NEP 50 — Promotion rules for Python scalars

Author:

Sebastian Berg

Status:

Draft

Type:

Standards Track

Created:

2021-05-25

Abstract

Since NumPy 1.7, promotion rules use so-called “safe casting” which relies on inspection of the values involved. This helped identify a number of edge cases for users, but was complex to implement and also made behavior hard to predict.

There are two kinds of confusing results:

1. Value-based promotion means that the value, for example of a Python integer, can determine output type as found by np.result_type:

   np.result_type(np.int8, 1) == np.int8
   np.result_type(np.int8, 255) == np.int16
   

   This logic arises because 1 can be represented by a uint8 or int8 while 255 cannot be represented by an int8 but only by by a uint8 or int16.

   This also holds when working with 0-D arrays (so-called “scalar arrays”):

   int64_0d_array = np.array(1, dtype=np.int64)
   np.result_type(np.int8, int64_0d_array) == np.int8
   

   Where the fact that int64_0d_array has an int64 dtype has no influence on the resulting dtype. The dtype=np.int64 is effectively ignored in this example since only its value matters.

2. For a Python int, float, or complex the value is inspected as previously shown. But surprisingly not when the NumPy object is a 0-D array or NumPy scalar:

   np.result_type(np.array(1, dtype=np.uint8), 1) == np.int64
   np.result_type(np.int8(1), 1) == np.int64
   

   The reason is that value-based promotion is disabled when all objects are scalars or 0-D arrays. NumPy thus returns the same type as np.array(1), which is usually an int64 (this depends on the system).

Note that the examples apply also to operations like multiplication, addition, comparisons, and their corresponding functions like np.multiply.

This NEP proposes to refactor the behaviour around two guiding principles:

1. Values must never influence result type.

2. NumPy scalars and 0-D arrays should behave consistently with their N-D counterparts.

We propose to remove all value-based logic and add special handling for Python scalars to preserve some convenient behaviors. Python scalars will be considered “weakly” typed. When a NumPy array/scalar is combined with a Python scalar, it will be converted to the NumPy dtype, such that:

np.array([1, 2, 3], dtype=np.uint8) + 1  # returns a uint8 array
np.array([1, 2, 3], dtype=np.float32) + 2.  # returns a float32 array

There will be no dependence on the Python value itself.

The proposed changes also apply to np.can_cast(100, np.int8), however, we expect that the behaviour in functions (promotion) will, in practice, be far more important than the casting change itself.

Note

As of the NumPy 1.24.x series, NumPy has preliminary and limited support to test this proposal.

It is further necessary to set the following environment variable:

export NPY_PROMOTION_STATE=weak

Valid values are weak, weak_and_warn, and legacy. Note that weak_and_warn implements the optional warnings proposed in this NEP and is expected to be very noisy. We recommend starting using the weak option and use weak_and_warn mainly to understand a specific observed change in behaviour.

The following additional API exists:

• np._set_promotion_state() and np._get_promotion_state() which is equivalent to the environment variable. (Not thread/context safe.)

• with np._no_nep50_warning(): allows to suppress warnings when weak_and_warn promotion is used. (Thread and context safe.)

At this time overflow warnings on integer power are missing. Further, np.can_cast fails to give warnings in the weak_and_warn mode. Its behavior with respect to Python scalar input may still be in flux (this should affect very few users).

Schema of the new proposed promotion rules

After the change, the promotions in NumPy will follow the schema below. Promotion always occurs along the green lines: from left to right within their kind and to a higher kind only when necessary. The result kind is always the largest kind of the inputs. Note that float32 has a lower precision than int32 or uint32 and is thus sorted slightly to the left in the schematic. This is because float32 cannot represent all int32 values exactly. However, for practical reasons, NumPy allows promoting int64 to float64 effectively considering them to have the same precision.

The Python scalars are inserted at the very left of each “kind” and the Python integer does not distinguish signed and unsigned. NumPy promotion thus uses the following, ordered, kind categories:

• boolean

• integral: signed or unsigned integers

• inexact: floating point numbers and complex floating point numbers

When promoting a Python scalar with a dtype of lower kind category (boolean < integral < inexact) with a higher one, we use the minimum/default precision: that is float64, complex128 or int64 (int32 is used on some systems, e.g. windows).

See the next section for examples which clarify the proposed behavior. Further examples with a comparison to the current behavior can be found in the table below.

Examples of new behaviour

To make interpretation of above text and figure easier, we provide a few examples of the new behaviour. Below, the Python integer has no influence on the result type:

np.uint8(1) + 1 == np.uint8(2)
np.int16(2) + 2 == np.int16(4)

In the following the Python float and complex are “inexact”, but the NumPy value is integral, so we use at least float64/complex128:

np.uint16(3) + 3.0 == np.float64(6.0)
np.int16(4) + 4j == np.complex128(4+4j)

But this does not happen for float to complex promotions, where float32 and complex64 have the same precision:

np.float32(5) + 5j == np.complex64(5+5j)

Note that the schematic omits bool. It is set below “integral”, so that the following hold:

np.bool_(True) + 1 == np.int64(2)
True + np.uint8(2) == np.uint8(3)

Note that while this NEP uses simple operators as example, the rules described generally apply to all of NumPy operations.

Table comparing new and old behaviour

The following table lists relevant changes and unchanged behaviours. Please see the Old implementation for a detailed explanation of the rules that lead to the “Old result”, and the following sections for the rules detailing the new. The backwards compatibility section discusses how these changes are likely to impact users.

Note the important distinction between a 0-D array like array(2) and arrays that are not 0-D, such as array([2]).

Table of changed behaviours

Expression	Old result	New result
uint8(1) + 2	int64(3)	uint8(3) [T1]
array([1], uint8) + int64(1) or array([1], uint8) + array(1, int64)	array([2], unit8)	array([2], int64) [T2]
array([1.], float32) + float64(1.) or array([1.], float32) + array(1., float64)	array([2.], float32)	array([2.], float64)
array([1], uint8) + 1	array([2], uint8)	unchanged
array([1], uint8) + 200	array([201], np.uint8)	unchanged
array([100], uint8) + 200	array([ 44], uint8)	unchanged [T3]
array([1], uint8) + 300	array([301], uint16)	Exception [T4]
uint8(1) + 300	int64(301)	Exception [T5]
uint8(100) + 200	int64(301)	uint8(44) and RuntimeWarning [T6]
float32(1) + 3e100	float64(3e100)	float32(Inf) and RuntimeWarning [T7]
array([1.0], float32) + 1e-14 == 1.0 [T8]	array([True])	unchanged
array(1.0, float32) + 1e-14 == 1.0 [T8]	False	True
array([1.], float32) + 3	array([4.], float32)	unchanged
array([1.], float32) + int64(3)	array([4.], float32)	array([4.], float64) [T9]
(3j + array(3, complex64)).dtype	complex128	complex64 [T10]
(float32(1) + 1j)).dtype	complex128	complex64 [T11]
(int32(1) + 5j).dtype	complex128	unchanged [T12]

[T1]

New behaviour honours the dtype of the uint8 scalar.

[T2]

Current NumPy ignores the precision of 0-D arrays or NumPy scalars when combined with arrays.

[T3]

Current NumPy ignores the precision of 0-D arrays or NumPy scalars when combined with arrays.

[T4]

Old behaviour uses uint16 because 300 does not fit uint8, new behaviour raises an error for the same reason.

[T5]

300 cannot be converted to uint8.

[T6]

One of the most dangerous changes maybe. Retaining the type leads to overflow. A RuntimeWarning indicating overflow is given for the NumPy scalars.

[T7]

np.float32(3e100) overflows to infinity with a warning.

[T8] (1,2)

1 + 1e-14 loses precision when done in float32 but not in float64. The old behavior was casting the scalar argument to float32 or float64 differently depending on the dimensionality of the array; with the new behavior the computation is always done in the array precision (float32 in this case).

[T9]

NumPy promotes float32 and int64 to float64. The old behaviour ignored the int64 here.

[T10]

The new behavior is consistent between array(3, complex64) and array([3], complex64): the dtype of the result is that of the array argument.

[T11]

The new behavior uses the complex dtype of the precision compatible with the array argument, float32.

[T12]

Since the array kind is integer, the result uses the default complex precision, which is complex128.

Motivation and scope

The motivation for changing the behaviour with respect to inspecting the value of Python scalars and NumPy scalars/0-D arrays is three-fold:

1. The special handling of NumPy scalars/0-D arrays as well as the value inspection can be very surprising to users,

2. The value-inspection logic is much harder to explain and implement. It is further harder to make it available to user-defined DTypes through NEP 42. Currently, this leads to a dual implementation of a new and an old (value sensitive) system. Fixing this will greatly simplify the internal logic and make results more consistent.

3. It largely aligns with the choice of other projects like JAX and data-apis.org (see also Related Work).

We believe that the proposal of “weak” Python scalars will help users by providing a clear mental model for which datatype an operation will result in. This model fits well with the preservation of array precisions that NumPy currently often follows, and also uses for in-place operations:

arr += value

Preserves precision as long as “kind” boundaries are not crossed (otherwise an error is raised).

While some users will potentially miss the value inspecting behavior, even for those cases where it seems useful it quickly leads to surprises. This may be expected:

np.array([100], dtype=np.uint8) + 1000 == np.array([1100], dtype=np.uint16)

But the following will then be a surprise:

np.array([100], dtype=np.uint8) + 200 == np.array([44], dtype=np.uint8)

Considering that the proposal aligns with the behavior of in-place operands and avoids the surprising switch in behavior that only sometimes avoids overflow in the result, we believe that the proposal follows the “principle of least surprise”.

Usage and impact

This NEP is expected to be implemented with no transition period that warns for all changes. Such a transition period would create many (often harmless) warnings which would be difficult to silence. We expect that most users will benefit long term from the clearer promotion rules and that few are directly (negatively) impacted by the change. However, certain usage patterns may lead to problematic changes, these are detailed in the backwards compatibility section.

The solution to this will be an optional warning mode capable of notifying users of potential changes in behavior. This mode is expected to generate many harmless warnings, but provide a way to systematically vet code and track down changes if problems are observed.

Impact on can_cast

can_cast will never inspect the value anymore. So that the following results are expected to change from True to False:

np.can_cast(np.int64(100), np.uint8)
np.can_cast(np.array(100, dtype=np.int64), np.uint8)
np.can_cast(100, np.uint8)

We expect that the impact of this change will be small compared to that of the following changes.

Note

The last example where the input is a Python scalar _may_ be preserved since 100 can be represented by a uint8.

Impact on operators and functions involving NumPy arrays or scalars

The main impact on operations not involving Python scalars (float, int, complex) will be that operations on 0-D arrays and NumPy scalars will never depend on their values. This removes currently surprising cases. For example:

np.arange(10, dtype=np.uint8) + np.int64(1)
# and:
np.add(np.arange(10, dtype=np.uint8), np.int64(1))

Will return an int64 array in the future because the type of np.int64(1) is strictly honoured. Currently a uint8 array is returned.

Impact on operators involving Python int, float, and complex

This NEP attempts to preserve the convenience of the old behaviour when working with literal values. The current value-based logic had some nice properties when “untyped”, literal Python scalars are involved:

np.arange(10, dtype=np.int8) + 1  # returns an int8 array
np.array([1., 2.], dtype=np.float32) * 3.5  # returns a float32 array

But led to surprises when it came to “unrepresentable” values:

np.arange(10, dtype=np.int8) + 256  # returns int16
np.array([1., 2.], dtype=np.float32) * 1e200  # returns float64

The proposal is to preserve this behaviour for the most part. This is achieved by considering Python int, float, and complex to be “weakly” typed in operations. However, to avoid surprises, we plan to make conversion to the new type more strict: The results will be unchanged in the first two examples, but in the second one, it will change the following way:

np.arange(10, dtype=np.int8) + 256  # raises a TypeError
np.array([1., 2.], dtype=np.float32) * 1e200  # warning and returns infinity

The second one warns because np.float32(1e200) overflows to infinity. It will then continue to do the calculation with inf as usual.

Behaviour in other libraries

Overflowing in the conversion rather than raising an error is a choice; it is one that is the default in most C setups (similar to NumPy C can be set up to raise an error due to the overflow, however). It is also for example the behaviour of pytorch 1.10.

Particular behavior of Python integers

The NEPs promotion rules stated in terms of the resulting dtype which is typically also the operation dtype (in terms of result precision). This leads to what may seem like exceptions for Python integers: While uint8(3) + 1000 must be rejected because operating in uint8 is not possible, uint8(3) / 1000 returns a float64 and can convert both inputs to float64 to find the result.

In practice this means that arbitrary Python integer values are accepted in the following cases:

• All comparisons (==, <, etc.) between NumPy and Python integers are always well defined.

• Unary functions like np.sqrt that give a floating point result can and will convert the Python integer to a float.

• Division of integers returns floating point by casting input to float64.

Note that there may be additional functions where these exceptions could be applied but are not. In these cases it should be considered an improvement to allow them, but when the user impact is low we may not do so for simplicity.

Backward compatibility

In general, code which only uses the default dtypes float64, or int32/int64 or more precise ones should not be affected.

However, the proposed changes will modify results in quite a few cases where 0-D or scalar values (with non-default dtypes) are mixed. In many cases, these will be bug-fixes, however, there are certain changes which may be problematic to the end-user.

The most important possible failure is probably the following example:

arr = np.arange(100, dtype=np.uint8)  # storage array with low precision
value = arr[10]

# calculation continues with "value" without considering where it came from
value * 100

Where previously the value * 100 would cause an up-cast to int32/int64 (because value is a scalar). The new behaviour will preserve the lower precision unless explicitly dealt with (just as if value was an array). This can lead to integer overflows and thus incorrect results beyond precision. In many cases this may be silent, although NumPy usually gives warnings for the scalar operators.

Similarly, if the storage array is float32 a calculation may retain the lower float32 precision rather than use the default float64.

Further issues can occur. For example:

• Floating point comparisons, especially equality, may change when mixing precisions:

  np.float32(1/3) == 1/3  # was False, will be True.
  

• Certain operations are expected to start failing:

  np.array([1], np.uint8) * 1000
  np.array([1], np.uint8) == 1000  # possibly also
  

  to protect users in cases where previous value-based casting led to an upcast. (Failures occur when converting 1000 to a uint8.)

• Floating point overflow may occur in odder cases:

  np.float32(1e-30) * 1e50  # will return ``inf`` and a warning
  

  Because np.float32(1e50) returns inf. Previously, this would return a double precision result even if the 1e50 was not a 0-D array

In other cases, increased precision may occur. For example:

np.multiple(float32_arr, 2.)
float32_arr * np.float64(2.)

Will both return a float64 rather than float32. This improves precision but slightly changes results and uses double the memory.

Changes due to the integer “ladder of precision”

When creating an array from a Python integer, NumPy will try the following types in order, with the result depending on the value:

long (usually int64) → int64 → uint64 -> object

which is subtly different from the promotion described above.

This NEP currently does not include changing this ladder (although it may be suggested in a separate document). However, in mixed operations, this ladder will be ignored, since the value will be ignored. This means, that operations will never silently use the object dtype:

np.array([3]) + 2**100  # Will error

The user will have to write one of:

np.array([3]) + np.array(2**100)
np.array([3]) + np.array(2**100, dtype=object)

As such implicit conversion to object should be rare and the work-around is clear, we expect that the backwards compatibility concerns are fairly small.

Detailed description

The following provides some additional details on the current “value based” promotion logic, and then on the “weak scalar” promotion and how it is handled internally.

Old implementation of “values based” promotion

This section reviews how the current value-based logic works in practice, please see the following section for examples on how it can be useful.

When NumPy sees a “scalar” value, which can be a Python int, float, complex, a NumPy scalar or an array:

1000  # Python scalar
int32(1000)  # NumPy scalar
np.array(1000, dtype=int64)  # zero dimensional

Or the float/complex equivalents, NumPy will ignore the precision of the dtype and find the smallest possible dtype that can hold the value. That is, it will try the following dtypes:

• Integral: uint8, int8, uint16, int16, uint32, int32, uint64, int64.

• Floating: float16, float32, float64, longdouble.

• Complex: complex64, complex128, clongdouble.

Note that e.g. for the integer value of 10, the smallest dtype can be either uint8 or int8.

NumPy never applied this rule when all arguments are scalar values:

np.int64(1) + np.int32(2) == np.int64(3)

For integers, whether a value fits is decided precisely by whether it can be represented by the dtype. For float and complex, the a dtype is considered sufficient if:

• float16: -65000 < value < 65000 (or NaN/Inf)

• float32: -3.4e38 < value < 3.4e38 (or NaN/Inf)

• float64: -1.7e308 < value < 1.7e308 (or Nan/Inf)

• longdouble: (largest range, so no limit)

for complex these bounds were applied to the real and imaginary component. These values roughly correspond to np.finfo(np.float32).max. (NumPy did never force the use of float64 for a value of float32(3.402e38) though, but it will for a Python value of 3.402e38.)

State of the current “value based” promotion

Before we can propose alternatives to the current datatype system, it is helpful to review how “value based promotion” is used and can be useful. Value based promotion allows for the following code to work:

# Create uint8 array, as this is sufficient:
uint8_arr = np.array([1, 2, 3], dtype=np.uint8)
result = uint8_arr + 4
result.dtype == np.uint8

result = uint8_arr * (-1)
result.dtype == np.int16  # upcast as little as possible.

Where especially the first part can be useful: The user knows that the input is an integer array with a specific precision. Considering that plain + 4 retaining the previous datatype is intuitive. Replacing this example with np.float32 is maybe even more clear, as float will rarely have overflows. Without this behaviour, the above example would require writing np.uint8(4) and lack of the behaviour would make the following surprising:

result = np.array([1, 2, 3], dtype=np.float32) * 2.
result.dtype == np.float32

where lack of a special case would cause float64 to be returned.

It is important to note that the behaviour also applies to universal functions and zero dimensional arrays:

# This logic is also used for ufuncs:
np.add(uint8_arr, 4).dtype == np.uint8
# And even if the other array is explicitly typed:
np.add(uint8_arr, np.array(4, dtype=np.int64)).dtype == np.uint8

To review, if we replace 4 with [4] to make it one dimensional, the result will be different:

# This logic is also used for ufuncs:
np.add(uint8_arr, [4]).dtype == np.int64  # platform dependent
# And even if the other array is explicitly typed:
np.add(uint8_arr, np.array([4], dtype=np.int64)).dtype == np.int64

Proposed weak promotion

This proposal uses a “weak scalar” logic. This means that Python int, float, and complex are not assigned one of the typical dtypes, such as float64 or int64. Rather, they are assigned a special abstract DType, similar to the “scalar” hierarchy names: Integral, Floating, ComplexFloating.

When promotion occurs (as it does for ufuncs if no exact loop matches), the other DType is able to decide how to regard the Python scalar. E.g. a UInt16 promoting with an Integral will give UInt16.

Note

A default will most likely be provided in the future for user-defined DTypes. Most likely this will end up being the default integer/float, but in principle more complex schemes could be implemented.

At no time is the value used to decide the result of this promotion. The value is only considered when it is converted to the new dtype; this may raise an error.

Related work

Different Python projects that fill a similar space to NumPy prefer the weakly typed Python scalars as proposed in this NEP. Details of these may differ or be unspecified though:

• JAX promotion also uses the weak-scalar concept. However, it makes use of it also for most functions. JAX further stores the “weak-type” information on the array: jnp.array(1) remains weakly typed.

• data-apis.org also suggests this weak-scalar logic for the Python scalars.

Implementation

Implementing this NEP requires some additional machinery to be added to all binary operators (or ufuncs), so that they attempt to use the “weak” logic if possible. There are two possible approaches to this:

1. The binary operator simply tries to call np.result_type() if this situation arises and converts the Python scalar to the result-type (if defined).

2. The binary operator indicates that an input was a Python scalar, and the ufunc dispatching/promotion machinery is used for the rest (see NEP 42). This allows more flexibility, but requires some additional logic in the ufunc machinery.

Note

As of now, it is not quite clear which approach is better, either will give fairly equivalent results and 1. could be extended by 2. in the future if necessary.

It further requires removing all current special value-based code paths.

Unintuitively, a larger step in the implementation may be to implement a solution to allow an error to be raised in the following example:

np.arange(10, dtype=np.uint8) + 1000

Even though np.uint8(1000) returns the same value as np.uint8(232).

Note

See alternatives, we may yet decide that this silent overflow is acceptable or at least a separate issue.

Alternatives

There are several design axes where different choices are possible. The below sections outline these.

Use strongly-typed scalars or a mix of both

The simplest solution to the value-based promotion/casting issue would be to use strongly typed Python scalars, i.e. Python floats are considered double precision and Python integers are always considered the same as the default integer dtype.

This would be the simplest solution, however, it would lead to many upcasts when working with arrays of float32 or int16, etc. The solution for these cases would be to rely on in-place operations. We currently believe that while less dangerous, this change would affect many users and would be surprising more often than not (although expectations differ widely).

In principle, the weak vs. strong behaviour need not be uniform. It would also be possible to make Python floats use the weak behaviour, but Python integers use the strong one, since integer overflows are far more surprising.

Do not use weak scalar logic in functions

One alternative to this NEPs proposal is to narrow the use of weak types to Python operators.

This has advantages and disadvantages:

• The main advantage is that limiting it to Python operators means that these “weak” types/dtypes are clearly ephemeral to short Python statements.

• A disadvantage is that np.multiply and * are less interchangeable.

• Using “weak” promotion only for operators means that libraries do not have to worry about whether they want to “remember” that an input was a Python scalar initially. On the other hand, it would add a the need for slightly different (or additional) logic for Python operators. (Technically, probably as a flag to the ufunc dispatching mechanism to toggle the weak logic.)

• __array_ufunc__ is often used on its own to provide Python operator support for array-likes implementing it. If operators are special, these array-likes may need a mechanism to match NumPy (e.g. a kwarg to ufuncs to enable weak promotion.)

NumPy scalars could be special

Many users expect that NumPy scalars should be different from NumPy arrays, in that np.uint8(3) + 3 should return an int64 (or Python integer), when uint8_arr + 3 preserves the uint8 dtype.

This alternative would be very close to the current behaviour for NumPy scalars but it would cement a distinction between arrays and scalars (NumPy arrays are “stronger” than Python scalars, but NumPy scalars are not).

Such a distinction is very much possible, however, at this time NumPy will often (and silently) convert 0-D arrays to scalars. It may thus make sense, to only consider this alternative if we also change this silent conversion (sometimes referred to as “decay”) behaviour.

Handling conversion of scalars when unsafe

Cases such as:

np.arange(10, dtype=np.uint8) + 1000

should raise an error as per this NEP. This could be relaxed to give a warning or even ignore the “unsafe” conversion which (on all relevant hardware) would lead to np.uint8(1000) == np.uint8(232) being used.

Allowing weakly typed arrays

One problem with having weakly typed Python scalars, but not weakly typed arrays is that in many cases np.asarray() is called indiscriminately on inputs. To solve this issue JAX will consider the result of np.asarray(1) also to be weakly typed. There are, however, two difficulties with this:

1. JAX noticed that it can be confusing that:

   np.broadcast_to(np.asarray(1), (100, 100))
   

   is a non 0-D array that “inherits” the weak typing. [2]

2. Unlike JAX tensors, NumPy arrays are mutable, so assignment may need to cause it to be strongly typed?

A flag will likely be useful as an implementation detail (e.g. in ufuncs), however, as of now we do not expect to have this as user API. The main reason is that such a flag may be surprising for users if it is passed out as a result from a function, rather than used only very localized.

TODO

Before accepting the NEP it may be good to discuss this issue further. Libraries may need clearer patterns to “propagate” the “weak” type, this could just be an np.asarray_or_literal() to preserve Python scalars, or a pattern of calling np.result_type() before np.asarray().

Keep using value-based logic for Python scalars

Some of the main issues with the current logic arise, because we apply it to NumPy scalars and 0-D arrays, rather than the application to Python scalars. We could thus consider to keep inspecting the value for Python scalars.

We reject this idea on the grounds that it will not remove the surprises given earlier:

np.uint8(100) + 1000 == np.uint16(1100)
np.uint8(100) + 200 == np.uint8(44)

And adapting the precision based on the result value rather than the input value might be possible for scalar operations, but is not feasible for array operations. This is because array operations need to allocate the result array before performing the calculation.

Discussion

• numpy/numpy#2878

• https://mail.python.org/archives/list/numpy-discussion@python.org/thread/R7D65SNGJW4PD6V7N3CEI4NJUHU6QP2I/#RB3JLIYJITVO3BWUPGLN4FJUUIKWKZIW

• https://mail.python.org/archives/list/numpy-discussion@python.org/thread/NA3UBE3XAUTXFYBX6HPIOCNCTNF3PWSZ/#T5WAYQPRMI5UCK7PKPCE3LGK7AQ5WNGH

• Poll about the desired future behavior: https://discuss.scientific-python.org/t/poll-future-numpy-behavior-when-mixing-arrays-numpy-scalars-and-python-scalars/202

References and footnotes

[1]

Each NEP must either be explicitly labeled as placed in the public domain (see this NEP as an example) or licensed under the Open Publication License.

[2]

numpy/numpy#files

Copyright

This document has been placed in the public domain. [1]