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%sdivide by zerooverflowunderflowinvalid valuetest__radd__subtract__rsub__multiply__rmul____rdiv__true_divide__rtruediv__floor_divide__rfloordiv__remainder__rmod__power__rpow__left_shift__rlshift__right_shift__rrshift__bitwise_and__rand__bitwise_xor__rxor__bitwise_or__ror____array_prepare__function not supportedNot implemented for this type__array_wrap__(OO)duplicate value in 'axis'reduceOOOOO|OO&O&O|OO&O&iO(O)itoo many values for 'axis''axis' entry is out of boundscannot %s on a scalaraccumulatereduceatexpect ','expect ',' or ')'expect dimension nameexpect '('expect '->'%s at position %d in "%s"umathnumpy.core.multiarray_ARRAY_API_ARRAY_API not found_ARRAY_API is NULL pointer_UFUNC_API0.4.0__version__DO NOT USE, ONLY FOR TESTING_arg_ones_likeabsoluteaddarccosarccosharcsinarcsinharctanarctan2arctanhceilconjugatecopysigndeg2raddegreesexpm1fabsfloorfmaxfminfmodgreatergreater_equalhypotinvertisfiniteisinfisnanlessless_equalloglog10log1plog2logaddexplogaddexp2logical_andlogical_notlogical_orlogical_xormaximumminimummodfnegativenextafternot_equalrad2degradiansreciprocalrintsignbitspacingsqrtsquaretruncfrexpCompute y = x1 * 2**x2.ldexppiERR_IGNOREERR_WARNERR_CALLERR_RAISEERR_PRINTERR_LOGERR_DEFAULTERR_DEFAULT2SHIFT_DIVIDEBYZEROSHIFT_OVERFLOWSHIFT_UNDERFLOWSHIFT_INVALIDFPE_DIVIDEBYZEROFPE_OVERFLOWFPE_UNDERFLOWFPE_INVALIDFLOATING_POINT_SUPPORTUFUNC_PYVALS_NAMEUFUNC_BUFSIZE_DEFAULTPINFNINFPZERONZERONANconjcannot load umath module.axiskeepdimsindicesfrompyfuncseterrobjgeterrobj__doc__ninnoutnargsntypesidentitysignatureouternumpy.ufuncCannot cast ufunc %s input from Cannot cast ufunc %s output from the ufunc default masked inner loop selector doesn't yet support wrapping the new inner loop selector, it still only wraps the legacy inner loop selectoronly boolean masks are supported in ufunc inner loops presentlypython callback specified for %s (in  %s) but no function found.Warning: %s encountered in %s
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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_likeCalculate the absolute value element-wise.

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

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 }`.

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(-10, 10, 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])
>>> plt.show()Add arguments element-wise.

Parameters
----------
x1, x2 : array_like
    The arrays to be added.  If ``x1.shape != x2.shape``, they must be
    broadcastable to a common shape (which may be the shape of one or
    the other).

Returns
-------
add : ndarray or scalar
    The sum of `x1` and `x2`, element-wise.  Returns a scalar if
    both  `x1` and `x2` are scalars.

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.]])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].

out : ndarray, optional
    Array of the same shape as `a`, to store results in. See
    `doc.ufuncs` (Section "Output arguments") for more details.

Returns
-------
angle : ndarray
    The angle of the ray intersecting the unit circle at the given
    `x`-coordinate in radians [0, pi]. If `x` is a scalar then a
    scalar is returned, otherwise an array of the same shape as `x`
    is returned.

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. http://www.math.sfu.ca/~cbm/aands/

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()Inverse hyperbolic cosine, elementwise.

Parameters
----------
x : array_like
    Input array.
out : ndarray, optional
    Array of the same shape as `x`, to store results in.
    See `doc.ufuncs` (Section "Output arguments") for details.


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

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. http://www.math.sfu.ca/~cbm/aands/
.. [2] Wikipedia, "Inverse hyperbolic function",
       http://en.wikipedia.org/wiki/Arccosh

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

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

out : ndarray, optional
    Array of the same shape as `x`, in which to store the results.
    See `doc.ufuncs` (Section "Output arguments") for more details.

Returns
-------
angle : ndarray
    The inverse sine of each element in `x`, in radians and in the
    closed interval ``[-pi/2, pi/2]``.  If `x` is a scalar, a scalar
    is returned, otherwise an array.

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.
http://www.math.sfu.ca/~cbm/aands/

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

Parameters
----------
x : array_like
    Input array.
out : ndarray, optional
    Array into which the output is placed. Its type is preserved and it
    must be of the right shape to hold the output. See `doc.ufuncs`.

Returns
-------
out : ndarray
    Array of of the same shape as `x`.

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. http://www.math.sfu.ca/~cbm/aands/
.. [2] Wikipedia, "Inverse hyperbolic function",
       http://en.wikipedia.org/wiki/Arcsinh

Examples
--------
>>> np.arcsinh(np.array([np.e, 10.0]))
array([ 1.72538256,  2.99822295])Trigonometric inverse tangent, element-wise.

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

Parameters
----------
x : array_like
    Input values.  `arctan` is applied to each element of `x`.

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

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.
http://www.math.sfu.ca/~cbm/aands/

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()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. `x2` must be broadcastable to match the shape of
    `x1` or vice versa.

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

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])Inverse hyperbolic tangent elementwise.

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

Returns
-------
out : ndarray
    Array of the same shape as `x`.

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. http://www.math.sfu.ca/~cbm/aands/
.. [2] Wikipedia, "Inverse hyperbolic function",
       http://en.wikipedia.org/wiki/Arctanh

Examples
--------
>>> np.arctanh([0, -0.5])
array([ 0.        , -0.54930614])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 types are handled (including booleans).

Returns
-------
out : array_like
    Result.

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], dtype=bool)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 types are handled (including booleans).
out : ndarray, optional
    Array into which the output is placed. Its type is preserved and it
    must be of the right shape to hold the output. See doc.ufuncs.

Returns
-------
out : array_like
    Result.

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 binaray 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, 2147483647L], dtype=np.int32),
...               np.array([4, 4, 4, 2147483647L], dtype=np.int32))
array([         6,          5,        255, 2147483647])
>>> np.bitwise_or([True, True], [False, True])
array([ True,  True], dtype=bool)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 types are handled (including booleans).

Returns
-------
out : array_like
    Result.

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], dtype=bool)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.

Returns
-------
y : {ndarray, scalar}
    The ceiling of each element in `x`, with `float` dtype.

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

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.])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.

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

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]])Change the sign of x1 to that of x2, element-wise.

If both arguments are arrays or sequences, they have to be of the same
length. 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`.
out : ndarray, optional
    Array into which the output is placed. Its type is preserved and it
    must be of the right shape to hold the output. See doc.ufuncs.

Returns
-------
out : array_like
    The values of `x1` with the sign of `x2`.

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.])Cosine elementwise.

Parameters
----------
x : array_like
    Input array in radians.
out : ndarray, optional
    Output array of same shape as `x`.

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

Raises
------
ValueError: invalid return array shape
    if `out` is provided and `out.shape` != `x.shape` (See Examples)

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
>>> 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: invalid return array shapeHyperbolic cosine, element-wise.

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

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

Returns
-------
out : ndarray
    Output array of same shape as `x`.

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()Convert angles from degrees to radians.

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

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

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)
3.1415926535897931Convert angles from radians to degrees.

Parameters
----------
x : array_like
    Input array in radians.
out : ndarray, optional
    Output array of same shape as x.

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

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 = degrees(rad, out)
>>> np.all(r == out)
TrueDivide arguments element-wise.

Parameters
----------
x1 : array_like
    Dividend array.
x2 : array_like
    Divisor array.
out : ndarray, optional
    Array into which the output is placed. Its type is preserved and it
    must be of the right shape to hold the output. See doc.ufuncs.

Returns
-------
y : {ndarray, scalar}
    The quotient `x1/x2`, element-wise. Returns a scalar if
    both  `x1` and `x2` are scalars.

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.

Behavior on division by zero can be changed using `seterr`.

When both `x1` and `x2` are of an integer type, `divide` will return
integers and throw away the fractional part. Moreover, division by zero
always yields zero in integer arithmetic.

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. ]])

Note the behavior with integer types:

>>> np.divide(2, 4)
0
>>> np.divide(2, 4.)
0.5

Division by zero always yields zero in integer arithmetic, and does not
raise an exception or a warning:

>>> np.divide(np.array([0, 1], dtype=int), np.array([0, 0], dtype=int))
array([0, 0])

Division by zero can, however, be caught using `seterr`:

>>> old_err_state = np.seterr(divide='raise')
>>> np.divide(1, 0)
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
FloatingPointError: divide by zero encountered in divide

>>> ignored_states = np.seterr(**old_err_state)
>>> np.divide(1, 0)
0Return (x1 == x2) element-wise.

Parameters
----------
x1, x2 : array_like
    Input arrays of the same shape.

Returns
-------
out : {ndarray, bool}
    Output array of bools, or a single bool if x1 and x2 are scalars.

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

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

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], dtype=bool)Calculate the exponential of all elements in the input array.

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

Returns
-------
out : ndarray
    Output array, element-wise exponential of `x`.

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",
       http://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,
       http://www.math.sfu.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])
>>> 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])
>>> plt.title('Phase (angle) of exp(x)')
>>> plt.show()Calculate `2**p` for all `p` in the input array.

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

out : ndarray, optional
    Array to insert results into.

Returns
-------
out : ndarray
    Element-wise 2 to the power `x`.

See Also
--------
exp : calculate x**p.

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



Examples
--------
>>> np.exp2([2, 3])
array([ 4.,  8.])Calculate ``exp(x) - 1`` for all elements in the array.

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

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

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


Notes
-----
This function provides greater precision than the formula ``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-10Compute the absolute values elementwise.

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.
out : ndarray, optional
    Array into which the output is placed. Its type is preserved and it
    must be of the right shape to hold the output. See doc.ufuncs.

Returns
-------
y : {ndarray, scalar}
    The absolute values of `x`, the returned values are always floats.

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])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.

Returns
-------
y : {ndarray, scalar}
    The floor of each element in `x`.

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

Notes
-----
Some spreadsheet programs calculate the "floor-towards-zero", in other
words ``floor(-2.5) == -2``.  NumPy, however, uses the a definition of
`floor` such that `floor(-2.5) == -3`.

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.])Return the largest integer smaller or equal to the division of the inputs.

Parameters
----------
x1 : array_like
    Numerator.
x2 : array_like
    Denominator.

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


See Also
--------
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.])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. They must have
    the same shape.

Returns
-------
y : {ndarray, scalar}
    The minimum of `x1` and `x2`, element-wise.  Returns scalar if
    both  `x1` and `x2` are scalars.

See Also
--------
fmin :
  element-wise minimum that ignores nans unless both inputs are nans.
maximum :
  element-wise maximum that propagates nans.
minimum :
  element-wise minimum that propagates nans.

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])fmin(x1, x2[, out])

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. They must have
    the same shape.

Returns
-------
y : {ndarray, scalar}
    The minimum of `x1` and `x2`, element-wise.  Returns scalar if
    both  `x1` and `x2` are scalars.

See Also
--------
fmax :
  element-wise maximum that ignores nans unless both inputs are nans.
maximum :
  element-wise maximum that propagates nans.
minimum :
  element-wise minimum that propagates nans.

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([2, 5, 4])

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

>>> np.fmin([np.nan, 0, np.nan],[0, np.nan, np.nan])
array([  0.,   0.,  NaN])Return the element-wise remainder of division.

This is the NumPy implementation of the Python modulo operator `%`.

Parameters
----------
x1 : array_like
  Dividend.
x2 : array_like
  Divisor.

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

See Also
--------
remainder : Modulo operation where the quotient is `floor(x1/x2)`.
divide

Notes
-----
The result of the modulo operation for negative dividend and divisors is
bound by conventions. In `fmod`, the sign of the remainder is the sign of
the dividend. In `remainder`, the sign of the divisor does not affect the
sign of the result.

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]])Return the truth value of (x1 > x2) element-wise.

Parameters
----------
x1, x2 : array_like
    Input arrays.  If ``x1.shape != x2.shape``, they must be
    broadcastable to a common shape (which may be the shape of one or
    the other).

Returns
-------
out : bool or ndarray of bool
    Array of bools, or a single bool if `x1` and `x2` are scalars.


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

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

If the inputs are ndarrays, then np.greater is equivalent to '>'.

>>> a = np.array([4,2])
>>> b = np.array([2,2])
>>> a > b
array([ True, False], dtype=bool)Return the truth value of (x1 >= x2) element-wise.

Parameters
----------
x1, x2 : array_like
    Input arrays.  If ``x1.shape != x2.shape``, they must be
    broadcastable to a common shape (which may be the shape of one or
    the other).

Returns
-------
out : bool or ndarray of bool
    Array of bools, or a single bool if `x1` and `x2` are scalars.

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

Examples
--------
>>> np.greater_equal([4, 2, 1], [2, 2, 2])
array([ True, True, False], dtype=bool)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).
out : ndarray, optional
    Array into which the output is placed. Its type is preserved and it
    must be of the right shape to hold the output. See doc.ufuncs.

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

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.]])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
----------
x1 : array_like
    Only integer types are handled (including booleans).

Returns
-------
out : array_like
    Result.

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",
    http://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:

>>> np.invert(np.array([13], dtype=uint8))
array([242], dtype=uint8)
>>> np.binary_repr(x, width=8)
'00001101'
>>> np.binary_repr(242, width=8)
'11110010'

The result depends on the bit-width:

>>> np.invert(np.array([13], dtype=uint16))
array([65522], dtype=uint16)
>>> np.binary_repr(x, width=16)
'0000000000001101'
>>> np.binary_repr(65522, 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=int8))
array([-14], dtype=int8)
>>> np.binary_repr(-14, width=8)
'11110010'

Booleans are accepted as well:

>>> np.invert(array([True, False]))
array([False,  True], dtype=bool)Test element-wise for finite-ness (not infinity or not Not a Number).

The result is returned as a boolean array.

Parameters
----------
x : array_like
    Input values.
out : ndarray, optional
    Array into which the output is placed. Its type is preserved and it
    must be of the right shape to hold the output. See `doc.ufuncs`.

Returns
-------
y : ndarray, bool
    For scalar input, the result is a new boolean with value True
    if the input is finite; otherwise the value is False (input is
    either positive infinity, negative infinity or Not a Number).

    For array input, the result is a boolean array with the same
    dimensions as the input and the values are True if the corresponding
    element of the input is finite; otherwise the values are False (element
    is either positive infinity, negative infinity or Not a Number).

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], dtype=bool)

>>> 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])Test element-wise for positive or negative infinity.

Return a bool-type array, the same shape as `x`, True where ``x ==
+/-inf``, False everywhere else.

Parameters
----------
x : array_like
    Input values
out : array_like, optional
    An array with the same shape as `x` to store the result.

Returns
-------
y : bool (scalar) or bool-type ndarray
    For scalar input, the result is a new boolean with value True
    if the input is positive or negative infinity; otherwise the value
    is False.

    For array input, the result is a boolean array with the same
    shape as the input and the values are True where the
    corresponding element of the input is positive or negative
    infinity; elsewhere the values are False.  If a second argument
    was supplied the result is stored there.  If the type of that array
    is a numeric type the result is represented as zeros and ones, if
    the type is boolean then as False and True, respectively.
    The return value `y` is then a reference to that array.

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], dtype=bool)

>>> 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])Test element-wise for Not a Number (NaN), return result as a bool array.

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

Returns
-------
y : {ndarray, bool}
    For scalar input, the result is a new boolean with value True
    if the input is NaN; otherwise the value is False.

    For array input, the result is a boolean array with the same
    dimensions as the input and the values are True if the corresponding
    element of the input is NaN; otherwise the values are False.

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

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], dtype=bool)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.

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

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])Return the truth value of (x1 < x2) element-wise.

Parameters
----------
x1, x2 : array_like
    Input arrays.  If ``x1.shape != x2.shape``, they must be
    broadcastable to a common shape (which may be the shape of one or
    the other).

Returns
-------
out : bool or ndarray of bool
    Array of bools, or a single bool if `x1` and `x2` are scalars.

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

Examples
--------
>>> np.less([1, 2], [2, 2])
array([ True, False], dtype=bool)Return the truth value of (x1 =< x2) element-wise.

Parameters
----------
x1, x2 : array_like
    Input arrays.  If ``x1.shape != x2.shape``, they must be
    broadcastable to a common shape (which may be the shape of one or
    the other).

Returns
-------
out : bool or ndarray of bool
    Array of bools, or a single bool if `x1` and `x2` are scalars.

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

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

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

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.

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

Examples
--------
>>> np.log([1, np.e, np.e**2, 0])
array([  0.,   1.,   2., -Inf])Return the base 10 logarithm of the input array, element-wise.

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

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

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.

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

Examples
--------
>>> np.log10([1e-15, -3.])
array([-15.,  NaN])Return the natural logarithm of one plus the input array, element-wise.

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

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

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

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. http://www.math.sfu.ca/~cbm/aands/
.. [2] Wikipedia, "Logarithm". http://en.wikipedia.org/wiki/Logarithm

Examples
--------
>>> np.log1p(1e-99)
1e-99
>>> np.log(1 + 1e-99)
0.0Base-2 logarithm of `x`.

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

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

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.

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])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.

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

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-50Logarithm 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.
out : ndarray, optional
    Array to store results in.

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

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-50Compute the truth value of x1 AND x2 elementwise.

Parameters
----------
x1, x2 : array_like
    Input arrays. `x1` and `x2` must be of the same shape.


Returns
-------
y : {ndarray, bool}
    Boolean result with the same shape as `x1` and `x2` of the logical
    AND operation on corresponding elements of `x1` and `x2`.

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], dtype=bool)

>>> x = np.arange(5)
>>> np.logical_and(x>1, x<4)
array([False, False,  True,  True, False], dtype=bool)Compute the truth value of NOT x elementwise.

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

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

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], dtype=bool)

>>> x = np.arange(5)
>>> np.logical_not(x<3)
array([False, False, False,  True,  True], dtype=bool)Compute the truth value of x1 OR x2 elementwise.

Parameters
----------
x1, x2 : array_like
    Logical OR is applied to the elements of `x1` and `x2`.
    They have to be of the same shape.

Returns
-------
y : {ndarray, bool}
    Boolean result with the same shape as `x1` and `x2` of the logical
    OR operation on elements of `x1` and `x2`.

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], dtype=bool)

>>> x = np.arange(5)
>>> np.logical_or(x < 1, x > 3)
array([ True, False, False, False,  True], dtype=bool)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`.  They must
    be broadcastable to the same shape.

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 whether or not
    broadcasting of one or both arrays was required.

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], dtype=bool)

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

Simple example showing support of broadcasting

>>> np.logical_xor(0, np.eye(2))
array([[ True, False],
       [False,  True]], dtype=bool)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. They must have
    the same shape, or shapes that can be broadcast to a single shape.

Returns
-------
y : {ndarray, scalar}
    The maximum of `x1` and `x2`, element-wise.  Returns scalar if
    both  `x1` and `x2` are scalars.

See Also
--------
minimum :
  element-wise minimum

fmax :
  element-wise maximum that ignores nans unless both inputs are nans.

fmin :
  element-wise minimum that ignores nans unless both inputs are nans.

Notes
-----
Equivalent to ``np.where(x1 > x2, x1, x2)`` but 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])
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)
infElement-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. They must have
    the same shape, or shapes that can be broadcast to a single shape.

Returns
-------
y : {ndarray, scalar}
    The minimum of `x1` and `x2`, element-wise.  Returns scalar if
    both  `x1` and `x2` are scalars.

See Also
--------
maximum :
  element-wise minimum that propagates nans.
fmax :
  element-wise maximum that ignores nans unless both inputs are nans.
fmin :
  element-wise minimum that ignores nans unless both inputs are nans.

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])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.

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

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

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

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

Returns
-------
y : ndarray
    The product of `x1` and `x2`, element-wise. Returns a scalar if
    both  `x1` and `x2` are scalars.

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.]])Returns an array with the negative of each element of the original array.

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

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

Examples
--------
>>> np.negative([1.,-1.])
array([-1.,  1.])Return the next representable floating-point value after x1 in the direction
of 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`.
out : ndarray, optional
    Array into which the output is placed. Its type is preserved and it
    must be of the right shape to hold the output. See `doc.ufuncs`.

Returns
-------
out : array_like
    The next representable values of `x1` in the direction of `x2`.

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], dtype=bool)Return (x1 != x2) element-wise.

Parameters
----------
x1, x2 : array_like
  Input arrays.
out : ndarray, optional
  A placeholder the same shape as `x1` to store the result.
  See `doc.ufuncs` (Section "Output arguments") for more details.

Returns
-------
not_equal : ndarray bool, scalar bool
  For each element in `x1, x2`, return True if `x1` is not equal
  to `x2` and False otherwise.


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

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

Parameters
----------
x1 : array_like
    The bases.
x2 : array_like
    The exponents.

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

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

>>> x1 = range(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]])Convert angles from radians to degrees.

Parameters
----------
x : array_like
    Angle in radians.
out : ndarray, optional
    Array into which the output is placed. Its type is preserved and it
    must be of the right shape to hold the output. See doc.ufuncs.

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

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)
90.0Convert angles from degrees to radians.

Parameters
----------
x : array_like
    Input array in degrees.
out : ndarray, optional
    Output array of same shape as `x`.

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

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
TrueReturn the reciprocal of the argument, element-wise.

Calculates ``1/x``.

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

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

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])Return element-wise remainder of division.

Computes ``x1 - floor(x1 / x2) * x2``.

Parameters
----------
x1 : array_like
    Dividend array.
x2 : array_like
    Divisor array.
out : ndarray, optional
    Array into which the output is placed. Its type is preserved and it
    must be of the right shape to hold the output. See doc.ufuncs.

Returns
-------
y : ndarray
    The remainder of the quotient ``x1/x2``, element-wise. Returns a scalar
    if both  `x1` and `x2` are scalars.

See Also
--------
divide, floor

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

Examples
--------
>>> np.remainder([4, 7], [2, 3])
array([0, 1])
>>> np.remainder(np.arange(7), 5)
array([0, 1, 2, 3, 4, 0, 1])Shift the bits of an integer to the right.

Bits are shifted to the right by removing `x2` bits at the right of `x1`.
Since 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`.

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

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])Round elements of the array to the nearest integer.

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

Returns
-------
out : {ndarray, scalar}
    Output array is same shape and type as `x`.

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

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.])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``.

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

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

Examples
--------
>>> np.sign([-5., 4.5])
array([-1.,  1.])
>>> np.sign(0)
0Returns element-wise True where signbit is set (less than zero).

Parameters
----------
x: array_like
    The input value(s).
out : ndarray, optional
    Array into which the output is placed. Its type is preserved
    and it must be of the right shape to hold the output.
    See `doc.ufuncs`.

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

Examples
--------
>>> np.signbit(-1.2)
True
>>> np.signbit(np.array([1, -2.3, 2.1]))
array([False,  True, False], dtype=bool)Trigonometric sine, element-wise.

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

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

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()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.
out : ndarray, optional
    Output array of same shape as `x`.

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

Raises
------
ValueError: invalid return array shape
    if `out` is provided and `out.shape` != `x.shape` (See Examples)

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
>>> 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: invalid return array shapeReturn the distance between x and the nearest adjacent number.

Parameters
----------
x1: array_like
    Values to find the spacing of.

Returns
-------
out : array_like
    The spacing of values of `x1`.

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
TrueReturn the positive square-root of an array, element-wise.

Parameters
----------
x : array_like
    The values whose square-roots are required.
out : ndarray, optional
    Alternate array object in which to put the result; if provided, it
    must have the same shape as `x`

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.

See Also
--------
lib.scimath.sqrt
    A version which returns complex numbers when given negative reals.

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, numpy.inf])
array([  2.,  NaN,  Inf])Return the element-wise square of the input.

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

Returns
-------
out : ndarray
    Element-wise `x*x`, of the same shape and dtype as `x`.
    Returns scalar if `x` is a scalar.

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

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

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

Returns
-------
y : ndarray
    The difference of `x1` and `x2`, element-wise.  Returns a scalar if
    both  `x1` and `x2` are scalars.

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.]])Compute tangent element-wise.

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

Parameters
----------
x : array_like
  Input array.
out : ndarray, optional
    Output array of same shape as `x`.

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

Raises
------
ValueError: invalid return array shape
    if `out` is provided and `out.shape` != `x.shape` (See Examples)

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
>>> 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: invalid return array shapeCompute hyperbolic tangent element-wise.

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

Parameters
----------
x : array_like
    Input array.
out : ndarray, optional
    Output array of same shape as `x`.

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

Raises
------
ValueError: invalid return array shape
    if `out` is provided and `out.shape` != `x.shape` (See Examples)

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.
       http://www.math.sfu.ca/~cbm/aands/

.. [2] Wikipedia, "Hyperbolic function",
       http://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
>>> 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: invalid return array shapeReturns a true division of the inputs, element-wise.

Instead of the Python traditional 'floor division', this returns a true
division.  True division adjusts the output type to present the best
answer, regardless of input types.

Parameters
----------
x1 : array_like
    Dividend array.
x2 : array_like
    Divisor array.

Returns
-------
out : ndarray
    Result is scalar if both inputs are scalar, ndarray otherwise.

Notes
-----
The floor division operator ``//`` was added in Python 2.2 making ``//``
and ``/`` equivalent operators.  The default floor division operation of
``/`` can be replaced by true division with
``from __future__ import division``.

In Python 3.0, ``//`` is the floor division operator and ``/`` the
true division operator.  The ``true_divide(x1, x2)`` function is
equivalent to true division in Python.

Examples
--------
>>> x = np.arange(5)
>>> np.true_divide(x, 4)
array([ 0.  ,  0.25,  0.5 ,  0.75,  1.  ])

>>> x/4
array([0, 0, 0, 0, 1])
>>> x//4
array([0, 0, 0, 0, 1])

>>> from __future__ import division
>>> x/4
array([ 0.  ,  0.25,  0.5 ,  0.75,  1.  ])
>>> x//4
array([0, 0, 0, 0, 1])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.

Returns
-------
y : {ndarray, scalar}
    The truncated value of each element in `x`.

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

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)
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