ufunc_docstrings.py

"""
Docstrings for generated ufuncs

The syntax is designed to look like the function add_newdoc is being
called from numpy.lib, but in this file  add_newdoc puts the docstrings
in a dictionary. This dictionary is used in
numpy/core/code_generators/generate_umath_doc.py to generate the docstrings
as a C #definitions for the ufuncs in numpy.core at the C level when the
ufuncs are created at compile time.

"""
import textwrap

docdict = {}

# common parameter text to all ufuncs
subst = {
    'PARAMS': textwrap.dedent("""
        out : ndarray, None, or tuple of ndarray and None, optional
            A location into which the result is stored. If provided, it must have
            a shape that the inputs broadcast to. If not provided or None,
            a freshly-allocated array is returned. A tuple (possible only as a
            keyword argument) must have length equal to the number of outputs.
        where : array_like, optional
            This condition is broadcast over the input. At locations where the
            condition is True, the `out` array will be set to the ufunc result.
            Elsewhere, the `out` array will retain its original value.
            Note that if an uninitialized `out` array is created via the default
            ``out=None``, locations within it where the condition is False will
            remain uninitialized.
        **kwargs
            For other keyword-only arguments, see the
            :ref:`ufunc docs <ufuncs.kwargs>`.
    """).strip(),
    'BROADCASTABLE_2': ("If ``x1.shape != x2.shape``, they must be "
                        "broadcastable to a common\n    shape (which becomes "
                        "the shape of the output)."),
    'OUT_SCALAR_1': "This is a scalar if `x` is a scalar.",
    'OUT_SCALAR_2': "This is a scalar if both `x1` and `x2` are scalars.",
}

def add_newdoc(place, name, doc):
    doc = textwrap.dedent(doc).strip()

    skip = (
        # gufuncs do not use the OUT_SCALAR replacement strings
        'matmul',
        # clip has 3 inputs, which is not handled by this
        'clip',
    )
    if name[0] != '_' and name not in skip:
        if '\nx :' in doc:
            assert '$OUT_SCALAR_1' in doc, "in {}".format(name)
        elif '\nx2 :' in doc or '\nx1, x2 :' in doc:
            assert '$OUT_SCALAR_2' in doc, "in {}".format(name)
        else:
            assert False, "Could not detect number of inputs in {}".format(name)

    for k, v in subst.items():
        doc = doc.replace('$' + k, v)

    docdict['.'.join((place, name))] = doc


add_newdoc('numpy.core.umath', 'absolute',
    """
    Calculate the absolute value element-wise.

    ``np.abs`` is a shorthand for this function.

    Parameters
    ----------
    x : array_like
        Input array.
    $PARAMS

    Returns
    -------
    absolute : ndarray
        An ndarray containing the absolute value of
        each element in `x`.  For complex input, ``a + ib``, the
        absolute value is :math:`\\sqrt{ a^2 + b^2 }`.
        $OUT_SCALAR_1

    Examples
    --------
    >>> x = np.array([-1.2, 1.2])
    >>> np.absolute(x)
    array([ 1.2,  1.2])
    >>> np.absolute(1.2 + 1j)
    1.5620499351813308

    Plot the function over ``[-10, 10]``:

    >>> import matplotlib.pyplot as plt

    >>> x = np.linspace(start=-10, stop=10, num=101)
    >>> plt.plot(x, np.absolute(x))
    >>> plt.show()

    Plot the function over the complex plane:

    >>> xx = x + 1j * x[:, np.newaxis]
    >>> plt.imshow(np.abs(xx), extent=[-10, 10, -10, 10], cmap='gray')
    >>> plt.show()

    The `abs` function can be used as a shorthand for ``np.absolute`` on
    ndarrays.

    >>> x = np.array([-1.2, 1.2])
    >>> abs(x)
    array([1.2, 1.2])

    """)

add_newdoc('numpy.core.umath', 'add',
    """
    Add arguments element-wise.

    Parameters
    ----------
    x1, x2 : array_like
        The arrays to be added.
        $BROADCASTABLE_2
    $PARAMS

    Returns
    -------
    add : ndarray or scalar
        The sum of `x1` and `x2`, element-wise.
        $OUT_SCALAR_2

    Notes
    -----
    Equivalent to `x1` + `x2` in terms of array broadcasting.

    Examples
    --------
    >>> np.add(1.0, 4.0)
    5.0
    >>> x1 = np.arange(9.0).reshape((3, 3))
    >>> x2 = np.arange(3.0)
    >>> np.add(x1, x2)
    array([[  0.,   2.,   4.],
           [  3.,   5.,   7.],
           [  6.,   8.,  10.]])

    The ``+`` operator can be used as a shorthand for ``np.add`` on ndarrays.

    >>> x1 = np.arange(9.0).reshape((3, 3))
    >>> x2 = np.arange(3.0)
    >>> x1 + x2
    array([[ 0.,  2.,  4.],
           [ 3.,  5.,  7.],
           [ 6.,  8., 10.]])
    """)

add_newdoc('numpy.core.umath', 'arccos',
    """
    Trigonometric inverse cosine, element-wise.

    The inverse of `cos` so that, if ``y = cos(x)``, then ``x = arccos(y)``.

    Parameters
    ----------
    x : array_like
        `x`-coordinate on the unit circle.
        For real arguments, the domain is [-1, 1].
    $PARAMS

    Returns
    -------
    angle : ndarray
        The angle of the ray intersecting the unit circle at the given
        `x`-coordinate in radians [0, pi].
        $OUT_SCALAR_1

    See Also
    --------
    cos, arctan, arcsin, emath.arccos

    Notes
    -----
    `arccos` is a multivalued function: for each `x` there are infinitely
    many numbers `z` such that ``cos(z) = x``. The convention is to return
    the angle `z` whose real part lies in `[0, pi]`.

    For real-valued input data types, `arccos` always returns real output.
    For each value that cannot be expressed as a real number or infinity,
    it yields ``nan`` and sets the `invalid` floating point error flag.

    For complex-valued input, `arccos` is a complex analytic function that
    has branch cuts ``[-inf, -1]`` and `[1, inf]` and is continuous from
    above on the former and from below on the latter.

    The inverse `cos` is also known as `acos` or cos^-1.

    References
    ----------
    M. Abramowitz and I.A. Stegun, "Handbook of Mathematical Functions",
    10th printing, 1964, pp. 79.
    https://personal.math.ubc.ca/~cbm/aands/page_79.htm

    Examples
    --------
    We expect the arccos of 1 to be 0, and of -1 to be pi:

    >>> np.arccos([1, -1])
    array([ 0.        ,  3.14159265])

    Plot arccos:

    >>> import matplotlib.pyplot as plt
    >>> x = np.linspace(-1, 1, num=100)
    >>> plt.plot(x, np.arccos(x))
    >>> plt.axis('tight')
    >>> plt.show()

    """)

add_newdoc('numpy.core.umath', 'arccosh',
    """
    Inverse hyperbolic cosine, element-wise.

    Parameters
    ----------
    x : array_like
        Input array.
    $PARAMS

    Returns
    -------
    arccosh : ndarray
        Array of the same shape as `x`.
        $OUT_SCALAR_1

    See Also
    --------

    cosh, arcsinh, sinh, arctanh, tanh

    Notes
    -----
    `arccosh` is a multivalued function: for each `x` there are infinitely
    many numbers `z` such that `cosh(z) = x`. The convention is to return the
    `z` whose imaginary part lies in ``[-pi, pi]`` and the real part in
    ``[0, inf]``.

    For real-valued input data types, `arccosh` always returns real output.
    For each value that cannot be expressed as a real number or infinity, it
    yields ``nan`` and sets the `invalid` floating point error flag.

    For complex-valued input, `arccosh` is a complex analytical function that
    has a branch cut `[-inf, 1]` and is continuous from above on it.

    References
    ----------
    .. [1] M. Abramowitz and I.A. Stegun, "Handbook of Mathematical Functions",
           10th printing, 1964, pp. 86.
           https://personal.math.ubc.ca/~cbm/aands/page_86.htm
    .. [2] Wikipedia, "Inverse hyperbolic function",
           https://en.wikipedia.org/wiki/Arccosh

    Examples
    --------
    >>> np.arccosh([np.e, 10.0])
    array([ 1.65745445,  2.99322285])
    >>> np.arccosh(1)
    0.0

    """)

add_newdoc('numpy.core.umath', 'arcsin',
    """
    Inverse sine, element-wise.

    Parameters
    ----------
    x : array_like
        `y`-coordinate on the unit circle.
    $PARAMS

    Returns
    -------
    angle : ndarray
        The inverse sine of each element in `x`, in radians and in the
        closed interval ``[-pi/2, pi/2]``.
        $OUT_SCALAR_1

    See Also
    --------
    sin, cos, arccos, tan, arctan, arctan2, emath.arcsin

    Notes
    -----
    `arcsin` is a multivalued function: for each `x` there are infinitely
    many numbers `z` such that :math:`sin(z) = x`.  The convention is to
    return the angle `z` whose real part lies in [-pi/2, pi/2].

    For real-valued input data types, *arcsin* always returns real output.
    For each value that cannot be expressed as a real number or infinity,
    it yields ``nan`` and sets the `invalid` floating point error flag.

    For complex-valued input, `arcsin` is a complex analytic function that
    has, by convention, the branch cuts [-inf, -1] and [1, inf]  and is
    continuous from above on the former and from below on the latter.

    The inverse sine is also known as `asin` or sin^{-1}.

    References
    ----------
    Abramowitz, M. and Stegun, I. A., *Handbook of Mathematical Functions*,
    10th printing, New York: Dover, 1964, pp. 79ff.
    https://personal.math.ubc.ca/~cbm/aands/page_79.htm

    Examples
    --------
    >>> np.arcsin(1)     # pi/2
    1.5707963267948966
    >>> np.arcsin(-1)    # -pi/2
    -1.5707963267948966
    >>> np.arcsin(0)
    0.0

    """)

add_newdoc('numpy.core.umath', 'arcsinh',
    """
    Inverse hyperbolic sine element-wise.

    Parameters
    ----------
    x : array_like
        Input array.
    $PARAMS

    Returns
    -------
    out : ndarray or scalar
        Array of the same shape as `x`.
        $OUT_SCALAR_1

    Notes
    -----
    `arcsinh` is a multivalued function: for each `x` there are infinitely
    many numbers `z` such that `sinh(z) = x`. The convention is to return the
    `z` whose imaginary part lies in `[-pi/2, pi/2]`.

    For real-valued input data types, `arcsinh` always returns real output.
    For each value that cannot be expressed as a real number or infinity, it
    returns ``nan`` and sets the `invalid` floating point error flag.

    For complex-valued input, `arccos` is a complex analytical function that
    has branch cuts `[1j, infj]` and `[-1j, -infj]` and is continuous from
    the right on the former and from the left on the latter.

    The inverse hyperbolic sine is also known as `asinh` or ``sinh^-1``.

    References
    ----------
    .. [1] M. Abramowitz and I.A. Stegun, "Handbook of Mathematical Functions",
           10th printing, 1964, pp. 86.
           https://personal.math.ubc.ca/~cbm/aands/page_86.htm
    .. [2] Wikipedia, "Inverse hyperbolic function",
           https://en.wikipedia.org/wiki/Arcsinh

    Examples
    --------
    >>> np.arcsinh(np.array([np.e, 10.0]))
    array([ 1.72538256,  2.99822295])

    """)

add_newdoc('numpy.core.umath', 'arctan',
    """
    Trigonometric inverse tangent, element-wise.

    The inverse of tan, so that if ``y = tan(x)`` then ``x = arctan(y)``.

    Parameters
    ----------
    x : array_like
    $PARAMS

    Returns
    -------
    out : ndarray or scalar
        Out has the same shape as `x`.  Its real part is in
        ``[-pi/2, pi/2]`` (``arctan(+/-inf)`` returns ``+/-pi/2``).
        $OUT_SCALAR_1

    See Also
    --------
    arctan2 : The "four quadrant" arctan of the angle formed by (`x`, `y`)
        and the positive `x`-axis.
    angle : Argument of complex values.

    Notes
    -----
    `arctan` is a multi-valued function: for each `x` there are infinitely
    many numbers `z` such that tan(`z`) = `x`.  The convention is to return
    the angle `z` whose real part lies in [-pi/2, pi/2].

    For real-valued input data types, `arctan` always returns real output.
    For each value that cannot be expressed as a real number or infinity,
    it yields ``nan`` and sets the `invalid` floating point error flag.

    For complex-valued input, `arctan` is a complex analytic function that
    has [``1j, infj``] and [``-1j, -infj``] as branch cuts, and is continuous
    from the left on the former and from the right on the latter.

    The inverse tangent is also known as `atan` or tan^{-1}.

    References
    ----------
    Abramowitz, M. and Stegun, I. A., *Handbook of Mathematical Functions*,
    10th printing, New York: Dover, 1964, pp. 79.
    https://personal.math.ubc.ca/~cbm/aands/page_79.htm

    Examples
    --------
    We expect the arctan of 0 to be 0, and of 1 to be pi/4:

    >>> np.arctan([0, 1])
    array([ 0.        ,  0.78539816])

    >>> np.pi/4
    0.78539816339744828

    Plot arctan:

    >>> import matplotlib.pyplot as plt
    >>> x = np.linspace(-10, 10)
    >>> plt.plot(x, np.arctan(x))
    >>> plt.axis('tight')
    >>> plt.show()

    """)

add_newdoc('numpy.core.umath', 'arctan2',
    """
    Element-wise arc tangent of ``x1/x2`` choosing the quadrant correctly.

    The quadrant (i.e., branch) is chosen so that ``arctan2(x1, x2)`` 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`, `x1`).  (Note the role reversal: the
    "`y`-coordinate" is the first function parameter, the "`x`-coordinate"
    is the second.)  By IEEE convention, this function is defined for
    `x2` = +/-0 and for either or both of `x1` and `x2` = +/-inf (see
    Notes for specific values).

    This function is not defined for complex-valued arguments; for the
    so-called argument of complex values, use `angle`.

    Parameters
    ----------
    x1 : array_like, real-valued
        `y`-coordinates.
    x2 : array_like, real-valued
        `x`-coordinates.
        $BROADCASTABLE_2
    $PARAMS

    Returns
    -------
    angle : ndarray
        Array of angles in radians, in the range ``[-pi, pi]``.
        $OUT_SCALAR_2

    See Also
    --------
    arctan, tan, angle

    Notes
    -----
    *arctan2* is identical to the `atan2` function of the underlying
    C library.  The following special values are defined in the C
    standard: [1]_

    ====== ====== ================
    `x1`   `x2`   `arctan2(x1,x2)`
    ====== ====== ================
    +/- 0  +0     +/- 0
    +/- 0  -0     +/- pi
     > 0   +/-inf +0 / +pi
     < 0   +/-inf -0 / -pi
    +/-inf +inf   +/- (pi/4)
    +/-inf -inf   +/- (3*pi/4)
    ====== ====== ================

    Note that +0 and -0 are distinct floating point numbers, as are +inf
    and -inf.

    References
    ----------
    .. [1] ISO/IEC standard 9899:1999, "Programming language C."

    Examples
    --------
    Consider four points in different quadrants:

    >>> x = np.array([-1, +1, +1, -1])
    >>> y = np.array([-1, -1, +1, +1])
    >>> np.arctan2(y, x) * 180 / np.pi
    array([-135.,  -45.,   45.,  135.])

    Note the order of the parameters. `arctan2` is defined also when `x2` = 0
    and at several other special points, obtaining values in
    the range ``[-pi, pi]``:

    >>> np.arctan2([1., -1.], [0., 0.])
    array([ 1.57079633, -1.57079633])
    >>> np.arctan2([0., 0., np.inf], [+0., -0., np.inf])
    array([0.        , 3.14159265, 0.78539816])

    """)

add_newdoc('numpy.core.umath', '_arg',
    """
    DO NOT USE, ONLY FOR TESTING
    """)

add_newdoc('numpy.core.umath', 'arctanh',
    """
    Inverse hyperbolic tangent element-wise.

    Parameters
    ----------
    x : array_like
        Input array.
    $PARAMS

    Returns
    -------
    out : ndarray or scalar
        Array of the same shape as `x`.
        $OUT_SCALAR_1

    See Also
    --------
    emath.arctanh

    Notes
    -----
    `arctanh` is a multivalued function: for each `x` there are infinitely
    many numbers `z` such that ``tanh(z) = x``. The convention is to return
    the `z` whose imaginary part lies in `[-pi/2, pi/2]`.

    For real-valued input data types, `arctanh` always returns real output.
    For each value that cannot be expressed as a real number or infinity,
    it yields ``nan`` and sets the `invalid` floating point error flag.

    For complex-valued input, `arctanh` is a complex analytical function
    that has branch cuts `[-1, -inf]` and `[1, inf]` and is continuous from
    above on the former and from below on the latter.

    The inverse hyperbolic tangent is also known as `atanh` or ``tanh^-1``.

    References
    ----------
    .. [1] M. Abramowitz and I.A. Stegun, "Handbook of Mathematical Functions",
           10th printing, 1964, pp. 86.
           https://personal.math.ubc.ca/~cbm/aands/page_86.htm
    .. [2] Wikipedia, "Inverse hyperbolic function",
           https://en.wikipedia.org/wiki/Arctanh

    Examples
    --------
    >>> np.arctanh([0, -0.5])
    array([ 0.        , -0.54930614])

    """)

add_newdoc('numpy.core.umath', 'bitwise_and',
    """
    Compute the bit-wise AND of two arrays element-wise.

    Computes the bit-wise AND of the underlying binary representation of
    the integers in the input arrays. This ufunc implements the C/Python
    operator ``&``.

    Parameters
    ----------
    x1, x2 : array_like
        Only integer and boolean types are handled.
        $BROADCASTABLE_2
    $PARAMS

    Returns
    -------
    out : ndarray or scalar
        Result.
        $OUT_SCALAR_2

    See Also
    --------
    logical_and
    bitwise_or
    bitwise_xor
    binary_repr :
        Return the binary representation of the input number as a string.

    Examples
    --------
    The number 13 is represented by ``00001101``.  Likewise, 17 is
    represented by ``00010001``.  The bit-wise AND of 13 and 17 is
    therefore ``000000001``, or 1:

    >>> np.bitwise_and(13, 17)
    1

    >>> np.bitwise_and(14, 13)
    12
    >>> np.binary_repr(12)
    '1100'
    >>> np.bitwise_and([14,3], 13)
    array([12,  1])

    >>> np.bitwise_and([11,7], [4,25])
    array([0, 1])
    >>> np.bitwise_and(np.array([2,5,255]), np.array([3,14,16]))
    array([ 2,  4, 16])
    >>> np.bitwise_and([True, True], [False, True])
    array([False,  True])

    The ``&`` operator can be used as a shorthand for ``np.bitwise_and`` on
    ndarrays.

    >>> x1 = np.array([2, 5, 255])
    >>> x2 = np.array([3, 14, 16])
    >>> x1 & x2
    array([ 2,  4, 16])

    """)

add_newdoc('numpy.core.umath', 'bitwise_or',
    """
    Compute the bit-wise OR of two arrays element-wise.

    Computes the bit-wise OR of the underlying binary representation of
    the integers in the input arrays. This ufunc implements the C/Python
    operator ``|``.

    Parameters
    ----------
    x1, x2 : array_like
        Only integer and boolean types are handled.
        $BROADCASTABLE_2
    $PARAMS

    Returns
    -------
    out : ndarray or scalar
        Result.
        $OUT_SCALAR_2

    See Also
    --------
    logical_or
    bitwise_and
    bitwise_xor
    binary_repr :
        Return the binary representation of the input number as a string.

    Examples
    --------
    The number 13 has the binary representation ``00001101``. Likewise,
    16 is represented by ``00010000``.  The bit-wise OR of 13 and 16 is
    then ``000111011``, or 29:

    >>> np.bitwise_or(13, 16)
    29
    >>> np.binary_repr(29)
    '11101'

    >>> np.bitwise_or(32, 2)
    34
    >>> np.bitwise_or([33, 4], 1)
    array([33,  5])
    >>> np.bitwise_or([33, 4], [1, 2])
    array([33,  6])

    >>> np.bitwise_or(np.array([2, 5, 255]), np.array([4, 4, 4]))
    array([  6,   5, 255])
    >>> np.array([2, 5, 255]) | np.array([4, 4, 4])
    array([  6,   5, 255])
    >>> np.bitwise_or(np.array([2, 5, 255, 2147483647], dtype=np.int32),
    ...               np.array([4, 4, 4, 2147483647], dtype=np.int32))
    array([         6,          5,        255, 2147483647])
    >>> np.bitwise_or([True, True], [False, True])
    array([ True,  True])

    The ``|`` operator can be used as a shorthand for ``np.bitwise_or`` on
    ndarrays.

    >>> x1 = np.array([2, 5, 255])
    >>> x2 = np.array([4, 4, 4])
    >>> x1 | x2
    array([  6,   5, 255])

    """)

add_newdoc('numpy.core.umath', 'bitwise_xor',
    """
    Compute the bit-wise XOR of two arrays element-wise.

    Computes the bit-wise XOR of the underlying binary representation of
    the integers in the input arrays. This ufunc implements the C/Python
    operator ``^``.

    Parameters
    ----------
    x1, x2 : array_like
        Only integer and boolean types are handled.
        $BROADCASTABLE_2
    $PARAMS

    Returns
    -------
    out : ndarray or scalar
        Result.
        $OUT_SCALAR_2

    See Also
    --------
    logical_xor
    bitwise_and
    bitwise_or
    binary_repr :
        Return the binary representation of the input number as a string.

    Examples
    --------
    The number 13 is represented by ``00001101``. Likewise, 17 is
    represented by ``00010001``.  The bit-wise XOR of 13 and 17 is
    therefore ``00011100``, or 28:

    >>> np.bitwise_xor(13, 17)
    28
    >>> np.binary_repr(28)
    '11100'

    >>> np.bitwise_xor(31, 5)
    26
    >>> np.bitwise_xor([31,3], 5)
    array([26,  6])

    >>> np.bitwise_xor([31,3], [5,6])
    array([26,  5])
    >>> np.bitwise_xor([True, True], [False, True])
    array([ True, False])

    The ``^`` operator can be used as a shorthand for ``np.bitwise_xor`` on
    ndarrays.

    >>> x1 = np.array([True, True])
    >>> x2 = np.array([False, True])
    >>> x1 ^ x2
    array([ True, False])

    """)

add_newdoc('numpy.core.umath', 'ceil',
    """
    Return the ceiling of the input, element-wise.

    The ceil of the scalar `x` is the smallest integer `i`, such that
    ``i >= x``.  It is often denoted as :math:`\\lceil x \\rceil`.

    Parameters
    ----------
    x : array_like
        Input data.
    $PARAMS

    Returns
    -------
    y : ndarray or scalar
        The ceiling of each element in `x`, with `float` dtype.
        $OUT_SCALAR_1

    See Also
    --------
    floor, trunc, rint, fix

    Examples
    --------
    >>> a = np.array([-1.7, -1.5, -0.2, 0.2, 1.5, 1.7, 2.0])
    >>> np.ceil(a)
    array([-1., -1., -0.,  1.,  2.,  2.,  2.])

    """)

add_newdoc('numpy.core.umath', 'trunc',
    """
    Return the truncated value of the input, element-wise.

    The truncated value of the scalar `x` is the nearest integer `i` which
    is closer to zero than `x` is. In short, the fractional part of the
    signed number `x` is discarded.

    Parameters
    ----------
    x : array_like
        Input data.
    $PARAMS

    Returns
    -------
    y : ndarray or scalar
        The truncated value of each element in `x`.
        $OUT_SCALAR_1

    See Also
    --------
    ceil, floor, rint, fix

    Notes
    -----
    .. versionadded:: 1.3.0

    Examples
    --------
    >>> a = np.array([-1.7, -1.5, -0.2, 0.2, 1.5, 1.7, 2.0])
    >>> np.trunc(a)
    array([-1., -1., -0.,  0.,  1.,  1.,  2.])

    """)

add_newdoc('numpy.core.umath', 'conjugate',
    """
    Return the complex conjugate, element-wise.

    The complex conjugate of a complex number is obtained by changing the
    sign of its imaginary part.

    Parameters
    ----------
    x : array_like
        Input value.
    $PARAMS

    Returns
    -------
    y : ndarray
        The complex conjugate of `x`, with same dtype as `y`.
        $OUT_SCALAR_1

    Notes
    -----
    `conj` is an alias for `conjugate`:

    >>> np.conj is np.conjugate
    True

    Examples
    --------
    >>> np.conjugate(1+2j)
    (1-2j)

    >>> x = np.eye(2) + 1j * np.eye(2)
    >>> np.conjugate(x)
    array([[ 1.-1.j,  0.-0.j],
           [ 0.-0.j,  1.-1.j]])

    """)

add_newdoc('numpy.core.umath', 'cos',
    """
    Cosine element-wise.

    Parameters
    ----------
    x : array_like
        Input array in radians.
    $PARAMS

    Returns
    -------
    y : ndarray
        The corresponding cosine values.
        $OUT_SCALAR_1

    Notes
    -----
    If `out` is provided, the function writes the result into it,
    and returns a reference to `out`.  (See Examples)

    References
    ----------
    M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions.
    New York, NY: Dover, 1972.

    Examples
    --------
    >>> np.cos(np.array([0, np.pi/2, np.pi]))
    array([  1.00000000e+00,   6.12303177e-17,  -1.00000000e+00])
    >>>
    >>> # Example of providing the optional output parameter
    >>> out1 = np.array([0], dtype='d')
    >>> out2 = np.cos([0.1], out1)
    >>> out2 is out1
    True
    >>>
    >>> # Example of ValueError due to provision of shape mis-matched `out`
    >>> np.cos(np.zeros((3,3)),np.zeros((2,2)))
    Traceback (most recent call last):
      File "<stdin>", line 1, in <module>
    ValueError: operands could not be broadcast together with shapes (3,3) (2,2)

    """)

add_newdoc('numpy.core.umath', 'cosh',
    """
    Hyperbolic cosine, element-wise.

    Equivalent to ``1/2 * (np.exp(x) + np.exp(-x))`` and ``np.cos(1j*x)``.

    Parameters
    ----------
    x : array_like
        Input array.
    $PARAMS

    Returns
    -------
    out : ndarray or scalar
        Output array of same shape as `x`.
        $OUT_SCALAR_1

    Examples
    --------
    >>> np.cosh(0)
    1.0

    The hyperbolic cosine describes the shape of a hanging cable:

    >>> import matplotlib.pyplot as plt
    >>> x = np.linspace(-4, 4, 1000)
    >>> plt.plot(x, np.cosh(x))
    >>> plt.show()

    """)

add_newdoc('numpy.core.umath', 'degrees',
    """
    Convert angles from radians to degrees.

    Parameters
    ----------
    x : array_like
        Input array in radians.
    $PARAMS

    Returns
    -------
    y : ndarray of floats
        The corresponding degree values; if `out` was supplied this is a
        reference to it.
        $OUT_SCALAR_1

    See Also
    --------
    rad2deg : equivalent function

    Examples
    --------
    Convert a radian array to degrees

    >>> rad = np.arange(12.)*np.pi/6
    >>> np.degrees(rad)
    array([   0.,   30.,   60.,   90.,  120.,  150.,  180.,  210.,  240.,
            270.,  300.,  330.])

    >>> out = np.zeros((rad.shape))
    >>> r = np.degrees(rad, out)
    >>> np.all(r == out)
    True

    """)

add_newdoc('numpy.core.umath', 'rad2deg',
    """
    Convert angles from radians to degrees.

    Parameters
    ----------
    x : array_like
        Angle in radians.
    $PARAMS

    Returns
    -------
    y : ndarray
        The corresponding angle in degrees.
        $OUT_SCALAR_1

    See Also
    --------
    deg2rad : Convert angles from degrees to radians.
    unwrap : Remove large jumps in angle by wrapping.

    Notes
    -----
    .. versionadded:: 1.3.0

    rad2deg(x) is ``180 * x / pi``.

    Examples
    --------
    >>> np.rad2deg(np.pi/2)
    90.0

    """)

add_newdoc('numpy.core.umath', 'heaviside',
    """
    Compute the Heaviside step function.

    The Heaviside step function is defined as::

                              0   if x1 < 0
        heaviside(x1, x2) =  x2   if x1 == 0
                              1   if x1 > 0

    where `x2` is often taken to be 0.5, but 0 and 1 are also sometimes used.

    Parameters
    ----------
    x1 : array_like
        Input values.
    x2 : array_like
        The value of the function when x1 is 0.
        $BROADCASTABLE_2
    $PARAMS

    Returns
    -------
    out : ndarray or scalar
        The output array, element-wise Heaviside step function of `x1`.
        $OUT_SCALAR_2

    Notes
    -----
    .. versionadded:: 1.13.0

    References
    ----------
    .. Wikipedia, "Heaviside step function",
       https://en.wikipedia.org/wiki/Heaviside_step_function

    Examples
    --------
    >>> np.heaviside([-1.5, 0, 2.0], 0.5)
    array([ 0. ,  0.5,  1. ])
    >>> np.heaviside([-1.5, 0, 2.0], 1)
    array([ 0.,  1.,  1.])
    """)

add_newdoc('numpy.core.umath', 'divide',
    """
    Divide arguments element-wise.

    Parameters
    ----------
    x1 : array_like
        Dividend array.
    x2 : array_like
        Divisor array.
        $BROADCASTABLE_2
    $PARAMS

    Returns
    -------
    y : ndarray or scalar
        The quotient ``x1/x2``, element-wise.
        $OUT_SCALAR_2

    See Also
    --------
    seterr : Set whether to raise or warn on overflow, underflow and
             division by zero.

    Notes
    -----
    Equivalent to ``x1`` / ``x2`` in terms of array-broadcasting.

    The ``true_divide(x1, x2)`` function is an alias for
    ``divide(x1, x2)``.

    Examples
    --------
    >>> np.divide(2.0, 4.0)
    0.5
    >>> x1 = np.arange(9.0).reshape((3, 3))
    >>> x2 = np.arange(3.0)
    >>> np.divide(x1, x2)
    array([[nan, 1. , 1. ],
           [inf, 4. , 2.5],
           [inf, 7. , 4. ]])

    The ``/`` operator can be used as a shorthand for ``np.divide`` on
    ndarrays.

    >>> x1 = np.arange(9.0).reshape((3, 3))
    >>> x2 = 2 * np.ones(3)
    >>> x1 / x2
    array([[0. , 0.5, 1. ],
           [1.5, 2. , 2.5],
           [3. , 3.5, 4. ]])

    """)

add_newdoc('numpy.core.umath', 'equal',
    """
    Return (x1 == x2) element-wise.

    Parameters
    ----------
    x1, x2 : array_like
        Input arrays.
        $BROADCASTABLE_2
    $PARAMS

    Returns
    -------
    out : ndarray or scalar
        Output array, element-wise comparison of `x1` and `x2`.
        Typically of type bool, unless ``dtype=object`` is passed.
        $OUT_SCALAR_2

    See Also
    --------
    not_equal, greater_equal, less_equal, greater, less

    Examples
    --------
    >>> np.equal([0, 1, 3], np.arange(3))
    array([ True,  True, False])

    What is compared are values, not types. So an int (1) and an array of
    length one can evaluate as True:

    >>> np.equal(1, np.ones(1))
    array([ True])

    The ``==`` operator can be used as a shorthand for ``np.equal`` on
    ndarrays.

    >>> a = np.array([2, 4, 6])
    >>> b = np.array([2, 4, 2])
    >>> a == b
    array([ True,  True, False])

    """)

add_newdoc('numpy.core.umath', 'exp',
    """
    Calculate the exponential of all elements in the input array.

    Parameters
    ----------
    x : array_like
        Input values.
    $PARAMS

    Returns
    -------
    out : ndarray or scalar
        Output array, element-wise exponential of `x`.
        $OUT_SCALAR_1

    See Also
    --------
    expm1 : Calculate ``exp(x) - 1`` for all elements in the array.
    exp2  : Calculate ``2**x`` for all elements in the array.

    Notes
    -----
    The irrational number ``e`` is also known as Euler's number.  It is
    approximately 2.718281, and is the base of the natural logarithm,
    ``ln`` (this means that, if :math:`x = \\ln y = \\log_e y`,
    then :math:`e^x = y`. For real input, ``exp(x)`` is always positive.

    For complex arguments, ``x = a + ib``, we can write
    :math:`e^x = e^a e^{ib}`.  The first term, :math:`e^a`, is already
    known (it is the real argument, described above).  The second term,
    :math:`e^{ib}`, is :math:`\\cos b + i \\sin b`, a function with
    magnitude 1 and a periodic phase.

    References
    ----------
    .. [1] Wikipedia, "Exponential function",
           https://en.wikipedia.org/wiki/Exponential_function
    .. [2] M. Abramovitz and I. A. Stegun, "Handbook of Mathematical Functions
           with Formulas, Graphs, and Mathematical Tables," Dover, 1964, p. 69,
           https://personal.math.ubc.ca/~cbm/aands/page_69.htm

    Examples
    --------
    Plot the magnitude and phase of ``exp(x)`` in the complex plane:

    >>> import matplotlib.pyplot as plt

    >>> x = np.linspace(-2*np.pi, 2*np.pi, 100)
    >>> xx = x + 1j * x[:, np.newaxis] # a + ib over complex plane
    >>> out = np.exp(xx)

    >>> plt.subplot(121)
    >>> plt.imshow(np.abs(out),
    ...            extent=[-2*np.pi, 2*np.pi, -2*np.pi, 2*np.pi], cmap='gray')
    >>> plt.title('Magnitude of exp(x)')

    >>> plt.subplot(122)
    >>> plt.imshow(np.angle(out),
    ...            extent=[-2*np.pi, 2*np.pi, -2*np.pi, 2*np.pi], cmap='hsv')
    >>> plt.title('Phase (angle) of exp(x)')
    >>> plt.show()

    """)

add_newdoc('numpy.core.umath', 'exp2',
    """
    Calculate `2**p` for all `p` in the input array.

    Parameters
    ----------
    x : array_like
        Input values.
    $PARAMS

    Returns
    -------
    out : ndarray or scalar
        Element-wise 2 to the power `x`.
        $OUT_SCALAR_1

    See Also
    --------
    power

    Notes
    -----
    .. versionadded:: 1.3.0



    Examples
    --------
    >>> np.exp2([2, 3])
    array([ 4.,  8.])

    """)

add_newdoc('numpy.core.umath', 'expm1',
    """
    Calculate ``exp(x) - 1`` for all elements in the array.

    Parameters
    ----------
    x : array_like
        Input values.
    $PARAMS

    Returns
    -------
    out : ndarray or scalar
        Element-wise exponential minus one: ``out = exp(x) - 1``.
        $OUT_SCALAR_1

    See Also
    --------
    log1p : ``log(1 + x)``, the inverse of expm1.


    Notes
    -----
    This function provides greater precision than ``exp(x) - 1``
    for small values of ``x``.

    Examples
    --------
    The true value of ``exp(1e-10) - 1`` is ``1.00000000005e-10`` to
    about 32 significant digits. This example shows the superiority of
    expm1 in this case.

    >>> np.expm1(1e-10)
    1.00000000005e-10
    >>> np.exp(1e-10) - 1
    1.000000082740371e-10

    """)

add_newdoc('numpy.core.umath', 'fabs',
    """
    Compute the absolute values element-wise.

    This function returns the absolute values (positive magnitude) of the
    data in `x`. Complex values are not handled, use `absolute` to find the
    absolute values of complex data.

    Parameters
    ----------
    x : array_like
        The array of numbers for which the absolute values are required. If
        `x` is a scalar, the result `y` will also be a scalar.
    $PARAMS

    Returns
    -------
    y : ndarray or scalar
        The absolute values of `x`, the returned values are always floats.
        $OUT_SCALAR_1

    See Also
    --------
    absolute : Absolute values including `complex` types.

    Examples
    --------
    >>> np.fabs(-1)
    1.0
    >>> np.fabs([-1.2, 1.2])
    array([ 1.2,  1.2])

    """)

add_newdoc('numpy.core.umath', 'floor',
    """
    Return the floor of the input, element-wise.

    The floor of the scalar `x` is the largest integer `i`, such that
    `i <= x`.  It is often denoted as :math:`\\lfloor x \\rfloor`.

    Parameters
    ----------
    x : array_like
        Input data.
    $PARAMS

    Returns
    -------
    y : ndarray or scalar
        The floor of each element in `x`.
        $OUT_SCALAR_1

    See Also
    --------
    ceil, trunc, rint, fix

    Notes
    -----
    Some spreadsheet programs calculate the "floor-towards-zero", where
    ``floor(-2.5) == -2``.  NumPy instead uses the definition of
    `floor` where `floor(-2.5) == -3`. The "floor-towards-zero"
    function is called ``fix`` in NumPy.

    Examples
    --------
    >>> a = np.array([-1.7, -1.5, -0.2, 0.2, 1.5, 1.7, 2.0])
    >>> np.floor(a)
    array([-2., -2., -1.,  0.,  1.,  1.,  2.])

    """)

add_newdoc('numpy.core.umath', 'floor_divide',
    """
    Return the largest integer smaller or equal to the division of the inputs.
    It is equivalent to the Python ``//`` operator and pairs with the
    Python ``%`` (`remainder`), function so that ``a = a % b + b * (a // b)``
    up to roundoff.

    Parameters
    ----------
    x1 : array_like
        Numerator.
    x2 : array_like
        Denominator.
        $BROADCASTABLE_2
    $PARAMS

    Returns
    -------
    y : ndarray
        y = floor(`x1`/`x2`)
        $OUT_SCALAR_2

    See Also
    --------
    remainder : Remainder complementary to floor_divide.
    divmod : Simultaneous floor division and remainder.
    divide : Standard division.
    floor : Round a number to the nearest integer toward minus infinity.
    ceil : Round a number to the nearest integer toward infinity.

    Examples
    --------
    >>> np.floor_divide(7,3)
    2
    >>> np.floor_divide([1., 2., 3., 4.], 2.5)
    array([ 0.,  0.,  1.,  1.])

    The ``//`` operator can be used as a shorthand for ``np.floor_divide``
    on ndarrays.

    >>> x1 = np.array([1., 2., 3., 4.])
    >>> x1 // 2.5
    array([0., 0., 1., 1.])

    """)

add_newdoc('numpy.core.umath', 'fmod',
    """
    Returns the element-wise remainder of division.

    This is the NumPy implementation of the C library function fmod, the
    remainder has the same sign as the dividend `x1`. It is equivalent to
    the Matlab(TM) ``rem`` function and should not be confused with the
    Python modulus operator ``x1 % x2``.

    Parameters
    ----------
    x1 : array_like
        Dividend.
    x2 : array_like
        Divisor.
        $BROADCASTABLE_2
    $PARAMS

    Returns
    -------
    y : array_like
        The remainder of the division of `x1` by `x2`.
        $OUT_SCALAR_2

    See Also
    --------
    remainder : Equivalent to the Python ``%`` operator.
    divide

    Notes
    -----
    The result of the modulo operation for negative dividend and divisors
    is bound by conventions. For `fmod`, the sign of result is the sign of
    the dividend, while for `remainder` the sign of the result is the sign
    of the divisor. The `fmod` function is equivalent to the Matlab(TM)
    ``rem`` function.

    Examples
    --------
    >>> np.fmod([-3, -2, -1, 1, 2, 3], 2)
    array([-1,  0, -1,  1,  0,  1])
    >>> np.remainder([-3, -2, -1, 1, 2, 3], 2)
    array([1, 0, 1, 1, 0, 1])

    >>> np.fmod([5, 3], [2, 2.])
    array([ 1.,  1.])
    >>> a = np.arange(-3, 3).reshape(3, 2)
    >>> a
    array([[-3, -2],
           [-1,  0],
           [ 1,  2]])
    >>> np.fmod(a, [2,2])
    array([[-1,  0],
           [-1,  0],
           [ 1,  0]])

    """)

add_newdoc('numpy.core.umath', 'greater',
    """
    Return the truth value of (x1 > x2) element-wise.

    Parameters
    ----------
    x1, x2 : array_like
        Input arrays.
        $BROADCASTABLE_2
    $PARAMS

    Returns
    -------
    out : ndarray or scalar
        Output array, element-wise comparison of `x1` and `x2`.
        Typically of type bool, unless ``dtype=object`` is passed.
        $OUT_SCALAR_2


    See Also
    --------
    greater_equal, less, less_equal, equal, not_equal

    Examples
    --------
    >>> np.greater([4,2],[2,2])
    array([ True, False])

    The ``>`` operator can be used as a shorthand for ``np.greater`` on
    ndarrays.

    >>> a = np.array([4, 2])
    >>> b = np.array([2, 2])
    >>> a > b
    array([ True, False])

    """)

add_newdoc('numpy.core.umath', 'greater_equal',
    """
    Return the truth value of (x1 >= x2) element-wise.

    Parameters
    ----------
    x1, x2 : array_like
        Input arrays.
        $BROADCASTABLE_2
    $PARAMS

    Returns
    -------
    out : bool or ndarray of bool
        Output array, element-wise comparison of `x1` and `x2`.
        Typically of type bool, unless ``dtype=object`` is passed.
        $OUT_SCALAR_2

    See Also
    --------
    greater, less, less_equal, equal, not_equal

    Examples
    --------
    >>> np.greater_equal([4, 2, 1], [2, 2, 2])
    array([ True, True, False])

    The ``>=`` operator can be used as a shorthand for ``np.greater_equal``
    on ndarrays.

    >>> a = np.array([4, 2, 1])
    >>> b = np.array([2, 2, 2])
    >>> a >= b
    array([ True,  True, False])

    """)

add_newdoc('numpy.core.umath', 'hypot',
    """
    Given the "legs" of a right triangle, return its hypotenuse.

    Equivalent to ``sqrt(x1**2 + x2**2)``, element-wise.  If `x1` or
    `x2` is scalar_like (i.e., unambiguously cast-able to a scalar type),
    it is broadcast for use with each element of the other argument.
    (See Examples)

    Parameters
    ----------
    x1, x2 : array_like
        Leg of the triangle(s).
        $BROADCASTABLE_2
    $PARAMS

    Returns
    -------
    z : ndarray
        The hypotenuse of the triangle(s).
        $OUT_SCALAR_2

    Examples
    --------
    >>> np.hypot(3*np.ones((3, 3)), 4*np.ones((3, 3)))
    array([[ 5.,  5.,  5.],
           [ 5.,  5.,  5.],
           [ 5.,  5.,  5.]])

    Example showing broadcast of scalar_like argument:

    >>> np.hypot(3*np.ones((3, 3)), [4])
    array([[ 5.,  5.,  5.],
           [ 5.,  5.,  5.],
           [ 5.,  5.,  5.]])

    """)

add_newdoc('numpy.core.umath', 'invert',
    """
    Compute bit-wise inversion, or bit-wise NOT, element-wise.

    Computes the bit-wise NOT of the underlying binary representation of
    the integers in the input arrays. This ufunc implements the C/Python
    operator ``~``.

    For signed integer inputs, the two's complement is returned.  In a
    two's-complement system negative numbers are represented by the two's
    complement of the absolute value. This is the most common method of
    representing signed integers on computers [1]_. A N-bit
    two's-complement system can represent every integer in the range
    :math:`-2^{N-1}` to :math:`+2^{N-1}-1`.

    Parameters
    ----------
    x : array_like
        Only integer and boolean types are handled.
    $PARAMS

    Returns
    -------
    out : ndarray or scalar
        Result.
        $OUT_SCALAR_1

    See Also
    --------
    bitwise_and, bitwise_or, bitwise_xor
    logical_not
    binary_repr :
        Return the binary representation of the input number as a string.

    Notes
    -----
    `bitwise_not` is an alias for `invert`:

    >>> np.bitwise_not is np.invert
    True

    References
    ----------
    .. [1] Wikipedia, "Two's complement",
        https://en.wikipedia.org/wiki/Two's_complement

    Examples
    --------
    We've seen that 13 is represented by ``00001101``.
    The invert or bit-wise NOT of 13 is then:

    >>> x = np.invert(np.array(13, dtype=np.uint8))
    >>> x
    242
    >>> np.binary_repr(x, width=8)
    '11110010'

    The result depends on the bit-width:

    >>> x = np.invert(np.array(13, dtype=np.uint16))
    >>> x
    65522
    >>> np.binary_repr(x, width=16)
    '1111111111110010'

    When using signed integer types the result is the two's complement of
    the result for the unsigned type:

    >>> np.invert(np.array([13], dtype=np.int8))
    array([-14], dtype=int8)
    >>> np.binary_repr(-14, width=8)
    '11110010'

    Booleans are accepted as well:

    >>> np.invert(np.array([True, False]))
    array([False,  True])

    The ``~`` operator can be used as a shorthand for ``np.invert`` on
    ndarrays.

    >>> x1 = np.array([True, False])
    >>> ~x1
    array([False,  True])

    """)

add_newdoc('numpy.core.umath', 'isfinite',
    """
    Test element-wise for finiteness (not infinity and not Not a Number).

    The result is returned as a boolean array.

    Parameters
    ----------
    x : array_like
        Input values.
    $PARAMS

    Returns
    -------
    y : ndarray, bool
        True where ``x`` is not positive infinity, negative infinity,
        or NaN; false otherwise.
        $OUT_SCALAR_1

    See Also
    --------
    isinf, isneginf, isposinf, isnan

    Notes
    -----
    Not a Number, positive infinity and negative infinity are considered
    to be non-finite.

    NumPy uses the IEEE Standard for Binary Floating-Point for Arithmetic
    (IEEE 754). This means that Not a Number is not equivalent to infinity.
    Also that positive infinity is not equivalent to negative infinity. But
    infinity is equivalent to positive infinity.  Errors result if the
    second argument is also supplied when `x` is a scalar input, or if
    first and second arguments have different shapes.

    Examples
    --------
    >>> np.isfinite(1)
    True
    >>> np.isfinite(0)
    True
    >>> np.isfinite(np.nan)
    False
    >>> np.isfinite(np.inf)
    False
    >>> np.isfinite(np.NINF)
    False
    >>> np.isfinite([np.log(-1.),1.,np.log(0)])
    array([False,  True, False])

    >>> x = np.array([-np.inf, 0., np.inf])
    >>> y = np.array([2, 2, 2])
    >>> np.isfinite(x, y)
    array([0, 1, 0])
    >>> y
    array([0, 1, 0])

    """)

add_newdoc('numpy.core.umath', 'isinf',
    """
    Test element-wise for positive or negative infinity.

    Returns a boolean array of the same shape as `x`, True where ``x ==
    +/-inf``, otherwise False.

    Parameters
    ----------
    x : array_like
        Input values
    $PARAMS

    Returns
    -------
    y : bool (scalar) or boolean ndarray
        True where ``x`` is positive or negative infinity, false otherwise.
        $OUT_SCALAR_1

    See Also
    --------
    isneginf, isposinf, isnan, isfinite

    Notes
    -----
    NumPy uses the IEEE Standard for Binary Floating-Point for Arithmetic
    (IEEE 754).

    Errors result if the second argument is supplied when the first
    argument is a scalar, or if the first and second arguments have
    different shapes.

    Examples
    --------
    >>> np.isinf(np.inf)
    True
    >>> np.isinf(np.nan)
    False
    >>> np.isinf(np.NINF)
    True
    >>> np.isinf([np.inf, -np.inf, 1.0, np.nan])
    array([ True,  True, False, False])

    >>> x = np.array([-np.inf, 0., np.inf])
    >>> y = np.array([2, 2, 2])
    >>> np.isinf(x, y)
    array([1, 0, 1])
    >>> y
    array([1, 0, 1])

    """)

add_newdoc('numpy.core.umath', 'isnan',
    """
    Test element-wise for NaN and return result as a boolean array.

    Parameters
    ----------
    x : array_like
        Input array.
    $PARAMS

    Returns
    -------
    y : ndarray or bool
        True where ``x`` is NaN, false otherwise.
        $OUT_SCALAR_1

    See Also
    --------
    isinf, isneginf, isposinf, isfinite, isnat

    Notes
    -----
    NumPy uses the IEEE Standard for Binary Floating-Point for Arithmetic
    (IEEE 754). This means that Not a Number is not equivalent to infinity.

    Examples
    --------
    >>> np.isnan(np.nan)
    True
    >>> np.isnan(np.inf)
    False
    >>> np.isnan([np.log(-1.),1.,np.log(0)])
    array([ True, False, False])

    """)

add_newdoc('numpy.core.umath', 'isnat',
    """
    Test element-wise for NaT (not a time) and return result as a boolean array.

    .. versionadded:: 1.13.0

    Parameters
    ----------
    x : array_like
        Input array with datetime or timedelta data type.
    $PARAMS

    Returns
    -------
    y : ndarray or bool
        True where ``x`` is NaT, false otherwise.
        $OUT_SCALAR_1

    See Also
    --------
    isnan, isinf, isneginf, isposinf, isfinite

    Examples
    --------
    >>> np.isnat(np.datetime64("NaT"))
    True
    >>> np.isnat(np.datetime64("2016-01-01"))
    False
    >>> np.isnat(np.array(["NaT", "2016-01-01"], dtype="datetime64[ns]"))
    array([ True, False])

    """)

add_newdoc('numpy.core.umath', 'left_shift',
    """
    Shift the bits of an integer to the left.

    Bits are shifted to the left by appending `x2` 0s at the right of `x1`.
    Since the internal representation of numbers is in binary format, this
    operation is equivalent to multiplying `x1` by ``2**x2``.

    Parameters
    ----------
    x1 : array_like of integer type
        Input values.
    x2 : array_like of integer type
        Number of zeros to append to `x1`. Has to be non-negative.
        $BROADCASTABLE_2
    $PARAMS

    Returns
    -------
    out : array of integer type
        Return `x1` with bits shifted `x2` times to the left.
        $OUT_SCALAR_2

    See Also
    --------
    right_shift : Shift the bits of an integer to the right.
    binary_repr : Return the binary representation of the input number
        as a string.

    Examples
    --------
    >>> np.binary_repr(5)
    '101'
    >>> np.left_shift(5, 2)
    20
    >>> np.binary_repr(20)
    '10100'

    >>> np.left_shift(5, [1,2,3])
    array([10, 20, 40])

    Note that the dtype of the second argument may change the dtype of the
    result and can lead to unexpected results in some cases (see
    :ref:`Casting Rules <ufuncs.casting>`):

    >>> a = np.left_shift(np.uint8(255), 1) # Expect 254
    >>> print(a, type(a)) # Unexpected result due to upcasting
    510 <class 'numpy.int64'>
    >>> b = np.left_shift(np.uint8(255), np.uint8(1))
    >>> print(b, type(b))
    254 <class 'numpy.uint8'>

    The ``<<`` operator can be used as a shorthand for ``np.left_shift`` on
    ndarrays.

    >>> x1 = 5
    >>> x2 = np.array([1, 2, 3])
    >>> x1 << x2
    array([10, 20, 40])

    """)

add_newdoc('numpy.core.umath', 'less',
    """
    Return the truth value of (x1 < x2) element-wise.

    Parameters
    ----------
    x1, x2 : array_like
        Input arrays.
        $BROADCASTABLE_2
    $PARAMS

    Returns
    -------
    out : ndarray or scalar
        Output array, element-wise comparison of `x1` and `x2`.
        Typically of type bool, unless ``dtype=object`` is passed.
        $OUT_SCALAR_2

    See Also
    --------
    greater, less_equal, greater_equal, equal, not_equal

    Examples
    --------
    >>> np.less([1, 2], [2, 2])
    array([ True, False])

    The ``<`` operator can be used as a shorthand for ``np.less`` on ndarrays.

    >>> a = np.array([1, 2])
    >>> b = np.array([2, 2])
    >>> a < b
    array([ True, False])

    """)

add_newdoc('numpy.core.umath', 'less_equal',
    """
    Return the truth value of (x1 <= x2) element-wise.

    Parameters
    ----------
    x1, x2 : array_like
        Input arrays.
        $BROADCASTABLE_2
    $PARAMS

    Returns
    -------
    out : ndarray or scalar
        Output array, element-wise comparison of `x1` and `x2`.
        Typically of type bool, unless ``dtype=object`` is passed.
        $OUT_SCALAR_2

    See Also
    --------
    greater, less, greater_equal, equal, not_equal

    Examples
    --------
    >>> np.less_equal([4, 2, 1], [2, 2, 2])
    array([False,  True,  True])

    The ``<=`` operator can be used as a shorthand for ``np.less_equal`` on
    ndarrays.

    >>> a = np.array([4, 2, 1])
    >>> b = np.array([2, 2, 2])
    >>> a <= b
    array([False,  True,  True])

    """)

add_newdoc('numpy.core.umath', 'log',
    """
    Natural logarithm, element-wise.

    The natural logarithm `log` is the inverse of the exponential function,
    so that `log(exp(x)) = x`. The natural logarithm is logarithm in base
    `e`.

    Parameters
    ----------
    x : array_like
        Input value.
    $PARAMS

    Returns
    -------
    y : ndarray
        The natural logarithm of `x`, element-wise.
        $OUT_SCALAR_1

    See Also
    --------
    log10, log2, log1p, emath.log

    Notes
    -----
    Logarithm is a multivalued function: for each `x` there is an infinite
    number of `z` such that `exp(z) = x`. The convention is to return the
    `z` whose imaginary part lies in `(-pi, pi]`.

    For real-valued input data types, `log` always returns real output. For
    each value that cannot be expressed as a real number or infinity, it
    yields ``nan`` and sets the `invalid` floating point error flag.

    For complex-valued input, `log` is a complex analytical function that
    has a branch cut `[-inf, 0]` and is continuous from above on it. `log`
    handles the floating-point negative zero as an infinitesimal negative
    number, conforming to the C99 standard.
    
    In the cases where the input has a negative real part and a very small
    negative complex part (approaching 0), the result is so close to `-pi`
    that it evaluates to exactly `-pi`.

    References
    ----------
    .. [1] M. Abramowitz and I.A. Stegun, "Handbook of Mathematical Functions",
           10th printing, 1964, pp. 67.
           https://personal.math.ubc.ca/~cbm/aands/page_67.htm
    .. [2] Wikipedia, "Logarithm". https://en.wikipedia.org/wiki/Logarithm

    Examples
    --------
    >>> np.log([1, np.e, np.e**2, 0])
    array([  0.,   1.,   2., -Inf])

    """)

add_newdoc('numpy.core.umath', 'log10',
    """
    Return the base 10 logarithm of the input array, element-wise.

    Parameters
    ----------
    x : array_like
        Input values.
    $PARAMS

    Returns
    -------
    y : ndarray
        The logarithm to the base 10 of `x`, element-wise. NaNs are
        returned where x is negative.
        $OUT_SCALAR_1

    See Also
    --------
    emath.log10

    Notes
    -----
    Logarithm is a multivalued function: for each `x` there is an infinite
    number of `z` such that `10**z = x`. The convention is to return the
    `z` whose imaginary part lies in `(-pi, pi]`.

    For real-valued input data types, `log10` always returns real output.
    For each value that cannot be expressed as a real number or infinity,
    it yields ``nan`` and sets the `invalid` floating point error flag.

    For complex-valued input, `log10` is a complex analytical function that
    has a branch cut `[-inf, 0]` and is continuous from above on it.
    `log10` handles the floating-point negative zero as an infinitesimal
    negative number, conforming to the C99 standard.

    In the cases where the input has a negative real part and a very small
    negative complex part (approaching 0), the result is so close to `-pi`
    that it evaluates to exactly `-pi`.

    References
    ----------
    .. [1] M. Abramowitz and I.A. Stegun, "Handbook of Mathematical Functions",
           10th printing, 1964, pp. 67.
           https://personal.math.ubc.ca/~cbm/aands/page_67.htm
    .. [2] Wikipedia, "Logarithm". https://en.wikipedia.org/wiki/Logarithm

    Examples
    --------
    >>> np.log10([1e-15, -3.])
    array([-15.,  nan])

    """)

add_newdoc('numpy.core.umath', 'log2',
    """
    Base-2 logarithm of `x`.

    Parameters
    ----------
    x : array_like
        Input values.
    $PARAMS

    Returns
    -------
    y : ndarray
        Base-2 logarithm of `x`.
        $OUT_SCALAR_1

    See Also
    --------
    log, log10, log1p, emath.log2

    Notes
    -----
    .. versionadded:: 1.3.0

    Logarithm is a multivalued function: for each `x` there is an infinite
    number of `z` such that `2**z = x`. The convention is to return the `z`
    whose imaginary part lies in `(-pi, pi]`.

    For real-valued input data types, `log2` always returns real output.
    For each value that cannot be expressed as a real number or infinity,
    it yields ``nan`` and sets the `invalid` floating point error flag.

    For complex-valued input, `log2` is a complex analytical function that
    has a branch cut `[-inf, 0]` and is continuous from above on it. `log2`
    handles the floating-point negative zero as an infinitesimal negative
    number, conforming to the C99 standard.

    In the cases where the input has a negative real part and a very small
    negative complex part (approaching 0), the result is so close to `-pi`
    that it evaluates to exactly `-pi`.

    Examples
    --------
    >>> x = np.array([0, 1, 2, 2**4])
    >>> np.log2(x)
    array([-Inf,   0.,   1.,   4.])

    >>> xi = np.array([0+1.j, 1, 2+0.j, 4.j])
    >>> np.log2(xi)
    array([ 0.+2.26618007j,  0.+0.j        ,  1.+0.j        ,  2.+2.26618007j])

    """)

add_newdoc('numpy.core.umath', 'logaddexp',
    """
    Logarithm of the sum of exponentiations of the inputs.

    Calculates ``log(exp(x1) + exp(x2))``. This function is useful in
    statistics where the calculated probabilities of events may be so small
    as to exceed the range of normal floating point numbers.  In such cases
    the logarithm of the calculated probability is stored. This function
    allows adding probabilities stored in such a fashion.

    Parameters
    ----------
    x1, x2 : array_like
        Input values.
        $BROADCASTABLE_2
    $PARAMS

    Returns
    -------
    result : ndarray
        Logarithm of ``exp(x1) + exp(x2)``.
        $OUT_SCALAR_2

    See Also
    --------
    logaddexp2: Logarithm of the sum of exponentiations of inputs in base 2.

    Notes
    -----
    .. versionadded:: 1.3.0

    Examples
    --------
    >>> prob1 = np.log(1e-50)
    >>> prob2 = np.log(2.5e-50)
    >>> prob12 = np.logaddexp(prob1, prob2)
    >>> prob12
    -113.87649168120691
    >>> np.exp(prob12)
    3.5000000000000057e-50

    """)

add_newdoc('numpy.core.umath', 'logaddexp2',
    """
    Logarithm of the sum of exponentiations of the inputs in base-2.

    Calculates ``log2(2**x1 + 2**x2)``. This function is useful in machine
    learning when the calculated probabilities of events may be so small as
    to exceed the range of normal floating point numbers.  In such cases
    the base-2 logarithm of the calculated probability can be used instead.
    This function allows adding probabilities stored in such a fashion.

    Parameters
    ----------
    x1, x2 : array_like
        Input values.
        $BROADCASTABLE_2
    $PARAMS

    Returns
    -------
    result : ndarray
        Base-2 logarithm of ``2**x1 + 2**x2``.
        $OUT_SCALAR_2

    See Also
    --------
    logaddexp: Logarithm of the sum of exponentiations of the inputs.

    Notes
    -----
    .. versionadded:: 1.3.0

    Examples
    --------
    >>> prob1 = np.log2(1e-50)
    >>> prob2 = np.log2(2.5e-50)
    >>> prob12 = np.logaddexp2(prob1, prob2)
    >>> prob1, prob2, prob12
    (-166.09640474436813, -164.77447664948076, -164.28904982231052)
    >>> 2**prob12
    3.4999999999999914e-50

    """)

add_newdoc('numpy.core.umath', 'log1p',
    """
    Return the natural logarithm of one plus the input array, element-wise.

    Calculates ``log(1 + x)``.

    Parameters
    ----------
    x : array_like
        Input values.
    $PARAMS

    Returns
    -------
    y : ndarray
        Natural logarithm of `1 + x`, element-wise.
        $OUT_SCALAR_1

    See Also
    --------
    expm1 : ``exp(x) - 1``, the inverse of `log1p`.

    Notes
    -----
    For real-valued input, `log1p` is accurate also for `x` so small
    that `1 + x == 1` in floating-point accuracy.

    Logarithm is a multivalued function: for each `x` there is an infinite
    number of `z` such that `exp(z) = 1 + x`. The convention is to return
    the `z` whose imaginary part lies in `[-pi, pi]`.

    For real-valued input data types, `log1p` always returns real output.
    For each value that cannot be expressed as a real number or infinity,
    it yields ``nan`` and sets the `invalid` floating point error flag.

    For complex-valued input, `log1p` is a complex analytical function that
    has a branch cut `[-inf, -1]` and is continuous from above on it.
    `log1p` handles the floating-point negative zero as an infinitesimal
    negative number, conforming to the C99 standard.

    References
    ----------
    .. [1] M. Abramowitz and I.A. Stegun, "Handbook of Mathematical Functions",
           10th printing, 1964, pp. 67.
           https://personal.math.ubc.ca/~cbm/aands/page_67.htm
    .. [2] Wikipedia, "Logarithm". https://en.wikipedia.org/wiki/Logarithm

    Examples
    --------
    >>> np.log1p(1e-99)
    1e-99
    >>> np.log(1 + 1e-99)
    0.0

    """)

add_newdoc('numpy.core.umath', 'logical_and',
    """
    Compute the truth value of x1 AND x2 element-wise.

    Parameters
    ----------
    x1, x2 : array_like
        Input arrays.
        $BROADCASTABLE_2
    $PARAMS

    Returns
    -------
    y : ndarray or bool
        Boolean result of the logical AND operation applied to the elements
        of `x1` and `x2`; the shape is determined by broadcasting.
        $OUT_SCALAR_2

    See Also
    --------
    logical_or, logical_not, logical_xor
    bitwise_and

    Examples
    --------
    >>> np.logical_and(True, False)
    False
    >>> np.logical_and([True, False], [False, False])
    array([False, False])

    >>> x = np.arange(5)
    >>> np.logical_and(x>1, x<4)
    array([False, False,  True,  True, False])


    The ``&`` operator can be used as a shorthand for ``np.logical_and`` on
    boolean ndarrays.

    >>> a = np.array([True, False])
    >>> b = np.array([False, False])
    >>> a & b
    array([False, False])

    """)

add_newdoc('numpy.core.umath', 'logical_not',
    """
    Compute the truth value of NOT x element-wise.

    Parameters
    ----------
    x : array_like
        Logical NOT is applied to the elements of `x`.
    $PARAMS

    Returns
    -------
    y : bool or ndarray of bool
        Boolean result with the same shape as `x` of the NOT operation
        on elements of `x`.
        $OUT_SCALAR_1

    See Also
    --------
    logical_and, logical_or, logical_xor

    Examples
    --------
    >>> np.logical_not(3)
    False
    >>> np.logical_not([True, False, 0, 1])
    array([False,  True,  True, False])

    >>> x = np.arange(5)
    >>> np.logical_not(x<3)
    array([False, False, False,  True,  True])

    """)

add_newdoc('numpy.core.umath', 'logical_or',
    """
    Compute the truth value of x1 OR x2 element-wise.

    Parameters
    ----------
    x1, x2 : array_like
        Logical OR is applied to the elements of `x1` and `x2`.
        $BROADCASTABLE_2
    $PARAMS

    Returns
    -------
    y : ndarray or bool
        Boolean result of the logical OR operation applied to the elements
        of `x1` and `x2`; the shape is determined by broadcasting.
        $OUT_SCALAR_2

    See Also
    --------
    logical_and, logical_not, logical_xor
    bitwise_or

    Examples
    --------
    >>> np.logical_or(True, False)
    True
    >>> np.logical_or([True, False], [False, False])
    array([ True, False])

    >>> x = np.arange(5)
    >>> np.logical_or(x < 1, x > 3)
    array([ True, False, False, False,  True])

    The ``|`` operator can be used as a shorthand for ``np.logical_or`` on
    boolean ndarrays.

    >>> a = np.array([True, False])
    >>> b = np.array([False, False])
    >>> a | b
    array([ True, False])

    """)

add_newdoc('numpy.core.umath', 'logical_xor',
    """
    Compute the truth value of x1 XOR x2, element-wise.

    Parameters
    ----------
    x1, x2 : array_like
        Logical XOR is applied to the elements of `x1` and `x2`.
        $BROADCASTABLE_2
    $PARAMS

    Returns
    -------
    y : bool or ndarray of bool
        Boolean result of the logical XOR operation applied to the elements
        of `x1` and `x2`; the shape is determined by broadcasting.
        $OUT_SCALAR_2

    See Also
    --------
    logical_and, logical_or, logical_not, bitwise_xor

    Examples
    --------
    >>> np.logical_xor(True, False)
    True
    >>> np.logical_xor([True, True, False, False], [True, False, True, False])
    array([False,  True,  True, False])

    >>> x = np.arange(5)
    >>> np.logical_xor(x < 1, x > 3)
    array([ True, False, False, False,  True])

    Simple example showing support of broadcasting

    >>> np.logical_xor(0, np.eye(2))
    array([[ True, False],
           [False,  True]])

    """)

add_newdoc('numpy.core.umath', 'maximum',
    """
    Element-wise maximum of array elements.

    Compare two arrays and returns a new array containing the element-wise
    maxima. If one of the elements being compared is a NaN, then that
    element is returned. If both elements are NaNs then the first is
    returned. The latter distinction is important for complex NaNs, which
    are defined as at least one of the real or imaginary parts being a NaN.
    The net effect is that NaNs are propagated.

    Parameters
    ----------
    x1, x2 : array_like
        The arrays holding the elements to be compared.
        $BROADCASTABLE_2
    $PARAMS

    Returns
    -------
    y : ndarray or scalar
        The maximum of `x1` and `x2`, element-wise.
        $OUT_SCALAR_2

    See Also
    --------
    minimum :
        Element-wise minimum of two arrays, propagates NaNs.
    fmax :
        Element-wise maximum of two arrays, ignores NaNs.
    amax :
        The maximum value of an array along a given axis, propagates NaNs.
    nanmax :
        The maximum value of an array along a given axis, ignores NaNs.

    fmin, amin, nanmin

    Notes
    -----
    The maximum is equivalent to ``np.where(x1 >= x2, x1, x2)`` when
    neither x1 nor x2 are nans, but it is faster and does proper
    broadcasting.

    Examples
    --------
    >>> np.maximum([2, 3, 4], [1, 5, 2])
    array([2, 5, 4])

    >>> np.maximum(np.eye(2), [0.5, 2]) # broadcasting
    array([[ 1. ,  2. ],
           [ 0.5,  2. ]])

    >>> np.maximum([np.nan, 0, np.nan], [0, np.nan, np.nan])
    array([nan, nan, nan])
    >>> np.maximum(np.Inf, 1)
    inf

    """)

add_newdoc('numpy.core.umath', 'minimum',
    """
    Element-wise minimum of array elements.

    Compare two arrays and returns a new array containing the element-wise
    minima. If one of the elements being compared is a NaN, then that
    element is returned. If both elements are NaNs then the first is
    returned. The latter distinction is important for complex NaNs, which
    are defined as at least one of the real or imaginary parts being a NaN.
    The net effect is that NaNs are propagated.

    Parameters
    ----------
    x1, x2 : array_like
        The arrays holding the elements to be compared.
        $BROADCASTABLE_2
    $PARAMS

    Returns
    -------
    y : ndarray or scalar
        The minimum of `x1` and `x2`, element-wise.
        $OUT_SCALAR_2

    See Also
    --------
    maximum :
        Element-wise maximum of two arrays, propagates NaNs.
    fmin :
        Element-wise minimum of two arrays, ignores NaNs.
    amin :
        The minimum value of an array along a given axis, propagates NaNs.
    nanmin :
        The minimum value of an array along a given axis, ignores NaNs.

    fmax, amax, nanmax

    Notes
    -----
    The minimum is equivalent to ``np.where(x1 <= x2, x1, x2)`` when
    neither x1 nor x2 are NaNs, but it is faster and does proper
    broadcasting.

    Examples
    --------
    >>> np.minimum([2, 3, 4], [1, 5, 2])
    array([1, 3, 2])

    >>> np.minimum(np.eye(2), [0.5, 2]) # broadcasting
    array([[ 0.5,  0. ],
           [ 0. ,  1. ]])

    >>> np.minimum([np.nan, 0, np.nan],[0, np.nan, np.nan])
    array([nan, nan, nan])
    >>> np.minimum(-np.Inf, 1)
    -inf

    """)

add_newdoc('numpy.core.umath', 'fmax',
    """
    Element-wise maximum of array elements.

    Compare two arrays and returns a new array containing the element-wise
    maxima. If one of the elements being compared is a NaN, then the
    non-nan element is returned. If both elements are NaNs then the first
    is returned.  The latter distinction is important for complex NaNs,
    which are defined as at least one of the real or imaginary parts being
    a NaN. The net effect is that NaNs are ignored when possible.

    Parameters
    ----------
    x1, x2 : array_like
        The arrays holding the elements to be compared.
        $BROADCASTABLE_2
    $PARAMS

    Returns
    -------
    y : ndarray or scalar
        The maximum of `x1` and `x2`, element-wise.
        $OUT_SCALAR_2

    See Also
    --------
    fmin :
        Element-wise minimum of two arrays, ignores NaNs.
    maximum :
        Element-wise maximum of two arrays, propagates NaNs.
    amax :
        The maximum value of an array along a given axis, propagates NaNs.
    nanmax :
        The maximum value of an array along a given axis, ignores NaNs.

    minimum, amin, nanmin

    Notes
    -----
    .. versionadded:: 1.3.0

    The fmax is equivalent to ``np.where(x1 >= x2, x1, x2)`` when neither
    x1 nor x2 are NaNs, but it is faster and does proper broadcasting.

    Examples
    --------
    >>> np.fmax([2, 3, 4], [1, 5, 2])
    array([ 2.,  5.,  4.])

    >>> np.fmax(np.eye(2), [0.5, 2])
    array([[ 1. ,  2. ],
           [ 0.5,  2. ]])

    >>> np.fmax([np.nan, 0, np.nan],[0, np.nan, np.nan])
    array([ 0.,  0., nan])

    """)

add_newdoc('numpy.core.umath', 'fmin',
    """
    Element-wise minimum of array elements.

    Compare two arrays and returns a new array containing the element-wise
    minima. If one of the elements being compared is a NaN, then the
    non-nan element is returned. If both elements are NaNs then the first
    is returned.  The latter distinction is important for complex NaNs,
    which are defined as at least one of the real or imaginary parts being
    a NaN. The net effect is that NaNs are ignored when possible.

    Parameters
    ----------
    x1, x2 : array_like
        The arrays holding the elements to be compared.
        $BROADCASTABLE_2
    $PARAMS

    Returns
    -------
    y : ndarray or scalar
        The minimum of `x1` and `x2`, element-wise.
        $OUT_SCALAR_2

    See Also
    --------
    fmax :
        Element-wise maximum of two arrays, ignores NaNs.
    minimum :
        Element-wise minimum of two arrays, propagates NaNs.
    amin :
        The minimum value of an array along a given axis, propagates NaNs.
    nanmin :
        The minimum value of an array along a given axis, ignores NaNs.

    maximum, amax, nanmax

    Notes
    -----
    .. versionadded:: 1.3.0

    The fmin is equivalent to ``np.where(x1 <= x2, x1, x2)`` when neither
    x1 nor x2 are NaNs, but it is faster and does proper broadcasting.

    Examples
    --------
    >>> np.fmin([2, 3, 4], [1, 5, 2])
    array([1, 3, 2])

    >>> np.fmin(np.eye(2), [0.5, 2])
    array([[ 0.5,  0. ],
           [ 0. ,  1. ]])

    >>> np.fmin([np.nan, 0, np.nan],[0, np.nan, np.nan])
    array([ 0.,  0., nan])

    """)

add_newdoc('numpy.core.umath', 'clip',
    """
    Clip (limit) the values in an array.

    Given an interval, values outside the interval are clipped to
    the interval edges.  For example, if an interval of ``[0, 1]``
    is specified, values smaller than 0 become 0, and values larger
    than 1 become 1.

    Equivalent to but faster than ``np.minimum(np.maximum(a, a_min), a_max)``.

    Parameters
    ----------
    a : array_like
        Array containing elements to clip.
    a_min : array_like
        Minimum value.
    a_max : array_like
        Maximum value.
    out : ndarray, optional
        The results will be placed in this array. It may be the input
        array for in-place clipping.  `out` must be of the right shape
        to hold the output.  Its type is preserved.
    $PARAMS

    See Also
    --------
    numpy.clip :
        Wrapper that makes the ``a_min`` and ``a_max`` arguments optional,
        dispatching to one of `~numpy.core.umath.clip`,
        `~numpy.core.umath.minimum`, and `~numpy.core.umath.maximum`.

    Returns
    -------
    clipped_array : ndarray
        An array with the elements of `a`, but where values
        < `a_min` are replaced with `a_min`, and those > `a_max`
        with `a_max`.
    """)

add_newdoc('numpy.core.umath', 'matmul',
    """
    Matrix product of two arrays.

    Parameters
    ----------
    x1, x2 : array_like
        Input arrays, scalars not allowed.
    out : ndarray, optional
        A location into which the result is stored. If provided, it must have
        a shape that matches the signature `(n,k),(k,m)->(n,m)`. If not
        provided or None, a freshly-allocated array is returned.
    **kwargs
        For other keyword-only arguments, see the
        :ref:`ufunc docs <ufuncs.kwargs>`.

        .. versionadded:: 1.16
           Now handles ufunc kwargs

    Returns
    -------
    y : ndarray
        The matrix product of the inputs.
        This is a scalar only when both x1, x2 are 1-d vectors.

    Raises
    ------
    ValueError
        If the last dimension of `x1` is not the same size as
        the second-to-last dimension of `x2`.

        If a scalar value is passed in.

    See Also
    --------
    vdot : Complex-conjugating dot product.
    tensordot : Sum products over arbitrary axes.
    einsum : Einstein summation convention.
    dot : alternative matrix product with different broadcasting rules.

    Notes
    -----

    The behavior depends on the arguments in the following way.

    - If both arguments are 2-D they are multiplied like conventional
      matrices.
    - If either argument is N-D, N > 2, it is treated as a stack of
      matrices residing in the last two indexes and broadcast accordingly.
    - If the first argument is 1-D, it is promoted to a matrix by
      prepending a 1 to its dimensions. After matrix multiplication
      the prepended 1 is removed.
    - If the second argument is 1-D, it is promoted to a matrix by
      appending a 1 to its dimensions. After matrix multiplication
      the appended 1 is removed.

    ``matmul`` differs from ``dot`` in two important ways:

    - Multiplication by scalars is not allowed, use ``*`` instead.
    - Stacks of matrices are broadcast together as if the matrices
      were elements, respecting the signature ``(n,k),(k,m)->(n,m)``:

      >>> a = np.ones([9, 5, 7, 4])
      >>> c = np.ones([9, 5, 4, 3])
      >>> np.dot(a, c).shape
      (9, 5, 7, 9, 5, 3)
      >>> np.matmul(a, c).shape
      (9, 5, 7, 3)
      >>> # n is 7, k is 4, m is 3

    The matmul function implements the semantics of the ``@`` operator
    introduced in Python 3.5 following :pep:`465`.

    It uses an optimized BLAS library when possible (see `numpy.linalg`).

    Examples
    --------
    For 2-D arrays it is the matrix product:

    >>> a = np.array([[1, 0],
    ...               [0, 1]])
    >>> b = np.array([[4, 1],
    ...               [2, 2]])
    >>> np.matmul(a, b)
    array([[4, 1],
           [2, 2]])

    For 2-D mixed with 1-D, the result is the usual.

    >>> a = np.array([[1, 0],
    ...               [0, 1]])
    >>> b = np.array([1, 2])
    >>> np.matmul(a, b)
    array([1, 2])
    >>> np.matmul(b, a)
    array([1, 2])


    Broadcasting is conventional for stacks of arrays

    >>> a = np.arange(2 * 2 * 4).reshape((2, 2, 4))
    >>> b = np.arange(2 * 2 * 4).reshape((2, 4, 2))
    >>> np.matmul(a,b).shape
    (2, 2, 2)
    >>> np.matmul(a, b)[0, 1, 1]
    98
    >>> sum(a[0, 1, :] * b[0 , :, 1])
    98

    Vector, vector returns the scalar inner product, but neither argument
    is complex-conjugated:

    >>> np.matmul([2j, 3j], [2j, 3j])
    (-13+0j)

    Scalar multiplication raises an error.

    >>> np.matmul([1,2], 3)
    Traceback (most recent call last):
    ...
    ValueError: matmul: Input operand 1 does not have enough dimensions ...

    The ``@`` operator can be used as a shorthand for ``np.matmul`` on
    ndarrays.

    >>> x1 = np.array([2j, 3j])
    >>> x2 = np.array([2j, 3j])
    >>> x1 @ x2
    (-13+0j)

    .. versionadded:: 1.10.0
    """)

add_newdoc('numpy.core.umath', 'modf',
    """
    Return the fractional and integral parts of an array, element-wise.

    The fractional and integral parts are negative if the given number is
    negative.

    Parameters
    ----------
    x : array_like
        Input array.
    $PARAMS

    Returns
    -------
    y1 : ndarray
        Fractional part of `x`.
        $OUT_SCALAR_1
    y2 : ndarray
        Integral part of `x`.
        $OUT_SCALAR_1

    Notes
    -----
    For integer input the return values are floats.

    See Also
    --------
    divmod : ``divmod(x, 1)`` is equivalent to ``modf`` with the return values
             switched, except it always has a positive remainder.

    Examples
    --------
    >>> np.modf([0, 3.5])
    (array([ 0. ,  0.5]), array([ 0.,  3.]))
    >>> np.modf(-0.5)
    (-0.5, -0)

    """)

add_newdoc('numpy.core.umath', 'multiply',
    """
    Multiply arguments element-wise.

    Parameters
    ----------
    x1, x2 : array_like
        Input arrays to be multiplied.
        $BROADCASTABLE_2
    $PARAMS

    Returns
    -------
    y : ndarray
        The product of `x1` and `x2`, element-wise.
        $OUT_SCALAR_2

    Notes
    -----
    Equivalent to `x1` * `x2` in terms of array broadcasting.

    Examples
    --------
    >>> np.multiply(2.0, 4.0)
    8.0

    >>> x1 = np.arange(9.0).reshape((3, 3))
    >>> x2 = np.arange(3.0)
    >>> np.multiply(x1, x2)
    array([[  0.,   1.,   4.],
           [  0.,   4.,  10.],
           [  0.,   7.,  16.]])

    The ``*`` operator can be used as a shorthand for ``np.multiply`` on
    ndarrays.

    >>> x1 = np.arange(9.0).reshape((3, 3))
    >>> x2 = np.arange(3.0)
    >>> x1 * x2
    array([[  0.,   1.,   4.],
           [  0.,   4.,  10.],
           [  0.,   7.,  16.]])

    """)

add_newdoc('numpy.core.umath', 'negative',
    """
    Numerical negative, element-wise.

    Parameters
    ----------
    x : array_like or scalar
        Input array.
    $PARAMS

    Returns
    -------
    y : ndarray or scalar
        Returned array or scalar: `y = -x`.
        $OUT_SCALAR_1

    Examples
    --------
    >>> np.negative([1.,-1.])
    array([-1.,  1.])

    The unary ``-`` operator can be used as a shorthand for ``np.negative`` on
    ndarrays.

    >>> x1 = np.array(([1., -1.]))
    >>> -x1
    array([-1.,  1.])

    """)

add_newdoc('numpy.core.umath', 'positive',
    """
    Numerical positive, element-wise.

    .. versionadded:: 1.13.0

    Parameters
    ----------
    x : array_like or scalar
        Input array.

    Returns
    -------
    y : ndarray or scalar
        Returned array or scalar: `y = +x`.
        $OUT_SCALAR_1

    Notes
    -----
    Equivalent to `x.copy()`, but only defined for types that support
    arithmetic.

    Examples
    --------

    >>> x1 = np.array(([1., -1.]))
    >>> np.positive(x1)
    array([ 1., -1.])

    The unary ``+`` operator can be used as a shorthand for ``np.positive`` on
    ndarrays.

    >>> x1 = np.array(([1., -1.]))
    >>> +x1
    array([ 1., -1.])

    """)

add_newdoc('numpy.core.umath', 'not_equal',
    """
    Return (x1 != x2) element-wise.

    Parameters
    ----------
    x1, x2 : array_like
        Input arrays.
        $BROADCASTABLE_2
    $PARAMS

    Returns
    -------
    out : ndarray or scalar
        Output array, element-wise comparison of `x1` and `x2`.
        Typically of type bool, unless ``dtype=object`` is passed.
        $OUT_SCALAR_2

    See Also
    --------
    equal, greater, greater_equal, less, less_equal

    Examples
    --------
    >>> np.not_equal([1.,2.], [1., 3.])
    array([False,  True])
    >>> np.not_equal([1, 2], [[1, 3],[1, 4]])
    array([[False,  True],
           [False,  True]])

    The ``!=`` operator can be used as a shorthand for ``np.not_equal`` on
    ndarrays.

    >>> a = np.array([1., 2.])
    >>> b = np.array([1., 3.])
    >>> a != b
    array([False,  True])


    """)

add_newdoc('numpy.core.umath', '_ones_like',
    """
    This function used to be the numpy.ones_like, but now a specific
    function for that has been written for consistency with the other
    *_like functions. It is only used internally in a limited fashion now.

    See Also
    --------
    ones_like

    """)

add_newdoc('numpy.core.umath', 'power',
    """
    First array elements raised to powers from second array, element-wise.

    Raise each base in `x1` to the positionally-corresponding power in
    `x2`.  `x1` and `x2` must be broadcastable to the same shape.

    An integer type raised to a negative integer power will raise a
    ``ValueError``.

    Negative values raised to a non-integral value will return ``nan``.
    To get complex results, cast the input to complex, or specify the
    ``dtype`` to be ``complex`` (see the example below).

    Parameters
    ----------
    x1 : array_like
        The bases.
    x2 : array_like
        The exponents.
        $BROADCASTABLE_2
    $PARAMS

    Returns
    -------
    y : ndarray
        The bases in `x1` raised to the exponents in `x2`.
        $OUT_SCALAR_2

    See Also
    --------
    float_power : power function that promotes integers to float

    Examples
    --------
    Cube each element in an array.

    >>> x1 = np.arange(6)
    >>> x1
    [0, 1, 2, 3, 4, 5]
    >>> np.power(x1, 3)
    array([  0,   1,   8,  27,  64, 125])

    Raise the bases to different exponents.

    >>> x2 = [1.0, 2.0, 3.0, 3.0, 2.0, 1.0]
    >>> np.power(x1, x2)
    array([  0.,   1.,   8.,  27.,  16.,   5.])

    The effect of broadcasting.

    >>> x2 = np.array([[1, 2, 3, 3, 2, 1], [1, 2, 3, 3, 2, 1]])
    >>> x2
    array([[1, 2, 3, 3, 2, 1],
           [1, 2, 3, 3, 2, 1]])
    >>> np.power(x1, x2)
    array([[ 0,  1,  8, 27, 16,  5],
           [ 0,  1,  8, 27, 16,  5]])

    The ``**`` operator can be used as a shorthand for ``np.power`` on
    ndarrays.

    >>> x2 = np.array([1, 2, 3, 3, 2, 1])
    >>> x1 = np.arange(6)
    >>> x1 ** x2
    array([ 0,  1,  8, 27, 16,  5])

    Negative values raised to a non-integral value will result in ``nan``
    (and a warning will be generated).

    >>> x3 = np.array([-1.0, -4.0])
    >>> with np.errstate(invalid='ignore'):
    ...     p = np.power(x3, 1.5)
    ...
    >>> p
    array([nan, nan])

    To get complex results, give the argument ``dtype=complex``.

    >>> np.power(x3, 1.5, dtype=complex)
    array([-1.83697020e-16-1.j, -1.46957616e-15-8.j])

    """)

add_newdoc('numpy.core.umath', 'float_power',
    """
    First array elements raised to powers from second array, element-wise.

    Raise each base in `x1` to the positionally-corresponding power in `x2`.
    `x1` and `x2` must be broadcastable to the same shape. This differs from
    the power function in that integers, float16, and float32  are promoted to
    floats with a minimum precision of float64 so that the result is always
    inexact.  The intent is that the function will return a usable result for
    negative powers and seldom overflow for positive powers.

    Negative values raised to a non-integral value will return ``nan``.
    To get complex results, cast the input to complex, or specify the
    ``dtype`` to be ``complex`` (see the example below).

    .. versionadded:: 1.12.0

    Parameters
    ----------
    x1 : array_like
        The bases.
    x2 : array_like
        The exponents.
        $BROADCASTABLE_2
    $PARAMS

    Returns
    -------
    y : ndarray
        The bases in `x1` raised to the exponents in `x2`.
        $OUT_SCALAR_2

    See Also
    --------
    power : power function that preserves type

    Examples
    --------
    Cube each element in a list.

    >>> x1 = range(6)
    >>> x1
    [0, 1, 2, 3, 4, 5]
    >>> np.float_power(x1, 3)
    array([   0.,    1.,    8.,   27.,   64.,  125.])

    Raise the bases to different exponents.

    >>> x2 = [1.0, 2.0, 3.0, 3.0, 2.0, 1.0]
    >>> np.float_power(x1, x2)
    array([  0.,   1.,   8.,  27.,  16.,   5.])

    The effect of broadcasting.

    >>> x2 = np.array([[1, 2, 3, 3, 2, 1], [1, 2, 3, 3, 2, 1]])
    >>> x2
    array([[1, 2, 3, 3, 2, 1],
           [1, 2, 3, 3, 2, 1]])
    >>> np.float_power(x1, x2)
    array([[  0.,   1.,   8.,  27.,  16.,   5.],
           [  0.,   1.,   8.,  27.,  16.,   5.]])

    Negative values raised to a non-integral value will result in ``nan``
    (and a warning will be generated).

    >>> x3 = np.array([-1, -4])
    >>> with np.errstate(invalid='ignore'):
    ...     p = np.float_power(x3, 1.5)
    ...
    >>> p
    array([nan, nan])

    To get complex results, give the argument ``dtype=complex``.

    >>> np.float_power(x3, 1.5, dtype=complex)
    array([-1.83697020e-16-1.j, -1.46957616e-15-8.j])

    """)

add_newdoc('numpy.core.umath', 'radians',
    """
    Convert angles from degrees to radians.

    Parameters
    ----------
    x : array_like
        Input array in degrees.
    $PARAMS

    Returns
    -------
    y : ndarray
        The corresponding radian values.
        $OUT_SCALAR_1

    See Also
    --------
    deg2rad : equivalent function

    Examples
    --------
    Convert a degree array to radians

    >>> deg = np.arange(12.) * 30.
    >>> np.radians(deg)
    array([ 0.        ,  0.52359878,  1.04719755,  1.57079633,  2.0943951 ,
            2.61799388,  3.14159265,  3.66519143,  4.1887902 ,  4.71238898,
            5.23598776,  5.75958653])

    >>> out = np.zeros((deg.shape))
    >>> ret = np.radians(deg, out)
    >>> ret is out
    True

    """)

add_newdoc('numpy.core.umath', 'deg2rad',
    """
    Convert angles from degrees to radians.

    Parameters
    ----------
    x : array_like
        Angles in degrees.
    $PARAMS

    Returns
    -------
    y : ndarray
        The corresponding angle in radians.
        $OUT_SCALAR_1

    See Also
    --------
    rad2deg : Convert angles from radians to degrees.
    unwrap : Remove large jumps in angle by wrapping.

    Notes
    -----
    .. versionadded:: 1.3.0

    ``deg2rad(x)`` is ``x * pi / 180``.

    Examples
    --------
    >>> np.deg2rad(180)
    3.1415926535897931

    """)

add_newdoc('numpy.core.umath', 'reciprocal',
    """
    Return the reciprocal of the argument, element-wise.

    Calculates ``1/x``.

    Parameters
    ----------
    x : array_like
        Input array.
    $PARAMS

    Returns
    -------
    y : ndarray
        Return array.
        $OUT_SCALAR_1

    Notes
    -----
    .. note::
        This function is not designed to work with integers.

    For integer arguments with absolute value larger than 1 the result is
    always zero because of the way Python handles integer division.  For
    integer zero the result is an overflow.

    Examples
    --------
    >>> np.reciprocal(2.)
    0.5
    >>> np.reciprocal([1, 2., 3.33])
    array([ 1.       ,  0.5      ,  0.3003003])

    """)

add_newdoc('numpy.core.umath', 'remainder',
    """
    Returns the element-wise remainder of division.

    Computes the remainder complementary to the `floor_divide` function.  It is
    equivalent to the Python modulus operator``x1 % x2`` and has the same sign
    as the divisor `x2`. The MATLAB function equivalent to ``np.remainder``
    is ``mod``.

    .. warning::

        This should not be confused with:

        * Python 3.7's `math.remainder` and C's ``remainder``, which
          computes the IEEE remainder, which are the complement to
          ``round(x1 / x2)``.
        * The MATLAB ``rem`` function and or the C ``%`` operator which is the
          complement to ``int(x1 / x2)``.

    Parameters
    ----------
    x1 : array_like
        Dividend array.
    x2 : array_like
        Divisor array.
        $BROADCASTABLE_2
    $PARAMS

    Returns
    -------
    y : ndarray
        The element-wise remainder of the quotient ``floor_divide(x1, x2)``.
        $OUT_SCALAR_2

    See Also
    --------
    floor_divide : Equivalent of Python ``//`` operator.
    divmod : Simultaneous floor division and remainder.
    fmod : Equivalent of the MATLAB ``rem`` function.
    divide, floor

    Notes
    -----
    Returns 0 when `x2` is 0 and both `x1` and `x2` are (arrays of)
    integers.
    ``mod`` is an alias of ``remainder``.

    Examples
    --------
    >>> np.remainder([4, 7], [2, 3])
    array([0, 1])
    >>> np.remainder(np.arange(7), 5)
    array([0, 1, 2, 3, 4, 0, 1])

    The ``%`` operator can be used as a shorthand for ``np.remainder`` on
    ndarrays.

    >>> x1 = np.arange(7)
    >>> x1 % 5
    array([0, 1, 2, 3, 4, 0, 1])

    """)

add_newdoc('numpy.core.umath', 'divmod',
    """
    Return element-wise quotient and remainder simultaneously.

    .. versionadded:: 1.13.0

    ``np.divmod(x, y)`` is equivalent to ``(x // y, x % y)``, but faster
    because it avoids redundant work. It is used to implement the Python
    built-in function ``divmod`` on NumPy arrays.

    Parameters
    ----------
    x1 : array_like
        Dividend array.
    x2 : array_like
        Divisor array.
        $BROADCASTABLE_2
    $PARAMS

    Returns
    -------
    out1 : ndarray
        Element-wise quotient resulting from floor division.
        $OUT_SCALAR_2
    out2 : ndarray
        Element-wise remainder from floor division.
        $OUT_SCALAR_2

    See Also
    --------
    floor_divide : Equivalent to Python's ``//`` operator.
    remainder : Equivalent to Python's ``%`` operator.
    modf : Equivalent to ``divmod(x, 1)`` for positive ``x`` with the return
           values switched.

    Examples
    --------
    >>> np.divmod(np.arange(5), 3)
    (array([0, 0, 0, 1, 1]), array([0, 1, 2, 0, 1]))

    The `divmod` function can be used as a shorthand for ``np.divmod`` on
    ndarrays.

    >>> x = np.arange(5)
    >>> divmod(x, 3)
    (array([0, 0, 0, 1, 1]), array([0, 1, 2, 0, 1]))

    """)

add_newdoc('numpy.core.umath', 'right_shift',
    """
    Shift the bits of an integer to the right.

    Bits are shifted to the right `x2`.  Because the internal
    representation of numbers is in binary format, this operation is
    equivalent to dividing `x1` by ``2**x2``.

    Parameters
    ----------
    x1 : array_like, int
        Input values.
    x2 : array_like, int
        Number of bits to remove at the right of `x1`.
        $BROADCASTABLE_2
    $PARAMS

    Returns
    -------
    out : ndarray, int
        Return `x1` with bits shifted `x2` times to the right.
        $OUT_SCALAR_2

    See Also
    --------
    left_shift : Shift the bits of an integer to the left.
    binary_repr : Return the binary representation of the input number
        as a string.

    Examples
    --------
    >>> np.binary_repr(10)
    '1010'
    >>> np.right_shift(10, 1)
    5
    >>> np.binary_repr(5)
    '101'

    >>> np.right_shift(10, [1,2,3])
    array([5, 2, 1])

    The ``>>`` operator can be used as a shorthand for ``np.right_shift`` on
    ndarrays.

    >>> x1 = 10
    >>> x2 = np.array([1,2,3])
    >>> x1 >> x2
    array([5, 2, 1])

    """)

add_newdoc('numpy.core.umath', 'rint',
    """
    Round elements of the array to the nearest integer.

    Parameters
    ----------
    x : array_like
        Input array.
    $PARAMS

    Returns
    -------
    out : ndarray or scalar
        Output array is same shape and type as `x`.
        $OUT_SCALAR_1

    See Also
    --------
    fix, ceil, floor, trunc

    Notes
    -----
    For values exactly halfway between rounded decimal values, NumPy
    rounds to the nearest even value. Thus 1.5 and 2.5 round to 2.0,
    -0.5 and 0.5 round to 0.0, etc.

    Examples
    --------
    >>> a = np.array([-1.7, -1.5, -0.2, 0.2, 1.5, 1.7, 2.0])
    >>> np.rint(a)
    array([-2., -2., -0.,  0.,  2.,  2.,  2.])

    """)

add_newdoc('numpy.core.umath', 'sign',
    """
    Returns an element-wise indication of the sign of a number.

    The `sign` function returns ``-1 if x < 0, 0 if x==0, 1 if x > 0``.  nan
    is returned for nan inputs.

    For complex inputs, the `sign` function returns
    ``sign(x.real) + 0j if x.real != 0 else sign(x.imag) + 0j``.

    complex(nan, 0) is returned for complex nan inputs.

    Parameters
    ----------
    x : array_like
        Input values.
    $PARAMS

    Returns
    -------
    y : ndarray
        The sign of `x`.
        $OUT_SCALAR_1

    Notes
    -----
    There is more than one definition of sign in common use for complex
    numbers.  The definition used here is equivalent to :math:`x/\\sqrt{x*x}`
    which is different from a common alternative, :math:`x/|x|`.

    Examples
    --------
    >>> np.sign([-5., 4.5])
    array([-1.,  1.])
    >>> np.sign(0)
    0
    >>> np.sign(5-2j)
    (1+0j)

    """)

add_newdoc('numpy.core.umath', 'signbit',
    """
    Returns element-wise True where signbit is set (less than zero).

    Parameters
    ----------
    x : array_like
        The input value(s).
    $PARAMS

    Returns
    -------
    result : ndarray of bool
        Output array, or reference to `out` if that was supplied.
        $OUT_SCALAR_1

    Examples
    --------
    >>> np.signbit(-1.2)
    True
    >>> np.signbit(np.array([1, -2.3, 2.1]))
    array([False,  True, False])

    """)

add_newdoc('numpy.core.umath', 'copysign',
    """
    Change the sign of x1 to that of x2, element-wise.

    If `x2` is a scalar, its sign will be copied to all elements of `x1`.

    Parameters
    ----------
    x1 : array_like
        Values to change the sign of.
    x2 : array_like
        The sign of `x2` is copied to `x1`.
        $BROADCASTABLE_2
    $PARAMS

    Returns
    -------
    out : ndarray or scalar
        The values of `x1` with the sign of `x2`.
        $OUT_SCALAR_2

    Examples
    --------
    >>> np.copysign(1.3, -1)
    -1.3
    >>> 1/np.copysign(0, 1)
    inf
    >>> 1/np.copysign(0, -1)
    -inf

    >>> np.copysign([-1, 0, 1], -1.1)
    array([-1., -0., -1.])
    >>> np.copysign([-1, 0, 1], np.arange(3)-1)
    array([-1.,  0.,  1.])

    """)

add_newdoc('numpy.core.umath', 'nextafter',
    """
    Return the next floating-point value after x1 towards x2, element-wise.

    Parameters
    ----------
    x1 : array_like
        Values to find the next representable value of.
    x2 : array_like
        The direction where to look for the next representable value of `x1`.
        $BROADCASTABLE_2
    $PARAMS

    Returns
    -------
    out : ndarray or scalar
        The next representable values of `x1` in the direction of `x2`.
        $OUT_SCALAR_2

    Examples
    --------
    >>> eps = np.finfo(np.float64).eps
    >>> np.nextafter(1, 2) == eps + 1
    True
    >>> np.nextafter([1, 2], [2, 1]) == [eps + 1, 2 - eps]
    array([ True,  True])

    """)

add_newdoc('numpy.core.umath', 'spacing',
    """
    Return the distance between x and the nearest adjacent number.

    Parameters
    ----------
    x : array_like
        Values to find the spacing of.
    $PARAMS

    Returns
    -------
    out : ndarray or scalar
        The spacing of values of `x`.
        $OUT_SCALAR_1

    Notes
    -----
    It can be considered as a generalization of EPS:
    ``spacing(np.float64(1)) == np.finfo(np.float64).eps``, and there
    should not be any representable number between ``x + spacing(x)`` and
    x for any finite x.

    Spacing of +- inf and NaN is NaN.

    Examples
    --------
    >>> np.spacing(1) == np.finfo(np.float64).eps
    True

    """)

add_newdoc('numpy.core.umath', 'sin',
    """
    Trigonometric sine, element-wise.

    Parameters
    ----------
    x : array_like
        Angle, in radians (:math:`2 \\pi` rad equals 360 degrees).
    $PARAMS

    Returns
    -------
    y : array_like
        The sine of each element of x.
        $OUT_SCALAR_1

    See Also
    --------
    arcsin, sinh, cos

    Notes
    -----
    The sine is one of the fundamental functions of trigonometry (the
    mathematical study of triangles).  Consider a circle of radius 1
    centered on the origin.  A ray comes in from the :math:`+x` axis, makes
    an angle at the origin (measured counter-clockwise from that axis), and
    departs from the origin.  The :math:`y` coordinate of the outgoing
    ray's intersection with the unit circle is the sine of that angle.  It
    ranges from -1 for :math:`x=3\\pi / 2` to +1 for :math:`\\pi / 2.`  The
    function has zeroes where the angle is a multiple of :math:`\\pi`.
    Sines of angles between :math:`\\pi` and :math:`2\\pi` are negative.
    The numerous properties of the sine and related functions are included
    in any standard trigonometry text.

    Examples
    --------
    Print sine of one angle:

    >>> np.sin(np.pi/2.)
    1.0

    Print sines of an array of angles given in degrees:

    >>> np.sin(np.array((0., 30., 45., 60., 90.)) * np.pi / 180. )
    array([ 0.        ,  0.5       ,  0.70710678,  0.8660254 ,  1.        ])

    Plot the sine function:

    >>> import matplotlib.pylab as plt
    >>> x = np.linspace(-np.pi, np.pi, 201)
    >>> plt.plot(x, np.sin(x))
    >>> plt.xlabel('Angle [rad]')
    >>> plt.ylabel('sin(x)')
    >>> plt.axis('tight')
    >>> plt.show()

    """)

add_newdoc('numpy.core.umath', 'sinh',
    """
    Hyperbolic sine, element-wise.

    Equivalent to ``1/2 * (np.exp(x) - np.exp(-x))`` or
    ``-1j * np.sin(1j*x)``.

    Parameters
    ----------
    x : array_like
        Input array.
    $PARAMS

    Returns
    -------
    y : ndarray
        The corresponding hyperbolic sine values.
        $OUT_SCALAR_1

    Notes
    -----
    If `out` is provided, the function writes the result into it,
    and returns a reference to `out`.  (See Examples)

    References
    ----------
    M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions.
    New York, NY: Dover, 1972, pg. 83.

    Examples
    --------
    >>> np.sinh(0)
    0.0
    >>> np.sinh(np.pi*1j/2)
    1j
    >>> np.sinh(np.pi*1j) # (exact value is 0)
    1.2246063538223773e-016j
    >>> # Discrepancy due to vagaries of floating point arithmetic.

    >>> # Example of providing the optional output parameter
    >>> out1 = np.array([0], dtype='d')
    >>> out2 = np.sinh([0.1], out1)
    >>> out2 is out1
    True

    >>> # Example of ValueError due to provision of shape mis-matched `out`
    >>> np.sinh(np.zeros((3,3)),np.zeros((2,2)))
    Traceback (most recent call last):
      File "<stdin>", line 1, in <module>
    ValueError: operands could not be broadcast together with shapes (3,3) (2,2)

    """)

add_newdoc('numpy.core.umath', 'sqrt',
    """
    Return the non-negative square-root of an array, element-wise.

    Parameters
    ----------
    x : array_like
        The values whose square-roots are required.
    $PARAMS

    Returns
    -------
    y : ndarray
        An array of the same shape as `x`, containing the positive
        square-root of each element in `x`.  If any element in `x` is
        complex, a complex array is returned (and the square-roots of
        negative reals are calculated).  If all of the elements in `x`
        are real, so is `y`, with negative elements returning ``nan``.
        If `out` was provided, `y` is a reference to it.
        $OUT_SCALAR_1

    See Also
    --------
    emath.sqrt
        A version which returns complex numbers when given negative reals.
        Note: 0.0 and -0.0 are handled differently for complex inputs.

    Notes
    -----
    *sqrt* has--consistent with common convention--as its branch cut the
    real "interval" [`-inf`, 0), and is continuous from above on it.
    A branch cut is a curve in the complex plane across which a given
    complex function fails to be continuous.

    Examples
    --------
    >>> np.sqrt([1,4,9])
    array([ 1.,  2.,  3.])

    >>> np.sqrt([4, -1, -3+4J])
    array([ 2.+0.j,  0.+1.j,  1.+2.j])

    >>> np.sqrt([4, -1, np.inf])
    array([ 2., nan, inf])

    """)

add_newdoc('numpy.core.umath', 'cbrt',
    """
    Return the cube-root of an array, element-wise.

    .. versionadded:: 1.10.0

    Parameters
    ----------
    x : array_like
        The values whose cube-roots are required.
    $PARAMS

    Returns
    -------
    y : ndarray
        An array of the same shape as `x`, containing the cube
        cube-root of each element in `x`.
        If `out` was provided, `y` is a reference to it.
        $OUT_SCALAR_1


    Examples
    --------
    >>> np.cbrt([1,8,27])
    array([ 1.,  2.,  3.])

    """)

add_newdoc('numpy.core.umath', 'square',
    """
    Return the element-wise square of the input.

    Parameters
    ----------
    x : array_like
        Input data.
    $PARAMS

    Returns
    -------
    out : ndarray or scalar
        Element-wise `x*x`, of the same shape and dtype as `x`.
        $OUT_SCALAR_1

    See Also
    --------
    numpy.linalg.matrix_power
    sqrt
    power

    Examples
    --------
    >>> np.square([-1j, 1])
    array([-1.-0.j,  1.+0.j])

    """)

add_newdoc('numpy.core.umath', 'subtract',
    """
    Subtract arguments, element-wise.

    Parameters
    ----------
    x1, x2 : array_like
        The arrays to be subtracted from each other.
        $BROADCASTABLE_2
    $PARAMS

    Returns
    -------
    y : ndarray
        The difference of `x1` and `x2`, element-wise.
        $OUT_SCALAR_2

    Notes
    -----
    Equivalent to ``x1 - x2`` in terms of array broadcasting.

    Examples
    --------
    >>> np.subtract(1.0, 4.0)
    -3.0

    >>> x1 = np.arange(9.0).reshape((3, 3))
    >>> x2 = np.arange(3.0)
    >>> np.subtract(x1, x2)
    array([[ 0.,  0.,  0.],
           [ 3.,  3.,  3.],
           [ 6.,  6.,  6.]])

    The ``-`` operator can be used as a shorthand for ``np.subtract`` on
    ndarrays.

    >>> x1 = np.arange(9.0).reshape((3, 3))
    >>> x2 = np.arange(3.0)
    >>> x1 - x2
    array([[0., 0., 0.],
           [3., 3., 3.],
           [6., 6., 6.]])

    """)

add_newdoc('numpy.core.umath', 'tan',
    """
    Compute tangent element-wise.

    Equivalent to ``np.sin(x)/np.cos(x)`` element-wise.

    Parameters
    ----------
    x : array_like
        Input array.
    $PARAMS

    Returns
    -------
    y : ndarray
        The corresponding tangent values.
        $OUT_SCALAR_1

    Notes
    -----
    If `out` is provided, the function writes the result into it,
    and returns a reference to `out`.  (See Examples)

    References
    ----------
    M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions.
    New York, NY: Dover, 1972.

    Examples
    --------
    >>> from math import pi
    >>> np.tan(np.array([-pi,pi/2,pi]))
    array([  1.22460635e-16,   1.63317787e+16,  -1.22460635e-16])
    >>>
    >>> # Example of providing the optional output parameter illustrating
    >>> # that what is returned is a reference to said parameter
    >>> out1 = np.array([0], dtype='d')
    >>> out2 = np.cos([0.1], out1)
    >>> out2 is out1
    True
    >>>
    >>> # Example of ValueError due to provision of shape mis-matched `out`
    >>> np.cos(np.zeros((3,3)),np.zeros((2,2)))
    Traceback (most recent call last):
      File "<stdin>", line 1, in <module>
    ValueError: operands could not be broadcast together with shapes (3,3) (2,2)

    """)

add_newdoc('numpy.core.umath', 'tanh',
    """
    Compute hyperbolic tangent element-wise.

    Equivalent to ``np.sinh(x)/np.cosh(x)`` or ``-1j * np.tan(1j*x)``.

    Parameters
    ----------
    x : array_like
        Input array.
    $PARAMS

    Returns
    -------
    y : ndarray
        The corresponding hyperbolic tangent values.
        $OUT_SCALAR_1

    Notes
    -----
    If `out` is provided, the function writes the result into it,
    and returns a reference to `out`.  (See Examples)

    References
    ----------
    .. [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions.
           New York, NY: Dover, 1972, pg. 83.
           https://personal.math.ubc.ca/~cbm/aands/page_83.htm

    .. [2] Wikipedia, "Hyperbolic function",
           https://en.wikipedia.org/wiki/Hyperbolic_function

    Examples
    --------
    >>> np.tanh((0, np.pi*1j, np.pi*1j/2))
    array([ 0. +0.00000000e+00j,  0. -1.22460635e-16j,  0. +1.63317787e+16j])

    >>> # Example of providing the optional output parameter illustrating
    >>> # that what is returned is a reference to said parameter
    >>> out1 = np.array([0], dtype='d')
    >>> out2 = np.tanh([0.1], out1)
    >>> out2 is out1
    True

    >>> # Example of ValueError due to provision of shape mis-matched `out`
    >>> np.tanh(np.zeros((3,3)),np.zeros((2,2)))
    Traceback (most recent call last):
      File "<stdin>", line 1, in <module>
    ValueError: operands could not be broadcast together with shapes (3,3) (2,2)

    """)

add_newdoc('numpy.core.umath', 'frexp',
    """
    Decompose the elements of x into mantissa and twos exponent.

    Returns (`mantissa`, `exponent`), where ``x = mantissa * 2**exponent``.
    The mantissa lies in the open interval(-1, 1), while the twos
    exponent is a signed integer.

    Parameters
    ----------
    x : array_like
        Array of numbers to be decomposed.
    out1 : ndarray, optional
        Output array for the mantissa. Must have the same shape as `x`.
    out2 : ndarray, optional
        Output array for the exponent. Must have the same shape as `x`.
    $PARAMS

    Returns
    -------
    mantissa : ndarray
        Floating values between -1 and 1.
        $OUT_SCALAR_1
    exponent : ndarray
        Integer exponents of 2.
        $OUT_SCALAR_1

    See Also
    --------
    ldexp : Compute ``y = x1 * 2**x2``, the inverse of `frexp`.

    Notes
    -----
    Complex dtypes are not supported, they will raise a TypeError.

    Examples
    --------
    >>> x = np.arange(9)
    >>> y1, y2 = np.frexp(x)
    >>> y1
    array([ 0.   ,  0.5  ,  0.5  ,  0.75 ,  0.5  ,  0.625,  0.75 ,  0.875,
            0.5  ])
    >>> y2
    array([0, 1, 2, 2, 3, 3, 3, 3, 4])
    >>> y1 * 2**y2
    array([ 0.,  1.,  2.,  3.,  4.,  5.,  6.,  7.,  8.])

    """)

add_newdoc('numpy.core.umath', 'ldexp',
    """
    Returns x1 * 2**x2, element-wise.

    The mantissas `x1` and twos exponents `x2` are used to construct
    floating point numbers ``x1 * 2**x2``.

    Parameters
    ----------
    x1 : array_like
        Array of multipliers.
    x2 : array_like, int
        Array of twos exponents.
        $BROADCASTABLE_2
    $PARAMS

    Returns
    -------
    y : ndarray or scalar
        The result of ``x1 * 2**x2``.
        $OUT_SCALAR_2

    See Also
    --------
    frexp : Return (y1, y2) from ``x = y1 * 2**y2``, inverse to `ldexp`.

    Notes
    -----
    Complex dtypes are not supported, they will raise a TypeError.

    `ldexp` is useful as the inverse of `frexp`, if used by itself it is
    more clear to simply use the expression ``x1 * 2**x2``.

    Examples
    --------
    >>> np.ldexp(5, np.arange(4))
    array([ 5., 10., 20., 40.], dtype=float16)

    >>> x = np.arange(6)
    >>> np.ldexp(*np.frexp(x))
    array([ 0.,  1.,  2.,  3.,  4.,  5.])

    """)

add_newdoc('numpy.core.umath', 'gcd',
    """
    Returns the greatest common divisor of ``|x1|`` and ``|x2|``

    Parameters
    ----------
    x1, x2 : array_like, int
        Arrays of values.
        $BROADCASTABLE_2

    Returns
    -------
    y : ndarray or scalar
        The greatest common divisor of the absolute value of the inputs
        $OUT_SCALAR_2

    See Also
    --------
    lcm : The lowest common multiple

    Examples
    --------
    >>> np.gcd(12, 20)
    4
    >>> np.gcd.reduce([15, 25, 35])
    5
    >>> np.gcd(np.arange(6), 20)
    array([20,  1,  2,  1,  4,  5])

    """)

add_newdoc('numpy.core.umath', 'lcm',
    """
    Returns the lowest common multiple of ``|x1|`` and ``|x2|``

    Parameters
    ----------
    x1, x2 : array_like, int
        Arrays of values.
        $BROADCASTABLE_2

    Returns
    -------
    y : ndarray or scalar
        The lowest common multiple of the absolute value of the inputs
        $OUT_SCALAR_2

    See Also
    --------
    gcd : The greatest common divisor

    Examples
    --------
    >>> np.lcm(12, 20)
    60
    >>> np.lcm.reduce([3, 12, 20])
    60
    >>> np.lcm.reduce([40, 12, 20])
    120
    >>> np.lcm(np.arange(6), 20)
    array([ 0, 20, 20, 60, 20, 20])

    """)

add_newdoc('numpy.core.umath', 'bit_count',
    """
    Computes the number of 1-bits in the absolute value of ``x``.
    Analogous to the builtin `int.bit_count` or ``popcount`` in C++.

    Parameters
    ----------
    x : array_like, unsigned int
        Input array.
    $PARAMS

    Returns
    -------
    y : ndarray
        The corresponding number of 1-bits in the input.
        $OUT_SCALAR_1

    References
    ----------
    .. [1] https://stackoverflow.com/a/109025/5671364

    .. [2] Wikipedia, "Hamming weight",
           https://en.wikipedia.org/wiki/Hamming_weight

    Examples
    --------
    >>> np.bit_count(1023)
    10
    >>> a = np.array([2**i - 1 for i in range(16)])
    >>> np.bit_count(a)
    array([ 0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15])

    """)