fp8.py

# Copyright (c) 2022-2024, NVIDIA CORPORATION & AFFILIATES. All rights reserved.
#
# See LICENSE for license information.

"""FP8 utilities for TransformerEngine"""
import os
from contextlib import contextmanager
from collections import deque
from typing import Callable, List, Optional, Dict, Any, Tuple, Union

import torch
import transformer_engine_torch as tex
from transformer_engine.common.recipe import DelayedScaling, Format

from .constants import dist_group_type
from .utils import get_device_compute_capability
from .jit import jit_fuser


__all__ = ["fp8_autocast", "fp8_model_init"]


def check_fp8_support() -> Tuple[bool, str]:
    """Return if fp8 support is available"""
    if get_device_compute_capability() >= (9, 0):  # hopper and above
        return True, ""
    if get_device_compute_capability() < (8, 9):  # pre-ada
        return False, "Device compute capability 8.9 or higher required for FP8 execution."
    if tex.get_cublasLt_version() < 120103:
        return False, "CublasLt version 12.1.3.x or higher required for FP8 execution on Ada."
    if float(torch.version.cuda) < 12.1:
        return False, "Cuda version 12.1 or higher required for FP8 execution on Ada."
    return True, ""


def get_default_fp8_recipe() -> DelayedScaling:
    """FP8 recipe with default args."""
    return DelayedScaling()


def get_fp8_torch_dtype(fp8_recipe: DelayedScaling, fprop_tensor: bool = True) -> torch.dtype:
    """Get fp8 data type according to recipe and tensor"""
    if fp8_recipe.fp8_format == Format.E4M3 or (
        fp8_recipe.fp8_format == Format.HYBRID and fprop_tensor
    ):
        return torch.float8_e4m3fn
    return torch.float8_e5m2fn


def get_fp8_te_dtype(fp8_recipe: DelayedScaling, fprop_tensor: bool = True) -> tex.DType:
    """Get fp8 data type according to recipe and tensor"""
    if fp8_recipe.fp8_format == Format.E4M3 or (
        fp8_recipe.fp8_format == Format.HYBRID and fprop_tensor
    ):
        return tex.DType.kFloat8E4M3
    return tex.DType.kFloat8E5M2


def get_fp8_max(fp8_recipe: DelayedScaling, fprop_tensor: bool = True) -> tex.DType:
    """Get max representible FP8 value."""
    if fp8_recipe.fp8_format == Format.E4M3 or (
        fp8_recipe.fp8_format == Format.HYBRID and fprop_tensor
    ):
        return Format.E4M3.value.max_fwd
    return Format.E5M2.value.max_fwd


class FP8GlobalStateManager:
    """Class to keep track of and manipulate the global
    FP8 state at different stages of execution.
    """

    FP8_ENABLED = False
    FP8_CALIBRATION = False
    FP8_RECIPE = None
    FP8_DISTRIBUTED_GROUP = None
    FP8_PARAMETERS = False
    IS_FIRST_FP8_MODULE = False
    FP8_GRAPH_CAPTURING = False
    FP8_AUTOCAST_DEPTH = 0
    global_amax_buffer = {}
    global_amax_history_buffer = {}
    global_scale_buffer = {}
    global_scale_inv_buffer = {}
    fp8_tensors_recompute_buffer = []
    fp8_available = None
    reason_for_no_fp8 = ""
    autocast_arguments = {}
    autocast_to_fp8_params = {}
    fp8_param_to_autocast = {}
    skip_fp8_weight_update_tensor = None

    @classmethod
    def reset(cls) -> None:
        """Reset the global state"""
        cls.FP8_ENABLED = False
        cls.FP8_CALIBRATION = False
        cls.FP8_RECIPE = None
        cls.FP8_DISTRIBUTED_GROUP = None
        cls.FP8_PARAMETERS = False
        cls.IS_FIRST_FP8_MODULE = False
        cls.FP8_GRAPH_CAPTURING = False
        cls.FP8_AUTOCAST_DEPTH = 0
        cls.global_amax_buffer = {}
        cls.global_amax_history_buffer = {}
        cls.global_scale_buffer = {}
        cls.global_scale_inv_buffer = {}
        cls.fp8_tensors_recompute_buffer = []
        cls.fp8_available = None
        cls.reason_for_no_fp8 = ""
        cls.autocast_arguments = {}
        cls.autocast_to_fp8_params = {}
        cls.fp8_param_to_autocast = {}
        cls.skip_fp8_weight_update_tensor = None

    @classmethod
    def set_skip_fp8_weight_update_tensor(cls, skip: bool) -> None:
        """`skip_fp8_weight_update_tensor` inplace setter."""
        if cls.skip_fp8_weight_update_tensor is None:
            cls.skip_fp8_weight_update_tensor = torch.empty(1, dtype=torch.float32, device="cuda")
        cls.skip_fp8_weight_update_tensor.fill_(skip)

    @classmethod
    def get_skip_fp8_weight_update_tensor(cls) -> None:
        """`skip_fp8_weight_update_tensor` getter."""
        return cls.skip_fp8_weight_update_tensor

    @classmethod
    def is_fp8_available(cls) -> Tuple[bool, str]:
        """Return if fp8 support is available"""
        if cls.fp8_available is None:
            cls.fp8_available, cls.reason_for_no_fp8 = check_fp8_support()
        return cls.fp8_available, cls.reason_for_no_fp8

    @staticmethod
    def get_meta_tensor_key(forward: bool = True) -> str:
        """Returns scaling key in `fp8_meta`."""
        if forward:
            return "scaling_fwd"
        return "scaling_bwd"

    @staticmethod
    def get_fwd_bwd_key(forward: bool = True) -> str:
        """Convert bool `forward` to string."""
        return "forward" if forward else "backward"

    @classmethod
    def get_buffer_info(cls) -> str:
        """
        Returns a key for `fp8_meta` that stores the module's index
        in the global buffers along with autocast information.
        """
        return "buffer_index_and_autocast_key"

    @classmethod
    def get_key_in_buffer(
        cls,
        forward: bool,
        fp8_weights: bool,
        fp8_recipe: DelayedScaling,
        fp8_group: dist_group_type,
    ) -> str:
        """Returns a key into the global FP8 buffers."""
        autocast_key = cls.get_unique_autocast_key(fp8_recipe, fp8_group)
        fwd_bwd_key = cls.get_fwd_bwd_key(forward)
        return f"{fwd_bwd_key}_{fp8_weights}_{autocast_key}"

    @classmethod
    def split_key_in_buffer(cls, key: str) -> Tuple[bool, bool, str]:
        """Splits buffer key into relevant parts."""
        forward, fp8_weights, autocast_key = key.split("_", 2)
        forward = forward == "forward"
        fp8_weights = fp8_weights == "True"
        return forward, fp8_weights, autocast_key

    @classmethod
    def add_fp8_tensors_to_global_buffer(
        cls,
        fp8_meta: Dict[str, Any],
        fp8_weights: Optional[List[torch.Tensor]] = None,
    ) -> None:
        """
        The amax reduction process happens completely outside the FP8 modules.
        To participate in the reduction, the only role played by a module is
        to call this function in order to append it's FP8 tensor into a global
        buffer. There are 5 global buffers maintained, one each for amax, amax
        history, scale, scale-inverse, and non-weight-mask. Each buffer has
        keys that hold FP8 tensors. Keys have a `forward_` or `backward_` prefix
        to indicate the type of FP8 tensor, since the forward and backward
        reductions happen separately.

        Note: For CG capture, this method is called from the graphed
        wrapper. For non CG case, it's called from within the module.
        """

        # Every module must call this function exactly once since
        # the amax tensors are static. Ensures that compatibility
        # with non-graphed modules is maintained.
        index_in_buffer = cls.get_buffer_info()  # Same index for fwd/bwd fp8 tensors.
        if index_in_buffer in fp8_meta:
            return

        fp8_meta[index_in_buffer] = []
        for forward in (True, False):
            # This algorithm creates a two-way map with `autocast_to_fp8_params` and
            # `fp8_param_to_autocast`. This is used for keeping track of FP8 weights
            # in an autocasted region and cross reference them in `float8_tensor.py`
            # to perform the forward amax reduction.
            fp8_meta_tensor_key = cls.get_meta_tensor_key(forward=forward)
            if fp8_meta_tensor_key not in fp8_meta:
                # Handles non-parameter FP8 modules, e.g. DPA.
                continue

            if forward and fp8_weights is not None:
                autocast_key = cls.get_unique_autocast_key(
                    fp8_meta["recipe"], fp8_meta["fp8_group"]
                )
                fp8_weight_set = {id(w._data) for w in fp8_weights}
                if autocast_key not in cls.autocast_to_fp8_params:
                    cls.autocast_to_fp8_params[autocast_key] = fp8_weight_set
                else:
                    cls.autocast_to_fp8_params[autocast_key] = cls.autocast_to_fp8_params[
                        autocast_key
                    ].union(fp8_weight_set)
                # Identify correct autocast key for a given param.
                for w in fp8_weight_set:
                    cls.fp8_param_to_autocast[w] = autocast_key

            key = cls.get_key_in_buffer(
                forward, fp8_weights is not None, fp8_meta["recipe"], fp8_meta["fp8_group"]
            )

            if key not in cls.global_amax_buffer:
                cls.global_amax_buffer[key] = [fp8_meta[fp8_meta_tensor_key].amax_history[0]]
                cls.global_amax_history_buffer[key] = [fp8_meta[fp8_meta_tensor_key].amax_history]
                cls.global_scale_buffer[key] = [fp8_meta[fp8_meta_tensor_key].scale]
                cls.global_scale_inv_buffer[key] = [fp8_meta[fp8_meta_tensor_key].scale_inv]
            else:
                cls.global_amax_buffer[key].append(fp8_meta[fp8_meta_tensor_key].amax_history[0])
                cls.global_amax_history_buffer[key].append(
                    fp8_meta[fp8_meta_tensor_key].amax_history
                )
                cls.global_scale_buffer[key].append(fp8_meta[fp8_meta_tensor_key].scale)
                cls.global_scale_inv_buffer[key].append(fp8_meta[fp8_meta_tensor_key].scale_inv)
            fp8_meta[index_in_buffer].append(len(cls.global_amax_buffer[key]) - 1)
            fp8_meta[index_in_buffer].append(key)

    @classmethod
    def is_fp8_enabled(cls) -> bool:
        """Is FP8 enabled"""
        return cls.FP8_ENABLED

    @classmethod
    def is_fp8_calibration(cls) -> bool:
        """Is FP8 calibration"""
        return cls.FP8_CALIBRATION

    @classmethod
    def with_fp8_parameters(cls) -> bool:
        """Should the parameters be stored as FP8"""
        return cls.FP8_PARAMETERS

    @classmethod
    def fp8_graph_capturing(cls) -> bool:
        """Is CUDA graph capture under way?"""
        return cls.FP8_GRAPH_CAPTURING or torch.cuda.is_current_stream_capturing()

    @classmethod
    def is_first_fp8_module(cls):
        """Returns `True` only the first time when called multiple
        times from within the same `fp8_autocast` context.
        """
        tmp = cls.IS_FIRST_FP8_MODULE
        cls.IS_FIRST_FP8_MODULE = False
        return tmp

    @classmethod
    def get_fp8_recipe(cls) -> DelayedScaling:
        """Return the fp8 recipe"""
        if cls.FP8_RECIPE is not None:
            return cls.FP8_RECIPE
        return get_default_fp8_recipe()

    @classmethod
    def get_fp8_group(cls) -> Union[dist_group_type, None]:
        """Return the fp8 group for scale/amax comm"""
        return cls.FP8_DISTRIBUTED_GROUP

    @classmethod
    def get_fp8_autocast_state(cls) -> Tuple[bool, bool, DelayedScaling, dist_group_type, bool]:
        """FP8 autocast state getter"""
        return (
            cls.FP8_ENABLED,
            cls.FP8_CALIBRATION,
            cls.FP8_RECIPE,
            cls.FP8_DISTRIBUTED_GROUP,
            cls.IS_FIRST_FP8_MODULE,
            cls.FP8_GRAPH_CAPTURING,
        )

    @classmethod
    def set_fp8_autocast_state(
        cls, fp8_state: Tuple[bool, bool, DelayedScaling, dist_group_type, bool]
    ) -> None:
        """FP8 autocast state setter"""
        (
            cls.FP8_ENABLED,
            cls.FP8_CALIBRATION,
            cls.FP8_RECIPE,
            cls.FP8_DISTRIBUTED_GROUP,
            cls.IS_FIRST_FP8_MODULE,
            cls.FP8_GRAPH_CAPTURING,
        ) = fp8_state

    @staticmethod
    def reduce_tensor_across_group_op_max(tensor: torch.Tensor, group: dist_group_type) -> None:
        """Reduce tensor across given group."""
        if torch.distributed.is_initialized():
            torch.distributed.all_reduce(
                tensor,
                op=torch.distributed.ReduceOp.MAX,
                group=group,
                async_op=False,
            )

    @classmethod
    def reduce_and_update_fp8_tensors(
        cls,
        forward: bool = True,
        fp8_weights: bool = False,
    ) -> None:
        """Concatenate, reduce, and split amaxes in the global buffer."""
        for buffer_key, amax_buffer in cls.global_amax_buffer.items():
            # Check for forward or backward reduction.
            fwd_update, fp8_weights_update, autocast_key = cls.split_key_in_buffer(buffer_key)
            if fwd_update != forward:
                continue
            # Only skip a forward update when `fp8_weights` is explicitly set to `True`
            # (inside optimizer) and the current key is not an `fp8_weight_update` key.
            # For other cases, we need to reduce because of activation tensors.
            # TODO(ksivaman) consider separate weight and activation fp8_tensors.
            if fwd_update and fp8_weights and not fp8_weights_update:
                continue
            if len(amax_buffer) == 0:
                continue

            # Retrieve autocast specific args and concat amaxes.
            recipe, group = cls.autocast_arguments[autocast_key]
            contiguous_amax = torch.cat(amax_buffer)

            # Reduction.
            if (
                recipe.reduce_amax
                and torch.distributed.is_initialized()
                and torch.distributed.get_world_size(group=group) > 1
            ):
                cls.reduce_tensor_across_group_op_max(contiguous_amax, group)

            # Amax and scale update.
            unfused_update = (
                bool(int(os.getenv("NVTE_UNFUSED_FP8_UPDATE", "0")))
                or callable(recipe.amax_compute_algo)
                or callable(recipe.scaling_factor_compute_algo)
            )

            if not unfused_update:
                tex.fused_amax_and_scale_update_after_reduction(
                    contiguous_amax,
                    cls.global_amax_history_buffer[buffer_key],
                    cls.global_scale_buffer[buffer_key],
                    cls.global_scale_inv_buffer[buffer_key],
                    recipe.amax_compute_algo,
                    get_fp8_te_dtype(recipe, forward),
                    recipe.margin,
                )
            else:
                split_and_copy(contiguous_amax, amax_buffer, [x.numel() for x in amax_buffer])

                for amax_history, scale, scale_inv in zip(
                    cls.global_amax_history_buffer[buffer_key],
                    cls.global_scale_buffer[buffer_key],
                    cls.global_scale_inv_buffer[buffer_key],
                ):
                    _amax_and_scale_update(
                        amax_history, scale, scale_inv, get_fp8_max(recipe, forward), recipe
                    )

    @classmethod
    def get_unique_autocast_key(
        cls,
        recipe: Optional[DelayedScaling] = None,
        group: Optional[dist_group_type] = None,
    ):
        """
        For FP8, each autocast can be uniquely identified by the recipe and fp8 group.
        Safely using `hash` as we never cross checkpoint boundaries.
        """
        return f"{str(recipe)}:{hash(group)}"

    @classmethod
    def fp8_autocast_enter(
        cls,
        enabled: bool = False,
        calibrating: bool = False,
        fp8_recipe: Optional[DelayedScaling] = None,
        fp8_group: Optional[dist_group_type] = None,
        _graph: bool = False,
    ) -> None:
        """Set state and tracking variables for entry into FP8 region."""

        fp8_recipe = get_default_fp8_recipe() if fp8_recipe is None else fp8_recipe
        autocast_key = cls.get_unique_autocast_key(fp8_recipe, fp8_group)
        cls.autocast_arguments[autocast_key] = (fp8_recipe, fp8_group)

        cls.FP8_ENABLED = enabled
        cls.FP8_CALIBRATION = calibrating
        cls.FP8_RECIPE = fp8_recipe
        cls.FP8_DISTRIBUTED_GROUP = fp8_group
        cls.FP8_GRAPH_CAPTURING = _graph

        if cls.FP8_AUTOCAST_DEPTH == 0:
            cls.IS_FIRST_FP8_MODULE = True
        cls.FP8_AUTOCAST_DEPTH += 1

        if enabled:
            fp8_available, reason_for_no_fp8 = cls.is_fp8_available()
            assert fp8_available, reason_for_no_fp8

    @classmethod
    def fp8_autocast_exit(cls, enabled: bool, _graph: bool) -> None:
        """Set state and tracking variables for exit from FP8 region."""
        cls.FP8_AUTOCAST_DEPTH -= 1
        # Reduce only the non-FP8 weight modules here.
        # FP8 weight modules are reduced at the end of the optimizer
        # step after the weight amax is populated.
        if enabled and cls.FP8_AUTOCAST_DEPTH == 0 and not _graph and torch.is_grad_enabled():
            cls.reduce_and_update_fp8_tensors(forward=True, fp8_weights=False)

    @classmethod
    def copy_forward_fp8_meta_tensors_for_recompute(cls, fp8_meta: Dict[str, Any]) -> None:
        """Copy the scaling factors and amaxes for recompute forward phase
        to ensure both forward steps are numerically same.
        """
        buffer_position_key = "global_fp8_buffer_pos_fwd_recompute"

        to_copy = [
            fp8_meta["scaling_fwd"].amax_history.clone(),
            fp8_meta["scaling_fwd"].scale.clone(),
            fp8_meta["scaling_fwd"].scale_inv.clone(),
        ]

        if buffer_position_key in fp8_meta:
            cls.fp8_tensors_recompute_buffer[fp8_meta[buffer_position_key]].append(to_copy)
        else:
            if len(cls.fp8_tensors_recompute_buffer) == 0:
                cls.fp8_tensors_recompute_buffer = [deque()]
            else:
                cls.fp8_tensors_recompute_buffer.append(deque())
            cls.fp8_tensors_recompute_buffer[-1].append(to_copy)
            fp8_meta[buffer_position_key] = len(cls.fp8_tensors_recompute_buffer) - 1

    @classmethod
    def get_old_fp8_meta_tensors_for_recompute(cls, fp8_meta: Dict[str, Any]) -> None:
        """Switch to the copied scaling factors and amaxes from phase
        1 forward for indentical numerical outputs.
        """

        # Store updated amaxes and scales from phase 1 post forward.
        fp8_meta["updated_amax_history_fwd"] = fp8_meta["scaling_fwd"].amax_history
        fp8_meta["updated_scale_fwd"] = fp8_meta["scaling_fwd"].scale
        fp8_meta["updated_scale_inv_fwd"] = fp8_meta["scaling_fwd"].scale_inv

        # Retrieve stashed amaxes and scales from phase 1 pre forward.
        buffer_position_key = "global_fp8_buffer_pos_fwd_recompute"
        stashed_fp8_meta = cls.fp8_tensors_recompute_buffer[fp8_meta[buffer_position_key]].popleft()

        # Replace amaxes and scales with stashed values for phase 2 forward
        fp8_meta["scaling_fwd"].amax_history = stashed_fp8_meta[0]
        fp8_meta["scaling_fwd"].scale = stashed_fp8_meta[1]
        fp8_meta["scaling_fwd"].scale_inv = stashed_fp8_meta[2]

    @staticmethod
    def restore_fp8_meta_tensors(fp8_meta: Dict[str, Any]) -> None:
        """Restore latest scaling factors and amaxes after recompute forward run."""
        fp8_meta["scaling_fwd"].amax_history = fp8_meta["updated_amax_history_fwd"]
        fp8_meta["scaling_fwd"].scale = fp8_meta["updated_scale_fwd"]
        fp8_meta["scaling_fwd"].scale_inv = fp8_meta["updated_scale_inv_fwd"]


@contextmanager
def fp8_model_init(enabled: bool = True) -> None:
    """
    Context manager for FP8 initialization of parameters.

    Example usage:

    .. code-block:: python

        with fp8_model_init(enabled=True):
            model = transformer_engine.pytorch.Linear(768, 768)

    Parameters
    ----------
    enabled: bool, default = `True`
             when enabled, Transformer Engine modules created inside this `fp8_model_init`
             region will hold only FP8 copies of its parameters, as opposed to the default
             behavior where both higher precision and FP8 copies are present. Setting this
             option to `True` may result in lower memory consumption and is especially
             useful for scenarios like:

             * full model training using optimizer with master weights, where the high
               precision copies of weights are already present in the optimizer.
             * inference, where only the FP8 copies of the parameters are used.
             * LoRA-like fine-tuning, where the main parameters of the model do not change.

             This functionality is *EXPERIMENTAL*.
    """
    _fp8_parameters = FP8GlobalStateManager.FP8_PARAMETERS
    FP8GlobalStateManager.FP8_PARAMETERS = enabled
    try:
        yield
    finally:
        FP8GlobalStateManager.FP8_PARAMETERS = _fp8_parameters


@contextmanager
def fp8_autocast(
    enabled: bool = True,
    calibrating: bool = False,
    fp8_recipe: Optional[DelayedScaling] = None,
    fp8_group: Optional[dist_group_type] = None,
    _graph: bool = False,
) -> None:
    """
    Context manager for FP8 usage.

    .. code-block:: python

        with fp8_autocast(enabled=True):
            out = model(inp)

    .. note::

        Support for FP8 in the Linear layer of Transformer Engine is currently limited to tensors
        with shapes where both dimensions are divisible by 16. In terms of the input to the full
        Transformer network, this typically requires padding sequence length to be multiple of 16.

    .. note::

        When :attr:`fp8_recipe.reduce_amax==True`, any module must not be invoked more than once
        inside a single `fp8_autocast` region. This is unsupported behavior because the amax
        reduction is handled during the exit of the `fp8_autocast` context. Calling the same
        module more than once inside an `fp8_autocast` region overrides the amax tensors
        before reduction can occur.

    Parameters
    ----------
    enabled: bool, default = `True`
             whether or not to enable fp8
    calibrating: bool, default = `False`
                 calibration mode allows collecting statistics such as amax and scale
                 data of fp8 tensors even when executing without fp8 enabled. This is
                 useful for saving an inference ready fp8 checkpoint while training
                 using a higher precision.
    fp8_recipe: recipe.DelayedScaling, default = `None`
                recipe used for FP8 training.
    fp8_group: torch._C._distributed_c10d.ProcessGroup, default = `None`
               distributed group over which amaxes for the fp8 tensors
               are reduced at the end of each training step.
    """
    fp8_state = FP8GlobalStateManager.get_fp8_autocast_state()
    FP8GlobalStateManager.fp8_autocast_enter(
        enabled=enabled,
        calibrating=calibrating,
        fp8_recipe=fp8_recipe,
        fp8_group=fp8_group,
        _graph=_graph,
    )
    try:
        yield
    finally:
        FP8GlobalStateManager.set_fp8_autocast_state(fp8_state)
        FP8GlobalStateManager.fp8_autocast_exit(enabled, _graph=_graph)


def _update_amax_history(amax_history: torch.Tensor) -> torch.Tensor:
    """Update amax history and set next amax to zero."""
    if amax_history.shape[0] > 1:
        new_amax_history = torch.roll(amax_history, -1, 0)
        amax_history.copy_(new_amax_history)
    amax_history[0].fill_(0.0)
    return amax_history


@torch.jit.script
def _default_get_amax_and_update_history(
    amax_history: torch.Tensor,
    amax_compute_algo: str,
) -> Tuple[torch.Tensor, torch.Tensor]:
    """Default function to obtain amax from history."""
    if amax_compute_algo == "max":
        amax = torch.max(amax_history, dim=0).values
    else:  # amax_compute_algo == "most_recent"
        amax = amax_history[0].clone()

    amax_history = _update_amax_history(amax_history)
    return amax_history, amax


@jit_fuser
def _default_sf_compute(
    amax: torch.Tensor,
    scale: torch.Tensor,
    fp8_max: float,
    margin: int,
    _fp32_max: float = torch.finfo(torch.float32).max,  # finfo not available in jitter
) -> torch.Tensor:
    """Default function to convert amax to scaling factor.
    Computing the scaling factor requires consideration of the following scenarios:
    1. amax == 0:
       No action is possible, set scale to the previous scale (or 1).
    2. 0 < amax < tiny_amax
       The amax is too tiny that the scale becomes infinite in FP32.
       Set scale = FP32_max
    3. tiny_amax <= amax < FP32_max:
       Set scale = FP8_max (or scaled_max) / amax
    4. When amax == inf or amax == nan:
       No action is possible, set scale to the previous scale (or 1).
    """
    sf = (fp8_max / amax) / (2**margin)
    sf = torch.where(amax > 0.0, sf, scale)
    sf = torch.where(torch.isfinite(amax), sf, scale)
    sf = torch.where(torch.isinf(sf), torch.full_like(sf, _fp32_max), sf)
    scale.copy_(sf)
    return scale


def _compute_amax_and_update_history(
    amax_history: torch.Tensor,
    amax_compute_algo: Union[Callable, str],
) -> Tuple[torch.Tensor, torch.Tensor]:
    """Obtain the amax from the history."""

    if callable(amax_compute_algo):
        amax = amax_compute_algo(amax_history)
        amax_history = _update_amax_history(amax_history)
        return amax_history, amax
    return _default_get_amax_and_update_history(
        amax_history,
        amax_compute_algo,
    )


def _compute_scaling_factor(
    amax: torch.Tensor,
    scale: torch.Tensor,
    fp8_max: float,
    recipe: DelayedScaling,
) -> torch.Tensor:
    """Convert amax to scaling factor."""

    if recipe.scaling_factor_compute_algo is None:
        return _default_sf_compute(
            amax,
            scale,
            fp8_max,
            recipe.margin,
        )
    return recipe.scaling_factor_compute_algo(amax, scale, fp8_max, recipe)


def _amax_and_scale_update(
    amax_history: torch.Tensor,
    scale: torch.Tensor,
    scale_inv: torch.Tensor,
    fp8_max: float,
    recipe: DelayedScaling,
) -> None:
    """Updates FP8 meta tensors."""
    new_amax_history, amax = _compute_amax_and_update_history(
        amax_history,
        recipe.amax_compute_algo,
    )
    new_scale = _compute_scaling_factor(amax, scale, fp8_max, recipe)
    scale.copy_(new_scale)
    scale_inv.copy_(1.0 / new_scale)
    amax_history.copy_(new_amax_history)


def split_and_copy(
    buffer: torch.Tensor,
    outputs: List[torch.Tensor],
    chunk_sizes: List[int],
) -> None:
    """Split `buffer` by `chunk_sizes` and copy into `outputs`."""
    splits = buffer.split(chunk_sizes)
    torch._foreach_copy_(outputs, splits)