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.

Abs

See also XlaBuilder::Abs.

Element-wise abs x -> |x|.

Abs(operand)

For StableHLO information see StableHLO - abs.

Add

See also XlaBuilder::Add.

Performs element-wise addition of lhs and rhs.

Add(lhs, rhs)

The arguments' shapes have to be either similar or compatible. See the broadcasting documentation about what it means for shapes to be compatible. The result of an operation has a shape which is the result of broadcasting the two input arrays. In this variant, operations between arrays of different ranks are not supported, unless one of the operands is a scalar.

An alternative variant with different-dimensional broadcasting support exists for Add:

Add(lhs,rhs, broadcast_dimensions)

This variant of the operation should be used for arithmetic operations between arrays of different ranks (such as adding a matrix to a vector).

The additional broadcast_dimensions operand is a slice of integers specifying the dimensions to use for broadcasting the operands. The semantics are described in detail on the broadcasting page.

For StableHLO information see StableHLO - add.

AddDependency

See also HloInstruction::AddDependency.

AddDependency may appear in HLO dumps, but they are not intended to be constructed manually by end users.

AfterAll

See also XlaBuilder::AfterAll.

AfterAll takes a variadic number of tokens and produces a single token. Tokens are primitive types which can be threaded between side-effecting operations to enforce ordering. AfterAll can be used as a join of tokens for ordering an operation after a set of operations.

AfterAll(tokens)

For StableHLO information see StableHLO - after_all.

AllGather

See also XlaBuilder::AllGather.

Performs concatenation across replicas.

AllGather(operand, all_gather_dimension, shard_count, replica_groups, channel_id, layout, use_global_device_ids)

• replica_groups is a list of replica groups between which the concatenation is performed (replica id for the current replica can be retrieved using ReplicaId). The order of replicas in each group determines the order in which their inputs are located in the result. replica_groups must either be empty (in which case all replicas belong to a single group, ordered from 0 to N - 1), or contain the same number of elements as the number of replicas. For example, replica_groups = {0, 2}, {1, 3} performs concatenation between the replicas 0 and 2, and 1 and 3.
• shard_count is the size of each replica group. We need this in cases where replica_groups are empty.
• channel_id is used for cross-module communication: only all-gather operations with the same channel_id can communicate to each other.
• use_global_device_ids Returns true if the ids in the ReplicaGroup config represent a global id of (replica_id * partition_count + partition_id) instead of a replica id. This enables more flexible grouping of devices if this all-reduce is both cross-partition and cross-replica.

The output shape is the input shape with the all_gather_dimension made shard_count times larger. For example, if there are two replicas and the operand has the value [1.0, 2.5] and [3.0, 5.25] respectively on the two replicas, then the output value from this op where all_gather_dim is 0 will be [1.0, 2.5, 3.0,5.25] on both replicas.

The API of AllGather is internally decomposed into 2 HLO instructions (AllGatherStart and AllGatherDone).

See also HloInstruction::CreateAllGatherStart.

AllGatherStart, AllGatherDone serve as primitives in HLO. These ops may appear in HLO dumps, but they are not intended to be constructed manually by end users.

For StableHLO information see StableHLO - all_gather.

AllReduce

See also XlaBuilder::AllReduce.

Performs a custom computation across replicas.

AllReduce(operand, computation, replica_groups, channel_id, shape_with_layout, use_global_device_ids)

• When operand is a tuple of arrays, the all-reduce is performed on each element of the tuple.
• replica_groups is a list of replica groups between which the reduction is performed (replica id for the current replica can be retrieved using ReplicaId). replica_groups must either be empty (in which case all replicas belong to a single group), or contain the same number of elements as the number of replicas. For example, replica_groups = {0, 2}, {1, 3} performs reduction between the replicas 0 and 2, and 1 and 3.
• channel_id is used for cross-module communication: only all-reduce operations with the same channel_id can communicate to each other.
• shape_with_layout: forces the layout of the AllReduce to the given layout. This is used to guarantee the same layout for a group of AllReduce ops compiled separately.
• use_global_device_ids Returns true if the ids in the ReplicaGroup config represent a global id of (replica_id * partition_count + partition_id) instead of a replica id. This enables more flexible grouping of devices if this all-reduce is both cross-partition and cross-replica.

The output shape is the same as the input shape. For example, if there are two replicas and the operand has the value [1.0, 2.5] and [3.0, 5.25] respectively on the two replicas, then the output value from this op and summation computation will be [4.0, 7.75] on both replicas. If the input is a tuple, the output is a tuple as well.

Computing the result of AllReduce requires having one input from each replica, so if one replica executes an AllReduce node more times than another, then the former replica will wait forever. Since the replicas are all running the same program, there are not a lot of ways for that to happen, but it is possible when a while loop's condition depends on data from infeed and the data that is infeed causes the while loop to iterate more times on one replica than another.

The API of AllReduce is internally decomposed into 2 HLO instructions (AllReduceStart and AllReduceDone).

See also HloInstruction::CreateAllReduceStart.

AllReduceStart and AllReduceDone serve as primitives in HLO. These ops may appear in HLO dumps, but they are not intended to be constructed manually by end users.

CrossReplicaSum

See also XlaBuilder::CrossReplicaSum.

Performs AllReduce with a summation computation.

CrossReplicaSum(operand, replica_groups)

Returns the sum of the operand value within each subgroup of replicas. All replicas supply one input to the sum and all replicas receive the resulting sum for each subgroup.

AllToAll

See also XlaBuilder::AllToAll.

AllToAll is a collective operation that sends data from all cores to all cores. It has two phases:

1. The scatter phase. On each core, the operand is split into split_count number of blocks along the split_dimensions, and the blocks are scattered to all cores, e.g., the ith block is sent to the ith core.
2. The gather phase. Each core concatenates the received blocks along the concat_dimension.

The participating cores can be configured by:

• replica_groups: each ReplicaGroup contains a list of replica ids participating in the computation (the replica id for the current replica can be retrieved using ReplicaId). AllToAll will be applied within subgroups in the specified order. For example, replica_groups = { {1,2,3}, {4,5,0} } means that an AllToAll will be applied within replicas {1, 2, 3}, and in the gather phase, and the received blocks will be concatenated in the same order of 1, 2, 3. Then, another AllToAll will be applied within replicas 4, 5, 0, and the concatenation order is also 4, 5, 0. If replica_groups is empty, all replicas belong to one group, in the concatenation order of their appearance.

Prerequisites:

• The dimension size of the operand on the split_dimension is divisible by split_count.
• The operand's shape is not tuple.

AllToAll(operand, split_dimension, concat_dimension, split_count, replica_groups, layout, channel_id)

See xla::shapes for more information on shapes and layouts..

For StableHLO information see StableHLO - all_to_all.

AllToAll - Example 1.

XlaBuilder b("alltoall");
auto x = Parameter(&b, 0, ShapeUtil::MakeShape(F32, {4, 16}), "x");
AllToAll(
    x,
    /*split_dimension=*/ 1,
    /*concat_dimension=*/ 0,
    /*split_count=*/ 4);

In the above example, there are 4 cores participating in the Alltoall. On each core, the operand is split into 4 parts along dimension 1, so each part has shape f32[4,4]. The 4 parts are scattered to all cores. Then each core concatenates the received parts along dimension 0, in the order of core 0-4. So the output on each core has shape f32[16,4].

AllToAll - Example 2 - StableHLO

In the above example, there are 2 replicas participating in the AllToAll. On each replica, the operand has shape f32[2,4]. The operand is split into 2 parts along dimension 1, so each part has shape f32[2,2]. The 2 parts are then exchanged across the replicas according to their position in the replica group. Each replica collects its corresponding part from both operands and concatenates them along dimension 0. As a result, the output on each replica has shape f32[4,2].

RaggedAllToAll

See also XlaBuilder::RaggedAllToAll.

RaggedAllToAll performs a collective all-to-all operation, where the input and output are ragged tensors.

RaggedAllToAll(input, input_offsets, send_sizes, output, output_offsets, recv_sizes, replica_groups, channel_id)

Ragged tensors are defined by a set of three tensors:

• data: the datatensor is “ragged” along its outermost dimension, along which each indexed element has variable size.
• offsets': the offsets tensor indexes the outermost dimension of the data tensor, and represents the starting offset of each ragged element of the data tensor.
• sizes: the sizes tensor represents the size of each ragged element of the data tensor, where the size is specified in units of sub-elements. A sub-element is defined as the suffix of the ‘data’ tensor shape obtained by removing the outermost “ragged” dimension.
• The offsets and sizes tensors must have the same size.

An example ragged tensor:

data: [8,3] =
{ {a,b,c},{d,e,f},{g,h,i},{j,k,l},{m,n,o},{p,q,r},{s,t,u},{v,w,x} }

offsets: [3] = {0, 1, 4}

sizes: [3] = {1, 3, 4}

// Index 'data' at 'offsets'[0], 'sizes'[0]' // {a,b,c}

// Index 'data' at 'offsets'[1], 'sizes'[1]' // {d,e,f},{g,h,i},{j,k,l}

// Index 'data' at 'offsets'[2], 'sizes'[2]' // {m,n,o},{p,q,r},{s,t,u},{v,w,x}

output_offsets must be sharded in a way that each replica has offsets in the target replica output perspective.

For i-th output offset, the current replica will send input[input_offsets[i]:input_offsets[i]+input_sizes[i]] update to i-th replica that will be written to output_i[output_offsets[i]:output_offsets[i]+send_sizes[i]] in i-th replica output.

For example, if we have 2 replicas:

replica 0:
input: [1, 2, 2]
output:[0, 0, 0, 0]
input_offsets: [0, 1]
send_sizes: [1, 2]
output_offsets: [0, 0]
recv_sizes: [1, 1]

replica 1:
input: [3, 4, 0]
output: [0, 0, 0, 0]
input_offsets: [0, 1]
send_sizes: [1, 1]
output_offsets: [1, 2]
recv_sizes: [2, 1]

// replica 0's result will be: [1, 3, 0, 0]
// replica 1's result will be: [2, 2, 4, 0]

The ragged all-to-all HLO has the following arguments:

• input: ragged input data tensor.
• output: ragged output data tensor.
• input_offsets: ragged input offsets tensor.
• send_sizes: ragged send sizes tensor.
• output_offsets: array of ragged offsets in the target replica output.
• recv_sizes: ragged recv sizes tensor.

The *_offsets and *_sizes tensors must all have the same shape.

Two shapes are supported for the *_offsets and *_sizes tensors:

• [num_devices] where ragged-all-to-all may send at most one update to each remote device in the replica group. For example:

for (remote_device_id : replica_group) {
     SEND input[input_offsets[remote_device_id]],
     output[output_offsets[remote_device_id]],
     send_sizes[remote_device_id] }

• [num_devices, num_updates] where ragged-all-to-all may send up to num_updates updates the same remote device (each at different offsets), for each remote device in the replica group.

For example:

for (remote_device_id : replica_group) {
    for (update_idx : num_updates) {
        SEND input[input_offsets[remote_device_id][update_idx]],
        output[output_offsets[remote_device_id][update_idx]]],
        send_sizes[remote_device_id][update_idx] } }

And

See also XlaBuilder::And.

Performs element-wise AND of two tensors lhs and rhs.

And(lhs, rhs)

The arguments' shapes have to be either similar or compatible. See the broadcasting documentation about what it means for shapes to be compatible. The result of an operation has a shape which is the result of broadcasting the two input arrays. In this variant, operations between arrays of different ranks are not supported, unless one of the operands is a scalar.

An alternative variant with different-dimensional broadcasting support exists for And:

And(lhs,rhs, broadcast_dimensions)

This variant of the operation should be used for arithmetic operations between arrays of different ranks (such as adding a matrix to a vector).

The additional broadcast_dimensions operand is a slice of integers specifying the dimensions to use for broadcasting the operands. The semantics are described in detail on the broadcasting page.

For StableHLO information see StableHLO - and.

Async

See also HloInstruction::CreateAsyncStart, HloInstruction::CreateAsyncUpdate, HloInstruction::CreateAsyncDone.

AsyncDone, AsyncStart, and AsyncUpdate are internal HLO instructions used for Asynchronous operations and serve as primitives in HLO. These ops may appear in HLO dumps but they are not intended to be constructed manually by end users.

Atan2

See also XlaBuilder::Atan2.

Performs element-wise atan2 operation on lhs and rhs.

Atan2(lhs, rhs)

The arguments' shapes have to be either similar or compatible. See the broadcasting documentation about what it means for shapes to be compatible. The result of an operation has a shape which is the result of broadcasting the two input arrays. In this variant, operations between arrays of different ranks are not supported, unless one of the operands is a scalar.

An alternative variant with different-dimensional broadcasting support exists for Atan2:

Atan2(lhs,rhs, broadcast_dimensions)

This variant of the operation should be used for arithmetic operations between arrays of different ranks (such as adding a matrix to a vector).

The additional broadcast_dimensions operand is a slice of integers specifying the dimensions to use for broadcasting the operands. The semantics are described in detail on the broadcasting page.

For StableHLO information see StableHLO - atan2.

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, batch_mean, batch_var, grad_output, epsilon, feature_index)

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):

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

The inputs batch_mean and batch_var represent moments values across batch and spatial dimensions.

The output type is a tuple of three handles:

For StableHLO information see StableHLO - batch_norm_grad.

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)

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.

For StableHLO information see StableHLO - batch_norm_inference.

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)

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.

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 a 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 \(\sigma^2_l\): $\sigma^2l=\frac{1}{mwh}\sum{i=1}^m\sum{j=1}^w\sum{k=1}^h (x_{ijkl} - \mu_l)^2$

• 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:

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

For StableHLO information see StableHLO - batch_norm_training.

Bitcast

See also HloInstruction::CreateBitcast.

Bitcast may appear in HLO dumps, but they are not intended to be constructed manually by end users.

BitcastConvertType

See also XlaBuilder::BitcastConvertType.

Similar to a tf.bitcast in TensorFlow, performs an element-wise bitcast operation from a data shape to a target shape. The input and output size must match: e.g. s32 elements become f32 elements via bitcast routine, and one s32 element will become four s8 elements. Bitcast is implemented as a low-level cast, so machines with different floating-point representations will give different results.

BitcastConvertType(operand, new_element_type)

The dimensions of the operand and the target shape must match, apart from the last dimension which will change by the ratio of the primitive size before and after the conversion.

The source and destination element types must not be tuples.

For StableHLO information see StableHLO - bitcast_convert.

Bitcast-converting to primitive type of different width

BitcastConvert HLO instruction supports the case where the size of the output element type T' is not equal to the size of the input element T. As the whole operation is conceptually a bitcast and does not change the underlying bytes, the shape of the output element has to change. For B = sizeof(T), B' = sizeof(T'), there are two possible cases.

First, when B > B', the output shape gets a new minor-most dimension of size B/B'. For example:

  f16[10,2]{1,0} %output = f16[10,2]{1,0} bitcast-convert(f32[10]{0} %input)

The rule remains the same for effective scalars:

  f16[2]{0} %output = f16[2]{0} bitcast-convert(f32[] %input)

Alternatively, for B' > B the instruction requires the last logical dimension of the input shape to be equal to B'/B, and this dimension is dropped during the conversion:

  f32[10]{0} %output = f32[10]{0} bitcast-convert(f16[10,2]{1,0} %input)

Note that conversions between different bitwidths are not elementwise.

Broadcast

See also XlaBuilder::Broadcast.

Adds dimensions to an array by duplicating the data in the array.

Broadcast(operand, broadcast_sizes)

The new dimensions are inserted on the left, i.e. if broadcast_sizes has values {a0, ..., aN} and the operand shape has dimensions {b0, ..., bM} then the shape of the output has dimensions {a0, ..., aN, b0, ..., bM}.

The new dimensions index into copies of the operand, i.e.

output[i0, ..., iN, j0, ..., jM] = operand[j0, ..., jM]

For example, if operand is a scalar f32 with value 2.0f, and broadcast_sizes is {2, 3}, then the result will be an array with shape f32[2, 3] and all the values in the result will be 2.0f.

For StableHLO information see StableHLO - broadcast.

BroadcastInDim

See also XlaBuilder::BroadcastInDim.

Expands the size and number of dimensions of an array by duplicating the data in the array.

BroadcastInDim(operand, out_dim_size, broadcast_dimensions)

Similar to Broadcast, but allows adding dimensions anywhere and expanding existing dimensions with size 1.

The operand is broadcast to the shape described by out_dim_size. broadcast_dimensions maps the dimensions of operand to the dimensions of the target shape, i.e. the i'th dimension of the operand is mapped to the broadcast_dimension[i]'th dimension of the output shape. The dimensions of operand must have size 1 or be the same size as the dimension in the output shape they are mapped to. The remaining dimensions are filled with dimensions of size 1. Degenerate-dimension broadcasting then broadcasts along these degenerate dimensions to reach the output shape. The semantics are described in detail on the broadcasting page.

Call

See also XlaBuilder::Call.

Invokes a computation with the given arguments.

Call(computation, operands...)

The arity and types of the operands must match the parameters of the computation. It is allowed to have no operands.

CompositeCall

See also XlaBuilder::CompositeCall.

Encapsulates an operation made up (composed) of other StableHLO operations, taking inputs and composite_attributes and producing results. The semantics of the op are implemented by the decomposition attribute. The composite op can be replaced with its decomposition without changing program semantics. In cases where inlining the decomposition does not provide the same op semantics, prefer using custom_call.

The version field (defaults to 0) is used to denote when a composite's semantics change.

This op is implemented as a kCall with attribute is_composite=true. The decomposition field is specified by the computation attribute. The frontend attributes store the remaining attributes prefixed with composite..

Example CompositeCall op:

f32[] call(f32[] %cst), to_apply=%computation, is_composite=true,
frontend_attributes = {
  composite.name="foo.bar",
  composite.attributes={n = 1 : i32, tensor = dense<1> : tensor<i32>},
  composite.version="1"
}

CompositeCall(computation, operands..., name, attributes, version)

An op’s decomposition isn’t a field called, but instead appears as a to_apply attribute that points to the function which contains the lower-level implementation, i.e. to_apply=%funcname

More information on composite and decomposition can be found on StableHLO Specification.

Cbrt

See also XlaBuilder::Cbrt.

Element-wise cubic root operation x -> cbrt(x).

Cbrt(operand)

Cbrt also supports the optional result_accuracy argument:

Cbrt(operand, result_accuracy)

For more information on result_accuracy see Result Accuracy.

For StableHLO information see StableHLO - cbrt.

Ceil

See also XlaBuilder::Ceil.

Element-wise ceil x -> ⌈x⌉.

Ceil(operand)

For StableHLO information see StableHLO - ceil.

Cholesky

See also XlaBuilder::Cholesky.

Computes the Cholesky decomposition of a batch of symmetric (Hermitian) positive definite matrices.

Cholesky(a, lower)

If lower is true, computes lower-triangular matrices l such that $a = l . l^T$. If lower is false, computes upper-triangular matrices u such that \(a = u^T . u\).

Input data is read only from the lower/upper triangle of a, depending on the value of lower. Values from the other triangle are ignored. Output data is returned in the same triangle; the values in the other triangle are implementation-defined and may be anything.

If a has greater than 2 dimensions, a is treated as a batch of matrices, where all except the minor 2 dimensions are batch dimensions.

If a is not symmetric (Hermitian) positive definite, the result is implementation-defined.

For StableHLO information see StableHLO - cholesky.

Clamp

See also XlaBuilder::Clamp.

Clamps an operand to within the range between a minimum and maximum value.

Clamp(min, operand, max)

Given an operand and minimum and maximum values, returns the operand if it is in the range between the minimum and maximum, else returns the minimum value if the operand is below this range or the maximum value if the operand is above this range. That is, clamp(a, x, b) = min(max(a, x), b).

All three arrays must be the same shape. Alternatively, as a restricted form of broadcasting, min and/or max can be a scalar of type T.

Example with scalar min and max:

let operand: s32[3] = {-1, 5, 9};
let min: s32 = 0;
let max: s32 = 6;
==>
Clamp(min, operand, max) = s32[3]{0, 5, 6};

For StableHLO information see StableHLO - clamp.

Collapse

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

Collapses dimensions of an array into one dimension.

Collapse(operand, 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 dimensions, 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} };

Clz

See also XlaBuilder::Clz.

Element-wise count leading zeros.

Clz(operand)

CollectiveBroadcast

See also XlaBuilder::CollectiveBroadcast.

Broadcasts data across replicas. Data is sent from the first replica id in each group to the other ids in the same group. If a replica id is not in any replica group, the output on that replica is a tensor consisting of 0(s) in shape.

CollectiveBroadcast(operand, replica_groups, channel_id)

For StableHLO information see StableHLO - collective_broadcast.

CollectivePermute

See also XlaBuilder::CollectivePermute.

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

CollectivePermute(operand, source_target_pairs, channel_id, inplace)

Note that there are the following restrictions on the source_target_pairs:

• 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 consisting of 0(s) with the same shape as the input.

The API of CollectivePermute operation is internally decomposed into 2 HLO instructions (CollectivePermuteStart and CollectivePermuteDone).

See also HloInstruction::CreateCollectivePermuteStart.

CollectivePermuteStart and CollectivePermuteDone serve as primitives in HLO. These ops may appear in HLO dumps, but they are not intended to be constructed manually by end users.

For StableHLO information see StableHLO - collective_permute.

Compare

See also XlaBuilder::Compare.

Performs element-wise comparison of lhs and rhs of the following:

Eq

See also XlaBuilder::Eq.

Performs element-wise equal-to comparison of lhs and rhs.

\(lhs = rhs\)

Eq(lhs, rhs)

The arguments' shapes have to be either similar or compatible. See the broadcasting documentation about what it means for shapes to be compatible. The result of an operation has a shape which is the result of broadcasting the two input arrays. In this variant, operations between arrays of different ranks are not supported, unless one of the operands is a scalar.

An alternative variant with different-dimensional broadcasting support exists for Eq:

Eq(lhs,rhs, broadcast_dimensions)

This variant of the operation should be used for arithmetic operations between arrays of different ranks (such as adding a matrix to a vector).

The additional broadcast_dimensions operand is a slice of integers specifying the dimensions to use for broadcasting the operands. The semantics are described in detail on the broadcasting page.

Support a total order over the floating point numbers exists for Eq, by enforcing:

\[-NaN < -Inf < -Finite < -0 < +0 < +Finite < +Inf < +NaN.\]

EqTotalOrder(lhs,rhs, broadcast_dimensions)

For StableHLO information see StableHLO - compare.

Ne

See also XlaBuilder::Ne.

Performs element-wise not equal-to comparison of lhs and rhs.

\(lhs != rhs\)

Ne(lhs, rhs)

The arguments' shapes have to be either similar or compatible. See the broadcasting documentation about what it means for shapes to be compatible. The result of an operation has a shape which is the result of broadcasting the two input arrays. In this variant, operations between arrays of different ranks are not supported, unless one of the operands is a scalar.

An alternative variant with different-dimensional broadcasting support exists for Ne:

Ne(lhs,rhs, broadcast_dimensions)

This variant of the operation should be used for arithmetic operations between arrays of different ranks (such as adding a matrix to a vector).

The additional broadcast_dimensions operand is a slice of integers specifying the dimensions to use for broadcasting the operands. The semantics are described in detail on the broadcasting page.

Support a total order over the floating point numbers exists for Ne, by enforcing:

\[-NaN < -Inf < -Finite < -0 < +0 < +Finite < +Inf < +NaN.\]

NeTotalOrder(lhs,rhs, broadcast_dimensions)

For StableHLO information see StableHLO - compare.

Ge

See also XlaBuilder::Ge.

Performs element-wise greater-or-equal-than comparison of lhs and rhs.

\(lhs >= rhs\)

Ge(lhs, rhs)

The arguments' shapes have to be either similar or compatible. See the broadcasting documentation about what it means for shapes to be compatible. The result of an operation has a shape which is the result of broadcasting the two input arrays. In this variant, operations between arrays of different ranks are not supported, unless one of the operands is a scalar.

An alternative variant with different-dimensional broadcasting support exists for Ge:

Ge(lhs,rhs, broadcast_dimensions)

This variant of the operation should be used for arithmetic operations between arrays of different ranks (such as adding a matrix to a vector).

The additional broadcast_dimensions operand is a slice of integers specifying the dimensions to use for broadcasting the operands. The semantics are described in detail on the broadcasting page.

Support a total order over the floating point numbers exists for Gt, by enforcing:

\[-NaN < -Inf < -Finite < -0 < +0 < +Finite < +Inf < +NaN.\]

GtTotalOrder(lhs,rhs, broadcast_dimensions)

For StableHLO information see StableHLO - compare.

Gt

See also XlaBuilder::Gt.

Performs element-wise greater-than comparison of lhs and rhs.

\(lhs > rhs\)

Gt(lhs, rhs)

The arguments' shapes have to be either similar or compatible. See the broadcasting documentation about what it means for shapes to be compatible. The result of an operation has a shape which is the result of broadcasting the two input arrays. In this variant, operations between arrays of different ranks are not supported, unless one of the operands is a scalar.

An alternative variant with different-dimensional broadcasting support exists for Gt:

Gt(lhs,rhs, broadcast_dimensions)

This variant of the operation should be used for arithmetic operations between arrays of different ranks (such as adding a matrix to a vector).

The additional broadcast_dimensions operand is a slice of integers specifying the dimensions to use for broadcasting the operands. The semantics are described in detail on the broadcasting page.

For StableHLO information see StableHLO - compare.

Le

See also XlaBuilder::Le.

Performs element-wise less-or-equal-than comparison of lhs and rhs.

\(lhs <= rhs\)

Le(lhs, rhs)

The arguments' shapes have to be either similar or compatible. See the broadcasting documentation about what it means for shapes to be compatible. The result of an operation has a shape which is the result of broadcasting the two input arrays. In this variant, operations between arrays of different ranks are not supported, unless one of the operands is a scalar.

An alternative variant with different-dimensional broadcasting support exists for Le:

Le(lhs,rhs, broadcast_dimensions)

This variant of the operation should be used for arithmetic operations between arrays of different ranks (such as adding a matrix to a vector).

The additional broadcast_dimensions operand is a slice of integers specifying the dimensions to use for broadcasting the operands. The semantics are described in detail on the broadcasting page.

Support a total order over the floating point numbers exists for Le, by enforcing:

\[-NaN < -Inf < -Finite < -0 < +0 < +Finite < +Inf < +NaN.\]

LeTotalOrder(lhs,rhs, broadcast_dimensions)

For StableHLO information see StableHLO - compare.

Lt

See also XlaBuilder::Lt.

Performs element-wise less-than comparison of lhs and rhs.

\(lhs < rhs\)

Lt(lhs, rhs)

The arguments' shapes have to be either similar or compatible. See the broadcasting documentation about what it means for shapes to be compatible. The result of an operation has a shape which is the result of broadcasting the two input arrays. In this variant, operations between arrays of different ranks are not supported, unless one of the operands is a scalar.

An alternative variant with different-dimensional broadcasting support exists for Lt:

Lt(lhs,rhs, broadcast_dimensions)

This variant of the operation should be used for arithmetic operations between arrays of different ranks (such as adding a matrix to a vector).

The additional broadcast_dimensions operand is a slice of integers specifying the dimensions to use for broadcasting the operands. The semantics are described in detail on the broadcasting page.

Support a total order over the floating point numbers exists for Lt, by enforcing:

\[-NaN < -Inf < -Finite < -0 < +0 < +Finite < +Inf < +NaN.\]

LtTotalOrder(lhs,rhs, broadcast_dimensions)

For StableHLO information see StableHLO - compare.

Complex

See also XlaBuilder::Complex.

Performs element-wise conversion to a complex value from a pair of real and imaginary values, lhs and rhs.

Complex(lhs, rhs)

The arguments' shapes have to be either similar or compatible. See the broadcasting documentation about what it means for shapes to be compatible. The result of an operation has a shape which is the result of broadcasting the two input arrays. In this variant, operations between arrays of different ranks are not supported, unless one of the operands is a scalar.

An alternative variant with different-dimensional broadcasting support exists for Complex:

Complex(lhs,rhs, broadcast_dimensions)

This variant of the operation should be used for arithmetic operations between arrays of different ranks (such as adding a matrix to a vector).

The additional broadcast_dimensions operand is a slice of integers specifying the dimensions to use for broadcasting the operands. The semantics are described in detail on the broadcasting page.

For StableHLO information see StableHLO - complex.

ConcatInDim (Concatenate)

See also XlaBuilder::ConcatInDim.

Concatenate composes an array from multiple array operands. The array has the same number of dimensions as each of the input array operands (which must have the same number of dimensions as each other) and contains the arguments in the order that they were specified.

Concatenate(operands..., dimension)

With the exception of dimension all dimensions must be the same. This is because XLA does not support "ragged" arrays. Also note that 0-dimensional values cannot be concatenated (as it's impossible to name the dimension along which the concatenation occurs).

1-dimensional example:

Concat({ {2, 3}, {4, 5}, {6, 7} }, 0)
//Output:  {2, 3, 4, 5, 6, 7}

2-dimensional example:

let a = { {1, 2},
         {3, 4},
         {5, 6} };

let b = { {7, 8} };

Concat({a, b}, 0)

//Output:  { {1, 2},
//          {3, 4},
//          {5, 6},
//          {7, 8} }

Diagram:

For StableHLO information see StableHLO - concatenate.

Conditional

See also XlaBuilder::Conditional.

Conditional(predicate, true_operand, true_computation, false_operand, false_computation)

Executes true_computation if predicate is true, false_computation if predicate is false, and returns the result.

The true_computation must take in a single argument of type \(T_0\) and will be invoked with true_operand which must be of the same type. The false_computation must take in a single argument of type \(T_1\) and will be invoked with false_operand which must be of the same type. The type of the returned value of true_computation and false_computation must be the same.

Note that only one of true_computation and false_computation will be executed depending on the value of predicate.

Conditional(branch_index, branch_computations, branch_operands)

Executes branch_computations[branch_index], and returns the result. If branch_index is an S32 which is < 0 or >= N, then branch_computations[N-1] is executed as the default branch.

Each branch_computations[b] must take in a single argument of type \(T_b\) and will be invoked with branch_operands[b] which must be of the same type. The type of the returned value of each branch_computations[b] must be the same.

Note that only one of the branch_computations will be executed depending on the value of branch_index.

For StableHLO information see StableHLO - if.

Constant

See also XlaBuilder::ConstantLiteral.

Produces an output from a constant literal.

Constant(literal)

For StableHLO information see StableHLO - constant.

ConvertElementType

See also XlaBuilder::ConvertElementType.

Similar to an element-wise static_cast in C++, ConvertElementType 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)

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.

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

For StableHLO information see StableHLO - convert.

Conv (Convolution)

See also XlaBuilder::Conv.

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.

Conv Enqueues a convolution instruction onto the computation, which uses the default convolution dimension numbers with no dilation.

The padding is specified in a short-hand way as either SAME or VALID. SAME padding pads the input (lhs) with zeroes so that the output has the same shape as the input when not taking striding into account. VALID padding simply means no padding.

Conv(lhs, rhs, window_strides, padding, feature_group_count, batch_group_count, precision_config, preferred_element_type)

Increasing levels of controls are available for Conv:

• ConvWithGeneralPadding
• ConvWithGeneralDimensions
• ConvGeneral
• ConvGeneralDilated

Let n be the number of spatial dimensions. The lhs argument is an (n+2)-dimensional 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 an (n+2)-dimensional 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 many feature_group_count 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 for 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 figure above shows how the batch_group_count field works. Effectively, we slice each lhs batch into batch_group_count groups, and do the same for the output features. Then, for each of these groups we do pairwise convolutions and concatenate the output along the output feature dimension. The operational semantics of all the other dimensions (feature and spatial) remain the same.

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;
}

precision_config is used to indicate the precision configuration. The level dictates whether hardware should attempt to generate more machine code instructions to provide more accurate dtype emulation when needed (i.e. emulating f32 on a TPU that only supports bf16 matmuls). Values may be DEFAULT, HIGH, HIGHEST. Additional details in the MXU sections.

preferred_element_type is a scalar element of higher/lower precision output types used for accumulation. preferred_element_type recommends the accumulation type for the given operation, however it is not guaranteed. This allows for some hardware backends to instead accumulate in a different type and convert to the preferred output type.

For StableHLO information see StableHLO - convolution.

ConvWithGeneralPadding

See also XlaBuilder::ConvWithGeneralPadding.

ConvWithGeneralPadding(lhs, rhs, window_strides, padding, feature_group_count, batch_group_count, precision_config, preferred_element_type)

Same as Conv where padding configuration is explicit.

ConvWithGeneralDimensions

See also XlaBuilder::ConvWithGeneralDimensions.

ConvWithGeneralDimensions(lhs, rhs, window_strides, padding, dimension_numbers, feature_group_count, batch_group_count, precision_config, preferred_element_type)

Same as Conv where dimension numbers are explicit.

ConvGeneral

See also XlaBuilder::ConvGeneral.

ConvGeneral(lhs, rhs, window_strides, padding, dimension_numbers, feature_group_count, batch_group_count, precision_config, preferred_element_type)

Same as Conv where dimension numbers and padding configuration is explicit

ConvGeneralDilated

See also XlaBuilder::ConvGeneralDilated.

ConvGeneralDilated(lhs, rhs, window_strides, padding, lhs_dilation, rhs_dilation, dimension_numbers, feature_group_count, batch_group_count, precision_config, preferred_element_type, window_reversal)

Same as Conv where padding configuration, dilation factors, and dimension numbers are explicit.

Copy

See also HloInstruction::CreateCopyStart.

Copy is internally decomposed into 2 HLO instructions CopyStart and CopyDone. Copy along with CopyStart and CopyDone serve as primitives in HLO. These ops may appear in HLO dumps, but they are not intended to be constructed manually by end users.

Cos

See alsoXlaBuilder::Cos.

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

Cos(operand)

Cos also supports the optional result_accuracy argument:

Cos(operand, result_accuracy)

For more information on result_accuracy see Result Accuracy.

For StableHLO information see StableHLO - cosine.

Cosh

See also XlaBuilder::Cosh.

Element-wise hyperbolic cosine x -> cosh(x).

Cosh(operand)

Cosh also supports the optional result_accuracy argument:

Cosh(operand, result_accuracy)

For more information on result_accuracy see Result Accuracy.

CustomCall

See also XlaBuilder::CustomCall.

Call a user-provided function within a computation.

CustomCall documentation is provided in Developer details - XLA Custom Calls

For StableHLO information see StableHLO - custom_call.

Div

See also XlaBuilder::Div.

Performs element-wise division of dividend lhs and divisor rhs.

Div(lhs, rhs)

Integer division overflow (signed/unsigned division/remainder by zero or signed division/remainder of INT_SMIN with -1) produces an implementation defined value.

The arguments' shapes have to be either similar or compatible. See the broadcasting documentation about what it means for shapes to be compatible. The result of an operation has a shape which is the result of broadcasting the two input arrays. In this variant, operations between arrays of different ranks are not supported, unless one of the operands is a scalar.

An alternative variant with different-dimensional broadcasting support exists for Div:

Div(lhs,rhs, broadcast_dimensions)

This variant of the operation should be used for arithmetic operations between arrays of different ranks (such as adding a matrix to a vector).

The additional broadcast_dimensions operand is a slice of integers specifying the dimensions to use for broadcasting the operands. The semantics are described in detail on the broadcasting page.

For StableHLO information see StableHLO - divide.

Domain

See also HloInstruction::CreateDomain.

Domain may appear in HLO dumps, but it is not intended to be constructed manually by end users.

Dot

See also XlaBuilder::Dot.

Dot(lhs, rhs, precision_config, preferred_element_type)

The exact semantics of this operation depend on the ranks of the operands:

The operation performs sum of products over the second dimension of lhs (or the first if it has 1 dimension) and the first dimension of rhs. These are the "contracted" dimensions. The contracted dimensions of lhs and rhs must be of the same size. In practice, it can be used to perform dot products between vectors, vector/matrix multiplications or matrix/matrix multiplications.

precision_config is used to indicate the precision configuration. The level dictates whether hardware should attempt to generate more machine code instructions to provide more accurate dtype emulation when needed (i.e. emulating f32 on a TPU that only supports bf16 matmuls). Values may be DEFAULT, HIGH, HIGHEST. Additional details in the MXU sections.

preferred_element_type is a scalar element of higher/lower precision output types used for accumulation. preferred_element_type recommends the accumulation type for the given operation, however it is not guaranteed. This allows for some hardware backends to instead accumulate in a different type and convert to the preferred output type.

For StableHLO information see StableHLO - dot.

DotGeneral

See also XlaBuilder::DotGeneral.

DotGeneral(lhs, rhs, dimension_numbers, precision_config, preferred_element_type)

Similar to Dot, but allows contracting and batch dimension numbers to be specified for both the lhs and rhs.

DotGeneral performs the sum of products over contracting dimensions specified in dimension_numbers.

Associated contracting dimension numbers from the lhs and rhs do not need to be the same but must have the same dimension sizes.

Example with contracting dimension numbers:

lhs = { {1.0, 2.0, 3.0},
        {4.0, 5.0, 6.0} }

rhs = { {1.0, 1.0, 1.0},
        {2.0, 2.0, 2.0} }

DotDimensionNumbers dnums;
dnums.add_lhs_contracting_dimensions(1);
dnums.add_rhs_contracting_dimensions(1);

DotGeneral(lhs, rhs, dnums) -> { { 6.0, 12.0},
                                 {15.0, 30.0} }

Associated batch dimension numbers from the lhs and rhs must have the same dimension sizes.

Example with batch dimension numbers (batch size 2, 2x2 matrices):

lhs = { { {1.0, 2.0},
          {3.0, 4.0} },
        { {5.0, 6.0},
          {7.0, 8.0} } }

rhs = { { {1.0, 0.0},
          {0.0, 1.0} },
        { {1.0, 0.0},
          {0.0, 1.0} } }

DotDimensionNumbers dnums;
dnums.add_lhs_contracting_dimensions(2);
dnums.add_rhs_contracting_dimensions(1);
dnums.add_lhs_batch_dimensions(0);
dnums.add_rhs_batch_dimensions(0);

DotGeneral(lhs, rhs, dnums) -> {
    { {1.0, 2.0},
      {3.0, 4.0} },
    { {5.0, 6.0},
      {7.0, 8.0} } }

It follows that the resulting dimension number starts with the batch dimension, then the lhs non-contracting/non-batch dimension, and finally the rhs non-contracting/non-batch dimension.

precision_config is used to indicate the precision configuration. The level dictates whether hardware should attempt to generate more machine code instructions to provide more accurate dtype emulation when needed (i.e. emulating f32 on a TPU that only supports bf16 matmuls). Values may be DEFAULT, HIGH, HIGHEST. Additional details can be found in the MXU sections.

preferred_element_type is a scalar element of higher/lower precision output types used for accumulation. preferred_element_type recommends the accumulation type for the given operation, however it is not guaranteed. This allows for some hardware backends to instead accumulate in a different type and convert to the preferred output type.

For StableHLO information see StableHLO - dot_general.

ScaledDot

See also XlaBuilder::ScaledDot.

ScaledDot(lhs, lhs_scale, rhs, rhs_scale, dimension_number, precision_config,preferred_element_type)

Similar to DotGeneral.

Creates a scaled dot op with operands 'lhs', 'lhs_scale', 'rhs', and 'rhs_scale', with contracting and batch dimensions specified in 'dimension_numbers'.

RaggedDot

See also XlaBuilder::RaggedDot.

For a breakdown of RaggedDot computation see StableHLO - chlo.ragged_dot

DynamicReshape

See also XlaBuilder::DynamicReshape.

This operation is functionally identical to reshape, but the result shape is specified dynamically via output_shape.

DynamicReshape(operand, dim_sizes, new_size_bounds, dims_are_dynamic)

For StableHLO information see StableHLO - dynamic_reshape.

DynamicSlice

See also XlaBuilder::DynamicSlice.

DynamicSlice extracts a sub-array from the input array at dynamic start_indices. The size of the slice in each dimension is passed in size_indices, which specify the end point of exclusive slice intervals in each dimension: [start, start + size). The shape of start_indices must be 1-dimensional, with dimension size equal to the number of dimensions of operand.

DynamicSlice(operand, start_indices, slice_sizes)

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] - slice_sizes[i])

This ensures that the extracted 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 s = {2};

DynamicSlice(a, s, {2});
// Result: {2.0, 3.0}

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 s = {2, 1}

DynamicSlice(b, s, {2, 2});
//Result:
// { { 7.0,  8.0},
//   {10.0, 11.0} }

For StableHLO information see StableHLO - dynamic_slice.

DynamicUpdateSlice

See also XlaBuilder::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 1-dimensional, with dimension size equal to the number of dimensions of operand.

DynamicUpdateSlice(operand, update, start_indices)

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)
// Result: {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)
// Result:
// { {0.0,  1.0,  2.0},
//   {3.0, 12.0, 13.0},
//   {6.0, 14.0, 15.0},
//   {9.0, 16.0, 17.0} }

For StableHLO information see StableHLO - dynamic_update_slice.

Erf

See also XlaBuilder::Erf.

Element-wise error function x -> erf(x) where:

\(\text{erf}(x) = \frac{2}{\sqrt{\pi} }\int_0^x e^{-t^2} \, dt\).

Erf(operand)

Erf also supports the optional result_accuracy argument:

Erf(operand, result_accuracy)

For more information on result_accuracy see Result Accuracy.

Exp

See also XlaBuilder::Exp.

Element-wise natural exponential x -> e^x.

Exp(operand)

Exp also supports the optional result_accuracy argument:

Exp(operand, result_accuracy)

For more information on result_accuracy see Result Accuracy.

For StableHLO information see StableHLO - exponential.

Expm1

See also XlaBuilder::Expm1.

Element-wise natural exponential minus one x -> e^x - 1.

Expm1(operand)

Expm1 also supports the optional result_accuracy argument:

Expm1(operand, result_accuracy)

For more information on result_accuracy see Result Accuracy.

For StableHLO information see StableHLO - exponential_minus_one.

Fft

See also XlaBuilder::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.

Fft(operand, ftt_type, fft_length)

For StableHLO information see StableHLO - fft.

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.

Floor

See also XlaBuilder::Floor.

Element-wise floor x -> ⌊x⌋.

Floor(operand)

For StableHLO information see StableHLO - floor.

Fusion

See also HloInstruction::CreateFusion.

Fusion operation represents HLO instructions and serves as a primitive in HLO. This op may appear in HLO dumps but is not intended to be constructed manually by end users.

Gather

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

For StableHLO information see StableHLO - gather.

General Semantics

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

gather(operand, start_indices, dimension_numbers, slice_sizes, indices_are_sorted)

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

The output is an array with batch_dims.size + offset_dims.size dimensions.

The operand.rank must equal the sum of offset_dims.size and collapsed_slice_dims.size. 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:

1. 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).

2. 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:

1. 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.

2. Create a starting index, Sin, into operand using S by scattering S using start_index_map. More precisely:

   1. Sin[start_index_map[k]] = S[k] if k < start_index_map.size.

   2. Sin[_] = 0 otherwise.

3. Create an index Oin into operand by scattering the indices at the offset dimensions in Out according to the collapsed_slice_dims set. More precisely:

   1. Oin[remapped_offset_dims(k)] = Out[offset_dims[k]] if k < offset_dims.size (remapped_offset_dims is defined below).

   2. Oin[_] = 0 otherwise.

4. In is Oin + Sin where + is element-wise addition.

remapped_offset_dims is a monotonic function with domain [0, offset_dims.size) and range [0, operand.rank) \ collapsed_slice_dims. So if, e.g., offset_dims.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 order, after scattering its values according to start_index_map) by the user. If they are not then the semantics are 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:

1. 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.

2. The number of output offset dimensions explicitly present in the output shape may be smaller than the input number of dimensions. 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.

3. The slice extracted from the "Gather Indices" array ((X, Y) in the last example) may have fewer elements than the input array's number of dimensions, and an explicit mapping dictates how the index should be expanded to have the same number of dimensions 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],O0] 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

See also XlaBuilder::GetDimensionSize.

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

GetDimensionSize(operand, dimension)

For StableHLO information see StableHLO - get_dimension_size.

GetTupleElement

See also XlaBuilder::GetTupleElement.

Indexes into a tuple with a compile-time-constant value.

The value must be a compile-time-constant so that shape inference can determine the type of the resulting value.

This is analogous to std::get<int N>(t) in C++. Conceptually:

let v: f32[10] = f32[10]{0, 1, 2, 3, 4, 5, 6, 7, 8, 9};
let s: s32 = 5;
let t: (f32[10], s32) = tuple(v, s);
let element_1: s32 = gettupleelement(t, 1); // Inferred shape matches s32.

See also tf.tuple.

GetTupleElement(tuple_data, index)

For StableHLO information see StableHLO - get_tuple_element.

Imag

See also XlaBuilder::Imag.

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

Imag(operand)

For StableHLO information see StableHLO - imag.

Infeed

See also XlaBuilder::Infeed.

Infeed(shape, config)

Reads a single data item from the implicit Infeed streaming interface of the device, interpreting the data as the given shape and its layout, and returns a XlaOp of the data. Multiple Infeed operations are allowed in a computation, but there must be a total order among the Infeed operations. For example, two Infeed's in the code below have a total order since there is a dependency between the while loops.

result1 = while (condition, init = init_value) {
  Infeed(shape)
  }

result2 = while (condition, init = result1) {
  Infeed(shape)
  }

Nested tuple shapes are not supported. For an empty tuple shape, the Infeed operation is effectively a no-op and proceeds without reading any data from the Infeed of the device.

For StableHLO information see StableHLO - infeed.

Iota

See also XlaBuilder::Iota.

Iota(shape, iota_dimension)

Builds a constant literal on device rather than a potentially large host transfer. Creates an array that has specified shape and holds values starting at zero and incrementing by one along the specified dimension. 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.

For example, Iota(s32[4, 8], 0) returns

[[0, 0, 0, 0, 0, 0, 0, 0 ],
 [1, 1, 1, 1, 1, 1, 1, 1 ],
 [2, 2, 2, 2, 2, 2, 2, 2 ],
 [3, 3, 3, 3, 3, 3, 3, 3 ]]

Iota(s32[4, 8], 1) returns

[[0, 1, 2, 3, 4, 5, 6, 7 ],
 [0, 1, 2, 3, 4, 5, 6, 7 ],
 [0, 1, 2, 3, 4, 5, 6, 7 ],
 [0, 1, 2, 3, 4, 5, 6, 7 ]]

For StableHLO information see StableHLO - iota.

IsFinite

See also XlaBuilder::IsFinite.

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.

IsFinite(operand)

For StableHLO information see StableHLO - is_finite.

Log

See also XlaBuilder::Log.

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

Log(operand)

Log also supports the optional result_accuracy argument:

Log(operand, result_accuracy)

For more information on result_accuracy see Result Accuracy.

For StableHLO information see StableHLO - log.

Log1p

See also XlaBuilder::Log1p.

Element-wise shifted natural logarithm x -> ln(1+x).

Log1p(operand)

Log1p also supports the optional result_accuracy argument:

Log1p(operand, result_accuracy)

For more information on result_accuracy see Result Accuracy.

For StableHLO information see StableHLO - log_plus_one.

Logistic

See also XlaBuilder::Logistic.

Element-wise logistic function computation x -> logistic(x).

Logistic(operand)

Logistic also supports the optional result_accuracy argument:

Logistic(operand, result_accuracy)

For more information on result_accuracy see Result Accuracy.

For StableHLO information see StableHLO - logistic.

Map

See also XlaBuilder::Map.

Map(operands..., computation, dimensions)

Applies a scalar function over the given operands arrays, producing an array of the same dimensions where each element is the result of the mapped function applied to the corresponding elements in the input arrays.

The mapped function is an arbitrary computation with the restriction that it has N inputs of scalar type T and a single output with type S. The output has the same dimensions as the operands except that the element type T is replaced with S.

For example: Map(op1, op2, op3, computation, par1) maps elem_out <- computation(elem1, elem2, elem3, par1) at each (multi-dimensional) index in the input arrays to produce the output array.

For StableHLO information see StableHLO - map.

Max

See also XlaBuilder::Max.

Performs element-wise max operation on tensors lhs and rhs.

Max(lhs, rhs)

The arguments' shapes have to be either similar or compatible. See the broadcasting documentation about what it means for shapes to be compatible. The result of an operation has a shape which is the result of broadcasting the two input arrays. In this variant, operations between arrays of different ranks are not supported, unless one of the operands is a scalar.

An alternative variant with different-dimensional broadcasting support exists for Max:

Max(lhs,rhs, broadcast_dimensions)

This variant of the operation should be used for arithmetic operations between arrays of different ranks (such as adding a matrix to a vector).

The additional broadcast_dimensions operand is a slice of integers specifying the dimensions to use for broadcasting the operands. The semantics are described in detail on the broadcasting page.

For StableHLO information see StableHLO - maximum.

Min

See also XlaBuilder::Min.

Performs element-wise min operation on lhs and rhs.

Min(lhs, rhs)

The arguments' shapes have to be either similar or compatible. See the broadcasting documentation about what it means for shapes to be compatible. The result of an operation has a shape which is the result of broadcasting the two input arrays. In this variant, operations between arrays of different ranks are not supported, unless one of the operands is a scalar.

An alternative variant with different-dimensional broadcasting support exists for Min:

Min(lhs,rhs, broadcast_dimensions)

This variant of the operation should be used for arithmetic operations between arrays of different ranks (such as adding a matrix to a vector).

The additional broadcast_dimensions operand is a slice of integers specifying the dimensions to use for broadcasting the operands. The semantics are described in detail on the broadcasting page.

For StableHLO information see StableHLO - minimum.

Mul

See also XlaBuilder::Mul.

Performs element-wise product of lhs and rhs.

Mul(lhs, rhs)

The arguments' shapes have to be either similar or compatible. See the broadcasting documentation about what it means for shapes to be compatible. The result of an operation has a shape which is the result of broadcasting the two input arrays. In this variant, operations between arrays of different ranks are not supported, unless one of the operands is a scalar.

An alternative variant with different-dimensional broadcasting support exists for Mul:

Mul(lhs,rhs, broadcast_dimensions)

This variant of the operation should be used for arithmetic operations between arrays of different ranks (such as adding a matrix to a vector).

The additional broadcast_dimensions operand is a slice of integers specifying the dimensions to use for broadcasting the operands. The semantics are described in detail on the broadcasting page.

For StableHLO information see StableHLO - multiply.

Neg

See also XlaBuilder::Neg.

Element-wise negation x -> -x.

Neg(operand)

For StableHLO information see StableHLO - negate

Not

See also XlaBuilder::Not.

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

Not(operand)

For StableHLO information see StableHLO - not.

OptimizationBarrier

See also XlaBuilder::OptimizationBarrier.

Blocks any optimization pass from moving computations across the barrier.

OptimizationBarrier(operand)

Ensures that all inputs are evaluated before any operators that depend on the barrier's outputs.

For StableHLO information see StableHLO - optimization_barrier.

Or

See also XlaBuilder::Or.

Performs element-wise OR of lhs and rhs .

Or(lhs, rhs)

The arguments' shapes have to be either similar or compatible. See the broadcasting documentation about what it means for shapes to be compatible. The result of an operation has a shape which is the result of broadcasting the two input arrays. In this variant, operations between arrays of different ranks are not supported, unless one of the operands is a scalar.

An alternative variant with different-dimensional broadcasting support exists for Or:

Or(lhs,rhs, broadcast_dimensions)

This variant of the operation should be used for arithmetic operations between arrays of different ranks (such as adding a matrix to a vector).

The additional broadcast_dimensions operand is a slice of integers specifying the dimensions to use for broadcasting the operands. The semantics are described in detail on the broadcasting page.

For StableHLO information see StableHLO - or.

Outfeed

See also XlaBuilder::Outfeed.

Writes inputs to the outfeed.

Outfeed(operand, shape_with_layout, outfeed_config)

shape_with_layout communicates the laid out shape that we want to outfeed.

For StableHLO information see StableHLO - outfeed.

Pad

See also XlaBuilder::Pad.

Pad(operand, padding_value, padding_config)

Expands the given operand array by padding around the array as well as between the elements of the array with the given padding_value. padding_config specifies the amount of edge padding and the interior padding for each dimension.

PaddingConfig is a repeated field of PaddingConfigDimension, which contains three fields for each dimension: edge_padding_low, edge_padding_high, and interior_padding.

edge_padding_low and edge_padding_high specify the amount of padding added at the low-end (next to index 0) and the high-end (next to the highest index) of each dimension respectively. The amount of edge padding can be negative -- the absolute value of negative padding indicates the number of elements to remove from the specified dimension.

interior_padding specifies the amount of padding added between any two elements in each dimension; it may not be negative. Interior padding occurs logically before edge padding, so in the case of negative edge padding, elements are removed from the interior-padded operand.

This operation is a no-op if the edge padding pairs are all (0, 0) and the interior padding values are all 0. The figure below shows examples of different edge_padding and interior_padding values for a two-dimensional array.

For StableHLO information see StableHLO - pad.

Parameter

See also XlaBuilder::Parameter.

Parameter represents an argument input to a computation.

PartitionID

See also XlaBuilder::BuildPartitionId.

Produces partition_id of the current process.

PartitionID(shape)

PartitionID may appear in HLO dumps but it is not intended to be constructed manually by end users.

For StableHLO information see StableHLO - partition_id.

PopulationCount

See also XlaBuilder::PopulationCount.

Computes the number of bits set in each element of operand.

PopulationCount(operand)

For StableHLO information see StableHLO - popcnt.

Pow

See also XlaBuilder::Pow.

Performs element-wise exponentiation of lhs by rhs.

Pow(lhs, rhs)

The arguments' shapes have to be either similar or compatible. See the broadcasting documentation about what it means for shapes to be compatible. The result of an operation has a shape which is the result of broadcasting the two input arrays. In this variant, operations between arrays of different ranks are not supported, unless one of the operands is a scalar.

An alternative variant with different-dimensional broadcasting support exists for Pow:

Pow(lhs,rhs, broadcast_dimensions)

This variant of the operation should be used for arithmetic operations between arrays of different ranks (such as adding a matrix to a vector).

The additional broadcast_dimensions operand is a slice of integers specifying the dimensions to use for broadcasting the operands. The semantics are described in detail on the broadcasting page.

For StableHLO information see StableHLO - power.

Real

See also XlaBuilder::Real.

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

Real(operand)

For StableHLO information see StableHLO - real.

Recv

See also XlaBuilder::Recv.

Recv, RecvWithTokens, and RecvToHost are operations that serve as communication primitives in HLO. These ops typically appear in HLO dumps as part of low-level input/output or cross-device transfer, but they are not intended to be constructed manually by end users.

Recv(shape, handle)

Receives data of the given shape from a Send instruction in another computation that shares the same channel handle. Returns a XlaOp for the received data.

For StableHLO information see StableHLO - recv.

RecvDone

See also HloInstruction::CreateRecv and HloInstruction::CreateRecvDone.

Similar to Send, the client API of Recv operation represents synchronous communication. However, the instruction is internally decomposed into 2 HLO instructions (Recv and RecvDone) to enable asynchronous data transfers.

Recv(const Shape& shape, int64 channel_id)

Allocates resources required to receive data from a Send instruction with the same channel_id. Returns a context for the allocated resources, which is used by a following RecvDone instruction to wait for the completion of the data transfer. The context is a tuple of {receive buffer (shape), request identifier (U32)} and it can only be used by a RecvDone instruction.

Given a context created by a Recv instruction, waits for the data transfer to complete and return the received data.

Reduce

See also XlaBuilder::Reduce.

Applies a reduction function to one or more arrays in parallel.

Reduce(operands..., init_values..., computation, dimensions_to_reduce)

Where:

• N is required to be greater or equal to 1.
• The computation has to be "roughly" associative (see below).
• All input arrays must have the same dimensions.
• All initial values have to form an identity under computation.
• If N = 1, Collate(T) is T.
• If N > 1, Collate(T_0, ..., T_{N-1}) is a tuple of N elements of type T.

This operation reduces one or more dimensions of each input array into scalars. The number of dimensions of each returned array is number_of_dimensions(operand) - len(dimensions). The output of the op is Collate(Q_0, ..., Q_N) where Q_i is an array of type T_i, the dimensions of which are described below.

Different backends are allowed to reassociate the reduction computation. This can lead to numerical differences, as some reduction functions like addition are not associative for floats. However, if the range of the data is limited, floating-point addition is close enough to be associative for most practical uses.

For StableHLO information see StableHLO - reduce.

Examples

When reducing across one dimension in a single 1D array with values [10, 11, 12, 13], with reduction function f (this is computation) then that could be computed as

f(10, f(11, f(12, f(init_value, 13)))

but there are also many other possibilities, e.g.

f(init_value, f(f(10, f(init_value, 11)), f(f(init_value, 12), f(init_value, 13))))

The following is a rough pseudo-code example of how reduction could be implemented, using summation as the reduction computation with an initial value of 0.

result_shape <- remove all dims in dimensions from operand_shape

# Iterate over all elements in result_shape. The number of r's here is equal
# to the number of dimensions of the result.
for r0 in range(result_shape[0]), r1 in range(result_shape[1]), ...:
  # Initialize this result element
  result[r0, r1...] <- 0

  # Iterate over all the reduction dimensions
  for d0 in range(dimensions[0]), d1 in range(dimensions[1]), ...:
    # Increment the result element with the value of the operand's element.
    # The index of the operand's element is constructed from all ri's and di's
    # in the right order (by construction ri's and di's together index over the
    # whole operand shape).
    result[r0, r1...] += operand[ri... di]

Here's an example of reducing a 2D array (matrix). The shape has 2 dimensions, dimension 0 of size 2 and dimension 1 of size 3:

Results of reducing dimensions 0 or 1 with an "add" function:

Note that both reduction results are 1D arrays. The diagram shows one as column and another as row just for visual convenience.

For a more complex example, here is a 3D array. Its number of dimensions is 3, dimension 0 of size 4, dimension 1 of size 2 and dimension 2 of size 3. For simplicity, the values 1 to 6 are replicated across dimension 0.

Similarly to the 2D example, we can reduce just one dimension. If we reduce dimension 0, for example, we get a 2-dimensional array where all values across dimension 0 were folded into a scalar:

|  4   8  12 |
| 16  20  24 |

If we reduce dimension 2, we also get a 2-dimensional array where all values across dimension 2 were folded into a scalar:

| 6  15 |
| 6  15 |
| 6  15 |
| 6  15 |

Note that the relative order between the remaining dimensions in the input is preserved in the output, but some dimensions may get assigned new numbers (since the number of dimensions changes).

We can also reduce multiple dimensions. Add-reducing dimensions 0 and 1 produces the 1D array [20, 28, 36].

Reducing the 3D array over all its dimensions produces the scalar 84.

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

See also XlaBuilder::ReducePrecision.

Models the effect of converting floating-point values to a lower-precision format (such as IEEE-FP16) and back to the original format. The number of exponent and mantissa bits in the lower-precision format can be specified arbitrarily, although all bit sizes may not be supported on all hardware implementations.

ReducePrecision(operand, exponent_bits, mantissa_bits)

The result is an array of type T. The input values are rounded to the nearest value representable with the given number of mantissa bits (using "ties to even" semantics), and any values that exceed the range specified by the number of exponent bits are clamped to positive or negative infinity. NaN values are retained, although they may be converted to canonical NaN values.

The lower-precision format must have at least one exponent bit (in order to distinguish a zero value from an infinity, since both have a zero mantissa), and must have a non-negative number of mantissa bits. The number of exponent or mantissa bits may exceed the corresponding value for type T; the corresponding portion of the conversion is then simply a no-op.

For StableHLO information see StableHLO - reduce_precision.

ReduceScatter

See also XlaBuilder::ReduceScatter.

ReduceScatter is a collective operation that effectively does an AllReduce and then scatters the result by splitting it into shard_count blocks along the scatter_dimension and replica i in the replica group receives the ith shard.

ReduceScatter(operand, computation, scatter_dimension, shard_count, replica_groups, channel_id, layout, use_global_device_ids)

• When operand is a tuple of arrays, the reduce-scatter is performed on each element of the tuple.
• replica_groups is a list of replica groups between which the reduction is performed (replica id for the current replica can be retrieved using ReplicaId). The order of replicas in each group determines the order in which the all-reduce result will be scattered. replica_groups must either be empty (in which case all replicas belong to a single group), or contain the same number of elements as the number of replicas. When there are more than one replica groups, they all must be of the same size. For example, replica_groups = {0, 2}, {1, 3} performs reduction between the replicas 0 and 2, and 1 and 3 and then scatters the result.
• shard_count is the size of each replica group. We need this in cases where replica_groups are empty. If replica_groups is not empty, shard_count must be equal to the size of each replica group.
• channel_id is used for cross-module communication: only reduce-scatter operations with the same channel_id can communicate with each other.
• layout See xla::shapes for more information on layouts.
• use_global_device_ids is a user-specified flag. When false(default) the numbers in replica_groups are ReplicaId when true the replica_groups represent a global id of (ReplicaID*partition_count + partition_id). For example:
  • With 2 replicas and 4 partitions,
  • replica_groups={ {0,1,4,5},{2,3,6,7} } and use_global_device_ids=true
  • group[0] = (0,0), (0,1), (1,0), (1,1)
  • group[1] = (0,2), (0,3), (1,2), (1,3)
  • where each pair is (replica_id, partition_id).

The output shape is the input shape with the scatter_dimension made shard_count times smaller. For example, if there are two replicas and the operand has the value [1.0, 2.25] and [3.0, 5.25] respectively on the two replicas, then the output value from this op where scatter_dim is 0 will be [4.0] for the first replica and [7.5] for the second replica.

For StableHLO information see StableHLO - reduce_scatter.

ReduceScatter - Example 1 - StableHLO

In the above example, there are 2 replicas participating in the ReduceScatter. On each replica, the operand has shape f32[2,4]. An all-reduce (sum) is performed across the replicas, producing a reduced value of shape f32[2,4] on each replica. This reduced value is then split into 2 parts along dimension 1, so each part has shape f32[2,2]. Each replica within the process group receives the part corresponding to its position in the group. As a result, the output on each replica has shape f32[2,2].

ReduceWindow

See also XlaBuilder::ReduceWindow.

Applies a reduction function to all elements in each window of a sequence of N multi-dimensional arrays, producing a single or a tuple of N multi-dimensional arrays as output. Each output array has the same number of elements as the number of valid positions of the window. A pooling layer can be expressed as a ReduceWindow. Similar to Reduce, the applied computation is always passed the init_values on the left-hand side.

ReduceWindow(operands..., init_values..., computation, window_dimensions, window_strides, padding)

Where:

• N is required to be greater or equal to 1.
• All input arrays must have the same dimensions.
• If N = 1, Collate(T) is T.
• If N > 1, Collate(T_0, ..., T_{N-1}) is a tuple of N elements of type (T0,...T{N-1}).

For StableHLO information see StableHLO - reduce_window.

ReduceWindow - Example 1

Input is a matrix of size [4x6] and both window_dimensions and window_stride_dimensions are [2x3].

// Create a computation for the reduction (maximum).
XlaComputation max;
{
  XlaBuilder builder(client_, "max");
  auto y = builder.Parameter(0, ShapeUtil::MakeShape(F32, {}), "y");
  auto x = builder.Parameter(1, ShapeUtil::MakeShape(F32, {}), "x");
  builder.Max(y, x);
  max = builder.Build().value();
}

// Create a ReduceWindow computation with the max reduction computation.
XlaBuilder builder(client_, "reduce_window_2x3");
auto shape = ShapeUtil::MakeShape(F32, {4, 6});
auto input = builder.Parameter(0, shape, "input");
builder.ReduceWindow(
    input,
    /*init_val=*/builder.ConstantLiteral(LiteralUtil::MinValue(F32)),
    *max,
    /*window_dimensions=*/{2, 3},
    /*window_stride_dimensions=*/{2, 3},
    Padding::kValid);