Operation Semantics

The following describes the semantics of operations defined in the XlaBuilder interface. Typically, these operations map one-to-one to operations defined in the RPC interface in xla_data.proto.

A note on nomenclature: the generalized data type XLA deals with is an N-dimensional array holding elements of some uniform type (such as 32-bit float). Throughout the documentation, array is used to denote an arbitrary-dimensional array. For convenience, special cases have more specific and familiar names; for example a vector is a 1-dimensional array and a matrix is a 2-dimensional array.

AfterAll

Arguments	Type	Semantics
`operands`	`XlaOp`	variadic number of tokens

AllReduce

Arguments	Type	Semantics
`operand`	`XlaOp`	Array or a non-empty tuple of arrays to reduce across replicas.
`computation`	`XlaComputation`	Reduction computation
`replica_groups`	vector of vectors of `int64`	Groups between which the reductions are performed
`channel_id`	optional `int64`	Optional channel ID for cross-module communication

AllToAll

BatchNormGrad

See also XlaBuilder::BatchNormGrad and the original batch normalization paper for a detailed description of the algorithm.

Calculates gradients of batch norm.

BatchNormGrad(operand, scale, mean, variance, grad_output, epsilon, feature_index)

Arguments	Type	Semantics
`operand`	`XlaOp`	n dimensional array to be normalized (x)
`scale`	`XlaOp`	1 dimensional array ($\gamma$)
`mean`	`XlaOp`	1 dimensional array ($\mu$)
`variance`	`XlaOp`	1 dimensional array ($\sigma^2$)
`grad_output`	`XlaOp`	Gradients passed to `BatchNormTraining` ($ \nabla y$)
`epsilon`	`float`	Epsilon value ($\epsilon$)
`feature_index`	`int64`	Index to feature dimension in `operand`

For each feature in the feature dimension (feature_index is the index for the feature dimension in operand), the operation calculates the gradients with respect to operand, offset and scale across all the other dimensions. The feature_index must be a valid index for the feature dimension in operand.

The three gradients are defined by the following formulas (assuming a 4-dimensional array as operand and with feature dimension index l, batch size m and spatial sizes w and h):

\[ \begin{split} c_l&= \frac{1}{mwh}\sum_{i=1}^m\sum_{j=1}^w\sum_{k=1}^h \left( \nabla y_{ijkl} \frac{x_{ijkl} - \mu_l}{\sigma^2_l+\epsilon} \right) \\\\ \nabla x_{ijkl} &= \frac{\gamma_{l} }{\sqrt{\sigma^2_{l}+\epsilon} } \left( \nabla y_{ijkl} - \mathrm{mean}(\nabla y) - c_l (x_{ijkl} - \mu_{l}) \right) \\\\ \nabla \gamma_l &= \sum_{i=1}^m\sum_{j=1}^w\sum_{k=1}^h \left( \nabla y_{ijkl} \frac{x_{ijkl} - \mu_l}{\sqrt{\sigma^2_{l}+\epsilon} } \right) \\\\\ \nabla \beta_l &= \sum_{i=1}^m\sum_{j=1}^w\sum_{k=1}^h \nabla y_{ijkl} \end{split} \]

The inputs mean and variance represent moments value across batch and spatial dimensions.

The output type is a tuple of three handles:

Outputs	Type	Semantics
`grad_operand`	`XlaOp`	gradient with respect to input `operand` ($ \nabla x$)
`grad_scale`	`XlaOp`	gradient with respect to input `scale` ($ \nabla \gamma$)
`grad_offset`	`XlaOp`	gradient with respect to input `offset`($ \nabla \beta$)

BatchNormInference

See also XlaBuilder::BatchNormInference and the original batch normalization paper for a detailed description of the algorithm.

Normalizes an array across batch and spatial dimensions.

BatchNormInference(operand, scale, offset, mean, variance, epsilon, feature_index)

Arguments	Type	Semantics
`operand`	`XlaOp`	n dimensional array to be normalized
`scale`	`XlaOp`	1 dimensional array
`offset`	`XlaOp`	1 dimensional array
`mean`	`XlaOp`	1 dimensional array
`variance`	`XlaOp`	1 dimensional array
`epsilon`	`float`	Epsilon value
`feature_index`	`int64`	Index to feature dimension in `operand`

For each feature in the feature dimension (feature_index is the index for the feature dimension in operand), the operation calculates the mean and variance across all the other dimensions and uses the mean and variance to normalize each element in operand. The feature_index must be a valid index for the feature dimension in operand.

BatchNormInference is equivalent to calling BatchNormTraining without computing mean and variance for each batch. It uses the input mean and variance instead as estimated values. The purpose of this op is to reduce latency in inference, hence the name BatchNormInference.

The output is an n-dimensional, normalized array with the same shape as input operand.

BatchNormTraining

See also XlaBuilder::BatchNormTraining and the original batch normalization paper for a detailed description of the algorithm.

Normalizes an array across batch and spatial dimensions.

BatchNormTraining(operand, scale, offset, epsilon, feature_index)

Arguments	Type	Semantics
`operand`	`XlaOp`	n dimensional array to be normalized (x)
`scale`	`XlaOp`	1 dimensional array ($\gamma$)
`offset`	`XlaOp`	1 dimensional array ($\beta$)
`epsilon`	`float`	Epsilon value ($\epsilon$)
`feature_index`	`int64`	Index to feature dimension in `operand`

The algorithm goes as follows for each batch in operand $x$ that contains m elements with w and h as the size of spatial dimensions (assuming operand is an 4 dimensional array):

Calculates batch mean $\mu_l$ for each feature l in feature dimension: $\mu_l=\frac{1}{mwh}\sum_{i=1}^m\sum_{j=1}^w\sum_{k=1}^h x_{ijkl}$
Calculates batch variance $\sigma^2_l$: $\sigma^2_l=\frac{1}{mwh}\sum_{i=1}^m\sum_{j=1}^w\sum_{k=1}^h (x_{ijkl} - \mu_l)^2$
Normalizes, scales and shifts: $y_{ijkl}=\frac{\gamma_l(x_{ijkl}-\mu_l)}{\sqrt[2]{\sigma^2_l+\epsilon} }+\beta_l$

The epsilon value, usually a small number, is added to avoid divide-by-zero errors.

The output type is a tuple of three XlaOps:

Outputs	Type	Semantics
`output`	`XlaOp`	n dimensional array with the same shape as input `operand` (y)
`batch_mean`	`XlaOp`	1 dimensional array ($\mu$)
`batch_var`	`XlaOp`	1 dimensional array ($\sigma^2$)

The batch_mean and batch_var are moments calculated across the batch and spatial dimensions using the formulas above.

BitcastConvertType

Similar to a tf.bitcast in TensorFlow, performs an element-wise bitcast operation from a data shape to a target shape. The dimensions must match, and the conversion is an element-wise one; e.g. s32 elements become f32 elements via bitcast routine. Bitcast is implemented as a low-level cast, so machines with different floating-point representations will give different results.

BitcastConvertType(operand, new_element_type)

Arguments	Type	Semantics
`operand`	`XlaOp`	array of type T with dims D
`new_element_type`	`PrimitiveType`	type U

The dimensions of the operand and the target shape must match. The bit-width of the source and destination element types must be equal. The source and destination element types must not be tuples.

Broadcast

Arguments	Type	Semantics
`operand`	`XlaOp`	The array to duplicate
`broadcast_sizes`	`ArraySlice<int64>`	The sizes of the new dimensions

BroadcastInDim

Arguments	Type	Semantics
`operand`	`XlaOp`	The array to duplicate
`out_dim_size`	`ArraySlice<int64>`	The sizes of the dimensions of the target shape
`broadcast_dimensions`	`ArraySlice<int64>`	Which dimension in the target shape each dimension of the operand shape corresponds to

Call

Arguments	Type	Semantics
`computation`	`XlaComputation`	computation of type `T_0, T_1, ..., T_N -> S` with N parameters of arbitrary type
`args`	sequence of N `XlaOp`s	N arguments of arbitrary type

Cholesky

Arguments	Type	Semantics
`a`	`XlaOp`	a rank > 2 array of a complex or floating-point type.
`lower`	`bool`	whether to use the upper or lower triangle of `a`.

Clamp

Arguments	Type	Semantics
`min`	`XlaOp`	array of type T
`operand`	`XlaOp`	array of type T
`max`	`XlaOp`	array of type T

Collapse

See also XlaBuilder::Collapse and the tf.reshape operation.

Collapses dimensions of an array into one dimension.

Collapse(operand, dimensions)

Arguments	Type	Semantics
`operand`	`XlaOp`	array of type T
`dimensions`	`int64` vector	in-order, consecutive subset of T's dimensions.

Collapse replaces the given subset of the operand's dimensions by a single dimension. The input arguments are an arbitrary array of type T and a compile-time-constant vector of dimension indices. The dimension indices must be an in-order (low to high dimension numbers), consecutive subset of T's dimensions. Thus, {0, 1, 2}, {0, 1}, or {1, 2} are all valid dimension sets, but {1, 0} or {0, 2} are not. They are replaced by a single new dimension, in the same position in the dimension sequence as those they replace, with the new dimension size equal to the product of original dimension sizes. The lowest dimension number in dimensions is the slowest varying dimension (most major) in the loop nest which collapses these dimension, and the highest dimension number is fastest varying (most minor). See the tf.reshape operator if more general collapse ordering is needed.

For example, let v be an array of 24 elements:

let v = f32[4x2x3] { { {10, 11, 12},  {15, 16, 17} },
{ {20, 21, 22},  {25, 26, 27} },
{ {30, 31, 32},  {35, 36, 37} },
{ {40, 41, 42},  {45, 46, 47} } };

// Collapse to a single dimension, leaving one dimension.
let v012 = Collapse(v, {0,1,2});
then v012 == f32[24] {10, 11, 12, 15, 16, 17,
20, 21, 22, 25, 26, 27,
30, 31, 32, 35, 36, 37,
40, 41, 42, 45, 46, 47};

// Collapse the two lower dimensions, leaving two dimensions.
let v01 = Collapse(v, {0,1});
then v01 == f32[4x6] { {10, 11, 12, 15, 16, 17},
{20, 21, 22, 25, 26, 27},
{30, 31, 32, 35, 36, 37},
{40, 41, 42, 45, 46, 47} };

// Collapse the two higher dimensions, leaving two dimensions.
let v12 = Collapse(v, {1,2});
then v12 == f32[8x3] { {10, 11, 12},
{15, 16, 17},
{20, 21, 22},
{25, 26, 27},
{30, 31, 32},
{35, 36, 37},
{40, 41, 42},
{45, 46, 47} };

CollectivePermute

CollectivePermute is a collective operation that sends and receives data cross replicas.

CollectivePermute(operand, source_target_pairs)

Arguments	Type	Semantics
`operand`	`XlaOp`	n dimensional input array
`source_target_pairs`	`<int64, int64>` vector	A list of (source_replica_id, target_replica_id) pairs. For each pair, the operand is sent from source replica to target replica.

Note that there are the following restrictions on the source_target_pair:

Any two pairs should not have the same target replica id, and they should not have the same source replica id.
If a replica id is not a target in any pair, then the output on that replica is a tensor consists of 0(s) with the same shape as the input.

Concatenate

Arguments	Type	Semantics
`operands`	sequence of N `XlaOp`	N arrays of type T with dimensions [L0, L1, ...]. Requires N >= 1.
`dimension`	`int64`	A value in the interval `[0, N)` that names the dimension to be concatenated between the `operands`.

Conditional

Arguments	Type	Semantics
`pred`	`XlaOp`	Scalar of type `PRED`
`true_operand`	`XlaOp`	Argument of type \(T_0\)
`true_computation`	`XlaComputation`	XlaComputation of type \(T_0 \to S\)
`false_operand`	`XlaOp`	Argument of type \(T_1\)
`false_computation`	`XlaComputation`	XlaComputation of type \(T_1 \to S\)

Arguments	Type	Semantics
`branch_index`	`XlaOp`	Scalar of type `S32`
`branch_computations`	sequence of N `XlaComputation`	XlaComputations of type \( T_0 \to S , T_1 \to S , ..., T_{N-1} \to S \)
`branch_operands`	sequence of N `XlaOp`	Arguments of type \( T_0 , T_1 , ..., T_{N-1} \)

Conv (convolution)

ConvWithGeneralPadding (convolution)

Computes a convolution of the kind used in neural networks. Here, a convolution can be thought of as a n-dimensional window moving across a n-dimensional base area and a computation is performed for each possible position of the window.

Arguments	Type	Semantics
`lhs`	`XlaOp`	rank n+2 array of inputs
`rhs`	`XlaOp`	rank n+2 array of kernel weights
`window_strides`	`ArraySlice<int64>`	n-d array of kernel strides
`padding`	`ArraySlice< pair<int64, int64>>`	n-d array of (low, high) padding
`lhs_dilation`	`ArraySlice<int64>`	n-d lhs dilation factor array
`rhs_dilation`	`ArraySlice<int64>`	n-d rhs dilation factor array
`feature_group_count`	int64	the number of feature groups
`batch_group_count`	int64	the number of batch groups

Let n be the number of spatial dimensions. The lhs argument is a rank n+2 array describing the base area. This is called the input, even though of course the rhs is also an input. In a neural network, these are the input activations. The n+2 dimensions are, in this order:

batch: Each coordinate in this dimension represents an independent input for which convolution is carried out.
z/depth/features: Each (y,x) position in the base area has a vector associated to it, which goes into this dimension.
spatial_dims: Describes the n spatial dimensions that define the base area that the window moves across.

The rhs argument is a rank n+2 array describing the convolutional filter/kernel/window. The dimensions are, in this order:

output-z: The z dimension of the output.
input-z: The size of this dimension times feature_group_count should equal the size of the z dimension in lhs.
spatial_dims: Describes the n spatial dimensions that define the n-d window that moves across the base area.

The window_strides argument specifies the stride of the convolutional window in the spatial dimensions. For example, if the stride in the first spatial dimension is 3, then the window can only be placed at coordinates where the first spatial index is divisible by 3.

The padding argument specifies the amount of zero padding to be applied to the base area. The amount of padding can be negative -- the absolute value of negative padding indicates the number of elements to remove from the specified dimension before doing the convolution. padding[0] specifies the padding for dimension y and padding[1] specifies the padding for dimension x. Each pair has the low padding as the first element and the high padding as the second element. The low padding is applied in the direction of lower indices while the high padding is applied in the direction of higher indices. For example, if padding[1] is (2,3) then there will be a padding by 2 zeroes on the left and by 3 zeroes on the right in the second spatial dimension. Using padding is equivalent to inserting those same zero values into the input (lhs) before doing the convolution.

The lhs_dilation and rhs_dilation arguments specify the dilation factor to be applied to the lhs and rhs, respectively, in each spatial dimension. If the dilation factor in a spatial dimension is d, then d-1 holes are implicitly placed between each of the entries in that dimension, increasing the size of the array. The holes are filled with a no-op value, which for convolution means zeroes.

Dilation of the rhs is also called atrous convolution. For more details, see tf.nn.atrous_conv2d. Dilation of the lhs is also called transposed convolution. For more details, see tf.nn.conv2d_transpose.

The feature_group_count argument (default value 1) can be used for grouped convolutions. feature_group_count needs to be a divisor of both the input and the output feature dimension. If feature_group_count is greater than 1, it means that conceptually the input and output feature dimension and the rhs output feature dimension are split evenly into feature_group_count many groups, each group consisting of a consecutive subsequence of features. The input feature dimension of rhs needs to be equal to the lhs input feature dimension divided by feature_group_count (so it already has the size of a group of input features). The i-th groups are used together to compute feature_group_count many separate convolutions. The results of these convolutions are concatenated together in the output feature dimension.

For depthwise convolution the feature_group_count argument would be set to the input feature dimension, and the filter would be reshaped from [filter_height, filter_width, in_channels, channel_multiplier] to [filter_height, filter_width, 1, in_channels * channel_multiplier]. For more details, see tf.nn.depthwise_conv2d.

The batch_group_count (default value 1) argument can be used for grouped filters during backpropagation. batch_group_count needs to be a divisor of the size of the lhs (input) batch dimension. If batch_group_count is greater than 1, it means that the output batch dimension should be of size input batch / batch_group_count. The batch_group_count must be a divisor of the output feature size.

The output shape has these dimensions, in this order:

batch: The size of this dimension times batch_group_count should equal the size of the batch dimension in lhs.
z: Same size as output-z on the kernel (rhs).
spatial_dims: One value for each valid placement of the convolutional window.

The valid placements of the convolutional window are determined by the strides and the size of the base area after padding.

To describe what a convolution does, consider a 2d convolution, and pick some fixed batch, z, y, x coordinates in the output. Then (y,x) is a position of a corner of the window within the base area (e.g. the upper left corner, depending on how you interpret the spatial dimensions). We now have a 2d window, taken from the base area, where each 2d point is associated to a 1d vector, so we get a 3d box. From the convolutional kernel, since we fixed the output coordinate z, we also have a 3d box. The two boxes have the same dimensions, so we can take the sum of the element-wise products between the two boxes (similar to a dot product). That is the output value.

Note that if output-z is e.g., 5, then each position of the window produces 5 values in the output into the z dimension of the output. These values differ in what part of the convolutional kernel is used - there is a separate 3d box of values used for each output-z coordinate. So you could think of it as 5 separate convolutions with a different filter for each of them.

Here is pseudo-code for a 2d convolution with padding and striding:

for (b, oz, oy, ox) {  // output coordinates
  value = 0;
  for (iz, ky, kx) {  // kernel coordinates and input z
    iy = oy*stride_y + ky - pad_low_y;
    ix = ox*stride_x + kx - pad_low_x;
    if ((iy, ix) inside the base area considered without padding) {
      value += input(b, iz, iy, ix) * kernel(oz, iz, ky, kx);
    }
  }
  output(b, oz, oy, ox) = value;
}

ConvertElementType

Similar to an element-wise static_cast in C++, performs an element-wise conversion operation from a data shape to a target shape. The dimensions must match, and the conversion is an element-wise one; e.g. s32 elements become f32 elements via an s32-to-f32 conversion routine.

ConvertElementType(operand, new_element_type)

Arguments	Type	Semantics
`operand`	`XlaOp`	array of type T with dims D
`new_element_type`	`PrimitiveType`	type U

The dimensions of the operand and the target shape must match. The source and destination element types must not be tuples.

A conversion such as T=s32 to U=f32 will perform a normalizing int-to-float conversion routine such as round-to-nearest-even.

Note: The precise float-to-int and visa-versa conversions are currently unspecified, but may become additional arguments to the convert operation in the future. Not all possible conversions have been implemented for all targets.

let a: s32[3] = {0, 1, 2};
let b: f32[3] = convert(a, f32);
then b == f32[3]{0.0, 1.0, 2.0}

CrossReplicaSum

Performs AllReduce with a summation computation.

CustomCall

Arguments	Type	Semantics
`target_name`	`string`	Name of the function. A call instruction will be emitted which targets this symbol name.
`args`	sequence of N `XlaOp`s	N arguments of arbitrary type, which will be passed to the function.
`shape`	`Shape`	Output shape of the function

Dot

Arguments	Type	Semantics
`lhs`	`XlaOp`	array of type T
`rhs`	`XlaOp`	array of type T

Input	Output	Semantics
vector [n] `dot` vector [n]	scalar	vector dot product
matrix [m x k] `dot` vector [k]	vector [m]	matrix-vector multiplication
matrix [m x k] `dot` matrix [k x n]	matrix [m x n]	matrix-matrix multiplication

DotGeneral

Arguments	Type	Semantics
`lhs`	`XlaOp`	array of type T
`rhs`	`XlaOp`	array of type T
`dimension_numbers`	`DotDimensionNumbers`	contracting and batch dimension numbers

DotDimensionNumbers Fields	Type	Semantics
'lhs_contracting_dimensions'	repeated int64	'lhs' contracting dimension numbers
'rhs_contracting_dimensions'	repeated int64	'rhs' contracting dimension numbers
'lhs_batch_dimensions'	repeated int64	'lhs' batch dimension numbers
'rhs_batch_dimensions'	repeated int64	'rhs' batch dimension numbers

Input	Output	Semantics
[b0, m, k] `dot` [b0, k, n]	[b0, m, n]	batch matmul
[b0, b1, m, k] `dot` [b0, b1, k, n]	[b0, b1, m, n]	batch matmul

DynamicSlice

DynamicUpdateSlice

DynamicUpdateSlice generates a result which is the value of the input array operand, with a slice update overwritten at start_indices. The shape of update determines the shape of the sub-array of the result which is updated. The shape of start_indices must be rank == 1, with dimension size equal to the rank of operand.

DynamicUpdateSlice(operand, update, start_indices)

Arguments	Type	Semantics
`operand`	`XlaOp`	N dimensional array of type T
`update`	`XlaOp`	N dimensional array of type T containing the slice update. Each dimension of update shape must be strictly greater than zero, and start + update must be less than or equal to the operand size for each dimension to avoid generating out-of-bounds update indices.
`start_indices`	sequence of N `XlaOp`	List of N scalar integers containing the starting indices of the slice for each dimension. Value must be greater than or equal to zero.

The effective slice indices are computed by applying the following transformation for each index i in [1, N) before performing the slice:

start_indices[i] = clamp(start_indices[i], 0, operand.dimension_size[i] - update.dimension_size[i])

This ensures that the updated slice is always in-bounds with respect to the operand array. If the slice is in-bounds before the transformation is applied, the transformation has no effect.

1-dimensional example:

let a = {0.0, 1.0, 2.0, 3.0, 4.0}
let u = {5.0, 6.0}
let s = {2}

DynamicUpdateSlice(a, u, s) produces:
{0.0, 1.0, 5.0, 6.0, 4.0}

2-dimensional example:

let b =
{ {0.0,  1.0,  2.0},
{3.0,  4.0,  5.0},
{6.0,  7.0,  8.0},
{9.0, 10.0, 11.0} }
let u =
{ {12.0,  13.0},
{14.0,  15.0},
{16.0,  17.0} }

let s = {1, 1}

DynamicUpdateSlice(b, u, s) produces:
{ {0.0,  1.0,  2.0},
{3.0, 12.0, 13.0},
{6.0, 14.0, 15.0},
{9.0, 16.0, 17.0} }

Element-wise binary arithmetic operations

Arguments	Type	Semantics
`lhs`	`XlaOp`	left-hand-side operand: array of type T
`rhs`	`XlaOp`	right-hand-side operand: array of type T

Element-wise comparison operations

Arguments	Type	Semantics
`lhs`	`XlaOp`	left-hand-side operand: array of type T
`rhs`	`XlaOp`	right-hand-side operand: array of type T

Element-wise unary functions

XlaBuilder supports these element-wise unary functions:

Abs(operand) Element-wise abs x -> |x|.

Ceil(operand) Element-wise ceil x -> ⌈x⌉.

Cos(operand) Element-wise cosine x -> cos(x).

Exp(operand) Element-wise natural exponential x -> e^x.

Floor(operand) Element-wise floor x -> ⌊x⌋.

Imag(operand) Element-wise imaginary part of a complex (or real) shape. x -> imag(x). If the operand is a floating point type, returns 0.

IsFinite(operand) Tests whether each element of operand is finite, i.e., is not positive or negative infinity, and is not NaN. Returns an array of PRED values with the same shape as the input, where each element is true if and only if the corresponding input element is finite.

Log(operand) Element-wise natural logarithm x -> ln(x).

LogicalNot(operand) Element-wise logical not x -> !(x).

PopulationCount(operand) Computes the number of bits set in each element of operand.

Neg(operand) Element-wise negation x -> -x.

Real(operand) Element-wise real part of a complex (or real) shape. x -> real(x). If the operand is a floating point type, returns the same value.

Rsqrt(operand) Element-wise reciprocal of square root operation x -> 1.0 / sqrt(x).

Sign(operand) Element-wise sign operation x -> sgn(x) where

$$\text{sgn}(x) = \begin{cases} -1 & x < 0\\ -0 & x = -0\\ NaN & x = NaN\\ +0 & x = +0\\ 1 & x > 0 \end{cases}$$

using the comparison operator of the element type of operand.

Sqrt(operand) Element-wise square root operation x -> sqrt(x).

Tanh(operand) Element-wise hyperbolic tangent x -> tanh(x).

Arguments	Type	Semantics
`operand`	`XlaOp`	The operand to the function

The function is applied to each element in the operand array, resulting in an array with the same shape. It is allowed for operand to be a scalar (rank 0).

Fft

The XLA FFT operation implements the forward and inverse Fourier Transforms for real and complex inputs/outputs. Multidimensional FFTs on up to 3 axes are supported, except on TPU, where only a single axis is supported (please file a github issue if you require higher order).

Arguments	Type	Semantics
`operand`	`XlaOp`	The array we are Fourier transforming.
`fft_type`	`FftType`	See the table below.
`fft_length`	`ArraySlice<int64>`	The time-domain lengths of the axes being transformed. This is needed in particular for IRFFT to right-size the innermost axis, since `RFFT(fft_length=[16])` has the same output shape as `RFFT(fft_length=[17])`.

`FftType`	Semantics
`FFT`	Forward complex-to-complex FFT. Shape is unchanged.
`IFFT`	Inverse complex-to-complex FFT. Shape is unchanged.
`RFFT`	Forward real-to-complex FFT. Shape of the innermost axis is reduced to `fft_length[-1] // 2 + 1` if `fft_length[-1]` is a non-zero value, omitting the reversed conjugate part of the transformed signal beyond the Nyquist frequency.
`IRFFT`	Inverse real-to-complex FFT (i.e. takes complex, returns real). Shape of the innermost axis is expanded to `fft_length[-1]` if `fft_length[-1]` is a non-zero value, inferring the part of the transformed signal beyond the Nyquist frequency from the reverse conjugate of the `1` to `fft_length[-1] // 2 + 1` entries.

Multidimensional FFT

When more than 1 fft_length is provided, this is equivalent to applying a cascade of FFT operations to each of the innermost axes. Note that for the real->complex and complex->real cases, the innermost axis transform is (effectively) performed first (RFFT; last for IRFFT), which is why the innermost axis is the one which changes size. Other axis transforms will then be complex->complex.

Implementation details

CPU FFT is backed by Eigen's TensorFFT. GPU FFT uses cuFFT.

Gather

The XLA gather operation stitches together several slices (each slice at a potentially different runtime offset) of an input array.

General Semantics

See also XlaBuilder::Gather. For a more intuitive description, see the "Informal Description" section below.

gather(operand, start_indices, offset_dims, collapsed_slice_dims, slice_sizes, start_index_map)

Arguments	Type	Semantics
`operand`	`XlaOp`	The array we’re gathering from.
`start_indices`	`XlaOp`	Array containing the starting indices of the slices we gather.
`index_vector_dim`	`int64`	The dimension in `start_indices` that "contains" the starting indices. See below for a detailed description.
`offset_dims`	`ArraySlice<int64>`	The set of dimensions in the output shape that offset into an array sliced from operand.
`slice_sizes`	`ArraySlice<int64>`	`slice_sizes[i]` is the bounds for the slice on dimension `i`.
`collapsed_slice_dims`	`ArraySlice<int64>`	The set of dimensions in each slice that are collapsed away. These dimensions must have size 1.
`start_index_map`	`ArraySlice<int64>`	A map that describes how to map indices in `start_indices` to legal indices into operand.
`indices_are_sorted`	`bool`	Whether the indices are guaranteed to be sorted by the caller.
`unique_indices`	`bool`	Whether the indices are guaranteed to be unique by the caller.

For convenience, we label dimensions in the output array not in offset_dims as batch_dims.

The output is an array of rank batch_dims.size + offset_dims.size.

The operand.rank must equal the sum of offset_dims.size and collapsed_slice_dims. Also, slice_sizes.size has to be equal to operand.rank.

If index_vector_dim is equal to start_indices.rank we implicitly consider start_indices to have a trailing 1 dimension (i.e. if start_indices was of shape [6,7] and index_vector_dim is 2 then we implicitly consider the shape of start_indices to be [6,7,1]).

The bounds for the output array along dimension i is computed as follows:

If i is present in batch_dims (i.e. is equal to batch_dims[k] for some k) then we pick the corresponding dimension bounds out of start_indices.shape, skipping index_vector_dim (i.e. pick start_indices.shape.dims[k] if k < index_vector_dim and start_indices.shape.dims[k+1] otherwise).
If i is present in offset_dims (i.e. equal to offset_dims[k] for some k) then we pick the corresponding bound out of slice_sizes after accounting for collapsed_slice_dims (i.e. we pick adjusted_slice_sizes[k] where adjusted_slice_sizes is slice_sizes with the bounds at indices collapsed_slice_dims removed).

Formally, the operand index In corresponding to a given output index Out is calculated as follows:

Let G = { Out[k] for k in batch_dims }. Use G to slice out a vector S such that S[i] = start_indices[Combine(G, i)] where Combine(A, b) inserts b at position index_vector_dim into A. Note that this is well defined even if G is empty -- if G is empty then S = start_indices.
Create a starting index, S_in, into operand using S by scattering S using start_index_map. More precisely:
1. S_in[start_index_map[k]] = S[k] if k < start_index_map.size.
2. S_in[_] = 0 otherwise.
Create an index O_in into operand by scattering the indices at the offset dimensions in Out according to the collapsed_slice_dims set. More precisely:
1. O_in[remapped_offset_dims(k)] = Out[offset_dims[k]] if k < offset_dims.size (remapped_offset_dims is defined below).
2. O_in[_] = 0 otherwise.
In is O_in + S_in where + is element-wise addition.

remapped_offset_dims is a monotonic function with domain [0, offset.size) and range [0, operand.rank) \ collapsed_slice_dims. So if, e.g., offset.size is 4, operand.rank is 6 and collapsed_slice_dims is {0, 2} then remapped_offset_dims is {0→1, 1→3, 2→4, 3→5}.

If indices_are_sorted is set to true then XLA can assume that start_indices are sorted (in ascending start_index_map order) by the user. If they are not then the semantics is implementation defined.

If unique_indices is set to true then XLA can assume that all element scattered to are unique. So XLA could use non-atomic operations. If unique_indices is set to true and the indices being scattered to are not unique then the semantics is implementation defined.

Informal Description and Examples

Informally, every index Out in the output array corresponds to an element E in the operand array, computed as follows:

We use the batch dimensions in Out to look up a starting index from start_indices.
We use start_index_map to map the starting index (whose size may be less than operand.rank) to a "full" starting index into the operand.
We dynamic-slice out a slice with size slice_sizes using the full starting index.
We reshape the slice by collapsing the collapsed_slice_dims dimensions. Since all collapsed slice dimensions must have a bound of 1, this reshape is always legal.
We use the offset dimensions in Out to index into this slice to get the input element, E, corresponding to output index Out.

index_vector_dim is set to start_indices.rank - 1 in all of the examples that follow. More interesting values for index_vector_dim do not change the operation fundamentally, but make the visual representation more cumbersome.

To get an intuition on how all of the above fits together, let's look at an example that gathers 5 slices of shape [8,6] from a [16,11] array. The position of a slice into the [16,11] array can be represented as an index vector of shape S64[2], so the set of 5 positions can be represented as a S64[5,2] array.

The behavior of the gather operation can then be depicted as an index transformation that takes [G,O₀,O₁], an index in the output shape, and maps it to an element in the input array in the following way:

We first select an (X,Y) vector from the gather indices array using G. The element in the output array at index [G,O₀,O₁] is then the element in the input array at index [X+O₀,Y+O₁].

slice_sizes is [8,6], which decides the range of O₀ and O₁, and this in turn decides the bounds of the slice.

This gather operation acts as a batch dynamic slice with G as the batch dimension.

The gather indices may be multidimensional. For instance, a more general version of the example above using a "gather indices" array of shape [4,5,2] would translate indices like this:

Again, this acts as a batch dynamic slice G₀ and G₁ as the batch dimensions. The slice size is still [8,6].

The gather operation in XLA generalizes the informal semantics outlined above in the following ways:

We can configure which dimensions in the output shape are the offset dimensions (dimensions containing O₀, O₁ in the last example). The output batch dimensions (dimensions containing G₀, G₁ in the last example) are defined to be the output dimensions that are not offset dimensions.
The number of output offset dimensions explicitly present in the output shape may be smaller than the input rank. These "missing" dimensions, which are listed explicitly as collapsed_slice_dims, must have a slice size of 1. Since they have a slice size of 1 the only valid index for them is 0 and eliding them does not introduce ambiguity.
The slice extracted from the "Gather Indices" array ((X, Y) in the last example) may have fewer elements than the input array rank, and an explicit mapping dictates how the index should be expanded to have the same rank as the input.

As a final example, we use (2) and (3) to implement tf.gather_nd:

G₀ and G₁ are used to slice out a starting index from the gather indices array as usual, except the starting index has only one element, X. Similarly, there is only one output offset index with the value O₀. However, before being used as indices into the input array, these are expanded in accordance to "Gather Index Mapping" (start_index_map in the formal description) and "Offset Mapping" (remapped_offset_dims in the formal description) into [X,0] and [0,O₀] respectively, adding up to [X,O₀]. In other words, the output index [G₀,G₁,O₀] maps to the input index [GatherIndices[G₀,G₁,0],X] which gives us the semantics for tf.gather_nd.

slice_sizes for this case is [1,11]. Intuitively this means that every index X in the gather indices array picks an entire row and the result is the concatenation of all these rows.

GetDimensionSize

Returns the size of the given dimension of the operand. The operand must be array shaped.

GetDimensionSize(operand, dimension)

Arguments	Type	Semantics
`operand`	`XlaOp`	n dimensional input array
`dimension`	`int64`	A value in the interval `[0, n)` that specifies the dimension

SetDimensionSize

Sets the dynamic size of XlaOp's given dimension. The operand must be array shaped.

SetDimensionSize(operand, size, dimension)

Arguments	Type	Semantics
`operand`	`XlaOp`	n dimensional input array.
`size`	`XlaOp`	int32 representing the runtime dynamic size.
`dimension`	`int64`	A value in the interval `[0, n)` that specifies the dimension.

Pass through the operand as result, with dynamic dimension tracked by the compiler.

Padded values will be ignored by downstream reduction ops.

let v: f32[10] = f32[10]{1, 2, 3, 4, 5, 6, 7, 8, 9, 10};
let five: s32 = 5;
let six: s32 = 6;

// Setting dynamic dimension size doesn't change the upper bound of the static
// shape.
let padded_v_five: f32[10] = set_dimension_size(v, five, /*dimension=*/0);
let padded_v_six: f32[10] = set_dimension_size(v, six, /*dimension=*/0);

// sum == 1 + 2 + 3 + 4 + 5
let sum:f32[] = reduce_sum(padded_v_five);
// product == 1 * 2 * 3 * 4 * 5
let product:f32[] = reduce_product(padded_v_five);

// Changing padding size will yield different result.
// sum == 1 + 2 + 3 + 4 + 5 + 6
let sum':f32[] = reduce_sum(padded_v_six);

GetTupleElement

Infeed

Argument	Type	Semantics
`shape`	`Shape`	Shape of the data read from the Infeed interface. The layout field of the shape must be set to match the layout of the data sent to the device; otherwise its behavior is undefined.

Iota

Iota()

Builds a constant literal on device rather than a potentially large host transfer. Creates a rank 1 array of values starting at zero and incrementing by one. For floating-point types, the produced array is equivalent to ConvertElementType(Iota(...)) where the Iota is of integral type and the conversion is to the floating-point type.

Arguments	Type	Semantics
`type`	`PrimitiveType`	type U
`size`	`int64`	The number of elements in the array.
`iota_dimension`	`int64`	The dimension to increment along.

Map

Arguments	Type	Semantics
`operands`	sequence of N `XlaOp`s	N arrays of types T0..T{N-1}
`computation`	`XlaComputation`	computation of type `T_0, T_1, ..., T_{N + M -1} -> S` with N parameters of type T and M of arbitrary type
`dimensions`	`int64` array	array of map dimensions

Pad

Arguments	Type	Semantics
`operand`	`XlaOp`	array of type `T`
`padding_value`	`XlaOp`	scalar of type `T` to fill in the added padding
`padding_config`	`PaddingConfig`	padding amount on both edges (low, high) and between the elements of each dimension

Recv

Arguments	Type	Semantics
`shape`	`Shape`	shape of the data to receive
`channel_handle`	`ChannelHandle`	unique identifier for each send/recv pair

Reduce

Arguments	Type	Semantics
`operands`	Sequence of N `XlaOp`	N arrays of types `T_0, ..., T_N`.
`init_values`	Sequence of N `XlaOp`	N scalars of types `T_0, ..., T_N`.
`computation`	`XlaComputation`	computation of type `T_0, ..., T_N, T_0, ..., T_N ->` `Collate(T_0, ..., T_N)`.
`dimensions`	`int64` array	unordered array of dimensions to reduce.

Variadic Reduce

When N > 1, reduce function application is slightly more complex, as it is applied simultaneously to all inputs. The operands are supplied to the computation in the following order:

Running reduced value for the first operand
...
Running reduced value for the N'th operand
Input value for the first operand
...
Input value for the N'th operand

For example, consider the following reduction function, which can be used to compute the max and the argmax of a 1-D array in parallel:

f: (Float, Int, Float, Int) -> Float, Int
f(max, argmax, value, index):
  if value >= max:
    return (value, index)
  else:
    return (max, argmax)

For 1-D Input arrays V = Float[N], K = Int[N], and init values I_V = Float, I_K = Int, the result f_(N-1) of reducing across the only input dimension is equivalent to the following recursive application:

f_0 = f(I_V, I_K, V_0, K_0)
f_1 = f(f_0.first, f_0.second, V_1, K_1)
...
f_(N-1) = f(f_(N-2).first, f_(N-2).second, V_(N-1), K_(N-1))

Applying this reduction to an array of values, and an array of sequential indices (i.e. iota), will co-iterate over the arrays, and return a tuple containing the maximal value and the matching index.

ReducePrecision

Arguments	Type	Semantics
`operand`	`XlaOp`	array of floating-point type `T`.
`exponent_bits`	`int32`	number of exponent bits in lower-precision format
`mantissa_bits`	`int32`	number of mantissa bits in lower-precision format

ReduceWindow

Arguments	Type	Semantics
`operand`	`XlaOp`	N dimensional array containing elements of type T. This is the base area on which the window is placed.
`init_value`	`XlaOp`	Starting value for the reduction. See Reduce for details.
`computation`	`XlaComputation`	Reduction function of type `T, T -> T`, to apply to all elements in each window
`window_dimensions`	`ArraySlice<int64>`	array of integers for window dimension values
`window_strides`	`ArraySlice<int64>`	array of integers for window stride values
`base_dilations`	`ArraySlice<int64>`	array of integers for base dilation values
`window_dilations`	`ArraySlice<int64>`	array of integers for window dilation values
`padding`	`Padding`	padding type for window (Padding::kSame, which pads so as to have the same output shape as input if the stride is 1, or Padding::kValid, which uses no padding and "stops" the window once it no longer fits)

ReplicaId

Reshape

See also XlaBuilder::Reshape and the Collapse operation.

Reshapes the dimensions of an array into a new configuration.

Reshape(operand, new_sizes) Reshape(operand, dimensions, new_sizes)

Arguments	Type	Semantics
`operand`	`XlaOp`	array of type T
`dimensions`	`int64` vector	order in which dimensions are collapsed
`new_sizes`	`int64` vector	vector of sizes of new dimensions

Conceptually, reshape first flattens an array into a one-dimensional vector of data values, and then refines this vector into a new shape. The input arguments are an arbitrary array of type T, a compile-time-constant vector of dimension indices, and a compile-time-constant vector of dimension sizes for the result. The values in the dimension vector, if given, must be a permutation of all of T's dimensions; the default if not given is {0, ..., rank - 1}. The order of the dimensions in dimensions is from slowest-varying dimension (most major) to fastest-varying dimension (most minor) in the loop nest which collapses the input array into a single dimension. The new_sizes vector determines the size of the output array. The value at index 0 in new_sizes is the size of dimension 0, the value at index 1 is the size of dimension 1, and so on. The product of the new_size dimensions must equal the product of the operand's dimension sizes. When refining the collapsed array into the multidimensional array defined by new_sizes, the dimensions in new_sizes are ordered from slowest varying (most major) and to fastest varying (most minor).

For example, let v be an array of 24 elements:

let v = f32[4x2x3] { { {10, 11, 12}, {15, 16, 17} },
                    { {20, 21, 22}, {25, 26, 27} },
                    { {30, 31, 32}, {35, 36, 37} },
                    { {40, 41, 42}, {45, 46, 47} } };

In-order collapse:
let v012_24 = Reshape(v, {0,1,2}, {24});
then v012_24 == f32[24] {10, 11, 12, 15, 16, 17, 20, 21, 22, 25, 26, 27,
                         30, 31, 32, 35, 36, 37, 40, 41, 42, 45, 46, 47};

let v012_83 = Reshape(v, {0,1,2}, {8,3});
then v012_83 == f32[8x3] { {10, 11, 12}, {15, 16, 17},
                          {20, 21, 22}, {25, 26, 27},
                          {30, 31, 32}, {35, 36, 37},
                          {40, 41, 42}, {45, 46, 47} };

Out-of-order collapse:
let v021_24 = Reshape(v, {1,2,0}, {24});
then v012_24 == f32[24]  {10, 20, 30, 40, 11, 21, 31, 41, 12, 22, 32, 42,
                          15, 25, 35, 45, 16, 26, 36, 46, 17, 27, 37, 47};

let v021_83 = Reshape(v, {1,2,0}, {8,3});
then v021_83 == f32[8x3] { {10, 20, 30}, {40, 11, 21},
                          {31, 41, 12}, {22, 32, 42},
                          {15, 25, 35}, {45, 16, 26},
                          {36, 46, 17}, {27, 37, 47} };


let v021_262 = Reshape(v, {1,2,0}, {2,6,2});
then v021_262 == f32[2x6x2] { { {10, 20}, {30, 40},
                              {11, 21}, {31, 41},
                              {12, 22}, {32, 42} },
                             { {15, 25}, {35, 45},
                              {16, 26}, {36, 46},
                              {17, 27}, {37, 47} } };

As a special case, reshape can transform a single-element array to a scalar and vice versa. For example,

Reshape(f32[1x1] { {5} }, {0,1}, {}) == 5;
Reshape(5, {}, {1,1}) == f32[1x1] { {5} };

Rev (reverse)

RngNormal

Arguments	Type	Semantics
`mu`	`XlaOp`	Scalar of type T specifying mean of generated numbers
`sigma`	`XlaOp`	Scalar of type T specifying standard deviation of generated numbers
`shape`	`Shape`	Output shape of type T

RngUniform

Arguments	Type	Semantics
`a`	`XlaOp`	Scalar of type T specifying lower limit of interval
`b`	`XlaOp`	Scalar of type T specifying upper limit of interval
`shape`	`Shape`	Output shape of type T

RngBitGenerator

Generates an output with a given shape filled with uniform random bits using the specified algorithm (or backend default) and returns an updated state (with the same shape as initial state) and the generated random data.

Initial state is the initial state of the current random number generation. It and the required shape and valid values are dependent on the algorithm used.

The output is guaranteed to be a deterministic function of the initial state but it is not guaranteed to be deterministic between backends and different compiler versions.

RngBitGenerator(algorithm, key, shape)

Arguments	Type	Semantics
`algorithm`	`RandomAlgorithm`	PRNG algorithm to be used.
`initial_state`	`XlaOp`	Initial state for the PRNG algorithm.
`shape`	`Shape`	Output shape for generated data.

Available values for algorithm:

rng_default: Backend specific algorithm with backend specific shape requirements.
rng_three_fry: ThreeFry counter-based PRNG algorithm. The initial_state shape is u64[2] with arbitrary values. Salmon et al. SC 2011. Parallel random numbers: as easy as 1, 2, 3.
rng_philox: Philox algorithm to generate random numbers in parallel. The initial_state shape is u64[3] with arbitrary values. Salmon et al. SC 2011. Parallel random numbers: as easy as 1, 2, 3.

Scatter

The XLA scatter operation generates a result which is the value of the input array operand, with several slices (at indices specified by scatter_indices) updated with the values in updates using update_computation.

Select

SelectAndScatter

This operation can be considered as a composite operation that first computes ReduceWindow on the operand array to select an element from each window, and then scatters the source array to the indices of the selected elements to construct an output array with the same shape as the operand array. The binary select function is used to select an element from each window by applying it across each window, and it is called with the property that the first parameter's index vector is lexicographically less than the second parameter's index vector. The select function returns true if the first parameter is selected and returns false if the second parameter is selected, and the function must hold transitivity (i.e., if select(a, b) and select(b, c) are true, then select(a, c) is also true) so that the selected element does not depend on the order of the elements traversed for a given window.

The function scatter is applied at each selected index in the output array. It takes two scalar parameters:

Current value at the selected index in the output array
The scatter value from source that applies to the selected index

It combines the two parameters and returns a scalar value that's used to update the value at the selected index in the output array. Initially, all indices of the output array are set to init_value.

The output array has the same shape as the operand array and the source array must have the same shape as the result of applying a ReduceWindow operation on the operand array. SelectAndScatter can be used to backpropagate the gradient values for a pooling layer in a neural network.

SelectAndScatter(operand, select, window_dimensions, window_strides, padding, source, init_value, scatter)

Arguments	Type	Semantics
`operand`	`XlaOp`	array of type T over which the windows slide
`select`	`XlaComputation`	binary computation of type `T, T -> PRED`, to apply to all elements in each window; returns `true` if the first parameter is selected and returns `false` if the second parameter is selected
`window_dimensions`	`ArraySlice<int64>`	array of integers for window dimension values
`window_strides`	`ArraySlice<int64>`	array of integers for window stride values
`padding`	`Padding`	padding type for window (Padding::kSame or Padding::kValid)
`source`	`XlaOp`	array of type T with the values to scatter
`init_value`	`XlaOp`	scalar value of type T for the initial value of the output array
`scatter`	`XlaComputation`	binary computation of type `T, T -> T`, to apply each scatter source element with its destination element

The figure below shows examples of using SelectAndScatter, with the select function computing the maximal value among its parameters. Note that when the windows overlap, as in the figure (2) below, an index of the operand array may be selected multiple times by different windows. In the figure, the element of value 9 is selected by both of the top windows (blue and red) and the binary addition scatter function produces the output element of value 8 (2 + 6).

The evaluation order of the scatter function is arbitrary and may be non-deterministic. Therefore, the scatter function should not be overly sensitive to reassociation. See the discussion about associativity in the context of Reduce for more details.

Arguments	Type	Semantics
`operand`	`XlaOp`	Array to be scattered into.
`scatter_indices`	`XlaOp`	Array containing the starting indices of the slices that must be scattered to.
`updates`	`XlaOp`	Array containing the values that must be used for scattering.
`update_computation`	`XlaComputation`	Computation to be used for combining the existing values in the input array and the updates during scatter. This computation should be of type `(T, T) -> T`.
`index_vector_dim`	`int64`	The dimension in `scatter_indices` that contains the starting indices.
`update_window_dims`	`ArraySlice<int64>`	The set of dimensions in `updates` shape that are window dimensions.
`inserted_window_dims`	`ArraySlice<int64>`	The set of window dimensions that must be inserted into `updates` shape.
`scatter_dims_to_operand_dims`	`ArraySlice<int64>`	A dimensions map from the scatter indices to the operand index space. This array is interpreted as mapping `i` to `scatter_dims_to_operand_dims[i]` . It has to be one-to-one and total.
`indices_are_sorted`	`bool`	Whether the indices are guaranteed to be sorted by the caller.

Arguments	Type	Semantics
`pred`	`XlaOp`	array of type PRED
`on_true`	`XlaOp`	array of type T
`on_false`	`XlaOp`	array of type T

Arguments	Type	Semantics
`operand`	`XlaOp`	data to send (array of type T)
`channel_handle`	`ChannelHandle`	unique identifier for each send/recv pair

Arguments	Type	Semantics
`operands`	`ArraySlice<XlaOp>`	The operands to sort.
`comparator`	`XlaComputation`	The comparator computation to use.
`dimension`	`int64`	The dimension along which to sort.
`is_stable`	`bool`	Whether stable sorting should be used.

Arguments	Type	Semantics
`operand`	`XlaOp`	The operand to transpose.
`permutation`	`ArraySlice<int64>`	How to permute the dimensions.

Arguments	Type	Semantics
`condition`	`XlaComputation`	XlaComputation of type `T -> PRED` which defines the termination condition of the loop.
`body`	`XlaComputation`	XlaComputation of type `T -> T` which defines the body of the loop.
`init`	`T`	Initial value for the parameter of `condition` and `body`.

Operation Semantics

AfterAll

AllReduce

AllToAll

BatchNormGrad

BatchNormInference

BatchNormTraining

BitcastConvertType

Broadcast

BroadcastInDim

Call

Cholesky

Clamp

Collapse

CollectivePermute

Concatenate

Conditional

Conv (convolution)

ConvWithGeneralPadding (convolution)

ConvertElementType

CrossReplicaSum

CustomCall

Dot

DotGeneral

DynamicSlice

DynamicUpdateSlice

Element-wise binary arithmetic operations

Element-wise comparison operations

Element-wise unary functions

Fft

Multidimensional FFT

Implementation details

Gather

General Semantics

Informal Description and Examples

GetDimensionSize

SetDimensionSize

GetTupleElement

Infeed

Iota

Map

Pad

Recv

Reduce

Variadic Reduce

ReducePrecision

ReduceWindow

ReplicaId

Reshape

Rev (reverse)

RngNormal

RngUniform

RngBitGenerator

Scatter

Select

SelectAndScatter

Send

Slice

Sort

Transpose

TriangularSolve

Tuple

While