torch/special/__init__.py - platform/external/pytorch - Git at Google

 import sys

 import torch
 from torch._C import _add_docstr, _special  # type: ignore[attr-defined]
 from torch._torch_docs import common_args, multi_dim_common

 Tensor = torch.Tensor

 entr = _add_docstr(_special.special_entr,
                    r"""
 entr(input, *, out=None) -> Tensor
 Computes the entropy on :attr:`input` (as defined below), elementwise.

 .. math::
     \begin{align}
     \text{entr(x)} = \begin{cases}
         -x * \ln(x)  & x > 0 \\
         0 &  x = 0.0 \\
         -\infty & x < 0
     \end{cases}
     \end{align}
 """ + """

 Args:
    input (Tensor): the input tensor.

 Keyword args:
     out (Tensor, optional): the output tensor.

 Example::
     >>> a = torch.arange(-0.5, 1, 0.5)
     >>> a
     tensor([-0.5000,  0.0000,  0.5000])
     >>> torch.special.entr(a)
     tensor([  -inf, 0.0000, 0.3466])
 """)

 psi = _add_docstr(_special.special_psi,
                   r"""
 psi(input, *, out=None) -> Tensor

 Alias for :func:`torch.special.digamma`.
 """)

 digamma = _add_docstr(_special.special_digamma,
                       r"""
 digamma(input, *, out=None) -> Tensor

 Computes the logarithmic derivative of the gamma function on `input`.

 .. math::
     \digamma(x) = \frac{d}{dx} \ln\left(\Gamma\left(x\right)\right) = \frac{\Gamma'(x)}{\Gamma(x)}
 """ + r"""
 Args:
     input (Tensor): the tensor to compute the digamma function on

 Keyword args:
     {out}

 .. note::  This function is similar to SciPy's `scipy.special.digamma`.

 .. note::  From PyTorch 1.8 onwards, the digamma function returns `-Inf` for `0`.
            Previously it returned `NaN` for `0`.

 Example::

     >>> a = torch.tensor([1, 0.5])
     >>> torch.special.digamma(a)
     tensor([-0.5772, -1.9635])

 """.format(**common_args))

 gammaln = _add_docstr(_special.special_gammaln,
                       r"""
 gammaln(input, *, out=None) -> Tensor

 Computes the natural logarithm of the absolute value of the gamma function on :attr:`input`.

 .. math::
     \text{out}_{i} = \ln \Gamma(|\text{input}_{i}|)
 """ + """
 Args:
     {input}

 Keyword args:
     {out}

 Example::

     >>> a = torch.arange(0.5, 2, 0.5)
     >>> torch.special.gammaln(a)
     tensor([ 0.5724,  0.0000, -0.1208])

 """.format(**common_args))

 erf = _add_docstr(_special.special_erf,
                   r"""
 erf(input, *, out=None) -> Tensor

 Computes the error function of :attr:`input`. The error function is defined as follows:

 .. math::
     \mathrm{erf}(x) = \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^2} dt
 """ + r"""
 Args:
     {input}

 Keyword args:
     {out}

 Example::

     >>> torch.special.erf(torch.tensor([0, -1., 10.]))
     tensor([ 0.0000, -0.8427,  1.0000])
 """.format(**common_args))

 erfc = _add_docstr(_special.special_erfc,
                    r"""
 erfc(input, *, out=None) -> Tensor

 Computes the complementary error function of :attr:`input`.
 The complementary error function is defined as follows:

 .. math::
     \mathrm{erfc}(x) = 1 - \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^2} dt
 """ + r"""
 Args:
     {input}

 Keyword args:
     {out}

 Example::

     >>> torch.special.erfc(torch.tensor([0, -1., 10.]))
     tensor([ 1.0000, 1.8427,  0.0000])
 """.format(**common_args))

 erfcx = _add_docstr(_special.special_erfcx,
                     r"""
 erfcx(input, *, out=None) -> Tensor

 Computes the scaled complementary error function for each element of :attr:`input`.
 The scaled complementary error function is defined as follows:

 .. math::
     \mathrm{erfcx}(x) = e^{x^2} \mathrm{erfc}(x)
 """ + r"""

 """ + r"""
 Args:
     {input}

 Keyword args:
     {out}

 Example::

     >>> torch.special.erfcx(torch.tensor([0, -1., 10.]))
     tensor([ 1.0000, 5.0090, 0.0561])
 """.format(**common_args))

 erfinv = _add_docstr(_special.special_erfinv,
                      r"""
 erfinv(input, *, out=None) -> Tensor

 Computes the inverse error function of :attr:`input`.
 The inverse error function is defined in the range :math:`(-1, 1)` as:

 .. math::
     \mathrm{erfinv}(\mathrm{erf}(x)) = x
 """ + r"""

 Args:
     {input}

 Keyword args:
     {out}

 Example::

     >>> torch.special.erfinv(torch.tensor([0, 0.5, -1.]))
     tensor([ 0.0000,  0.4769,    -inf])
 """.format(**common_args))

 logit = _add_docstr(_special.special_logit,
                     r"""
 logit(input, eps=None, *, out=None) -> Tensor

 Returns a new tensor with the logit of the elements of :attr:`input`.
 :attr:`input` is clamped to [eps, 1 - eps] when eps is not None.
 When eps is None and :attr:`input` < 0 or :attr:`input` > 1, the function will yields NaN.

 .. math::
     \begin{align}
     y_{i} &= \ln(\frac{z_{i}}{1 - z_{i}}) \\
     z_{i} &= \begin{cases}
         x_{i} & \text{if eps is None} \\
         \text{eps} & \text{if } x_{i} < \text{eps} \\
         x_{i} & \text{if } \text{eps} \leq x_{i} \leq 1 - \text{eps} \\
         1 - \text{eps} & \text{if } x_{i} > 1 - \text{eps}
     \end{cases}
     \end{align}
 """ + r"""
 Args:
     {input}
     eps (float, optional): the epsilon for input clamp bound. Default: ``None``

 Keyword args:
     {out}

 Example::

     >>> a = torch.rand(5)
     >>> a
     tensor([0.2796, 0.9331, 0.6486, 0.1523, 0.6516])
     >>> torch.special.logit(a, eps=1e-6)
     tensor([-0.9466,  2.6352,  0.6131, -1.7169,  0.6261])
 """.format(**common_args))

 logsumexp = _add_docstr(_special.special_logsumexp,
                         r"""
 logsumexp(input, dim, keepdim=False, *, out=None)

 Alias for :func:`torch.logsumexp`.
 """.format(**multi_dim_common))

 expit = _add_docstr(_special.special_expit,
                     r"""
 expit(input, *, out=None) -> Tensor

 Computes the expit (also known as the logistic sigmoid function) of the elements of :attr:`input`.

 .. math::
     \text{out}_{i} = \frac{1}{1 + e^{-\text{input}_{i}}}
 """ + r"""
 Args:
     {input}

 Keyword args:
     {out}

 Example::

     >>> t = torch.randn(4)
     >>> t
     tensor([ 0.9213,  1.0887, -0.8858, -1.7683])
     >>> torch.special.expit(t)
     tensor([ 0.7153,  0.7481,  0.2920,  0.1458])
 """.format(**common_args))

 exp2 = _add_docstr(_special.special_exp2,
                    r"""
 exp2(input, *, out=None) -> Tensor

 Computes the base two exponential function of :attr:`input`.

 .. math::
     y_{i} = 2^{x_{i}}

 """ + r"""
 Args:
     {input}

 Keyword args:
     {out}

 Example::

     >>> torch.special.exp2(torch.tensor([0, math.log2(2.), 3, 4]))
     tensor([ 1.,  2.,  8., 16.])
 """.format(**common_args))

 expm1 = _add_docstr(_special.special_expm1,
                     r"""
 expm1(input, *, out=None) -> Tensor

 Computes the exponential of the elements minus 1
 of :attr:`input`.

 .. math::
     y_{i} = e^{x_{i}} - 1

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

 """ + r"""
 Args:
     {input}

 Keyword args:
     {out}

 Example::

     >>> torch.special.expm1(torch.tensor([0, math.log(2.)]))
     tensor([ 0.,  1.])
 """.format(**common_args))

 xlog1py = _add_docstr(_special.special_xlog1py,
                       r"""
 xlog1py(input, other, *, out=None) -> Tensor

 Computes ``input * log1p(other)`` with the following cases.

 .. math::
     \text{out}_{i} = \begin{cases}
         \text{NaN} & \text{if } \text{other}_{i} = \text{NaN} \\
         0 & \text{if } \text{input}_{i} = 0.0 \text{ and } \text{other}_{i} != \text{NaN} \\
         \text{input}_{i} * \text{log1p}(\text{other}_{i})& \text{otherwise}
     \end{cases}

 Similar to SciPy's `scipy.special.xlog1py`.

 """ + r"""

 Args:
     input (Number or Tensor) : Multiplier
     other (Number or Tensor) : Argument

 .. note:: At least one of :attr:`input` or :attr:`other` must be a tensor.

 Keyword args:
     {out}

 Example::

     >>> x = torch.zeros(5,)
     >>> y = torch.tensor([-1, 0, 1, float('inf'), float('nan')])
     >>> torch.special.xlog1py(x, y)
     tensor([0., 0., 0., 0., nan])
     >>> x = torch.tensor([1, 2, 3])
     >>> y = torch.tensor([3, 2, 1])
     >>> torch.special.xlog1py(x, y)
     tensor([1.3863, 2.1972, 2.0794])
     >>> torch.special.xlog1py(x, 4)
     tensor([1.6094, 3.2189, 4.8283])
     >>> torch.special.xlog1py(2, y)
     tensor([2.7726, 2.1972, 1.3863])
 """.format(**common_args))

 xlogy = _add_docstr(_special.special_xlogy,
                     r"""
 xlogy(input, other, *, out=None) -> Tensor

 Computes ``input * log(other)`` with the following cases.

 .. math::
     \text{out}_{i} = \begin{cases}
         \text{NaN} & \text{if } \text{other}_{i} = \text{NaN} \\
         0 & \text{if } \text{input}_{i} = 0.0 \\
         \text{input}_{i} * \log{(\text{other}_{i})} & \text{otherwise}
     \end{cases}

 Similar to SciPy's `scipy.special.xlogy`.

 """ + r"""

 Args:
     input (Number or Tensor) : Multiplier
     other (Number or Tensor) : Argument

 .. note:: At least one of :attr:`input` or :attr:`other` must be a tensor.

 Keyword args:
     {out}

 Example::

     >>> x = torch.zeros(5,)
     >>> y = torch.tensor([-1, 0, 1, float('inf'), float('nan')])
     >>> torch.special.xlogy(x, y)
     tensor([0., 0., 0., 0., nan])
     >>> x = torch.tensor([1, 2, 3])
     >>> y = torch.tensor([3, 2, 1])
     >>> torch.special.xlogy(x, y)
     tensor([1.0986, 1.3863, 0.0000])
     >>> torch.special.xlogy(x, 4)
     tensor([1.3863, 2.7726, 4.1589])
     >>> torch.special.xlogy(2, y)
     tensor([2.1972, 1.3863, 0.0000])
 """.format(**common_args))

 i0 = _add_docstr(_special.special_i0,
                  r"""
 i0(input, *, out=None) -> Tensor

 Computes the zeroth order modified Bessel function of the first kind for each element of :attr:`input`.

 .. math::
     \text{out}_{i} = I_0(\text{input}_{i}) = \sum_{k=0}^{\infty} \frac{(\text{input}_{i}^2/4)^k}{(k!)^2}

 """ + r"""
 Args:
     input (Tensor): the input tensor

 Keyword args:
     {out}

 Example::

     >>> torch.i0(torch.arange(5, dtype=torch.float32))
     tensor([ 1.0000,  1.2661,  2.2796,  4.8808, 11.3019])

 """.format(**common_args))

 i0e = _add_docstr(_special.special_i0e,
                   r"""
 i0e(input, *, out=None) -> Tensor
 Computes the exponentially scaled zeroth order modified Bessel function of the first kind (as defined below)
 for each element of :attr:`input`.

 .. math::
     \text{out}_{i} = \exp(-|x|) * i0(x) = \exp(-|x|) * \sum_{k=0}^{\infty} \frac{(\text{input}_{i}^2/4)^k}{(k!)^2}

 """ + r"""
 Args:
     {input}

 Keyword args:
     {out}

 Example::
     >>> torch.special.i0e(torch.arange(5, dtype=torch.float32))
     tensor([1.0000, 0.4658, 0.3085, 0.2430, 0.2070])
 """.format(**common_args))

 i1 = _add_docstr(_special.special_i1,
                  r"""
 i1(input, *, out=None) -> Tensor
 Computes the first order modified Bessel function of the first kind (as defined below)
 for each element of :attr:`input`.

 .. math::
     \text{out}_{i} = \frac{(\text{input}_{i})}{2} * \sum_{k=0}^{\infty} \frac{(\text{input}_{i}^2/4)^k}{(k!) * (k+1)!}

 """ + r"""
 Args:
     {input}

 Keyword args:
     {out}

 Example::
     >>> torch.special.i1(torch.arange(5, dtype=torch.float32))
     tensor([0.0000, 0.5652, 1.5906, 3.9534, 9.7595])
 """.format(**common_args))

 i1e = _add_docstr(_special.special_i1e,
                   r"""
 i1e(input, *, out=None) -> Tensor
 Computes the exponentially scaled first order modified Bessel function of the first kind (as defined below)
 for each element of :attr:`input`.

 .. math::
     \text{out}_{i} = \exp(-|x|) * i1(x) =
         \exp(-|x|) * \frac{(\text{input}_{i})}{2} * \sum_{k=0}^{\infty} \frac{(\text{input}_{i}^2/4)^k}{(k!) * (k+1)!}

 """ + r"""
 Args:
     {input}

 Keyword args:
     {out}

 Example::
     >>> torch.special.i1e(torch.arange(5, dtype=torch.float32))
     tensor([0.0000, 0.2079, 0.2153, 0.1968, 0.1788])
 """.format(**common_args))

 ndtr = _add_docstr(_special.special_ndtr,
                    r"""
 ndtr(input, *, out=None) -> Tensor
 Computes the area under the standard Gaussian probability density function,
 integrated from minus infinity to :attr:`input`, elementwise.

 .. math::
     \text{ndtr}(x) = \frac{1}{\sqrt{2 \pi}}\int_{-\infty}^{x} e^{-\frac{1}{2}t^2} dt

 """ + r"""
 Args:
     {input}

 Keyword args:
     {out}

 Example::
     >>> torch.special.ndtr(torch.tensor([-3., -2, -1, 0, 1, 2, 3]))
     tensor([0.0013, 0.0228, 0.1587, 0.5000, 0.8413, 0.9772, 0.9987])
 """.format(**common_args))

 ndtri = _add_docstr(_special.special_ndtri,
                     r"""
 ndtri(input, *, out=None) -> Tensor
 Computes the argument, x, for which the area under the Gaussian probability density function
 (integrated from minus infinity to x) is equal to :attr:`input`, elementwise.

 .. math::
     \text{ndtri}(p) = \sqrt{2}\text{erf}^{-1}(2p - 1)

 .. note::
     Also known as quantile function for Normal Distribution.

 """ + r"""
 Args:
     {input}

 Keyword args:
     {out}

 Example::
     >>> torch.special.ndtri(torch.tensor([0, 0.25, 0.5, 0.75, 1]))
     tensor([   -inf, -0.6745,  0.0000,  0.6745,     inf])
 """.format(**common_args))

 log1p = _add_docstr(_special.special_log1p,
                     r"""
 log1p(input, *, out=None) -> Tensor

 Alias for :func:`torch.log1p`.
 """)

 sinc = _add_docstr(_special.special_sinc,
                    r"""
 sinc(input, *, out=None) -> Tensor

 Computes the normalized sinc of :attr:`input.`

 .. math::
     \text{out}_{i} =
     \begin{cases}
       1, & \text{if}\ \text{input}_{i}=0 \\
       \sin(\pi \text{input}_{i}) / (\pi \text{input}_{i}), & \text{otherwise}
     \end{cases}
 """ + r"""

 Args:
     {input}

 Keyword args:
     {out}

 Example::
     >>> t = torch.randn(4)
     >>> t
     tensor([ 0.2252, -0.2948,  1.0267, -1.1566])
     >>> torch.special.sinc(t)
     tensor([ 0.9186,  0.8631, -0.0259, -0.1300])
 """.format(**common_args))

 round = _add_docstr(_special.special_round,
                     r"""
 round(input, *, out=None) -> Tensor

 Alias for :func:`torch.round`.
 """)

 log_softmax = _add_docstr(_special.special_log_softmax,
                           r"""
 log_softmax(input, dim, *, dtype=None) -> Tensor
 Computes softmax followed by a logarithm.

 While mathematically equivalent to log(softmax(x)), doing these two
 operations separately is slower and numerically unstable. This function
 is computed as:

 .. math::
     \text{log\_softmax}(x_{i}) = \log\left(\frac{\exp(x_i) }{ \sum_j \exp(x_j)} \right)
 """ + r"""

 Args:
     input (Tensor): input
     dim (int): A dimension along which log_softmax will be computed.
     dtype (:class:`torch.dtype`, optional): the desired data type of returned tensor.
         If specified, the input tensor is cast to :attr:`dtype` before the operation
         is performed. This is useful for preventing data type overflows. Default: None.

 Example::
     >>> t = torch.ones(2, 2)
     >>> torch.special.log_softmax(t, 0)
     tensor([[-0.6931, -0.6931],
             [-0.6931, -0.6931]])
 """)

 zeta = _add_docstr(_special.special_zeta,
                    r"""
 zeta(input, other, *, out=None) -> Tensor

 Computes the Hurwitz zeta function, elementwise.

 .. math::
     \zeta(x, q) = \sum_{k=0}^{\infty} \frac{1}{(k + q)^x}

 """ + r"""
 Args:
     input (Tensor): the input tensor corresponding to `x`.
     other (Tensor): the input tensor corresponding to `q`.

 .. note::
     The Riemann zeta function corresponds to the case when `q = 1`

 Keyword args:
     {out}

 Example::
     >>> x = torch.tensor([2., 4.])
     >>> torch.special.zeta(x, 1)
     tensor([1.6449, 1.0823])
     >>> torch.special.zeta(x, torch.tensor([1., 2.]))
     tensor([1.6449, 0.0823])
     >>> torch.special.zeta(2, torch.tensor([1., 2.]))
     tensor([1.6449, 0.6449])
 """.format(**common_args))
	import sys

	import torch
	from torch._C import _add_docstr, _special # type: ignore[attr-defined]
	from torch._torch_docs import common_args, multi_dim_common

	Tensor = torch.Tensor

	entr = _add_docstr(_special.special_entr,
	r"""
	entr(input, *, out=None) -> Tensor
	Computes the entropy on :attr:`input` (as defined below), elementwise.

	.. math::
	\begin{align}
	\text{entr(x)} = \begin{cases}
	-x * \ln(x) & x > 0 \\
	0 & x = 0.0 \\
	-\infty & x < 0
	\end{cases}
	\end{align}
	""" + """

	Args:
	input (Tensor): the input tensor.

	Keyword args:
	out (Tensor, optional): the output tensor.

	Example::
	>>> a = torch.arange(-0.5, 1, 0.5)
	>>> a
	tensor([-0.5000, 0.0000, 0.5000])
	>>> torch.special.entr(a)
	tensor([ -inf, 0.0000, 0.3466])
	""")

	psi = _add_docstr(_special.special_psi,
	r"""
	psi(input, *, out=None) -> Tensor

	Alias for :func:`torch.special.digamma`.
	""")

	digamma = _add_docstr(_special.special_digamma,
	r"""
	digamma(input, *, out=None) -> Tensor

	Computes the logarithmic derivative of the gamma function on `input`.

	.. math::
	\digamma(x) = \frac{d}{dx} \ln\left(\Gamma\left(x\right)\right) = \frac{\Gamma'(x)}{\Gamma(x)}
	""" + r"""
	Args:
	input (Tensor): the tensor to compute the digamma function on

	Keyword args:
	{out}

	.. note:: This function is similar to SciPy's `scipy.special.digamma`.

	.. note:: From PyTorch 1.8 onwards, the digamma function returns `-Inf` for `0`.
	Previously it returned `NaN` for `0`.

	Example::

	>>> a = torch.tensor([1, 0.5])
	>>> torch.special.digamma(a)
	tensor([-0.5772, -1.9635])

	""".format(**common_args))

	gammaln = _add_docstr(_special.special_gammaln,
	r"""
	gammaln(input, *, out=None) -> Tensor

	Computes the natural logarithm of the absolute value of the gamma function on :attr:`input`.

	.. math::
	\text{out}_{i} = \ln \Gamma(\|\text{input}_{i}\|)
	""" + """
	Args:
	{input}

	Keyword args:
	{out}

	Example::

	>>> a = torch.arange(0.5, 2, 0.5)
	>>> torch.special.gammaln(a)
	tensor([ 0.5724, 0.0000, -0.1208])

	""".format(**common_args))

	erf = _add_docstr(_special.special_erf,
	r"""
	erf(input, *, out=None) -> Tensor

	Computes the error function of :attr:`input`. The error function is defined as follows:

	.. math::
	\mathrm{erf}(x) = \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^2} dt
	""" + r"""
	Args:
	{input}

	Keyword args:
	{out}

	Example::

	>>> torch.special.erf(torch.tensor([0, -1., 10.]))
	tensor([ 0.0000, -0.8427, 1.0000])
	""".format(**common_args))

	erfc = _add_docstr(_special.special_erfc,
	r"""
	erfc(input, *, out=None) -> Tensor

	Computes the complementary error function of :attr:`input`.
	The complementary error function is defined as follows:

	.. math::
	\mathrm{erfc}(x) = 1 - \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^2} dt
	""" + r"""
	Args:
	{input}

	Keyword args:
	{out}

	Example::

	>>> torch.special.erfc(torch.tensor([0, -1., 10.]))
	tensor([ 1.0000, 1.8427, 0.0000])
	""".format(**common_args))

	erfcx = _add_docstr(_special.special_erfcx,
	r"""
	erfcx(input, *, out=None) -> Tensor

	Computes the scaled complementary error function for each element of :attr:`input`.
	The scaled complementary error function is defined as follows:

	.. math::
	\mathrm{erfcx}(x) = e^{x^2} \mathrm{erfc}(x)
	""" + r"""

	""" + r"""
	Args:
	{input}

	Keyword args:
	{out}

	Example::

	>>> torch.special.erfcx(torch.tensor([0, -1., 10.]))
	tensor([ 1.0000, 5.0090, 0.0561])
	""".format(**common_args))

	erfinv = _add_docstr(_special.special_erfinv,
	r"""
	erfinv(input, *, out=None) -> Tensor

	Computes the inverse error function of :attr:`input`.
	The inverse error function is defined in the range :math:`(-1, 1)` as:

	.. math::
	\mathrm{erfinv}(\mathrm{erf}(x)) = x
	""" + r"""

	Args:
	{input}

	Keyword args:
	{out}

	Example::

	>>> torch.special.erfinv(torch.tensor([0, 0.5, -1.]))
	tensor([ 0.0000, 0.4769, -inf])
	""".format(**common_args))

	logit = _add_docstr(_special.special_logit,
	r"""
	logit(input, eps=None, *, out=None) -> Tensor

	Returns a new tensor with the logit of the elements of :attr:`input`.
	:attr:`input` is clamped to [eps, 1 - eps] when eps is not None.
	When eps is None and :attr:`input` < 0 or :attr:`input` > 1, the function will yields NaN.

	.. math::
	\begin{align}
	y_{i} &= \ln(\frac{z_{i}}{1 - z_{i}}) \\
	z_{i} &= \begin{cases}
	x_{i} & \text{if eps is None} \\
	\text{eps} & \text{if } x_{i} < \text{eps} \\
	x_{i} & \text{if } \text{eps} \leq x_{i} \leq 1 - \text{eps} \\
	1 - \text{eps} & \text{if } x_{i} > 1 - \text{eps}
	\end{cases}
	\end{align}
	""" + r"""
	Args:
	{input}
	eps (float, optional): the epsilon for input clamp bound. Default: ``None``

	Keyword args:
	{out}

	Example::

	>>> a = torch.rand(5)
	>>> a
	tensor([0.2796, 0.9331, 0.6486, 0.1523, 0.6516])
	>>> torch.special.logit(a, eps=1e-6)
	tensor([-0.9466, 2.6352, 0.6131, -1.7169, 0.6261])
	""".format(**common_args))

	logsumexp = _add_docstr(_special.special_logsumexp,
	r"""
	logsumexp(input, dim, keepdim=False, *, out=None)

	Alias for :func:`torch.logsumexp`.
	""".format(**multi_dim_common))

	expit = _add_docstr(_special.special_expit,
	r"""
	expit(input, *, out=None) -> Tensor

	Computes the expit (also known as the logistic sigmoid function) of the elements of :attr:`input`.

	.. math::
	\text{out}_{i} = \frac{1}{1 + e^{-\text{input}_{i}}}
	""" + r"""
	Args:
	{input}

	Keyword args:
	{out}

	Example::

	>>> t = torch.randn(4)
	>>> t
	tensor([ 0.9213, 1.0887, -0.8858, -1.7683])
	>>> torch.special.expit(t)
	tensor([ 0.7153, 0.7481, 0.2920, 0.1458])
	""".format(**common_args))

	exp2 = _add_docstr(_special.special_exp2,
	r"""
	exp2(input, *, out=None) -> Tensor

	Computes the base two exponential function of :attr:`input`.

	.. math::
	y_{i} = 2^{x_{i}}

	""" + r"""
	Args:
	{input}

	Keyword args:
	{out}

	Example::

	>>> torch.special.exp2(torch.tensor([0, math.log2(2.), 3, 4]))
	tensor([ 1., 2., 8., 16.])
	""".format(**common_args))

	expm1 = _add_docstr(_special.special_expm1,
	r"""
	expm1(input, *, out=None) -> Tensor

	Computes the exponential of the elements minus 1
	of :attr:`input`.

	.. math::
	y_{i} = e^{x_{i}} - 1

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

	""" + r"""
	Args:
	{input}

	Keyword args:
	{out}

	Example::

	>>> torch.special.expm1(torch.tensor([0, math.log(2.)]))
	tensor([ 0., 1.])
	""".format(**common_args))

	xlog1py = _add_docstr(_special.special_xlog1py,
	r"""
	xlog1py(input, other, *, out=None) -> Tensor

	Computes ``input * log1p(other)`` with the following cases.

	.. math::
	\text{out}_{i} = \begin{cases}
	\text{NaN} & \text{if } \text{other}_{i} = \text{NaN} \\
	0 & \text{if } \text{input}_{i} = 0.0 \text{ and } \text{other}_{i} != \text{NaN} \\
	\text{input}_{i} * \text{log1p}(\text{other}_{i})& \text{otherwise}
	\end{cases}

	Similar to SciPy's `scipy.special.xlog1py`.

	""" + r"""

	Args:
	input (Number or Tensor) : Multiplier
	other (Number or Tensor) : Argument

	.. note:: At least one of :attr:`input` or :attr:`other` must be a tensor.

	Keyword args:
	{out}

	Example::

	>>> x = torch.zeros(5,)
	>>> y = torch.tensor([-1, 0, 1, float('inf'), float('nan')])
	>>> torch.special.xlog1py(x, y)
	tensor([0., 0., 0., 0., nan])
	>>> x = torch.tensor([1, 2, 3])
	>>> y = torch.tensor([3, 2, 1])
	>>> torch.special.xlog1py(x, y)
	tensor([1.3863, 2.1972, 2.0794])
	>>> torch.special.xlog1py(x, 4)
	tensor([1.6094, 3.2189, 4.8283])
	>>> torch.special.xlog1py(2, y)
	tensor([2.7726, 2.1972, 1.3863])
	""".format(**common_args))

	xlogy = _add_docstr(_special.special_xlogy,
	r"""
	xlogy(input, other, *, out=None) -> Tensor

	Computes ``input * log(other)`` with the following cases.

	.. math::
	\text{out}_{i} = \begin{cases}
	\text{NaN} & \text{if } \text{other}_{i} = \text{NaN} \\
	0 & \text{if } \text{input}_{i} = 0.0 \\
	\text{input}_{i} * \log{(\text{other}_{i})} & \text{otherwise}
	\end{cases}

	Similar to SciPy's `scipy.special.xlogy`.

	""" + r"""

	Args:
	input (Number or Tensor) : Multiplier
	other (Number or Tensor) : Argument

	.. note:: At least one of :attr:`input` or :attr:`other` must be a tensor.

	Keyword args:
	{out}

	Example::

	>>> x = torch.zeros(5,)
	>>> y = torch.tensor([-1, 0, 1, float('inf'), float('nan')])
	>>> torch.special.xlogy(x, y)
	tensor([0., 0., 0., 0., nan])
	>>> x = torch.tensor([1, 2, 3])
	>>> y = torch.tensor([3, 2, 1])
	>>> torch.special.xlogy(x, y)
	tensor([1.0986, 1.3863, 0.0000])
	>>> torch.special.xlogy(x, 4)
	tensor([1.3863, 2.7726, 4.1589])
	>>> torch.special.xlogy(2, y)
	tensor([2.1972, 1.3863, 0.0000])
	""".format(**common_args))

	i0 = _add_docstr(_special.special_i0,
	r"""
	i0(input, *, out=None) -> Tensor

	Computes the zeroth order modified Bessel function of the first kind for each element of :attr:`input`.

	.. math::
	\text{out}_{i} = I_0(\text{input}_{i}) = \sum_{k=0}^{\infty} \frac{(\text{input}_{i}^2/4)^k}{(k!)^2}

	""" + r"""
	Args:
	input (Tensor): the input tensor

	Keyword args:
	{out}

	Example::

	>>> torch.i0(torch.arange(5, dtype=torch.float32))
	tensor([ 1.0000, 1.2661, 2.2796, 4.8808, 11.3019])

	""".format(**common_args))

	i0e = _add_docstr(_special.special_i0e,
	r"""
	i0e(input, *, out=None) -> Tensor
	Computes the exponentially scaled zeroth order modified Bessel function of the first kind (as defined below)
	for each element of :attr:`input`.

	.. math::
	\text{out}_{i} = \exp(-\|x\|) * i0(x) = \exp(-\|x\|) * \sum_{k=0}^{\infty} \frac{(\text{input}_{i}^2/4)^k}{(k!)^2}

	""" + r"""
	Args:
	{input}

	Keyword args:
	{out}

	Example::
	>>> torch.special.i0e(torch.arange(5, dtype=torch.float32))
	tensor([1.0000, 0.4658, 0.3085, 0.2430, 0.2070])
	""".format(**common_args))

	i1 = _add_docstr(_special.special_i1,
	r"""
	i1(input, *, out=None) -> Tensor
	Computes the first order modified Bessel function of the first kind (as defined below)
	for each element of :attr:`input`.

	.. math::
	\text{out}_{i} = \frac{(\text{input}_{i})}{2} * \sum_{k=0}^{\infty} \frac{(\text{input}_{i}^2/4)^k}{(k!) * (k+1)!}

	""" + r"""
	Args:
	{input}

	Keyword args:
	{out}

	Example::
	>>> torch.special.i1(torch.arange(5, dtype=torch.float32))
	tensor([0.0000, 0.5652, 1.5906, 3.9534, 9.7595])
	""".format(**common_args))

	i1e = _add_docstr(_special.special_i1e,
	r"""
	i1e(input, *, out=None) -> Tensor
	Computes the exponentially scaled first order modified Bessel function of the first kind (as defined below)
	for each element of :attr:`input`.

	.. math::
	\text{out}_{i} = \exp(-\|x\|) * i1(x) =
	\exp(-\|x\|) * \frac{(\text{input}_{i})}{2} * \sum_{k=0}^{\infty} \frac{(\text{input}_{i}^2/4)^k}{(k!) * (k+1)!}

	""" + r"""
	Args:
	{input}

	Keyword args:
	{out}

	Example::
	>>> torch.special.i1e(torch.arange(5, dtype=torch.float32))
	tensor([0.0000, 0.2079, 0.2153, 0.1968, 0.1788])
	""".format(**common_args))

	ndtr = _add_docstr(_special.special_ndtr,
	r"""
	ndtr(input, *, out=None) -> Tensor
	Computes the area under the standard Gaussian probability density function,
	integrated from minus infinity to :attr:`input`, elementwise.

	.. math::
	\text{ndtr}(x) = \frac{1}{\sqrt{2 \pi}}\int_{-\infty}^{x} e^{-\frac{1}{2}t^2} dt

	""" + r"""
	Args:
	{input}

	Keyword args:
	{out}

	Example::
	>>> torch.special.ndtr(torch.tensor([-3., -2, -1, 0, 1, 2, 3]))
	tensor([0.0013, 0.0228, 0.1587, 0.5000, 0.8413, 0.9772, 0.9987])
	""".format(**common_args))

	ndtri = _add_docstr(_special.special_ndtri,
	r"""
	ndtri(input, *, out=None) -> Tensor
	Computes the argument, x, for which the area under the Gaussian probability density function
	(integrated from minus infinity to x) is equal to :attr:`input`, elementwise.

	.. math::
	\text{ndtri}(p) = \sqrt{2}\text{erf}^{-1}(2p - 1)

	.. note::
	Also known as quantile function for Normal Distribution.

	""" + r"""
	Args:
	{input}

	Keyword args:
	{out}

	Example::
	>>> torch.special.ndtri(torch.tensor([0, 0.25, 0.5, 0.75, 1]))
	tensor([ -inf, -0.6745, 0.0000, 0.6745, inf])
	""".format(**common_args))

	log1p = _add_docstr(_special.special_log1p,
	r"""
	log1p(input, *, out=None) -> Tensor

	Alias for :func:`torch.log1p`.
	""")

	sinc = _add_docstr(_special.special_sinc,
	r"""
	sinc(input, *, out=None) -> Tensor

	Computes the normalized sinc of :attr:`input.`

	.. math::
	\text{out}_{i} =
	\begin{cases}
	1, & \text{if}\ \text{input}_{i}=0 \\
	\sin(\pi \text{input}_{i}) / (\pi \text{input}_{i}), & \text{otherwise}
	\end{cases}
	""" + r"""

	Args:
	{input}

	Keyword args:
	{out}

	Example::
	>>> t = torch.randn(4)
	>>> t
	tensor([ 0.2252, -0.2948, 1.0267, -1.1566])
	>>> torch.special.sinc(t)
	tensor([ 0.9186, 0.8631, -0.0259, -0.1300])
	""".format(**common_args))

	round = _add_docstr(_special.special_round,
	r"""
	round(input, *, out=None) -> Tensor

	Alias for :func:`torch.round`.
	""")

	log_softmax = _add_docstr(_special.special_log_softmax,
	r"""
	log_softmax(input, dim, *, dtype=None) -> Tensor
	Computes softmax followed by a logarithm.

	While mathematically equivalent to log(softmax(x)), doing these two
	operations separately is slower and numerically unstable. This function
	is computed as:

	.. math::
	\text{log\_softmax}(x_{i}) = \log\left(\frac{\exp(x_i) }{ \sum_j \exp(x_j)} \right)
	""" + r"""

	Args:
	input (Tensor): input
	dim (int): A dimension along which log_softmax will be computed.
	dtype (:class:`torch.dtype`, optional): the desired data type of returned tensor.
	If specified, the input tensor is cast to :attr:`dtype` before the operation
	is performed. This is useful for preventing data type overflows. Default: None.

	Example::
	>>> t = torch.ones(2, 2)
	>>> torch.special.log_softmax(t, 0)
	tensor([[-0.6931, -0.6931],
	[-0.6931, -0.6931]])
	""")

	zeta = _add_docstr(_special.special_zeta,
	r"""
	zeta(input, other, *, out=None) -> Tensor

	Computes the Hurwitz zeta function, elementwise.

	.. math::
	\zeta(x, q) = \sum_{k=0}^{\infty} \frac{1}{(k + q)^x}

	""" + r"""
	Args:
	input (Tensor): the input tensor corresponding to `x`.
	other (Tensor): the input tensor corresponding to `q`.

	.. note::
	The Riemann zeta function corresponds to the case when `q = 1`

	Keyword args:
	{out}

	Example::
	>>> x = torch.tensor([2., 4.])
	>>> torch.special.zeta(x, 1)
	tensor([1.6449, 1.0823])
	>>> torch.special.zeta(x, torch.tensor([1., 2.]))
	tensor([1.6449, 0.0823])
	>>> torch.special.zeta(2, torch.tensor([1., 2.]))
	tensor([1.6449, 0.6449])
	""".format(**common_args))