# Numerical accuracy¶

In modern computers, floating point numbers are represented using IEEE 754 standard. For more details on floating point arithmetics and IEEE 754 standard, please see Floating point arithmetic In particular, note that floating point provides limited accuracy (about 7 decimal digits for single precision floating point numbers, about 16 decimal digits for double precision floating point numbers) and that floating point addition and multiplication are not associative, so the order of the operations affects the results. Because of this, PyTorch is not guaranteed to produce bitwise identical results for floating point computations that are mathematically identical. Similarly, bitwise identical results are not guaranteed across PyTorch releases, individual commits, or different platforms. In particular, CPU and GPU results can be different even for bitwise-identical inputs and even after controlling for the sources of randomness.

## Batched computations or slice computations¶

Many operations in PyTorch support batched computation, where the same operation is performed for the elements of the batches of inputs. An example of this is torch.mm() and torch.bmm(). It is possible to implement batched computation as a loop over batch elements, and apply the necessary math operations to the individual batch elements, for efficiency reasons we are not doing that, and typically perform computation for the whole batch. The mathematical libraries that we are calling, and PyTorch internal implementations of operations can produces slightly different results in this case, compared to non-batched computations. In particular, let A and B be 3D tensors with the dimensions suitable for batched matrix multiplication. Then (A@B)[0] (the first element of the batched result) is not guaranteed to be bitwise identical to A[0]@B[0] (the matrix product of the first elements of the input batches) even though mathematically it’s an identical computation.

Similarly, an operation applied to a tensor slice is not guaranteed to produce results that are identical to the slice of the result of the same operation applied to the full tensor. E.g. let A be a 2-dimentional tensor. A.sum(-1)[0] is not guaranteed to be bitwise equal to A[:,0].sum().

## Extremal values¶

When inputs contain large values such that intermediate results may overflow the range of the used datatype, the end result may overflow too, even though it is representable in the original datatype. E.g.:

import torch
a=torch.tensor([1e20, 1e20]) # fp32 type by default
a.norm() # produces tensor(inf)
a.double().norm() # produces tensor(1.4142e+20, dtype=torch.float64), representable in fp32


## TensorFloat-32(TF32) on Nvidia Ampere devices¶

On Ampere Nvidia GPUs, PyTorch can use TensorFloat32 (TF32) to speed up mathematically intensive operations, in particular matrix multiplications and convolutions. When an operation is performed using TF32 tensor cores, only the first 10 bits of the input mantissa are read. This elision leads to less accurate results, and surprising results (e.g., multiplying a matrix by the identity matrix produces results that are different from the input). By default, this option is disabled for matrix multiplications and enabled for convolutions, although most neural network workloads have the same convergence behavior when using TF32 as they have with fp32. However, if better throughput is desired for matrix multiplications, TF32 can be turned on with torch.backends.cuda.matmul.allow_tf32 = True. Conversely, if better accuracy is desired for convolutions, TF32 can be turned off with torch.backends.cudnn.allow_tf32 = False.

## Reduced Precision Reduction for FP16 GEMMs¶

Half-precision GEMM operations are typically done with intermediate accumulations (reduction) in single-precision for numerical accuracy and improved resilience to overflow. For performance, certain GPU architectures, especially more recent ones, allow a few truncations of the intermediate accumulation results to the reduced precision (e.g., half-precision). This change is often benign from the perspective of model convergence, though it may lead to unexpected results (e.g., inf values when the final result should be be representable in half-precision). If reduced-precision reductions are problematic, they can be turned off with torch.backends.cuda.matmul.allow_fp16_reduced_precision_reduction = False

## Reduced Precision FP16 and BF16 GEMMs and Convolutions on AMD Instinct MI200 devices¶

On AMD Instinct MI200 GPUs, the FP16 and BF16 V_DOT2 and MFMA matrix instructions flush input and output denormal values to zero. FP32 and FP64 MFMA matrix instructions do not flush input and output denormal values to zero. The affected instructions are only used by rocBLAS (GEMM) and MIOpen (convolution) kernels; all other PyTorch operations will not encounter this behavior. All other supported AMD GPUs will not encounter this behavior.

rocBLAS and MIOpen provide alternate implementations for affected FP16 operations. Alternate implementations for BF16 operations are not provided; BF16 numbers have a larger dynamic range than FP16 numbers and are less likely to encounter denormal values. For the FP16 alternate implementations, FP16 input values are cast to an intermediate BF16 value and then cast back to FP16 output after the accumulate FP32 operations. In this way, the input and output types are unchanged.

When training using FP16 precision, some models may fail to converge with FP16 denorms flushed to zero. Denormal values more frequently occur in the backward pass of training during gradient calculation. PyTorch by default will use the rocBLAS and MIOpen alternate implementations during the backward pass. The default behavior can be overridden using environment variables, ROCBLAS_INTERNAL_FP16_ALT_IMPL and MIOPEN_DEBUG_CONVOLUTION_ATTRIB_FP16_ALT_IMPL. The behavior of these environment variables is as follows:

forward

backward

Env unset

original

alternate

Env set to 1

alternate

alternate

Env set to 0

original

original

The following is the list of operations where rocBLAS may be used:

• torch.bmm

• torch.mm

• torch.nn.GRUCell

• torch.nn.LSTMCell

• torch.nn.Linear

• the following torch._C._ConvBackend implementations:

• slowNd

• slowNd_transposed

• slowNd_dilated

• slowNd_dilated_transposed

The following is the list of operations where MIOpen may be used:

• torch.nn.Conv[Transpose]Nd

• the following torch._C._ConvBackend implementations:

• ConvBackend::Miopen

• ConvBackend::MiopenDepthwise

• ConvBackend::MiopenTranspose