---
title: Numpy Value Based Promotion Rules
author: Sebastian Berg
---

NumPy Value Based Scalar Casting and Promotion
==============================================

This document reviews some of the behaviours of the promotion rules within numpy. This is especially with respect to the promotion of scalars and 0D arrays which inspect the value to decide casting and promotion.

Other documents discussing these things:

* `from numpy.testing import print_coercion_tables` prints the current promotion tables including value based promotion for small positive/negative scalars.
* Allan Haldane's thoughts on changing casting/promotion to be more C-like and discussing things such as here:
https://gist.github.com/ahaldane/0f5ade49730e1a5d16ff6df4303f2e76
* Discussion around the problem of uint64 and int64 being promoted to float64: https://github.com/numpy/numpy/issues/12525 (lists many related issues).

Nomenclature and Defintions
---------------------------

* **dtype/type**: The data type of an array or scalar: `float32`, `float64`, `int8`, …

* **Category**: A category to which the data type belongs, in this context these are:
1. boolean
2. integer (unsigned and signed are not split up here, but are different "kinds")
3. floating point and complex (not split up here but are different "kinds")
5. All others

* **Casting**: converting from one dtype to another. There are four different rules of casting:
1. *"safe"* casting: All values are representable in the new data type. I.e. no information is lost during the conversion.
2. *"same kind"* casting: data loss may occur, but only within the same "kind". For example a float64 can be converted to float32 using "same kind" rules, an int64 can be converted to int16. This is although both lose precision or even produce incorrect values. Note that "kind" is different from "category" in that it distinguishes between signed and unsigned integers.
4. *"unsafe"* casting: Any conversion which can be defined, e.g. floating point to integer. For promotion this is fairly unimportant. (Some conversions such as string to integer, which not even work fall in this category, but could also be called coercions or conversions.)

* **Promotion**: The general process of finding a new dtype for multiple input dtypes. Will be used here to also denote any kind of casting/promotion done before a specific function is called. This can be more complex, because in rare cases a functions can for example take floating point numbers and integers as input at the same time (i.e. `np.ldexp`).

* **Common dtype**: A dtype which can represent all input data. In general this means that all inputs can be safely cast to this dtype. Within numpy this is the normal and simplest form of promotion.

* **`type1, type2 -> type3`**: Defines a promotion or signature. For example adding two integers: `np.int32(5) + np.int32(3)` gives `np.int32(8)`. The dtype signature for that example would be: `int32, int32 -> int32`. A short form for this is also `ii->i` using C-like type codes, this can be found for example in `np.ldexp.types` (and any numpy ufunc).

* **Scalar**: A numpy or python scalar or a **0-D array**. It is important to remember that zero dimensional arrays are treated just like scalars with respect to casting and promotion.

Current Situation in Numpy
--------------------------

The current situation can be understand mostly in terms of safe casting which is defined based on the type hierarchy and is sensitive to values for scalars.

This safe casting based approach is in contrast for example to promotion within C or Julia, which work based on category first. For example `int32` cannot be safely cast to `float32`, but C or Julia will use `int32, float32 -> float32` as the common type/promotion rule for example to decide on the output dtype for addition.

### Python Integers and Floats

Note that python integers are handled exactly like numpy ones. They are, however, special in that they do not have a dtype associated with them explicitly. Value based logic, as described here, seems useful for python integers and floats to allow:

arr = np.arange(10, dtype=np.int8)
arr += 1
# or:
res = arr + 1
res.dtype == np.int8

which ensures that no upcast (for example with higher memory usage) occurs.

### Safe Casting

Most safe casting is clearly defined based on whether or not any possible value is representable in the ouput dtype. Within numpy there is currently a single exception to this rule: `np.can_cast(np.int64, np.float64, casting="safe")` is considered to be true although float64 cannot represent some large integer values exactly. In contrast, `np.can_cast(np.int32, np.float32, casting="safe")` is `False` and `np.float64` would have to be used if a "safe" cast is desired.

This exception may be one thing that should be changed, however, concurrently the promotion rules have to be adapted to keep doing the same thing, or a larger behaviour change decided.

#### Scalar based rules

Unlike arrays, where inspection of all values is not feasable, for scalars (and 0-D arrays) the value is inspected. The casting becomes a two step process:
1. The minimal dtype capable of holding the value is found.
2. The normal casting rules are applied to the new dtype.

The first step uses the following rules by finding the minimal dtype within its category:

* Boolean: Dtype is already minimal

* Integers:
Casting is possible if output can hold the value. This includes uint8(127) casting to an int8.

* Floats and Complex
Scalars can be demoted based on value, roughly this avoids overflows:

    float16:     -65000 < value < 65000
    float32:    -3.4e38 < value < 3.4e38
    float64:   -1.7e308 < value < 1.7e308
    float128 (largest type, does not apply).

For complex, the logic is simply applied to both real and imaginary part. Complex numbers cannot be downcast to floating point.

* Others: Dtype is not modified.

This two step process means that `np.can_cast(np.int16(1024), np.float16)` is `False` even though float16 is capable of exactly representing the value 1024, since value based "demotion" to a lower dtype is used only within each category.

### Common Type Promotion

For most operations in numpy the output type is just the common type of the inputs, this holds for example for concatenation, as well as almost all math funcions (e.g. addition and multiplication have two identical inputs and need one ouput dtype). This operation is exposed as `np.result_type` which includes value based logic, and `np.promote_types` which only accepts dtypes as input.

Normal type promotion without value based/scalar logic finds the smallest type which both inputs can cast to safely. This will be the largest "kind" (bool < unsigned < integer < float < complex < other).

Note that type promotion is handled in a "reduce" manner from left to right. In rare cases this means it is not associatetive: `float32, uint16, int16 -> float32`, but `float32, (uint16, int16) -> float64`.

#### Scalar based rule

When there is a mix of scalars and arrays, numpy will usually allow the scalars to be handled in the same fashion as for "safe" casting rules.

The rules are as follows:

1. Value based logic is only applied if the "category" of any array is larger or equal to the category of all scalars. If this is not the case, the typical rules are used.
* Specifically, this means: `np.array([1, 2, 3], dtype=np.uint8) + np.float64(12.)` gives a `float64` result, because the `np.float64(12.)` is not considered for being demoted.

2. Promotion is applied as normally, however, instead of the original dtype, the minimal dtype is used. In the case where the minimal data type is unsigned (say uint8) but the value is small enough, the minimal type may in fact be either `uint8` or `int8` (127 can be both). This promotion is also applied in pairs (reduction-like) from left to right.

### General Promotion during Function Execution

General functions (read "ufuncs" such as `np.add`) may have a specific dtype signature which is (for most dtypes) stored e.g. as `np.add.types`. For many of these functions the common type promotion is used unchanged.

However, some functions will employ a slightly different method (which should be equivalent in most cases). They will loop through all loops listed in `np.add.types` in order and find the first one to which all inputs can be safely cast:

np.divide.types = ['ee->e', 'ff->f', 'dd->d', ...]

Thus, `np.divide(np.int16(4), np.float16(3)` will refuse the first `float16, float16 -> float16` (`'ee->e'`) loop because `int16` cannot be cast safely, and then pick the float32 (`'ff->f'`) one.

For simple functions, which commonly have two identical inputs, this should be identical, since normally a clear order exists for the dtypes (it does require checking int8 before uint8, etc.).

#### Scalar based rule

When scalars are involved, the "safe" cast logic based on values is applied *if and only if* rule 1. applies as before: That is there must be an array with a higher or equal category as all of the scalars.

In the above `np.divide` example, this means that `np.divide(np.int16(4), np.array([3], dtype=np.float16))` *will* use the `'ee->e'` loop, because the scalar `4` is of a lower or equal category than the array (integer <= float or complex). While checking, 4 is found to be safely castable to float16, since `(u)int8` is sufficient to hold 4 and that can be safely cast to `float16`.
However, `np.divide(np.int16(4), np.int16(3))` would use `float32` because both are scalars and thus value based logic is not used (Note that in reality numpy forces double output for an all integer input in divide).

In it is possible for ufuncs to have mixed type signatures (this is very rare within numy) and arbitrary inputs. In this case, in principle, the question is whether or not a clear ordering exists and if the rule of using value based logic is always clear. This is rather academical (I could not find any such function in numpy or `scipy.special` [^scipy-ufuncs]). But consider:

imaginary_ufunc.types:
    int32, float32 -> int32, float32
    int64, float32 -> int64, float32
    ...

it is not clear that `np.int64(5) + np.float32(3.)` should be able to demote the `5`. This is very theoretical of course

Footnotes
---------

[^scipy-ufuncs]: See for example these functions:

    import scipy.special
    for n, func in scipy.special.__dict__.items():
        if not isinstance(func, np.ufunc):
            continue

        if func.nin == 1:
            # a single input is not interesting
            continue

        # check if the signature is not uniform
        for types in func.types:
            if len(set(types[:func.nin])) != 1:
                break
        else:
            continue
        print(func, func.types)