is greater than ``0`` and ``x2_i`` is ``-0``, the result is ``-infinity``.
    - If ``x1_i`` is less than ``0`` and ``x2_i`` is ``+0``, the result is ``-infinity``.
    - If ``x1_i`` is less than ``0`` and ``x2_i`` is ``-0``, the result is ``+infinity``.
    - If ``x1_i`` is ``+infinity`` and ``x2_i`` is a positive (i.e., greater than ``0``) finite number, the result is ``+infinity``.
    - If ``x1_i`` is ``+infinity`` and ``x2_i`` is a negative (i.e., less than ``0``) finite number, the result is ``-infinity``.
    - If ``x1_i`` is ``-infinity`` and ``x2_i`` is a positive (i.e., greater than ``0``) finite number, the result is ``-infinity``.
    - If ``x1_i`` is ``-infinity`` and ``x2_i`` is a negative (i.e., less than ``0``) finite number, the result is ``+infinity``.
    - If ``x1_i`` is a positive (i.e., greater than ``0``) finite number and ``x2_i`` is ``+infinity``, the result is ``+0``.
    - If ``x1_i`` is a positive (i.e., greater than ``0``) finite number and ``x2_i`` is ``-infinity``, the result is ``-0``.
    - If ``x1_i`` is a negative (i.e., less than ``0``) finite number and ``x2_i`` is ``+infinity``, the result is ``-0``.
    - If ``x1_i`` is a negative (i.e., less than ``0``) finite number and ``x2_i`` is ``-infinity``, the result is ``+0``.
    - If ``x1_i`` and ``x2_i`` have the same mathematical sign and are both nonzero finite numbers, the result has a positive mathematical sign.
    - If ``x1_i`` and ``x2_i`` have different mathematical signs and are both nonzero finite numbers, the result has a negative mathematical sign.
    - In the remaining cases, where neither ``-infinity``, ``+0``, ``-0``, nor ``NaN`` is involved, the quotient must be computed and rounded to the nearest representable value according to IEEE 754-2019 and a supported rounding mode. If the magnitude is too large to represent, the operation overflows and the result is an ``infinity`` of appropriate mathematical sign. If the magnitude is too small to represent, the operation underflows and the result is a zero of appropriate mathematical sign.

    For complex floating-point operands, division is defined according to the following table. For real components ``a`` and ``c`` and imaginary components ``b`` and ``d``,

    +------------+----------------+-----------------+--------------------------+
    |            | c              | dj              | c + dj                   |
    +============+================+=================+==========================+
    | **a**      | a / c          | -(a/d)j         | special rules            |
    +------------+----------------+-----------------+--------------------------+
    | **bj**     | (b/c)j         | b/d             | special rules            |
    +------------+----------------+-----------------+--------------------------+
    | **a + bj** | (a/c) + (b/c)j | b/d - (a/d)j    | special rules            |
    +------------+----------------+-----------------+--------------------------+

    In general, for complex floating-point operands, real-valued floating-point special cases must independently apply to the real and imaginary component operations involving real numbers as described in the above table.

    When ``a``, ``b``, ``c``, or ``d`` are all finite numbers (i.e., a value other than ``NaN``, ``+infinity``, or ``-infinity``), division of complex floating-point operands should be computed as if calculated according to the textbook formula for complex number division

    .. math::
       \frac{a + bj}{c + dj} = \frac{(ac + bd) + (bc - ad)j}{c^2 + d^2}

    When at least one of ``a``, ``b``, ``c``, or ``d`` is ``NaN``, ``+infinity``, or ``-infinity``,

    - If ``a``, ``b``, ``c``, and ``d`` are all ``NaN``, the result is ``NaN + NaN j``.
    - In the remaining cases, the result is implementation dependent.

    .. note::
       For complex floating-point operands, the results of special cases may be implementation dependent depending on how an implementation chooses to model complex numbers and complex infinity (e.g., complex plane versus Riemann sphere). For those implementations following C99 and its one-infinity model, when at least one component is infinite, even if the other component is ``NaN``, the complex value is infinite, and the usual arithmetic rules do not apply to complex-complex division. In the interest of performance, other implementations may want to avoid the complex branching logic necessary to implement the one-infinity model and choose to implement all complex-complex division according to the textbook formula. Accordingly, special case behavior is unlikely to be consistent across implementations.

    .. versionchanged:: 2022.12
       Added complex data type support.

    .. versionchanged:: 2024.12
       Added scalar argument support.
    """


def equal(
    x1: Union[array, int, float, complex, bool],
    x2: Union[array, int, float, complex, bool],
    /,
) -> array:
    r"""
    Computes the truth value of ``x1_i == x2_i`` for each element ``x1_i`` of the input array ``x1`` with the respective element ``x2_i`` of the input array ``x2``.

    Parameters
    ----------
    x1: Union[array, int, float, complex, bool]
        first input array. May have any data type.
    x2: Union[array, int, float, complex, bool]
        second input array. Must be compatible with ``x1`` (see :ref:`broadcasting`). May have any data type.

    Returns
    -------
    out: array
        an array containing the element-wise results. The returned array must have a data type of ``bool``.

    Notes
    -----

    -   At least one of ``x1`` or ``x2`` must be an array.

    **Special Cases**

    For real-valued floating-point operands,

    - If ``x1_i`` is ``NaN`` or ``x2_i`` is ``NaN``, the result is ``False``.
    - If ``x1_i`` is ``+infinity`` and ``x2_i`` is ``+infinity``, the result is ``True``.
    - If ``x1_i`` is ``-infinity`` and ``x2_i`` is ``-infinity``, the result is ``True``.
    - If ``x1_i`` is ``-0`` and ``x2_i`` is either ``+0`` or ``-0``, the result is ``True``.
    - If ``x1_i`` is ``+0`` and ``x2_i`` is either ``+0`` or ``-0``, the result is ``True``.
    - If ``x1_i`` is a finite number, ``x2_i`` is a finite number, and ``x1_i`` equals ``x2_i``, the result is ``True``.
    - In the remaining cases, the result is ``False``.

    For complex floating-point operands, let ``a = real(x1_i)``, ``b = imag(x1_i)``, ``c = real(x2_i)``, ``d = imag(x2_i)``, and

    - If ``a``, ``b``, ``c``, or ``d`` is ``NaN``, the result is ``False``.
    - In the remaining cases, the result is the logical AND of the equality comparison between the real values ``a`` and ``c`` (real components) and between the real values ``b`` and ``d`` (imaginary components), as described above for real-valued floating-point operands (i.e., ``a == c AND b == d``).

    .. note::
       For discussion of complex number equality, see :ref:`complex-numbers`.

    .. note::
       Comparison of arrays without a corresponding promotable data type (see :ref:`type-promotion`) is undefined and thus implementation-dependent.

    .. versionchanged:: 2022.12
       Added complex data type support.

    .. versionchanged:: 2024.12
       Cross-kind comparisons are explicitly left unspecified.

    .. versionchanged:: 2024.12
       Added scalar argument support.
    """


def exp(x: array, /) -> array:
    """
    Calculates an implementation-dependent approximation to the exponential function for each element ``x_i`` of the input array ``x`` (``e`` raised to the power of ``x_i``, where ``e`` is the base of the natural logarithm).

    .. note::
       For complex floating-point operands, ``exp(conj(x))`` must equal ``conj(exp(x))``.

    .. note::
       The exponential function is an entire function in the complex plane and has no branch cuts.

    Parameters
    ----------
    x: array
        input array. Should have a floating-point data type.

    Returns
    -------
    out: array
        an array containing the evaluated exponential function result for each element in ``x``. The returned array must have a floating-point data type determined by :ref:`type-promotion`.

    Notes
    -----

    **Special cases**

    For real-valued floating-point operands,

    - If ``x_i`` is ``NaN``, the result is ``NaN``.
    - If ``x_i`` is ``+0``, the result is ``1``.
    - If ``x_i`` is ``-0``, the result is ``1``.
    - If ``x_i`` is ``+infinity``, the result is ``+infinity``.
    - If ``x_i`` is ``-infinity``, the result is ``+0``.

    For complex floating-point operands, let ``a = real(x_i)``, ``b = imag(x_i)``, and

    - If ``a`` is either ``+0`` or ``-0`` and ``b`` is ``+0``, the result is ``1 + 0j``.
    - If ``a`` is a finite number and ``b`` is ``+infinity``, the result is ``NaN + NaN j``.
    - If ``a`` is a finite number and ``b`` is ``NaN``, the result is ``NaN + NaN j``.
    - If ``a`` is ``+infinity`` and ``b`` is ``+0``, the result is ``infinity + 0j``.
    - If ``a`` is ``-infinity`` and ``b`` is a finite number, the result is ``+0 * cis(b)``.
    - If ``a`` is ``+infinity`` and ``b`` is a nonzero finite number, the result is ``+infinity * cis(b)``.
    - If ``a`` is ``-infinity`` and ``b`` is ``+infinity``, the result is ``0 + 0j`` (signs of real and imaginary components are unspecified).
    - If ``a`` is ``+infinity`` and ``b`` is ``+infinity``, the result is ``infinity + NaN j`` (sign of real component is unspecified).
    - If ``a`` is ``-infinity`` and ``b`` is ``NaN``, the result is ``0 + 0j`` (signs of real and imaginary components are unspecified).
    - If ``a`` is ``+infinity`` and ``b`` is ``NaN``, the result is ``infinity + NaN j`` (sign of real component is unspecified).
    - If ``a`` is ``NaN`` and ``b`` is ``+0``, the result is ``NaN + 0j``.
    - If ``a`` is ``NaN`` and ``b`` is not equal to ``0``, the result is ``NaN + NaN j``.
    - If ``a`` is ``NaN`` and ``b`` is ``NaN``, the result is ``NaN + NaN j``.

    where ``cis(v)`` is ``cos(v) + sin(v)*1j``.

    .. versionchanged:: 2022.12
       Added complex data type support.
    """


def expm1(x: array, /) -> array:
    """
    Calculates an implementation-dependent approximation to ``exp(x)-1`` for each element ``x_i`` of the input array ``x``.

    .. note::
       The purpose of this function is to calculate ``exp(x)-1.0`` more accurately when `x` is close to zero. Accordingly, conforming implementations should avoid implementing this function as simply ``exp(x)-1.0``. See FDLIBM, or some other IEEE 754-2019 compliant mathematical library, for a potential reference implementation.

    .. note::
       For complex floating-point operands, ``expm1(conj(x))`` must equal ``conj(expm1(x))``.

    .. note::
       The exponential function is an entire function in the complex plane and has no branch cuts.

    Parameters
    ----------
    x: array
        input array. Should have a floating-point data type.

    Returns
    -------
    out: array
        an array containing the evaluated result for each element in ``x``. The returned array must have a floating-point data type determined by :ref:`type-promotion`.

    Notes
    -----

    **Special cases**

    For real-valued floating-point operands,

    - If ``x_i`` is ``NaN``, the result is ``NaN``.
    - If ``x_i`` is ``+0``, the result is ``+0``.
    - If ``x_i`` is ``-0``, the result is ``-0``.
    - If ``x_i`` is ``+infinity``, the result is ``+infinity``.
    - If ``x_i`` is ``-infinity``, the result is ``-1``.

    For complex floating-point operands, let ``a = real(x_i)``, ``b = imag(x_i)``, and

    - If ``a`` is either ``+0`` or ``-0`` and ``b`` is ``+0``, the result is ``0 + 0j``.
    - If ``a`` is a finite number and ``b`` is ``+infinity``, the result is ``NaN + NaN j``.
    - If ``a`` is a finite number and ``b`` is ``NaN``, the result is ``NaN + NaN j``.
    - If ``a`` is ``+infinity`` and ``b`` is ``+0``, the result is ``+infinity + 0j``.
    - If ``a`` is ``-infinity`` and ``b`` is a finite number, the result is ``+0 * cis(b) - 1.0``.
    - If ``a`` is ``+infinity`` and ``b`` is a nonzero finite number, the result is ``+infinity * cis(b) - 1.0``.
    - If ``a`` is ``-infinity`` and ``b`` is ``+infinity``, the result is ``-1 + 0j`` (sign of imaginary component is unspecified).
    - If ``a`` is ``+infinity`` and ``b`` is ``+infinity``, the result is ``infinity + NaN j`` (sign of real component is unspecified).
    - If ``a`` is ``-infinity`` and ``b`` is ``NaN``, the result is ``-1 + 0j`` (sign of imaginary component is unspecified).
    - If ``a`` is ``+infinity`` and ``b`` is ``NaN``, the result is ``infinity + NaN j`` (sign of real component is unspecified).
    - If ``a`` is ``NaN`` and ``b`` is ``+0``, the result is ``NaN + 0j``.
    - If ``a`` is ``NaN`` and ``b`` is not equal to ``0``, the result is ``NaN + NaN j``.
    - If ``a`` is ``NaN`` and ``b`` is ``NaN``, the result is ``NaN + NaN j``.

    where ``cis(v)`` is ``cos(v) + sin(v)*1j``.

    .. versionchanged:: 2022.12
       Added complex data type support.
    """


def floor(x: array, /) -> array:
    """
    Rounds each element ``x_i`` of the input array ``x`` to the greatest (i.e., closest to ``+infinity``) integer-valued number that is not greater than ``x_i``.

    Parameters
    ----------
    x: array
        input array. Should have a real-valued data type.

    Returns
    -------
    out: array
        an array containing the rounded result for each element in ``x``. The returned array must have the same data type as ``x``.

    Notes
    -----

    **Special cases**

    - If ``x_i`` is already integer-valued, the result is ``x_i``.

    For floating-point operands,

    - If ``x_i`` is ``+infinity``, the result is ``+infinity``.
    - If ``x_i`` is ``-infinity``, the result is ``-infinity``.
    - If ``x_i`` is ``+0``, the result is ``+0``.
    - If ``x_i`` is ``-0``, the result is ``-0``.
    - If ``x_i`` is ``NaN``, the result is ``NaN``.
    """


def floor_divide(
    x1: Union[array, int, float], x2: Union[array, int, float], /
) -> array:
    r"""
    Rounds the result of dividing each element ``x1_i`` of the input array ``x1`` by the respective element ``x2_i`` of the input array ``x2`` to the greatest (i.e., closest to `+infinity`) integer-value number that is not greater than the division result.

    Parameters
    ----------
    x1: Union[array, int, float]
        dividend input array. Should have a real-valued data type.
    x2: Union[array, int, float]
        divisor input array. Must be compatible with ``x1`` (see :ref:`broadcasting`). Should have a real-valued data type.

    Returns
    -------
    out: array
        an array containing the element-wise results. The returned array must have a data type determined by :ref:`type-promotion`.

    Notes
    -----

    -   At least one of ``x1`` or ``x2`` must be an array.
    -   For input arrays which promote to an integer data type, the result of division by zero is unspecified and thus implementation-defined.

    **Special cases**

    .. note::
       Floor division was introduced in Python via `PEP 238 <https://www.python.org/dev/peps/pep-0238/>`_ with the goal to disambiguate "true division" (i.e., computing an approximation to the mathematical operation of division) from "floor division" (i.e., rounding the result of division toward negative infinity). The former was computed when one of the operands was a ``float``, while the latter was computed when both operands were ``int``\s. Overloading the ``/`` operator to support both behaviors led to subtle numerical bugs when integers are possible, but not expected.

       To resolve this ambiguity, ``/`` was designated for true division, and ``//`` was designated for floor division. Semantically, floor division was `defined <https://www.python.org/dev/peps/pep-0238/#semantics-of-floor-division>`_ as equivalent to ``a // b == floor(a/b)``; however, special floating-point cases were left ill-defined.

       Accordingly, floor division is not implemented consistently across array libraries for some of the special cases documented below. Namely, when one of the operands is ``infinity``, libraries may diverge with some choosing to strictly follow ``floor(a/b)`` and others choosing to pair ``//`` with ``%`` according to the relation ``b = a % b + b * (a // b)``. The special cases leading to divergent behavior are documented below.

       This specification prefers floor division to match ``floor(divide(x1, x2))`` in order to avoid surprising and unexpected results; however, array libraries may choose to more strictly follow Python behavior.

    For floating-point operands,

    - If either ``x1_i`` or ``x2_i`` is ``NaN``, the result is ``NaN``.
    - If ``x1_i`` is either ``+infinity`` or ``-infinity`` and ``x2_i`` is either ``+infinity`` or ``-infinity``, the result is ``NaN``.
    - If ``x1_i`` is either ``+0`` or ``-0`` and ``x2_i`` is either ``+0`` or ``-0``, the result is ``NaN``.
    - If ``x1_i`` is ``+0`` and ``x2_i`` is greater than ``0``, the result is ``+0``.
    - If ``x1_i`` is ``-0`` and ``x2_i`` is greater than ``0``, the result is ``-0``.
    - If ``x1_i`` is ``+0`` and ``x2_i`` is less than ``0``, the result is ``-0``.
    - If ``x1_i`` is ``-0`` and ``x2_i`` is less than ``0``, the result is ``+0``.
    - If ``x1_i`` is greater than ``0`` and ``x2_i`` is ``+0``, the result is ``+infinity``.
    - If ``x1_i`` is greater than ``0`` and ``x2_i`` is ``-0``, the result is ``-infinity``.
    - If ``x1_i`` is less than ``0`` and ``x2_i`` is ``+0``, the result is ``-infinity``.
    - If ``x1_i`` is less than ``0`` and ``x2_i`` is ``-0``, the result is ``+infinity``.
    - If ``x1_i`` is ``+infinity`` and ``x2_i`` is a positive (i.e., greater than ``0``) finite number, the result is ``+infinity``. (**note**: libraries may return ``NaN`` to match Python behavior.)
    - If ``x1_i`` is ``+infinity`` and ``x2_i`` is a negative (i.e., less than ``0``) finite number, the result is ``-infinity``. (**note**: libraries may return ``NaN`` to match Python behavior.)
    - If ``x1_i`` is ``-infinity`` and ``x2_i`` is a positive (i.e., greater than ``0``) finite number, the result is ``-infinity``. (**note**: libraries may return ``NaN`` to match Python behavior.)
    - If ``x1_i`` is ``-infinity`` and ``x2_i`` is a negative (i.e., less than ``0``) finite number, the result is ``+infinity``. (**note**: libraries may return ``NaN`` to match Python behavior.)
    - If ``x1_i`` is a positive (i.e., greater than ``0``) finite number and ``x2_i`` is ``+infinity``, the result is ``+0``.
    - If ``x1_i`` is a positive (i.e., greater than ``0``) finite number and ``x2_i`` is ``-infinity``, the result is ``-0``. (**note**: libraries may return ``-1.0`` to match Python behavior.)
    - If ``x1_i`` is a negative (i.e., less than ``0``) finite number and ``x2_i`` is ``+infinity``, the result is ``-0``. (**note**: libraries may return ``-1.0`` to match Python behavior.)
    - If ``x1_i`` is a negative (i.e., less than ``0``) finite number and ``x2_i`` is ``-infinity``, the result is ``+0``.
    - If ``x1_i`` and ``x2_i`` have the same mathematical sign and are both nonzero finite numbers, the result has a positive mathematical sign.
    - If ``x1_i`` and ``x2_i`` have different mathematical signs and are both nonzero finite numbers, the result has a negative mathematical sign.
    - In the remaining cases, where neither ``-infinity``, ``+0``, ``-0``, nor ``NaN`` is involved, the quotient must be computed and rounded to the greatest (i.e., closest to `+infinity`) representable integer-value number that is not greater than the division result. If the magnitude is too large to represent, the operation overflows and the result is an ``infinity`` of appropriate mathematical sign. If the magnitude is too small to represent, the operation underflows and the result is a zero of appropriate mathematical sign.

    .. versionchanged:: 2024.12
       Added scalar argument support.
    """


def greater(x1: Union[array, int, float], x2: Union[array, int, float], /) -> array:
    """
    Computes the truth value of ``x1_i > x2_i`` for each element ``x1_i`` of the input array ``x1`` with the respective element ``x2_i`` of the input array ``x2``.

    Parameters
    ----------
    x1: Union[array, int, float]
        first input array. Should have a real-valued data type.
    x2: Union[array, int, float]
        second input array. Must be compatible with ``x1`` (see :ref:`broadcasting`). Should have a real-valued data type.

    Returns
    -------
    out: array
        an array containing the element-wise results. The returned array must have a data type of ``bool``.

    Notes
    -----

    -   At least one of ``x1`` or ``x2`` must be an array.
    -   Comparison of arrays without a corresponding promotable data type (see :ref:`type-promotion`) is undefined and thus implementation-dependent.
    -   For backward compatibility, conforming implementations may support complex numbers; however, inequality comparison of complex numbers is unspecified and thus implementation-dependent (see :ref:`complex-number-ordering`).

    .. versionchanged:: 2024.12
       Cross-kind comparisons are explicitly left unspecified.

    .. versionchanged:: 2024.12
       Added scalar argument support.
    """


def greater_equal(
    x1: Union[array, int, float], x2: Union[array, int, float], /
) -> array:
    """
    Computes the truth value of ``x1_i >= x2_i`` for each element ``x1_i`` of the input array ``x1`` with the respective element ``x2_i`` of the input array ``x2``.

    Parameters
    ----------
    x1: Union[array, int, float]
        first input array. Should have a real-valued data type.
    x2: Union[array, int, float]
        second input array. Must be compatible with ``x1`` (see :ref:`broadcasting`). Should have a real-valued data type.

    Returns
    -------
    out: array
        an array containing the element-wise results. The returned array must have a data type of ``bool``.

    Notes
    -----

    -   At least one of ``x1`` or ``x2`` must be an array.
    -   Comparison of arrays without a corresponding promotable data type (see :ref:`type-promotion`) is undefined and thus implementation-dependent.
    -   For backward compatibility, conforming implementations may support complex numbers; however, inequality comparison of complex numbers is unspecified and thus implementation-dependent (see :ref:`complex-number-ordering`).

    .. versionchanged:: 2024.12
       Cross-kind comparisons are explicitly left unspecified.

    .. versionchanged:: 2024.12
       Added scalar argument support.
    """


def hypot(x1: Union[array, int, float], x2: Union[array, int, float], /) -> array:
    r"""
    Computes the square root of the sum of squares for each element ``x1_i`` of the input array ``x1`` with the respective element ``x2_i`` of the input array ``x2``.

    .. note::
       The value computed by this function may be interpreted as the length of the hypotenuse of a right-angled triangle with sides of length ``x1_i`` and ``x2_i``, the distance of a point ``(x1_i, x2_i)`` from the origin ``(0, 0)``, or the magnitude of a complex number ``x1_i + x2_i * 1j``.

    Parameters
    ----------
    x1: Union[array, int, float]
       first input array. Should have a real-valued floating-point data type.
    x2: Union[array, int, float]
       second input array. Must be compatible with ``x1`` (see :ref:`broadcasting`). Should have a real-valued floating-point data type.

    Returns
    -------
    out: array
       an array containing the element-wise results. The returned array must have a real-valued floating-point data type determined by :ref:`type-promotion`.

    Notes
    -----

    -   At least one of ``x1`` or ``x2`` must be an array.
    -   The purpose of this function is to avoid underflow and overflow during intermediate stages of computation. Accordingly, conforming implementations should not use naive implementations.

    **Special Cases**

    For real-valued floating-point operands,

    - If ``x1_i`` is ``+infinity`` or ``-infinity`` and ``x2_i`` is any value, including ``NaN``, the result is ``+infinity``.
    - If ``x2_i`` is ``+infinity`` or ``-infinity`` and ``x1_i`` is any value, including ``NaN``, the result is ``+infinity``.
    - If ``x1_i`` is either ``+0`` or ``-0``, the result is equivalent to ``abs(x2_i)``.
    - If ``x2_i`` is either ``+0`` or ``-0``, the result is equivalent to ``abs(x1_i)``.
    - If ``x1_i`` is a finite number or ``NaN`` and ``x2_i`` is ``NaN``, the result is ``NaN``.
    - If ``x2_i`` is a finite number or ``NaN`` and ``x1_i`` is ``NaN``, the result is ``NaN``.
    - Underflow may only occur when both arguments are subnormal and the correct result is also subnormal.

    For real-valued floating-point operands, ``hypot(x1, x2)`` must equal ``hypot(x2, x1)``, ``hypot(x1, -x2)``, ``hypot(-x1, x2)``, and ``hypot(-x1, -x2)``.

    .. note::
       IEEE 754-2019 requires support for subnormal (a.k.a., denormal) numbers, which are useful for supporting gradual underflow. However, hardware support for subnormal numbers is not universal, and many platforms (e.g., accelerators) and compilers support toggling denormals-are-zero (DAZ) and/or flush-to-zero (FTZ) behavior to increase performance and to guard against timing attacks.

       Accordingly, conforming implementations may vary in their support for subnormal numbers.

    .. versionadded:: 2023.12

    .. versionchanged:: 2024.12
       Added scalar argument support.
    """


def imag(x: array, /) -> array:
    """
    Returns the imaginary component of a complex number for each element ``x_i`` of the input array ``x``.

    Parameters
    ----------
    x: array
        input array. Should have a complex floating-point data type.

    Returns
    -------
    out: array
        an array containing the element-wise results. The returned array must have a floating-point data type with the same floating-point precision as ``x`` (e.g., if ``x`` is ``complex64``, the returned array must have the floating-point data type ``float32``).

    Notes
    -----

    .. versionadded:: 2022.12
    """


def isfinite(x: array, /) -> array:
    """
    Tests each element ``x_i`` of the input array ``x`` to determine if finite.

    Parameters
    ----------
    x: array
        input array. Should have a numeric data type.

    Returns
    -------
    out: array
        an array containing test results. The returned array must have a data type of ``bool``.

    Notes
    -----

    **Special Cases**

    For real-valued floating-point operands,

    - If ``x_i`` is either ``+infinity`` or ``-infinity``, the result is ``False``.
    - If ``x_i`` is ``NaN``, the result is ``False``.
    - If ``x_i`` is a finite number, the result is ``True``.

    For complex floating-point operands, let ``a = real(x_i)``, ``b = imag(x_i)``, and

    - If ``a`` is ``NaN`` or ``b`` is ``NaN``, the result is ``False``.
    - If ``a`` is either ``+infinity`` or ``-infinity`` and ``b`` is any value, the result is ``False``.
    - If ``a`` is any value and ``b`` is either ``+infinity`` or ``-infinity``, the result is ``False``.
    - If ``a`` is a finite number and ``b`` is a finite number, the result is ``True``.

    .. versionchanged:: 2022.12
       Added complex data type support.
    """


def isinf(x: array, /) -> array:
    """
    Tests each element ``x_i`` of the input array ``x`` to determine if equal to positive or negative infinity.

    Parameters
    ----------
    x: array
        input array. Should have a numeric data type.

    Returns
    -------
    out: array
        an array containing test results. The returned array must have a data type of ``bool``.

    Notes
    -----

    **Special Cases**

    For real-valued floating-point operands,

    - If ``x_i`` is either ``+infinity`` or ``-infinity``, the result is ``True``.
    - In the remaining cases, the result is ``False``.

    For complex floating-point operands, let ``a = real(x_i)``, ``b = imag(x_i)``, and

    - If ``a`` is either ``+infinity`` or ``-infinity`` and ``b`` is any value (including ``NaN``), the result is ``True``.
    - If ``a`` is either a finite number or ``NaN`` and ``b`` is either ``+infinity`` or ``-infinity``, the result is ``True``.
    - In the remaining cases, the result is ``False``.

    .. versionchanged:: 2022.12
       Added complex data type support.
    """


def isnan(x: array, /) -> array:
    """
    Tests each element ``x_i`` of the input array ``x`` to determine whether the element is ``NaN``.

    Parameters
    ----------
    x: array
        input array. Should have a numeric data type.

    Returns
    -------
    out: array
        an array containing test results. The returned array should have a data type of ``bool``.

    Notes
    -----

    **Special Cases**

    For real-valued floating-point operands,

    - If ``x_i`` is ``NaN``, the result is ``True``.
    - In the remaining cases, the result is ``False``.

    For complex floating-point operands, let ``a = real(x_i)``, ``b = imag(x_i)``, and

    - If ``a`` or ``b`` is ``NaN``, the result is ``True``.
    - In the remaining cases, the result is ``False``.

    .. versionchanged:: 2022.12
       Added complex data type support.
    """


def less(x1: Union[array, int, float], x2: Union[array, int, float], /) -> array:
    """
    Computes the truth value of ``x1_i < x2_i`` for each element ``x1_i`` of the input array ``x1`` with the respective element ``x2_i`` of the input array ``x2``.

    Parameters
    ----------
    x1: Union[array, int, float]
        first input array. Should have a real-valued data type.
    x2: Union[array, int, float]
        second input array. Must be compatible with ``x1`` (see :ref:`broadcasting`). Should have a real-valued data type.

    Returns
    -------
    out: array
        an array containing the element-wise results. The returned array must have a data type of ``bool``.

    Notes
    -----

    -   At least one of ``x1`` or ``x2`` must be an array.
    -   Comparison of arrays without a corresponding promotable data type (see :ref:`type-promotion`) is undefined and thus implementation-dependent.
    -   For backward compatibility, conforming implementations may support complex numbers; however, inequality comparison of complex numbers is unspecified and thus implementation-dependent (see :ref:`complex-number-ordering`).

    .. versionchanged:: 2024.12
       Cross-kind comparisons are explicitly left unspecified.

    .. versionchanged:: 2024.12
       Added scalar argument support.
    """


def less_equal(x1: Union[array, int, float], x2: Union[array, int, float], /) -> array:
    """
    Computes the truth value of ``x1_i <= x2_i`` for each element ``x1_i`` of the input array ``x1`` with the respective element ``x2_i`` of the input array ``x2``.

    Parameters
    ----------
    x1: Union[array, int, float]
        first input array. Should have a real-valued data type.
    x2: Union[array, int, float]
        second input array. Must be compatible with ``x1`` (see :ref:`broadcasting`). Should have a real-valued data type.

    Returns
    -------
    out: array
        an array containing the element-wise results. The returned array must have a data type of ``bool``.

    Notes
    -----

    -   At least one of ``x1`` or ``x2`` must be an array.
    -   Comparison of arrays without a corresponding promotable data type (see :ref:`type-promotion`) is undefined and thus implementation-dependent.
    -   For backward compatibility, conforming implementations may support complex numbers; however, inequality comparison of complex numbers is unspecified and thus implementation-dependent (see :ref:`complex-number-ordering`).

    .. versionchanged:: 2024.12
       Cross-kind comparisons are explicitly left unspecified.

    .. versionchanged:: 2024.12
       Added scalar argument support.
    """


def log(x: array, /) -> array:
    r"""
    Calculates an implementation-dependent approximation to the natural (base ``e``) logarithm for each element ``x_i`` of the input array ``x``.

    .. note::
       The natural logarithm of a complex number :math:`z` with polar coordinates :math:`(r,\theta)` equals :math:`\ln r + (\theta + 2n\pi)j` with principal value :math:`\ln r + \theta j`.

    .. note::
       For complex floating-point operands, ``log(conj(x))`` must equal ``conj(log(x))``.

    .. note::
       By convention, the branch cut of the natural logarithm is the negative real axis :math:`(-\infty, 0)`.

       The natural logarithm is a continuous function from above the branch cut, taking into account the sign of the imaginary component.

       Accordingly, for complex arguments, the function returns the natural logarithm in the range of a strip in the interval :math:`[-\pi j, +\pi j]` along the imaginary axis and mathematically unbounded along the real axis.

       *Note: branch cuts follow C99 and have provisional status* (see :ref:`branch-cuts`).

    Parameters
    ----------
    x: array
        input array. Should have a floating-point data type.

    Returns
    -------
    out: array
        an array containing the evaluated natural logarithm for each element in ``x``. The returned array must have a floating-point data type determined by :ref:`type-promotion`.

    Notes
    -----

    **Special cases**

    For real-valued floating-point operands,

    - If ``x_i`` is ``NaN``, the result is ``NaN``.
    - If ``x_i`` is less than ``0``, the result is ``NaN``.
    - If ``x_i`` is either ``+0`` or ``-0``, the result is ``-infinity``.
    - If ``x_i`` is ``1``, the result is ``+0``.
    - If ``x_i`` is ``+infinity``, the result is ``+infinity``.

    For complex floating-point operands, let ``a = real(x_i)``, ``b = imag(x_i)``, and

    - If ``a`` is ``-0`` and ``b`` is ``+0``, the result is ``-infinity + πj``.
    - If ``a`` is ``+0`` and ``b`` is ``+0``, the result is ``-infinity + 0j``.
    - If ``a`` is a finite number and ``b`` is ``+infinity``, the result is ``+infinity + πj/2``.
    - If ``a`` is a finite number and ``b`` is ``NaN``, the result is ``NaN + NaN j``.
    - If ``a`` is ``-infinity`` and ``b`` is a positive (i.e., greater than ``0``) finite number, the result is ``+infinity + πj``.
    - If ``a`` is ``+infinity`` and ``b`` is a positive (i.e., greater than ``0``) finite number, the result is ``+infinity + 0j``.
    - If ``a`` is ``-infinity`` and ``b`` is ``+infinity``, the result is ``+infinity + 3πj/4``.
    - If ``a`` is ``+infinity`` and ``b`` is ``+infinity``, the result is ``+infinity + πj/4``.
    - If ``a`` is either ``+infinity`` or ``-infinity`` and ``b`` is ``NaN``, the result is ``+infinity + NaN j``.
    - If ``a`` is ``NaN`` and ``b`` is a finite number, the result is ``NaN + NaN j``.
    - If ``a`` is ``NaN`` and ``b`` is ``+infinity``, the result is ``+infinity + NaN j``.
    - If ``a`` is ``NaN`` and ``b`` is ``NaN``, the result is ``NaN + NaN j``.

    .. versionchanged:: 2022.12
       Added complex data type support.
    """


def log1p(x: array, /) -> array:
    r"""
    Calculates an implementation-dependent approximation to ``log(1+x)``, where ``log`` refers to the natural (base ``e``) logarithm, for each element ``x_i`` of the input array ``x``.

    .. note::
       The purpose of this function is to calculate ``log(1+x)`` more accurately when `x` is close to zero. Accordingly, conforming implementations should avoid implementing this function as simply ``log(1+x)``. See FDLIBM, or some other IEEE 754-2019 compliant mathematical library, for a potential reference implementation.

    .. note::
       For complex floating-point operands, ``log1p(conj(x))`` must equal ``conj(log1p(x))``.

    .. note::
       By convention, the branch cut of the natural logarithm is the negative real axis :math:`(-\infty, 0)`.

       The natural logarithm is a continuous function from above the branch cut, taking into account the sign of the imaginary component.

       Accordingly, for complex arguments, the function returns the natural logarithm in the range of a strip in the interval :math:`[-\pi j, +\pi j]` along the imaginary axis and mathematically unbounded along the real axis.

       *Note: branch cuts follow C99 and have provisional status* (see :ref:`branch-cuts`).

    Parameters
    ----------
    x: array
        input array. Should have a floating-point data type.

    Returns
    -------
    out: array
        an array containing the evaluated result for each element in ``x``. The returned array must have a floating-point data type determined by :ref:`type-promotion`.

    Notes
    -----

    **Special cases**

    For real-valued floating-point operands,

    - If ``x_i`` is ``NaN``, the result is ``NaN``.
    - If ``x_i`` is less than ``-1``, the result is ``NaN``.
    - If ``x_i`` is ``-1``, the result is ``-infinity``.
    - If ``x_i`` is ``-0``, the result is ``-0``.
    - If ``x_i`` is ``+0``, the result is ``+0``.
    - If ``x_i`` is ``+infinity``, the result is ``+infinity``.

    For complex floating-point operands, let ``a = real(x_i)``, ``b = imag(x_i)``, and

    - If ``a`` is ``-1`` and ``b`` is ``+0``, the result is ``-infinity + 0j``.
    - If ``a`` is a finite number and ``b`` is ``+infinity``, the result is ``+infinity + πj/2``.
    - If ``a`` is a finite number and ``b`` is ``NaN``, the result is ``NaN + NaN j``.
    - If ``a`` is ``-infinity`` and ``b`` is a positive (i.e., greater than ``0``) finite number, the result is ``+infinity + πj``.
    - If ``a`` is ``+infinity`` and ``b`` is a positive (i.e., greater than ``0``) finite number, the result is ``+infinity + 0j``.
    - If ``a`` is ``-infinity`` and ``b`` is ``+infinity``, the result is ``+infinity + 3πj/4``.
    - If ``a`` is ``+infinity`` and ``b`` is ``+infinity``, the result is ``+infinity + πj/4``.
    - If ``a`` is either ``+infinity`` or ``-infinity`` and ``b`` is ``NaN``, the result is ``+infinity + NaN j``.
    - If ``a`` is ``NaN`` and ``b`` is a finite number, the result is ``NaN + NaN j``.
    - If ``a`` is ``NaN`` and ``b`` is ``+infinity``, the result is ``+infinity + NaN j``.
    - If ``a`` is ``NaN`` and ``b`` is ``NaN``, the result is ``NaN + NaN j``.

    .. versionchanged:: 2022.12
       Added complex data type support.
    """


def log2(x: array, /) -> array:
    r"""
    Calculates an implementation-dependent approximation to the base ``2`` logarithm for each element ``x_i`` of the input array ``x``.

    .. note::
       For complex floating-point operands, ``log2(conj(x))`` must equal ``conj(log2(x))``.

    Parameters
    ----------
    x: array
        input array. Should have a floating-point data type.

    Returns
    -------
    out: array
        an array containing the evaluated base ``2`` logarithm for each element in ``x``. The returned array must have a floating-point data type determined by :ref:`type-promotion`.

    Notes
    -----

    **Special cases**

    For real-valued floating-point operands,

    - If ``x_i`` is ``NaN``, the result is ``NaN``.
    - If ``x_i`` is less than ``0``, the result is ``NaN``.
    - If ``x_i`` is either ``+0`` or ``-0``, the result is ``-infinity``.
    - If ``x_i`` is ``1``, the result is ``+0``.
    - If ``x_i`` is ``+infinity``, the result is ``+infinity``.

    For complex floating-point operands, special cases must be handled as if the operation is implemented using the standard change of base formula

    .. math::
       \log_{2} x = \frac{\log_{e} x}{\log_{e} 2}

    where :math:`\log_{e}` is the natural logarithm, as implemented by :func:`~array_api.log`.

    .. versionchanged:: 2022.12
       Added complex data type support.
    """


def log10(x: array, /) -> array:
    r"""
    Calculates an implementation-dependent approximation to the base ``10`` logarithm for each element ``x_i`` of the input array ``x``.

    .. note::
       For complex floating-point operands, ``log10(conj(x))`` must equal ``conj(log10(x))``.

    Parameters
    ----------
    x: array
        input array. Should have a floating-point data type.

    Returns
    -------
    out: array
        an array containing the evaluated base ``10`` logarithm for each element in ``x``. The returned array must have a floating-point data type determined by :ref:`type-promotion`.

    Notes
    -----

    **Special cases**

    For real-valued floating-point operands,

    - If ``x_i`` is ``NaN``, the result is ``NaN``.
    - If ``x_i`` is less than ``0``, the result is ``NaN``.
    - If ``x_i`` is either ``+0`` or ``-0``, the result is ``-infinity``.
    - If ``x_i`` is ``1``, the result is ``+0``.
    - If ``x_i`` is ``+infinity``, the result is ``+infinity``.

    For complex floating-point operands, special cases must be handled as if the operation is implemented using the standard change of base formula

    .. math::
       \log_{10} x = \frac{\log_{e} x}{\log_{e} 10}

    where :math:`\log_{e}` is the natural logarithm, as implemented by :func:`~array_api.log`.

    .. versionchanged:: 2022.12
       Added complex data type support.
    """


def logaddexp(x1: Union[array, int, float], x2: Union[array, int, float], /) -> array:
    """
    Calculates the logarithm of the sum of exponentiations ``log(exp(x1) + exp(x2))`` for each element ``x1_i`` of the input array ``x1`` with the respective element ``x2_i`` of the input array ``x2``.

    Parameters
    ----------
    x1: Union[array, int, float]
        first input array. Should have a real-valued floating-point data type.
    x2: Union[array, int, float]
        second input array. Must be compatible with ``x1`` (see :ref:`broadcasting`). Should have a real-valued floating-point data type.

    Returns
    -------
    out: array
        an array containing the element-wise results. The returned array must have a real-valued floating-point data type determined by :ref:`type-promotion`.

    Notes
    -----

    -   At least one of ``x1`` or ``x2`` must be an array.

    **Special cases**

    For floating-point operands,

    - If either ``x1_i`` or ``x2_i`` is ``NaN``, the result is ``NaN``.
    - If ``x1_i`` is ``+infinity`` and ``x2_i`` is not ``NaN``, the result is ``+infinity``.
    - If ``x1_i`` is not ``NaN`` and ``x2_i`` is ``+infinity``, the result is ``+infinity``.

    .. versionchanged:: 2024.12
       Added scalar argument support.
    """


def logical_and(x1: Union[array, bool], x2: Union[array, bool], /) -> array:
    """
    Computes the logical AND for each element ``x1_i`` of the input array ``x1`` with the respective element ``x2_i`` of the input array ``x2``.

    .. note::
       While this specification recommends that this function only accept input arrays having a boolean data type, specification-compliant array libraries may choose to accept input arrays having real-valued data types. If non-boolean data types are supported, zeros must be considered the equivalent of ``False``, while non-zeros must be considered the equivalent of ``True``.

    Parameters
    ----------
    x1: Union[array, bool]
        first input array. Should have a boolean data type.
    x2: Union[array, bool]
        second input array. Must be compatible with ``x1`` (see :ref:`broadcasting`). Should have a boolean data type.

    Returns
    -------
    out: array
        an array containing the element-wise results. The returned array must have a data type of `bool`.

    Notes
    -----

    -   At least one of ``x1`` or ``x2`` must be an array.

    .. versionchanged:: 2024.12
       Added scalar argument support.
    """


def logical_not(x: array, /) -> array:
    """
    Computes the logical NOT for each element ``x_i`` of the input array ``x``.

    .. note::
       While this specification recommends that this function only accept input arrays having a boolean data type, specification-compliant array libraries may choose to accept input arrays having real-valued data types. If non-boolean data types are supported, zeros must be considered the equivalent of ``False``, while non-zeros must be considered the equivalent of ``True``.

    Parameters
    ----------
    x: array
        input array. Should have a boolean data type.

    Returns
    -------
    out: array
        an array containing the element-wise results. The returned array must have a data type of ``bool``.
    """


def logical_or(x1: Union[array, bool], x2: Union[array, bool], /) -> array:
    """
    Computes the logical OR for each element ``x1_i`` of the input array ``x1`` with the respective element ``x2_i`` of the input array ``x2``.

    .. note::
       While this specification recommends that this function only accept input arrays having a boolean data type, specification-compliant array libraries may choose to accept input arrays having real-valued data types. If non-boolean data types are supported, zeros must be considered the equivalent of ``False``, while non-zeros must be considered the equivalent of ``True``.

    Parameters
    ----------
    x1: Union[array, bool]
        first input array. Should have a boolean data type.
    x2: Union[array, bool]
        second input array. Must be compatible with ``x1`` (see :ref:`broadcasting`). Should have a boolean data type.

    Returns
    -------
    out: array
        an array containing the element-wise results. The returned array must have a data type of ``bool``.

    Notes
    -----

    -   At least one of ``x1`` or ``x2`` must be an array.

    .. versionchanged:: 2024.12
       Added scalar argument support.
    """


def logical_xor(x1: Union[array, bool], x2: Union[array, bool], /) -> array:
    """
    Computes the logical XOR for each element ``x1_i`` of the input array ``x1`` with the respective element ``x2_i`` of the input array ``x2``.

    .. note::
       While this specification recommends that this function only accept input arrays having a boolean data type, specification-compliant array libraries may choose to accept input arrays having real-valued data types. If non-boolean data types are supported, zeros must be considered the equivalent of ``False``, while non-zeros must be considered the equivalent of ``True``.

    Parameters
    ----------
    x1: Union[array, bool]
        first input array. Should have a boolean data type.
    x2: Union[array, bool]
        second input array. Must be compatible with ``x1`` (see :ref:`broadcasting`). Should have a boolean data type.

    Returns
    -------
    out: array
        an array containing the element-wise results. The returned array must have a data type of ``bool``.

    Notes
    -----

    -   At least one of ``x1`` or ``x2`` must be an array.

    .. versionchanged:: 2024.12
       Added scalar argument support.
    """


def maximum(x1: Union[array, int, float], x2: Union[array, int, float], /) -> array:
    r"""
    Computes the maximum value for each element ``x1_i`` of the input array ``x1`` relative to the respective element ``x2_i`` of the input array ``x2``.

    Parameters
    ----------
    x1: Union[array, int, float]
       first input array. Should have a real-valued data type.
    x2: Union[array, int, float]
       second input array. Must be compatible with ``x1`` (see :ref:`broadcasting`). Should have a real-valued data type.

    Returns
    -------
    out: array
       an array containing the element-wise maximum values. The returned array must have a data type determined by :ref:`type-promotion`.

    Notes
    -----

    -   At least one of ``x1`` or ``x2`` must be an array.
    -   The order of signed zeros is unspecified and thus implementation-defined. When choosing between ``-0`` or ``+0`` as a maximum value, specification-compliant libraries may choose to return either value.
    -   For backward compatibility, conforming implementations may support complex numbers; however, inequality comparison of complex numbers is unspecified and thus implementation-defined (see :ref:`complex-number-ordering`).

    **Special Cases**

    For floating-point operands,

    -   If either ``x1_i`` or ``x2_i`` is ``NaN``, the result is ``NaN``.

    .. versionadded:: 2023.12

    .. versionchanged:: 2024.12
       Added scalar argument support.
    """


def minimum(x1: Union[array, int, float], x2: Union[array, int, float], /) -> array:
    r"""
    Computes the minimum value for each element ``x1_i`` of the input array ``x1`` relative to the respective element ``x2_i`` of the input array ``x2``.

    Parameters
    ----------
    x1: Union[array, int, float]
       first input array. Should have a real-valued data type.
    x2: Union[array, int, float]
       second input array. Must be compatible with ``x1`` (see :ref:`broadcasting`). Should have a real-valued data type.

    Returns
    -------
    out: array
       an array containing the element-wise minimum values. The returned array must have a data type determined by :ref:`type-promotion`.

    Notes
    -----

    -   At least one of ``x1`` or ``x2`` must be an array.
    -   The order of signed zeros is unspecified and thus implementation-defined. When choosing between ``-0`` or ``+0`` as a minimum value, specification-compliant libraries may choose to return either value.
    -   For backward compatibility, conforming implementations may support complex numbers; however, inequality comparison of complex numbers is unspecified and thus implementation-defined (see :ref:`complex-number-ordering`).

    **Special Cases**

    For floating-point operands,

    -   If either ``x1_i`` or ``x2_i`` is ``NaN``, the result is ``NaN``.

    .. versionadded:: 2023.12

    .. versionchanged:: 2024.12
       Added scalar argument support.
    """


def multiply(
    x1: Union[array, int, float, complex], x2: Union[array, int, float, complex], /
) -> array:
    r"""
    Calculates the product for each element ``x1_i`` of the input array ``x1`` with the respective element ``x2_i`` of the input array ``x2``.

    .. note::
       Floating-point multiplication is not always associative due to finite precision.

    Parameters
    ----------
    x1: Union[array, int, float, complex]
        first input array. Should have a numeric data type.
    x2: Union[array, int, float, complex]
        second input array. Must be compatible with ``x1`` (see :ref:`broadcasting`). Should have a numeric data type.

    Returns
    -------
    out: array
        an array containing the element-wise products. The returned array must have a data type determined by :ref:`type-promotion`.

    Notes
    -----

    -   At least one of ``x1`` or ``x2`` must be an array.

    **Special cases**

    For real-valued floating-point operands,

    - If either ``x1_i`` or ``x2_i`` is ``NaN``, the result is ``NaN``.
    - If ``x1_i`` is either ``+infinity`` or ``-infinity`` and ``x2_i`` is either ``+0`` or ``-0``, the result is ``NaN``.
    - If ``x1_i`` is either ``+0`` or ``-0`` and ``x2_i`` is either ``+infinity`` or ``-infinity``, the result is ``NaN``.
    - If ``x1_i`` and ``x2_i`` have the same mathematical sign, the result has a positive mathematical sign, unless the result is ``NaN``. If the result is ``NaN``, the "sign" of ``NaN`` is implementation-defined.
    - If ``x1_i`` and ``x2_i`` have different mathematical signs, the result has a negative mathematical sign, unless the result is ``NaN``. If the result is ``NaN``, the "sign" of ``NaN`` is implementation-defined.
    - If ``x1_i`` is either ``+infinity`` or ``-infinity`` and ``x2_i`` is either ``+infinity`` or ``-infinity``, the result is a signed infinity with the mathematical sign determined by the rule already stated above.
    - If ``x1_i`` is either ``+infinity`` or ``-infinity`` and ``x2_i`` is a nonzero finite number, the result is a signed infinity with the mathematical sign determined by the rule already stated above.
    - If ``x1_i`` is a nonzero finite number and ``x2_i`` is either ``+infinity`` or ``-infinity``, the result is a signed infinity with the mathematical sign determined by the rule already stated above.
    - In the remaining cases, where neither ``infinity`` nor ``NaN`` is involved, the product must be computed and rounded to the nearest representable value according to IEEE 754-2019 and a supported rounding mode. If the magnitude is too large to represent, the result is an `infinity` of appropriate mathematical sign. If the magnitude is too small to represent, the result is a zero of appropriate mathematical sign.

    For complex floating-point operands, multiplication is defined according to the following table. For real components ``a`` and ``c`` and imaginary components ``b`` and ``d``,

    +------------+----------------+-----------------+--------------------------+
    |            | c              | dj              | c + dj                   |
    +============+================+=================+==========================+
    | **a**      | a * c          | (a*d)j          | (a*c) + (a*d)j           |
    +------------+----------------+-----------------+--------------------------+
    | **bj**     | (b*c)j         | -(b*d)          | -(b*d) + (b*c)j          |
    +------------+----------------+-----------------+--------------------------+
    | **a + bj** | (a*c) + (b*c)j | -(b*d) + (a*d)j | special rules            |
    +------------+----------------+-----------------+--------------------------+

    In general, for complex floating-point operands, real-valued floating-point special cases must independently apply to the real and imaginary component operations involving real numbers as described in the above table.

    When ``a``, ``b``, ``c``, or ``d`` are all finite numbers (i.e., a value other than ``NaN``, ``+infinity``, or ``-infinity``), multiplication of complex floating-point operands should be computed as if calculated according to the textbook formula for complex number multiplication

    .. math::
       (a + bj) \cdot (c + dj) = (ac - bd) + (bc + ad)j

    When at least one of ``a``, ``b``, ``c``, or ``d`` is ``NaN``, ``+infinity``, or ``-infinity``,

    - If ``a``, ``b``, ``c``, and ``d`` are all ``NaN``, the result is ``NaN + NaN j``.
    - In the remaining cases, the result is implementation dependent.

    .. note::
       For complex floating-point operands, the results of special cases may be implementation dependent depending on how an implementation chooses to model complex numbers and complex infinity (e.g., complex plane versus Riemann sphere). For those implementations following C99 and its one-infinity model, when at least one component is infinite, even if the other component is ``NaN``, the complex value is infinite, and the usual arithmetic rules do not apply to complex-complex multiplication. In the interest of performance, other implementations may want to avoid the complex branching logic necessary to implement the one-infinity model and choose to implement all complex-complex multiplication according to the textbook formula. Accordingly, special case behavior is unlikely to be consistent across implementations.

    .. versionchanged:: 2022.12
       Added complex data type support.

    .. versionchanged:: 2024.12
       Added scalar argument support.
    """


def negative(x: array, /) -> array:
    """
    Computes the numerical negative of each element ``x_i`` (i.e., ``y_i = -x_i``) of the input array ``x``.

    .. note::
       For signed integer data types, the numerical negative of the minimum representable integer is implementation-dependent.

    .. note::
       If ``x`` has a complex floating-point data type, both the real and imaginary components for each ``x_i`` must be negated (a result which follows from the rules of complex number multiplication).

    Parameters
    ----------
    x: array
        input array. Should have a numeric data type.

    Returns
    -------
    out: array
        an array containing the evaluated result for each element in ``x``. The returned array must have a data type determined by :ref:`type-promotion`.

    Notes
    -----

    .. versionchanged:: 2022.12
       Added complex data type support.
    """


def nextafter(x1: Union[array, int, float], x2: Union[array, int, float], /) -> array:
    """
    Returns the next representable floating-point value for each element ``x1_i`` of the input array ``x1`` in the direction of the respective element ``x2_i`` of the input array ``x2``.

    Parameters
    ----------
    x1: Union[array, int, float]
        first input array. Should have a real-valued floating-point data type.
    x2: Union[array, int, float]
        second input array. Must be compatible with ``x1`` (see :ref:`broadcasting`). Should have the same data type as ``x1``.

    Returns
    -------
    out: array
        an array containing the element-wise results. The returned array must have the same data type as ``x1``.

    Notes
    -----

    -   At least one of ``x1`` or ``x2`` must be an array.

    **Special cases**

    For real-valued floating-point operands,

    - If either ``x1_i`` or ``x2_i`` is ``NaN``, the result is ``NaN``.
    - If ``x1_i`` is ``-0`` and ``x2_i`` is ``+0``, the result is ``+0``.
    - If ``x1_i`` is ``+0`` and ``x2_i`` is ``-0``, the result is ``-0``.

    .. versionadded:: 2024.12
    """


def not_equal(
    x1: Union[array, int, float, complex, bool],
    x2: Union[array, int, float, complex, bool],
    /,
) -> array:
    """
    Computes the truth value of ``x1_i != x2_i`` for each element ``x1_i`` of the input array ``x1`` with the respective element ``x2_i`` of the input array ``x2``.

    Parameters
    ----------
    x1: Union[array, int, float, complex, bool]
        first input array. May have any data type.
    x2: Union[array, int, float, complex, bool]
        second input array. Must be compatible with ``x1`` (see :ref:`broadcasting`).

    Returns
    -------
    out: array
        an array containing the element-wise results. The returned array must have a data type of ``bool``.

    Notes
    -----

    -   At least one of ``x1`` or ``x2`` must be an array.

    **Special Cases**

    For real-valued floating-point operands,

    - If ``x1_i`` is ``NaN`` or ``x2_i`` is ``NaN``, the result is ``True``.
    - If ``x1_i`` is ``+infinity`` and ``x2_i`` is ``-infinity``, the result is ``True``.
    - If ``x1_i`` is ``-infinity`` and ``x2_i`` is ``+infinity``, the result is ``True``.
    - If ``x1_i`` is a finite number, ``x2_i`` is a finite number, and ``x1_i`` does not equal ``x2_i``, the result is ``True``.
    - In the remaining cases, the result is ``False``.

    For complex floating-point operands, let ``a = real(x1_i)``, ``b = imag(x1_i)``, ``c = real(x2_i)``, ``d = imag(x2_i)``, and

    - If ``a``, ``b``, ``c``, or ``d`` is ``NaN``, the result is ``True``.
    - In the remaining cases, the result is the logical OR of the equality comparison between the real values ``a`` and ``c`` (real components) and between the real values ``b`` and ``d`` (imaginary components), as described above for real-valued floating-point operands (i.e., ``a != c OR b != d``).

    .. note::
       For discussion of complex number equality, see :ref:`complex-numbers`.

    .. note::
       Comparison of arrays without a corresponding promotable data type (see :ref:`type-promotion`) is undefined and thus implementation-dependent.

    .. versionchanged:: 2022.12
       Added complex data type support.

    .. versionchanged:: 2024.12
       Cross-kind comparisons are explicitly left unspecified.

    .. versionchanged:: 2024.12
       Added scalar argument support.
    """


def positive(x: array, /) -> array:
    """
    Computes the numerical positive of each element ``x_i`` (i.e., ``y_i = +x_i``) of the input array ``x``.

    Parameters
    ----------
    x: array
        input array. Should have a numeric data type.

    Returns
    -------
    out: array
        an array containing the evaluated result for each element in ``x``. The returned array must have the same data type as ``x``.

    Notes
    -----

    .. versionchanged:: 2022.12
       Added complex data type support.
    """


def pow(
    x1: Union[array, int, float, complex], x2: Union[array, int, float, complex], /
) -> array:
    r"""
    Calculates an implementation-dependent approximation of exponentiation by raising each element ``x1_i`` (the base) of the input array ``x1`` to the power of ``x2_i`` (the exponent), where ``x2_i`` is the corresponding element of the input array ``x2``.

    Parameters
    ----------
    x1: Union[array, int, float, complex]
        first input array whose elements correspond to the exponentiation base. Should have a numeric data type.
    x2: Union[array, int, float, complex]
        second input array whose elements correspond to the exponentiation exponent. Must be compatible with ``x1`` (see :ref:`broadcasting`). Should have a numeric data type.

    Returns
    -------
    out: array
        an array containing the element-wise results. The returned array must have a data type determined by :ref:`type-promotion`.

    Notes
    -----

    -   At least one of ``x1`` or ``x2`` must be an array.

    -   If both ``x1`` and ``x2`` have integer data types, the result of ``pow`` when ``x2_i`` is negative (i.e., less than zero) is unspecified and thus implementation-dependent.

    -   If ``x1`` has an integer data type and ``x2`` has a floating-point data type, behavior is implementation-dependent (type promotion between data type "kinds" (integer versus floating-point) is unspecified).

    -   By convention, the branch cut of the natural logarithm is the negative real axis :math:`(-\infty, 0)`.

        The natural logarithm is a continuous function from above the branch cut, taking into account the sign of the imaginary component. As special cases involving complex floating-point operands should be handled according to ``exp(x2*log(x1))``, exponentiation has the same branch cut for ``x1`` as the natural logarithm (see :func:`~array_api.log`).

        *Note: branch cuts follow C99 and have provisional status* (see :ref:`branch-cuts`).

    **Special cases**

    For real-valued floating-point operands,

    - If ``x1_i`` is not equal to ``1`` and ``x2_i`` is ``NaN``, the result is ``NaN``.
    - If ``x2_i`` is ``+0``, the result is ``1``, even if ``x1_i`` is ``NaN``.
    - If ``x2_i`` is ``-0``, the result is ``1``, even if ``x1_i`` is ``NaN``.
    - If ``x1_i`` is ``NaN`` and ``x2_i`` is not equal to ``0``, the result is ``NaN``.
    - If ``abs(x1_i)`` is greater than ``1`` and ``x2_i`` is ``+infinity``, the result is ``+infinity``.
    - If ``abs(x1_i)`` is greater than ``1`` and ``x2_i`` is ``-infinity``, the result is ``+0``.
    - If ``abs(x1_i)`` is ``1`` and ``x2_i`` is ``+infinity``, the result is ``1``.
    - If ``abs(x1_i)`` is ``1`` and ``x2_i`` is ``-infinity``, the result is ``1``.
    - If ``x1_i`` is ``1`` and ``x2_i`` is not ``NaN``, the result is ``1``.
    - If ``abs(x1_i)`` is less than ``1`` and ``x2_i`` is ``+infinity``, the result is ``+0``.
    - If ``abs(x1_i)`` is less than ``1`` and ``x2_i`` is ``-infinity``, the result is ``+infinity``.
    - If ``x1_i`` is ``+infinity`` and ``x2_i`` is greater than ``0``, the result is ``+infinity``.
    - If ``x1_i`` is ``+infinity`` and ``x2_i`` is less than ``0``, the result is ``+0``.
    - If ``x1_i`` is ``-infinity``, ``x2_i`` is greater than ``0``, and ``x2_i`` is an odd integer value, the result is ``-infinity``.
    - If ``x1_i`` is ``-infinity``, ``x2_i`` is greater than ``0``, and ``x2_i`` is not an odd integer value, the result is ``+infinity``.
    - If ``x1_i`` is ``-infinity``, ``x2_i`` is less than ``0``, and ``x2_i`` is an odd integer value, the result is ``-0``.
    - If ``x1_i`` is ``-infinity``, ``x2_i`` is less than ``0``, and ``x2_i`` is not an odd integer value, the result is ``+0``.
    - If ``x1_i`` is ``+0`` and ``x2_i`` is greater than ``0``, the result is ``+0``.
    - If ``x1_i`` is ``+0`` and ``x2_i`` is less than ``0``, the result is ``+infinity``.
    - If ``x1_i`` is ``-0``, ``x2_i`` is greater than ``0``, and ``x2_i`` is an odd integer value, the result is ``-0``.
    - If ``x1_i`` is ``-0``, ``x2_i`` is greater than ``0``, and ``x2_i`` is not an odd integer value, the result is ``+0``.
    - If ``x1_i`` is ``-0``, ``x2_i`` is less than ``0``, and ``x2_i`` is an odd integer value, the result is ``-infinity``.
    - If ``x1_i`` is ``-0``, ``x2_i`` is less than ``0``, and ``x2_i`` is not an odd integer value, the result is ``+infinity``.
    - If ``x1_i`` is less than ``0``, ``x1_i`` is a finite number, ``x2_i`` is a finite number, and ``x2_i`` is not an integer value, the result is ``NaN``.

    For complex floating-point operands, special cases should be handled as if the operation is implemented as ``exp(x2*log(x1))``.

    .. note::
       Conforming implementations are allowed to treat special cases involving complex floating-point operands more carefully than as described in this specification.

    .. versionchanged:: 2022.12
       Added complex data type support.

    .. versionchanged:: 2024.12
       Added scalar argument support.
    """


def real(x: array, /) -> array:
    """
    Returns the real component of a complex number for each element ``x_i`` of the input array ``x``.

    Parameters
    ----------
    x: array
        input array. Must have a numeric data type.

    Returns
    -------
    out: array
        an array containing the element-wise results. The returned array must have a floating-point data type with the same floating-point precision as ``x`` (e.g., if ``x`` is ``complex64``, the returned array must have the floating-point data type ``float32``).

    Notes
    -----

    -   Whether the returned array and the input array share the same underlying memory is unspecified and thus implementation-defined.

    .. versionadded:: 2022.12

    .. versionchanged:: 2024.12
       Added support for real-valued arrays.
    """


def reciprocal(x: array, /) -> array:
    """
    Returns the reciprocal for each element ``x_i`` of the input array ``x``.

    Parameters
    ----------
    x: array
        input array. Should have a floating-point data type.

    Returns
    -------
    out: array
        an array containing the element-wise results. The returned array must have a floating-point data type determined by :ref:`type-promotion`.

    Notes
    -----

    **Special cases**

    For floating-point operands, special cases must be handled as if the operation is implemented as ``1.0 / x`` (see :func:`~array_api.divide`).

    .. versionadded:: 2024.12
    """


def remainder(x1: Union[array, int, float], x2: Union[array, int, float], /) -> array:
    """
    Returns the remainder of division for each element ``x1_i`` of the input array ``x1`` and the respective element ``x2_i`` of the input array ``x2``.

    .. note::
       This function is equivalent to the Python modulus operator ``x1_i % x2_i``.

    Parameters
    ----------
    x1: Union[array, int, float]
        dividend input array. Should have a real-valued data type.
    x2: Union[array, int, float]
        divisor input array. Must be compatible with ``x1`` (see :ref:`broadcasting`). Should have a real-valued data type.

    Returns
    -------
    out: array
        an array containing the element-wise results. Each element-wise result must have the same sign as the respective element ``x2_i``. The returned array must have a data type determined by :ref:`type-promotion`.

    Notes
    -----

    -   At least one of ``x1`` or ``x2`` must be an array.
    -   For input arrays which promote to an integer data type, the result of division by zero is unspecified and thus implementation-defined.

    **Special cases**

    .. note::
       In general, similar to Python's ``%`` operator, this function is **not** recommended for floating-point operands as semantics do not follow IEEE 754. That this function is specified to accept floating-point operands is primarily for reasons of backward compatibility.

    For floating-point operands,

    - If either ``x1_i`` or ``x2_i`` is ``NaN``, the result is ``NaN``.
    - If ``x1_i`` is either ``+infinity`` or ``-infinity`` and ``x2_i`` is either ``+infinity`` or ``-infinity``, the result is ``NaN``.
    - If ``x1_i`` is either ``+0`` or ``-0`` and ``x2_i`` is either ``+0`` or ``-0``, the result is ``NaN``.
    - If ``x1_i`` is ``+0`` and ``x2_i`` is greater than ``0``, the result is ``+0``.
    - If ``x1_i`` is ``-0`` and ``x2_i`` is greater than ``0``, the result is ``+0``.
    - If ``x1_i`` is ``+0`` and ``x2_i`` is less than ``0``, the result is ``-0``.
    - If ``x1_i`` is ``-0`` and ``x2_i`` is less than ``0``, the result is ``-0``.
    - If ``x1_i`` is greater than ``0`` and ``x2_i`` is ``+0``, the result is ``NaN``.
    - If ``x1_i`` is greater than ``0`` and ``x2_i`` is ``-0``, the result is ``NaN``.
    - If ``x1_i`` is less than ``0`` and ``x2_i`` is ``+0``, the result is ``NaN``.
    - If ``x1_i`` is less than ``0`` and ``x2_i`` is ``-0``, the result is ``NaN``.
    - If ``x1_i`` is ``+infinity`` and ``x2_i`` is a positive (i.e., greater than ``0``) finite number, the result is ``NaN``.
    - If ``x1_i`` is ``+infinity`` and ``x2_i`` is a negative (i.e., less than ``0``) finite number, the result is ``NaN``.
    - If ``x1_i`` is ``-infinity`` and ``x2_i`` is a positive (i.e., greater than ``0``) finite number, the result is ``NaN``.
    - If ``x1_i`` is ``-infinity`` and ``x2_i`` is a negative (i.e., less than ``0``) finite number, the result is ``NaN``.
    - If ``x1_i`` is a positive (i.e., greater than ``0``) finite number and ``x2_i`` is ``+infinity``, the result is ``x1_i``. (**note**: this result matches Python behavior.)
    - If ``x1_i`` is a positive (i.e., greater than ``0``) finite number and ``x2_i`` is ``-infinity``, the result is ``x2_i``. (**note**: this result matches Python behavior.)
    - If ``x1_i`` is a negative (i.e., less than ``0``) finite number and ``x2_i`` is ``+infinity``, the result is ``x2_i``. (**note**: this results matches Python behavior.)
    - If ``x1_i`` is a negative (i.e., less than ``0``) finite number and ``x2_i`` is ``-infinity``, the result is ``x1_i``. (**note**: this result matches Python behavior.)
    - In the remaining cases, the result must match that of the Python ``%`` operator.

    .. versionchanged:: 2024.12
       Added scalar argument support.
    """


def round(x: array, /) -> array:
    """
    Rounds each element ``x_i`` of the input array ``x`` to the nearest integer-valued number.

    .. note::
       For complex floating-point operands, real and imaginary components must be independently rounded to the nearest integer-valued number.

       Rounded real and imaginary components must be equal to their equivalent rounded real-valued floating-point counterparts (i.e., for complex-valued ``x``, ``real(round(x))`` must equal ``round(real(x)))`` and ``imag(round(x))`` must equal ``round(imag(x))``).

    Parameters
    ----------
    x: array
        input array. Should have a numeric data type.

    Returns
    -------
    out: array
        an array containing the rounded result for each element in ``x``. The returned array must have the same data type as ``x``.

    Notes
    -----

    **Special cases**

    .. note::
       For complex floating-point operands, the following special cases apply to real and imaginary components independently (e.g., if ``real(x_i)`` is ``NaN``, the rounded real component is ``NaN``).

    - If ``x_i`` is already integer-valued, the result is ``x_i``.

    For floating-point operands,

    - If ``x_i`` is ``+infinity``, the result is ``+infinity``.
    - If ``x_i`` is ``-infinity``, the result is ``-infinity``.
    - If ``x_i`` is ``+0``, the result is ``+0``.
    - If ``x_i`` is ``-0``, the result is ``-0``.
    - If ``x_i`` is ``NaN``, the result is ``NaN``.
    - If two integers are equally close to ``x_i``, the result is the even integer closest to ``x_i``.

    .. versionchanged:: 2022.12
       Added complex data type support.
    """


def sign(x: array, /) -> array:
    r"""
    Returns an indication of the sign of a number for each element ``x_i`` of the input array ``x``.

    The sign function (also known as the **signum function**) of a number :math:`x_i` is defined as

    .. math::
       \operatorname{sign}(x_i) = \begin{cases}
       0 & \textrm{if } x_i = 0 \\
       \frac{x_i}{|x_i|} & \textrm{otherwise}
       \end{cases}

    where :math:`|x_i|` is the absolute value of :math:`x_i`.

    Parameters
    ----------
    x: array
        input array. Should have a numeric data type.

    Returns
    -------
    out: array
        an array containing the evaluated result for each element in ``x``. The returned array must have the same data type as ``x``.

    Notes
    -----

    **Special cases**

    For real-valued operands,

    - If ``x_i`` is less than ``0``, the result is ``-1``.
    - If ``x_i`` is either ``-0`` or ``+0``, the result is ``0``.
    - If ``x_i`` is greater than ``0``, the result is ``+1``.
    - If ``x_i`` is ``NaN``, the result is ``NaN``.

    For complex floating-point operands, let ``a = real(x_i)``, ``b = imag(x_i)``, and

    - If ``a`` is either ``-0`` or ``+0`` and ``b`` is either ``-0`` or ``+0``, the result is ``0 + 0j``.
    - If ``a`` is ``NaN`` or ``b`` is ``NaN``, the result is ``NaN + NaN j``.
    - In the remaining cases, special cases must be handled according to the rules of complex number division (see :func:`~array_api.divide`).

    .. versionchanged:: 2022.12
       Added complex data type support.
    """


def signbit(x: array, /) -> array:
    r"""
    Determines whether the sign bit is set for each element ``x_i`` of the input array ``x``.

    The sign bit of a real-valued floating-point number ``x_i`` is set whenever ``x_i`` is either ``-0``, less than zero, or a signed ``NaN`` (i.e., a ``NaN`` value whose sign bit is ``1``).

    Parameters
    ----------
    x: array
        input array. Should have a real-valued floating-point data type.

    Returns
    -------
    out: array
        an array containing the evaluated result for each element in ``x``. The returned array must have a data type of ``bool``.

    Notes
    -----

    **Special cases**

    For real-valued floating-point operands,

    - If ``x_i`` is ``+0``, the result is ``False``.
    - If ``x_i`` is ``-0``, the result is ``True``.
    - If ``x_i`` is ``+infinity``, the result is ``False``.
    - If ``x_i`` is ``-infinity``, the result is ``True``.
    - If ``x_i`` is a positive (i.e., greater than ``0``) finite number, the result is ``False``.
    - If ``x_i`` is a negative (i.e., less than ``0``) finite number, the result is ``True``.
    - If ``x_i`` is ``NaN`` and the sign bit of ``x_i`` is ``0``, the result is ``False``.
    - If ``x_i`` is ``NaN`` and the sign bit of ``x_i`` is ``1``, the result is ``True``.

    .. versionadded:: 2023.12
    """


def sin(x: array, /) -> array:
    r"""
    Calculates an implementation-dependent approximation to the sine for each element ``x_i`` of the input array ``x``.

    Each element ``x_i`` is assumed to be expressed in radians.

    .. note::
       The sine is an entire function on the complex plane and has no branch cuts.

    .. note::
       For complex arguments, the mathematical definition of sine is

       .. math::
          \begin{align} \operatorname{sin}(x) &= \frac{e^{jx} - e^{-jx}}{2j} \\ &= \frac{\operatorname{sinh}(jx)}{j} \\ &= \frac{\operatorname{sinh}(jx)}{j} \cdot \frac{j}{j} \\ &= -j \cdot \operatorname{sinh}(jx) \end{align}

       where :math:`\operatorname{sinh}` is the hyperbolic sine.

    Parameters
    ----------
    x: array
        input array whose elements are each expressed in radians. Should have a floating-point data type.

    Returns
    -------
    out: array
        an array containing the sine of each element in ``x``. The returned array must have a floating-point data type determined by :ref:`type-promotion`.

    Notes
    -----

    **Special cases**

    For real-valued floating-point operands,

    - If ``x_i`` is ``NaN``, the result is ``NaN``.
    - If ``x_i`` is ``+0``, the result is ``+0``.
    - If ``x_i`` is ``-0``, the result is ``-0``.
    - If ``x_i`` is either ``+infinity`` or ``-infinity``, the result is ``NaN``.

    For complex floating-point operands, special cases must be handled as if the operation is implemented as ``-1j * sinh(x*1j)``.

    .. versionchanged:: 2022.12
       Added complex data type support.
    """


def sinh(x: array, /) -> array:
    r"""
    Calculates an implementation-dependent approximation to the hyperbolic sine for each element ``x_i`` of the input array ``x``.

    The mathematical definition of the hyperbolic sine is

    .. math::
       \operatorname{sinh}(x) = \frac{e^x - e^{-x}}{2}

    .. note::
       The hyperbolic sine is an entire function in the complex plane and has no branch cuts. The function is periodic, with period :math:`2\pi j`, with respect to the imaginary component.

    Parameters
    ----------
    x: array
        input array whose elements each represent a hyperbolic angle. Should have a floating-point data type.

    Returns
    -------
    out: array
        an array containing the hyperbolic sine of each element in ``x``. The returned array must have a floating-point data type determined by :ref:`type-promotion`.

    Notes
    -----

    **Special cases**

    .. note::
       For all operands, ``sinh(x)`` must equal ``-sinh(-x)``.

    For real-valued floating-point operands,

    - If ``x_i`` is ``NaN``, the result is ``NaN``.
    - If ``x_i`` is ``+0``, the result is ``+0``.
    - If ``x_i`` is ``-0``, the result is ``-0``.
    - If ``x_i`` is ``+infinity``, the result is ``+infinity``.
    - If ``x_i`` is ``-infinity``, the result is ``-infinity``.

    For complex floating-point operands, let ``a = real(x_i)``, ``b = imag(x_i)``, and

    .. note::
       For complex floating-point operands, ``sinh(conj(x))`` must equal ``conj(sinh(x))``.

    - If ``a`` is ``+0`` and ``b`` is ``+0``, the result is ``+0 + 0j``.
    - If ``a`` is ``+0`` and ``b`` is ``+infinity``, the result is ``0 + NaN j`` (sign of the real component is unspecified).
    - If ``a`` is ``+0`` and ``b`` is ``NaN``, the result is ``0 + NaN j`` (sign of the real component is unspecified).
    - If ``a`` is a positive (i.e., greater than ``0``) finite number and ``b`` is ``+infinity``, the result is ``NaN + NaN j``.
    - If ``a`` is a positive (i.e., greater than ``0``) finite number and ``b`` is ``NaN``, the result is ``NaN + NaN j``.
    - If ``a`` is ``+infinity`` and ``b`` is ``+0``, the result is ``+infinity + 0j``.
    - If ``a`` is ``+infinity`` and ``b`` is a positive finite number, the result is ``+infinity * cis(b)``.
    - If ``a`` is ``+infinity`` and ``b`` is ``+infinity``, the result is ``infinity + NaN j`` (sign of the real component is unspecified).
    - If ``a`` is ``+infinity`` and ``b`` is ``NaN``, the result is ``infinity + NaN j`` (sign of the real component is unspecified).
    - If ``a`` is ``NaN`` and ``b`` is ``+0``, the result is ``NaN + 0j``.
    - If ``a`` is ``NaN`` and ``b`` is a nonzero finite number, the result is ``NaN + NaN j``.
    - If ``a`` is ``NaN`` and ``b`` is ``NaN``, the result is ``NaN + NaN j``.

    where ``cis(v)`` is ``cos(v) + sin(v)*1j``.

    .. versionchanged:: 2022.12
       Added complex data type support.
    """


def square(x: array, /) -> array:
    r"""
    Squares each element ``x_i`` of the input array ``x``.

    The square of a number ``x_i`` is defined as

    .. math::
       x_i^2 = x_i \cdot x_i

    Parameters
    ----------
    x: array
        input array. Should have a numeric data type.

    Returns
    -------
    out: array
        an array containing the evaluated result for each element in ``x``. The returned array must have a data type determined by :ref:`type-promotion`.

    Notes
    -----

    **Special cases**

    For floating-point operands, special cases must be handled as if the operation is implemented as ``x * x`` (see :func:`~array_api.multiply`).

    .. versionchanged:: 2022.12
       Added complex data type support.
    """


def sqrt(x: array, /) -> array:
    r"""
    Calculates the principal square root for each element ``x_i`` of the input array ``x``.

    .. note::
       After rounding, each result must be indistinguishable from the infinitely precise result (as required by IEEE 754).

    .. note::
       For complex floating-point operands, ``sqrt(conj(x))`` must equal ``conj(sqrt(x))``.

    .. note::
       By convention, the branch cut of the square root is the negative real axis :math:`(-\infty, 0)`.

       The square root is a continuous function from above the branch cut, taking into account the sign of the imaginary component.

       Accordingly, for complex arguments, the function returns the square root in the range of the right half-plane, including the imaginary axis (i.e., the plane defined by :math:`[0, +\infty)` along the real axis and :math:`(-\infty, +\infty)` along the imaginary axis).

       *Note: branch cuts follow C99 and have provisional status* (see :ref:`branch-cuts`).

    Parameters
    ----------
    x: array
        input array. Should have a floating-point data type.

    Returns
    -------
    out: array
        an array containing the square root of each element in ``x``. The returned array must have a floating-point data type determined by :ref:`type-promotion`.

    Notes
    -----

    **Special cases**

    For real-valued floating-point operands,

    - If ``x_i`` is ``NaN``, the result is ``NaN``.
    - If ``x_i`` is less than ``0``, the result is ``NaN``.
    - If ``x_i`` is ``+0``, the result is ``+0``.
    - If ``x_i`` is ``-0``, the result is ``-0``.
    - If ``x_i`` is ``+infinity``, the result is ``+infinity``.

    For complex floating-point operands, let ``a = real(x_i)``, ``b = imag(x_i)``, and

    - If ``a`` is either ``+0`` or ``-0`` and ``b`` is ``+0``, the result is ``+0 + 0j``.
    - If ``a`` is any value (including ``NaN``) and ``b`` is ``+infinity``, the result is ``+infinity + infinity j``.
    - If ``a`` is a finite number and ``b`` is ``NaN``, the result is ``NaN + NaN j``.
    - If ``a`` ``-infinity`` and ``b`` is a positive (i.e., greater than ``0``) finite number, the result is ``NaN + NaN j``.
    - If ``a`` is ``+infinity`` and ``b`` is a positive (i.e., greater than ``0``) finite number, the result is ``+0 + infinity j``.
    - If ``a`` is ``-infinity`` and ``b`` is ``NaN``, the result is ``NaN + infinity j`` (sign of the imaginary component is unspecified).
    - If ``a`` is ``+infinity`` and ``b`` is ``NaN``, the result is ``+infinity + NaN j``.
    - If ``a`` is ``NaN`` and ``b`` is any value, the result is ``NaN + NaN j``.
    - If ``a`` is ``NaN`` and ``b`` is ``NaN``, the result is ``NaN + NaN j``.

    .. versionchanged:: 2022.12
       Added complex data type support.
    """


def subtract(
    x1: Union[array, int, float, complex], x2: Union[array, int, float, complex], /
) -> array:
    """
    Calculates the difference for each element ``x1_i`` of the input array ``x1`` with the respective element ``x2_i`` of the input array ``x2``.

    Parameters
    ----------
    x1: Union[array, int, float, complex]
        first input array. Should have a numeric data type.
    x2: Union[array, int, float, complex]
        second input array. Must be compatible with ``x1`` (see :ref:`broadcasting`). Should have a numeric data type.

    Returns
    -------
    out: array
        an array containing the element-wise differences. The returned array must have a data type determined by :ref:`type-promotion`.

    Notes
    -----

    -   At least one of ``x1`` or ``x2`` must be an array.
    -   The result of ``x1_i - x2_i`` must be the same as ``x1_i + (-x2_i)`` and must be governed by the same floating-point rules as addition (see :meth:`add`).

    .. versionchanged:: 2022.12
       Added complex data type support.

    .. versionchanged:: 2024.12
       Added scalar argument support.
    """


def tan(x: array, /) -> array:
    r"""
    Calculates an implementation-dependent approximation to the tangent for each element ``x_i`` of the input array ``x``.

    Each element ``x_i`` is assumed to be expressed in radians.

    .. note::
       Tangent is an analytical function on the complex plane and has no branch cuts. The function is periodic, with period :math:`\pi j`, with respect to the real component and has first order poles along the real line at coordinates :math:`(\pi (\frac{1}{2} + n), 0)`. However, IEEE 754 binary floating-point representation cannot represent the value :math:`\pi / 2` exactly, and, thus, no argument value is possible for which a pole error occurs.

    .. note::
       For complex arguments, the mathematical definition of tangent is

       .. math::
          \begin{align} \operatorname{tan}(x) &= \frac{j(e^{-jx} - e^{jx})}{e^{-jx} + e^{jx}} \\ &= (-1) \frac{j(e^{jx} - e^{-jx})}{e^{jx} + e^{-jx}} \\ &= -j \cdot \operatorname{tanh}(jx) \end{align}

       where :math:`\operatorname{tanh}` is the hyperbolic tangent.

    Parameters
    ----------
    x: array
        input array whose elements are expressed in radians. Should have a floating-point data type.

    Returns
    -------
    out: array
        an array containing the tangent of each element in ``x``. The returned array must have a floating-point data type determined by :ref:`type-promotion`.

    Notes
    -----

    **Special cases**

    For real-valued floating-point operands,

    - If ``x_i`` is ``NaN``, the result is ``NaN``.
    - If ``x_i`` is ``+0``, the result is ``+0``.
    - If ``x_i`` is ``-0``, the result is ``-0``.
    - If ``x_i`` is either ``+infinity`` or ``-infinity``, the result is ``NaN``.

    For complex floating-point operands, special cases must be handled as if the operation is implemented as ``-1j * tanh(x*1j)``.

    .. versionchanged:: 2022.12
       Added complex data type support.
    """


def tanh(x: array, /) -> array:
    r"""
    Calculates an implementation-dependent approximation to the hyperbolic tangent for each element ``x_i`` of the input array ``x``.

    The mathematical definition of the hyperbolic tangent is

    .. math::
       \begin{align} \operatorname{tanh}(x) &= \frac{\operatorname{sinh}(x)}{\operatorname{cosh}(x)} \\ &= \frac{e^x - e^{-x}}{e^x + e^{-x}} \end{align}

    where :math:`\operatorname{sinh}(x)` is the hyperbolic sine and :math:`\operatorname{cosh}(x)` is the hyperbolic cosine.

    .. note::
       The hyperbolic tangent is an analytical function on the complex plane and has no branch cuts. The function is periodic, with period :math:`\pi j`, with respect to the imaginary component and has first order poles along the imaginary line at coordinates :math:`(0, \pi (\frac{1}{2} + n))`. However, IEEE 754 binary floating-point representation cannot represent :math:`\pi / 2` exactly, and, thus, no argument value is possible such that a pole error occurs.

    Parameters
    ----------
    x: array
        input array whose elements each represent a hyperbolic angle. Should have a floating-point data type.

    Returns
    -------
    out: array
        an array containing the hyperbolic tangent of each element in ``x``. The returned array must have a floating-point data type determined by :ref:`type-promotion`.

    Notes
    -----

    **Special cases**

    .. note::
       For all operands, ``tanh(-x)`` must equal ``-tanh(x)``.

    For real-valued floating-point operands,

    - If ``x_i`` is ``NaN``, the result is ``NaN``.
    - If ``x_i`` is ``+0``, the result is ``+0``.
    - If ``x_i`` is ``-0``, the result is ``-0``.
    - If ``x_i`` is ``+infinity``, the result is ``+1``.
    - If ``x_i`` is ``-infinity``, the result is ``-1``.

    For complex floating-point operands, let ``a = real(x_i)``, ``b = imag(x_i)``, and

    .. note::
       For complex floating-point operands, ``tanh(conj(x))`` must equal ``conj(tanh(x))``.

    - If ``a`` is ``+0`` and ``b`` is ``+0``, the result is ``+0 + 0j``.
    - If ``a`` is a nonzero finite number and ``b`` is ``+infinity``, the result is ``NaN + NaN j``.
    - If ``a`` is ``+0`` and ``b`` is ``+infinity``, the result is ``+0 + NaN j``.
    - If ``a`` is a nonzero finite number and ``b`` is ``NaN``, the result is ``NaN + NaN j``.
    - If ``a`` is ``+0`` and ``b`` is ``NaN``, the result is ``+0 + NaN j``.
    - If ``a`` is ``+infinity`` and ``b`` is a positive (i.e., greater than ``0``) finite number, the result is ``1 + 0j``.
    - If ``a`` is ``+infinity`` and ``b`` is ``+infinity``, the result is ``1 + 0j`` (sign of the imaginary component is unspecified).
    - If ``a`` is ``+infinity`` and ``b`` is ``NaN``, the result is ``1 + 0j`` (sign of the imaginary component is unspecified).
    - If ``a`` is ``NaN`` and ``b`` is ``+0``, the result is ``NaN + 0j``.
    - If ``a`` is ``NaN`` and ``b`` is a nonzero number, the result is ``NaN + NaN j``.
    - If ``a`` is ``NaN`` and ``b`` is ``NaN``, the result is ``NaN + NaN j``.

    .. warning::
       For historical reasons stemming from the C standard, array libraries may not return the expected result when ``a`` is ``+0`` and ``b`` is either ``+infinity`` or ``NaN``. The result should be ``+0 + NaN j`` in both cases; however, for libraries compiled against older C versions, the result may be ``NaN + NaN j``.

       Array libraries are not required to patch these older C versions, and, thus, users are advised that results may vary across array library implementations for these special cases.

    .. versionchanged:: 2022.12
       Added complex data type support.
    """


def trunc(x: array, /) -> array:
    """
    Rounds each element ``x_i`` of the input array ``x`` to the nearest integer-valued number that is closer to zero than ``x_i``.

    Parameters
    ----------
    x: array
        input array. Should have a real-valued data type.

    Returns
    -------
    out: array
        an array containing the rounded result for each element in ``x``. The returned array must have the same data type as ``x``.

    Notes
    -----

    **Special cases**

    - If ``x_i`` is already integer-valued, the result is ``x_i``.

    For floating-point operands,

    - If ``x_i`` is ``+infinity``, the result is ``+infinity``.
    - If ``x_i`` is ``-infinity``, the result is ``-infinity``.
    - If ``x_i`` is ``+0``, the result is ``+0``.
    - If ``x_i`` is ``-0``, the result is ``-0``.
    - If ``x_i`` is ``NaN``, the result is ``NaN``.
    """