## lib/nx/random.ex

``````defmodule Nx.Random do
@moduledoc """
Pseudo-random number generators.

Unlike the stateful pseudo-random number generators (PRNGs)
that users of most programming languages and numerical libraries
may be accustomed to, Nx random functions require an explicit
PRNG key to be passed as a first argument. That key is defined by
an `Nx.Tensor` composed of 2 unsigned 32-bit integers, usually
generated by the `Nx.Random.key/1` function:

iex> Nx.Random.key(12)
#Nx.Tensor<
u32[2]
[0, 12]
>

This key can then be used in any of Nx’s random number generation
routines:

iex> key = Nx.Random.key(12)
iex> {uniform, _new_key} = Nx.Random.uniform(key)
iex> uniform
#Nx.Tensor<
f32
0.7691127061843872
>

Now, when generating a new random number, you pass the `new_key`
to get a different number.

The function in this module also have a `*_split` variant, which
is used when the key has been split before hand.

## Design and Context

In short, Nx's PRNGs are based on a Threefry counter PRNG
associated to a functional array-oriented splitting model.
To summarize, among other requirements, Nx's PRNG aims to:

1. Ensure reproducibility

2. Parallelize well, both in terms of vectorization
(generating array values) and multi-replica, multi-core
computation. In particular it should not use sequencing
constraints between random function calls.
"""

import Nx.Defn, only: [deftransform: 2, deftransformp: 2, defn: 2, defnp: 2]

@nbits 32

@doc """
Create a pseudo-random number generator (PRNG) key given an integer seed.

## Examples

iex> Nx.Random.key(12)
#Nx.Tensor<
u32[2]
[0, 12]
>

iex> Nx.Random.key(999999999999)
#Nx.Tensor<
u32[2]
[232, 3567587327]
>
"""
defn key(seed) do
k1 = Nx.right_shift(seed, 32)
k2 = Nx.bitwise_and(seed, 0xFFFFFFFF)

Nx.stack([k1, k2])
|> Nx.as_type(:u32)
end

@doc """
Splits a PRNG key into `num` new keys by adding a leading axis.

## Examples

iex> key = Nx.Random.key(1701)
iex> Nx.Random.split(key)
#Nx.Tensor<
u32[2][2]
[
[56197195, 1801093307],
[961309823, 1704866707]
]
>

iex> key = Nx.Random.key(999999999999)
iex> Nx.Random.split(key, parts: 4)
#Nx.Tensor<
u32[4][2]
[
[3959978897, 4079927650],
[3769699049, 3585271160],
[3182829676, 333122445],
[3185556048, 1258545461]
]
>
"""
defn split(key, opts \\ []) do
assert_key!(key)
opts = keyword!(opts, parts: 2)
end

@doc """
Folds in new data to a PRNG key.

## Examples

iex> key = Nx.Random.key(42)
iex> Nx.Random.fold_in(key, 99)
#Nx.Tensor<
u32[2]
[2015327502, 1351855566]
>

iex> key = Nx.Random.key(42)
iex> Nx.Random.fold_in(key, 1234)
#Nx.Tensor<
u32[2]
[1356445167, 2917756949]
>

iex> key = Nx.Random.key(42)
iex> Nx.Random.fold_in(key, Nx.tensor([[1, 99], [1234, 13]]))
#Nx.Tensor<
u32[2][2][2]
[
[
[64467757, 2916123636],
[2015327502, 1351855566]
],
[
[1356445167, 2917756949],
[3514951389, 229662949]
]
]
>
"""
defn fold_in(key, data) do
assert_key!(key)

k1 = Nx.right_shift(data, 32)
k2 = Nx.bitwise_and(data, 0xFFFFFFFF)

{x1, x2} =
Nx.stack([k1, k2])
|> Nx.reshape({2, :auto})
|> Nx.as_type(:u32)
|> threefry2x32_20_pair(key)

[x1, x2]
|> Nx.stack(axis: -1)
|> Nx.reshape(fold_shape(Nx.shape(data)))
end

deftransformp fold_shape(shape) do
Tuple.insert_at(shape, tuple_size(shape), 2)
end

defnp threefry2x32(key, shape) do
case shape |> Nx.size() |> rem(2) do
0 ->
Nx.iota({2, div(Nx.size(shape), 2)}, type: :u32)
|> threefry2x32_20_concat(key)
|> Nx.reshape(shape)

1 ->
Nx.concatenate([Nx.iota({Nx.size(shape)}, type: :u32), Nx.tensor([0], type: :u32)])
|> Nx.reshape({2, :auto})
|> threefry2x32_20_concat(key)
|> Access.get(0..-2//1)
|> Nx.reshape(shape)
end
end

defn threefry2x32_20_concat(xs, ks) do
{nx1, nx2} = threefry2x32_20_pair(xs, ks)
Nx.concatenate([nx1, nx2], axis: 0)
end

defnp threefry2x32_20_pair(xs, ks) do
rotations = {Nx.tensor([13, 15, 26, 6], type: :u8), Nx.tensor([17, 29, 16, 24], type: :u8)}

key1 = ks[0]
key2 = ks[1]
xs = {xs[0] + key1, xs[1] + key2}

ks = {
key2,
Nx.bitwise_xor(key1, key2)
|> Nx.bitwise_xor(0x1BD11BDA),
key1
}

state = {xs, ks, rotations}

{_, {{nx1, nx2}, _, _}} =
while {x = Nx.tensor(1, type: :u32), state}, x < 6 do
{x + Nx.tensor(1, type: :u32), rolled_loop_step(x, state)}
end

{nx1, nx2}
end

defnp apply_round({xs1, xs2}, rot) do
y1 = xs1 + xs2

y2 =
rotate_left(xs2, rot)
|> Nx.bitwise_xor(y1)

{y1, y2}
end

defnp rolled_loop_step(i, {{_xs1, _xs2} = xs, {k1, k2, k3}, {r1, r2}}) do
{xs1, xs2} =
while xs, r <- r1 do
apply_round(xs, r)
end

xs1 = k1 + xs1
xs2 = k2 + i + xs2

new_xs = {xs1, xs2}
new_ks = {k2, k3, k1}
new_rs = {r2, r1}

{new_xs, new_ks, new_rs}
end

defnp rotate_left(x, rot) do
x <<< rot ||| x >>> (Nx.tensor(@nbits, type: :u32) - rot)
end

defnp random_bits(key, opts \\ []) do
assert_key!(key)
opts = keyword!(opts, shape: {}, bit_width: 32)
shape = opts[:shape]
bit_width = opts[:bit_width]

case bit_width do
64 ->
bits =
threefry2x32(key, {2, Nx.size(shape)})
|> Nx.as_type({:u, 64})

bits = bits[0] <<< 32 ||| bits[1]
Nx.reshape(bits, shape)

32 ->
threefry2x32(key, shape)

_ ->
threefry2x32(key, shape)
|> Nx.as_type({:u, bit_width})
end
end

deftransformp mantissa_shift(nbits, type) do
mantissa =
case type do
{:bf, 16} -> 7
{:f, 16} -> 10
{:f, 32} -> 23
{:f, 64} -> 52
end

Nx.tensor(nbits - mantissa, type: {:u, nbits})
end

@doc """
Sample uniform random integer values in `[min_value, max_value)`.

## Options

* `:type` - the integer type for the returned tensor
* `:shape` - shape of the returned tensor
* `:names` - the names of the returned tensor

## Examples

iex> key = Nx.Random.key(1701)
iex> {randint, _new_key} = Nx.Random.randint(key, 1, 100)
iex> randint
#Nx.Tensor<
s64
66
>

iex> key = Nx.Random.key(1701)
iex> {randint, _new_key} = Nx.Random.randint(key, 1, 100, shape: {3, 2}, type: :u32)
iex> randint
#Nx.Tensor<
u32[3][2]
[
[9, 20],
[19, 6],
[71, 15]
]
>

"""
defn randint(key, min_val, max_val, opts \\ []) do
keys = split(key)
{randint_split(keys[1], min_val, max_val, opts), keys[0]}
end

@doc """
Same as `randint/4` but assumes the key has already been split.
"""
defn randint_split(key, min_val, max_val, opts \\ []) do
opts = keyword!(opts, [:names, :type, shape: {}])
assert_key!(key)

shape = opts[:shape]
type = {_, nbits} = infer_type(min_val, max_val, opts)

case type do
{:u, _} -> :ok
{:s, _} -> :ok
_ -> raise ArgumentError, "expected integer type, got type #{inspect(type)}"
end

random_bits = random_bits(key, shape: randint_random_bits_shape(shape), bit_width: nbits)

higher_bits = random_bits[0]
lower_bits = random_bits[1]

span = max_val - min_val

multiplier =
Nx.pow(2, Nx.quotient(nbits, 2))
|> Nx.remainder(span)
|> Nx.pow(2)
|> Nx.remainder(span)

offset =
higher_bits
|> Nx.remainder(span)
|> Nx.multiply(multiplier)
|> Nx.remainder(span)

(min_val + offset)
|> Nx.as_type(type)
|> Nx.reshape(shape, take_names(opts))
end

deftransformp randint_random_bits_shape(shape), do: Tuple.insert_at(shape, 0, 2)

@doc """
Shortcut for `uniform(key, 0.0, 1.0, opts)`.
"""
defn uniform(key, opts \\ []) do
uniform(key, 0.0, 1.0, opts)
end

@doc """
Sample uniform float values in `[min_val, max_val)`.

## Options

* `:type` - a float type for the returned tensor

* `:shape` - shape of the returned tensor

* `:names` - the names of the returned tensor

## Examples

iex> key = Nx.Random.key(1701)
iex> {uniform, _new_key} = Nx.Random.uniform(key)
iex> uniform
#Nx.Tensor<
f32
0.9728643894195557
>

iex> key = Nx.Random.key(1701)
iex> {uniform, _new_key} = Nx.Random.uniform(key, shape: {3, 2}, type: :f16)
iex> uniform
#Nx.Tensor<
f16[3][2]
[
[0.75390625, 0.6484375],
[0.7294921875, 0.21484375],
[0.09765625, 0.0693359375]
]
>

iex> key = Nx.Random.key(1701)
iex> {uniform, _new_key} = Nx.Random.uniform(key, shape: {2, 2}, type: :c64)
iex> uniform
#Nx.Tensor<
c64[2][2]
[
[0.18404805660247803+0.6546461582183838i, 0.5525915622711182+0.11568140983581543i],
[0.6074584722518921+0.8104375600814819i, 0.247686505317688+0.21975469589233398i]
]
>
"""
defn uniform(key, min_val, max_val, opts \\ []) do
keys = split(key)
{uniform_split(keys[1], min_val, max_val, opts), keys[0]}
end

@doc """
Same as `uniform/4` but assumes the key has already been split.
"""
defn uniform_split(key, min_value, max_value, opts \\ []) do
assert_key!(key)
opts = keyword!(opts, [:names, :type, shape: {}])
type = infer_float_type(min_value, max_value, opts)

float_or_complex(key, type, opts[:shape], fn key, {_type, nbits} = type, shape ->
u_one = Nx.tensor(1.0, type: type) |> Nx.bitcast({:u, nbits})

min_value = Nx.as_type(min_value, type)
max_value = Nx.as_type(max_value, type)

random_bits(key, shape: shape, bit_width: nbits)
|> Nx.right_shift(mantissa_shift(nbits, type))
|> Nx.bitwise_or(u_one)
|> Nx.bitcast(type)
|> Nx.subtract(Nx.tensor(1.0, type: type))
|> Nx.multiply(max_value - min_value)
|> Nx.max(min_value)
|> Nx.reshape(shape, take_names(opts))
end)
end

@doc """
Shortcut for `normal(key, 0.0, 1.0, opts)`.
"""
defn normal(key, opts \\ []) do
normal(key, 0.0, 1.0, opts)
end

@doc """
Returns a normal distribution with the given `mean` and `standard_deviation`.

## Options

* `:type` - a float or complex type for the returned tensor

* `:shape` - shape of the returned tensor

* `:names` - the names of the returned tensor

## Examples

iex> key = Nx.Random.key(42)
iex> {normal, _new_key} = Nx.Random.normal(key)
iex> normal
#Nx.Tensor<
f32
1.3694695234298706
>

iex> key = Nx.Random.key(42)
iex> {normal, _new_key} = Nx.Random.normal(key, 0, 1, shape: {3, 2}, type: :f16)
iex> normal
#Nx.Tensor<
f16[3][2]
[
[-0.32568359375, -0.77197265625],
[0.39208984375, 0.5341796875],
[0.270751953125, -2.080078125]
]
>

iex> key = Nx.Random.key(42)
iex> {normal, _new_key} = Nx.Random.normal(key, 0, 1, shape: {2, 2}, type: :c64)
iex> normal
#Nx.Tensor<
c64[2][2]
[
[-0.7632761001586914+0.8661127686500549i, -0.14282889664173126-0.7384796142578125i],
[0.678461492061615+0.4118310809135437i, -2.269538402557373-0.3689095079898834i]
]
>

iex> key = Nx.Random.key(1337)
iex> {normal, _new_key} = Nx.Random.normal(key, 10, 5, shape: {1_000})
iex> Nx.mean(normal)
#Nx.Tensor<
f32
9.70022201538086
>
iex> Nx.standard_deviation(normal)
#Nx.Tensor<
f32
5.051416397094727
>
"""
defn normal(key, mean, standard_deviation, opts \\ []) do
keys = split(key)
{normal_split(keys[1], mean, standard_deviation, opts), keys[0]}
end

@doc """
Same as `normal/4` but assumes the key has already been split.
"""
defn normal_split(key, mean, standard_deviation, opts \\ []) do
assert_key!(key)
opts = keyword!(opts, [:names, :type, shape: {}])
type = infer_float_type(mean, standard_deviation, opts)

float_or_complex(key, type, opts[:shape], fn key, type, shape ->
min_value = next_after_minus_1(type)
u = uniform_split(key, min_value, 1, opts |> put_type(type) |> put_shape(shape))

normal = Nx.sqrt(Nx.tensor(2, type: type)) * Nx.erf_inv(u)
Nx.as_type(standard_deviation, type) * normal + Nx.as_type(mean, type)
end)
end

@doc """
Randomly shuffles tensor elements along an axis.

## Options

* `:axis` - the axis along which to shuffle. Defaults to `0`

* `:independent` - a boolean that indicates wether the permutations
are independent along the given axis. Defaults to `false`

## Examples

iex> key = Nx.Random.key(42)
iex> {shuffled, _new_key} = Nx.Random.shuffle(key, Nx.iota({3, 4}, axis: 0))
iex> shuffled
#Nx.Tensor<
s64[3][4]
[
[2, 2, 2, 2],
[0, 0, 0, 0],
[1, 1, 1, 1]
]
>

iex> key = Nx.Random.key(10)
iex> {shuffled, _new_key} = Nx.Random.shuffle(key, Nx.iota({3, 4}, axis: 1), independent: true, axis: 1)
iex> shuffled
#Nx.Tensor<
s64[3][4]
[
[2, 1, 3, 0],
[3, 0, 1, 2],
[2, 3, 0, 1]
]
>
"""
defn shuffle(key, tensor, opts \\ []) do
opts = keyword!(opts, axis: 0, independent: false)
axis = opts[:axis]

if opts[:independent] do
shuffle_independent(key, tensor, axis: axis)
else
{idx, key} = shuffle_independent(key, Nx.iota({Nx.axis_size(tensor, axis)}), axis: 0)
{Nx.take(tensor, idx, axis: axis), key}
end
end

defnp shuffle_independent(key, tensor, opts) do
axis = opts[:axis]

exponent = 3
uint32max = Nx.Constants.max_finite(:u32)

num_rounds =
Nx.ceil(exponent * Nx.log(Nx.size(tensor)) / Nx.log(uint32max))
|> Nx.as_type(:u32)

{_, out, key} =
while {i = 0, tensor, key}, i < num_rounds do
keys = split(key)
sort_keys = random_bits(keys[1], shape: tensor.shape)
tensor = sort_key_val(tensor, sort_keys, axis: axis)
{i + 1, tensor, keys[0]}
end

{out, key}
end

defnp sort_key_val(tensor, sort_keys, opts \\ []) do
idx = Nx.argsort(sort_keys, axis: opts[:axis])
Nx.take_along_axis(tensor, idx, axis: opts[:axis])
end

@choice_options """
## Options

* `:samples` - The number of samples to take

* `:axis` - The axis along which to take samples.
If `nil`, the tensor is flattened beforehand.

* `:replace` - a boolean that specifies if samples will
be taken with or without replacement. Defaults to `true`.
"""
@doc """
Generates random samples from a tensor.

#{@choice_options}

## Examples

iex> k = Nx.Random.key(1)
iex> t = Nx.iota({4, 3})
iex> {result, _key} = Nx.Random.choice(k, t, samples: 4, axis: 0)
iex> result
#Nx.Tensor<
s64[4][3]
[
[6, 7, 8],
[3, 4, 5],
[6, 7, 8],
[3, 4, 5]
]
>
iex> {result, _key} = Nx.Random.choice(k, t, samples: 4, axis: 0, replace: false)
iex> result
#Nx.Tensor<
s64[4][3]
[
[3, 4, 5],
[9, 10, 11],
[6, 7, 8],
[0, 1, 2]
]
>

If no axis is specified, the tensor is flattened:

iex> k = Nx.Random.key(2)
iex> t = Nx.iota({3, 2})
iex> {result, _key} = Nx.Random.choice(k, t)
iex> result
#Nx.Tensor<
s64[1]
[3]
>
iex> {result, _key} = Nx.Random.choice(k, t, samples: 6, replace: false)
iex> result
#Nx.Tensor<
s64[6]
[2, 0, 4, 5, 1, 3]
>
"""
deftransform choice(key, tensor, opts \\ []) do
if is_list(opts) do
choice_no_probabilities(key, tensor, opts)
else
choice(key, tensor, opts, [])
end
end

defnp choice_no_probabilities(key, tensor, opts) do
{tensor_shape, n_inputs, n_draws, axis, replace} = validate_choice_opts(tensor, opts)
tensor = Nx.reshape(tensor, tensor_shape)

if replace do
{idx, key} = randint(key, 0, n_inputs, shape: {n_draws})
result = Nx.take(tensor, idx, axis: axis)
{result, key}
else
{shuffled, key} = shuffle(key, tensor, axis: axis)
result = Nx.slice_along_axis(shuffled, 0, n_draws, axis: axis)
{result, key}
end
end

@doc """
Generates random samples from a tensor with specified probabilities.

The probabilities tensor must have the same size as the axis along
which the samples are being taken. If no axis is given, the size
must be equal to the input tensor's size.

#{@choice_options}

## Examples

iex> k = Nx.Random.key(1)
iex> t = Nx.iota({4, 3})
iex> p = Nx.tensor([0.1, 0.7, 0.2])
iex> {result, _key} = Nx.Random.choice(k, t, p, samples: 3, axis: 1)
iex> result
#Nx.Tensor<
s64[4][3]
[
[1, 0, 1],
[4, 3, 4],
[7, 6, 7],
[10, 9, 10]
]
>
iex> {result, _key} = Nx.Random.choice(k, t, p, samples: 3, axis: 1, replace: false)
iex> result
#Nx.Tensor<
s64[4][3]
[
[1, 2, 0],
[4, 5, 3],
[7, 8, 6],
[10, 11, 9]
]
>

If no axis is specified, the tensor is flattened.
Notice that in the first case we get a higher occurence
of the entries with bigger probabilities, while in the
second case, without replacements, we get those samples
first.

iex> k = Nx.Random.key(2)
iex> t = Nx.iota({2, 3})
iex> p = Nx.tensor([0.01, 0.1, 0.19, 0.6, 0.05, 0.05])
iex> {result, _key} = Nx.Random.choice(k, t, p)
iex> result
#Nx.Tensor<
s64[1]
[3]
>
iex> {result, _key} = Nx.Random.choice(k, t, p, samples: 6)
iex> result
#Nx.Tensor<
s64[6]
[3, 3, 3, 0, 3, 3]
>
iex> {result, _key} = Nx.Random.choice(k, t, p, samples: 6, replace: false)
iex> result
#Nx.Tensor<
s64[6]
[3, 1, 2, 5, 4, 0]
>
"""
defn choice(key, tensor, p, opts) do
{tensor_shape, n_inputs, n_draws, axis, replace} = validate_choice_opts(tensor, opts)
tensor = Nx.reshape(tensor, tensor_shape)

case Nx.rank(p) do
1 -> :ok
r -> raise ArgumentError, "propability tensor must have rank 1, got: #{r}"
end

case {Nx.size(p), Nx.axis_size(tensor, axis)} do
{n, n} ->
:ok

{p_size, a_size} ->
raise ArgumentError,
"probability tensor of size #{p_size} is expected to have size #{a_size}"
end

if replace do
p_cumulative = Nx.cumulative_sum(p)
{uniform, key} = uniform(key, shape: {n_draws}, type: Nx.type(p_cumulative))
r = p_cumulative[-1] * (1 - uniform)

# naïve implementation of jax.numpy.searchsorted
p_cumulative = Nx.new_axis(p_cumulative, 0)
r = Nx.new_axis(r, 1)
idx = Nx.argmin(p_cumulative <= r, tie_break: :low, axis: 1)

result = Nx.take(tensor, idx, axis: axis)
{result, key}
else
{g, k} = gumbel(key, shape: {n_inputs}, type: Nx.type(p))
g = -g - Nx.log(p)
idx = g |> Nx.argsort() |> Nx.slice_along_axis(0, n_draws, axis: 0)

result = Nx.take(tensor, idx, axis: axis)
{result, k}
end
end

deftransformp validate_choice_opts(tensor, opts) do
opts = Keyword.validate!(opts, [:axis, samples: 1, replace: true])

{axis, tensor_shape} =
case opts[:axis] do
nil ->
{0, {Tuple.product(tensor.shape)}}

axis ->
{Nx.Shape.normalize_axis(tensor.shape, axis, tensor.names), tensor.shape}
end

if Nx.rank(tensor) < 1 do
raise ArgumentError, "tensor must have rank 1 or greater"
end

n_draws = opts[:samples]

if n_draws < 1 do
raise "must take at least one sample, got samples=#{n_draws}"
end

n_inputs =
case opts[:axis] do
nil -> Nx.size(tensor)
_ -> Nx.axis_size(tensor, axis)
end

replace = opts[:replace]

if not replace and n_draws > n_inputs do
raise ArgumentError, "cannot take more samples than the input size when replace: false"
end

{tensor_shape, n_inputs, n_draws, axis, replace}
end

@doc """
Sample Gumbel random values with given shape and float dtype.

## Options

* `:shape` - the shape of the output tensor containing the
random samples. Defaults to `{}`

* `:type` - the floating-point output type. Defaults to `{:f, 32}`

## Examples

iex> {result, _key} = Nx.Random.gumbel(Nx.Random.key(1))
iex> result
#Nx.Tensor<
f32
-0.7294610142707825
>

iex> {result, _key} = Nx.Random.gumbel(Nx.Random.key(1), shape: {2, 3})
iex> result
#Nx.Tensor<
f32[2][3]
[
[0.6247938275337219, -0.21740718185901642, 0.7678327560424805],
[0.7778404355049133, 4.0895304679870605, 0.3029090166091919]
]
>
"""
defn gumbel(key, opts \\ []) do
keys = split(key)
{gumbel_split(keys[1], opts), keys[0]}
end

@doc """
Same as `gumbel/2`, but assumes the key has been split beforehand.
"""
defn gumbel_split(key, opts \\ []) do
opts = keyword!(opts, shape: {}, type: {:f, 32})
type = opts[:type]
shape = opts[:shape]

if not Nx.Type.float?(type) do
raise ArgumentError, "output type must be floating-point, got: #{inspect(type)}"
end

u =
uniform_split(key, Nx.Constants.smallest_positive_normal(type), 1, shape: shape, type: type)

end

deftransformp next_after_minus_1({_, bits}) do
# Get the floating point representation of -1 and
# convert it to a big integer so the precision comes last (after exponent)
<<x::size(bits)-big>> = <<-1::float-size(bits)-big>>

# Decrement the precision by one (decrement because the sign is separate)
# and convert it back to a float
<<f::float-size(bits)-big>> = <<x - 1::integer-size(bits)-big>>

f
end

deftransformp float_or_complex(key, type, shape, fun) do
case type do
{:c, _} ->
type = Nx.Type.to_real(type)
data = fun.(key, type, Tuple.append(shape, 2))
to_complex = Nx.stack([1, Nx.Constants.i()])
Nx.dot(data, to_complex)

{t, _} when t == :f or t == :bf ->
fun.(key, type, shape)

_ ->
raise ArgumentError, "expected float or complex type, got type #{inspect(type)}"
end
end

deftransformp take_names(opts), do: Keyword.take(opts, [:names])

deftransformp infer_type(left, right, opts) do
if type = opts[:type] do
Nx.Type.normalize!(type)
else
Nx.Type.merge(Nx.type(left), Nx.type(right))
end
end

deftransformp infer_float_type(left, right, opts) do
if type = opts[:type] do
Nx.Type.normalize!(type)
else
Nx.Type.to_floating(Nx.Type.merge(Nx.type(left), Nx.type(right)))
end
end

deftransformp put_type(opts, type), do: Keyword.put(opts, :type, type)
deftransformp put_shape(opts, shape), do: Keyword.put(opts, :shape, shape)

defnp assert_key!(tensor) do
%{shape: shape, type: type} = tensor

case shape do
{2} ->
:ok

_ ->
raise ArgumentError,
"expected key to have shape {2}, got tensor with shape #{inspect(shape)}"
end

case type do
{:u, 32} ->
:ok

_ ->
raise ArgumentError,
"expected key with 32-bit unsigned integer type, got key with type #{inspect(type)}"
end
end
end
``````