[Numpy-discussion] (Value Based Promotion) Current Behaviour
Sebastian Berg
sebastian at sipsolutions.net
Wed Jun 12 15:49:55 EDT 2019
On Wed, 2019-06-12 at 12:03 -0500, Sebastian Berg wrote:
> On Tue, 2019-06-11 at 22:08 -0400, Marten van Kerkwijk wrote:
> > HI Sebastian,
> >
> > Thanks for the overview! In the value-based casting, what perhaps
> > surprises me most is that it is done within a kind; it would seem
> > an
> > improvement to check whether a given integer scalar is exactly
> > representable in a given float (your example of 1024 in `float16`).
> > If we switch to the python-only scalar values idea, I would suggest
> > to abandon this. That might make dealing with things like `Decimal`
> > or `Fraction` easier as well.
> >
>
> Yeah, one can argue that since we have this "safe casting" based
> approach, we should go all the way for the value based logic. I think
> I
> tend to agree, but I am not quite sure right now to be honest.
Just realized, one issue with this is that you get much more "special
cases" if you think of it in terms of "minimal dtype". Because
suddenly, not just the unsigned/signed integers such as "< 128" are
special, but even more values require special handling. An int16
"minimal dtype" may or may not be castable to float16.
For `can_cast` that does not matter much, but if we use the same logic
for promotion things may get uglier. Although, maybe it just gets
uglier implementation wise and is fairly logic on the user side...
- Sebastian
>
> Fractions and Decimals are very interesting in that they raise the
> question what happens to user dtypes [0]. Although, you would still
> need a "no lower category" rule, since you do not want 1024. or 12/3
> be
> demoted to an integer.
>
> For me right now, what is most interesting is what we should do with
> ufunc calls, and if we can simplify them. I feel right now we have to
> types of ufuncs:
>
> 1. Ufuncs which use a "common type", where we can find the minimal
> type
> before dispatching.
>
> 2. More complex ufuncs, for which finding the minimal type is
> trickier
> [1]. And while I could not find any weird enough ufunc, I am not sure
> that blind promotion is a good idea for general ufuncs.
>
> Best,
>
> Sebastian
>
>
> [0] A python fraction could be converted to int64/int64 or
> int32/int32,
> etc. depending on the value, in principle. If we want such things to
> work in principle, we need machinery (although I expect one could tag
> that on later).
> [1] It is not impossible, but we need to insert non-existing types
> into
> the type hierarchy.
>
>
>
> PS: Another interesting issue is that if we try to move away from
> value
> based casting for numpy scalars, that initial `np.asarray(...)` call
> may lose the information that a python integer was passed in. So to
> support such things, we might need a whole new machinery.
>
>
>
>
> > All the best,
> >
> > Marten
> >
> > On Tue, Jun 11, 2019 at 8:46 PM Sebastian Berg <
> > sebastian at sipsolutions.net> wrote:
> > > Hi all,
> > >
> > > strange, something went wrong sending that email, but in any
> > > case...
> > >
> > > I tried to "summarize" the current behaviour of promotion and
> > > value
> > > based promotion in numpy (correcting a small error in what I
> > > wrote
> > > earlier). Since it got a bit long, you can find it here (also
> > > copy
> > > pasted at the end):
> > >
> > > https://hackmd.io/NF7Jz3ngRVCIQLU6IZrufA
> > >
> > > Allan's document which I link in there is also very interesting.
> > > One
> > > thing I had not really thought about before was the problem of
> > > commutativity.
> > >
> > > I do not have any specific points I want to discuss based on it
> > > (but
> > > those are likely to come up again later).
> > >
> > > All the Best,
> > >
> > > Sebastian
> > >
> > >
> > > -----------------------------
> > >
> > > PS: Below a copy of what I wrote:
> > >
> > > ---
> > > 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:
> > > ```python
> > > 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)
> > > ```
> > > _______________________________________________
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> > > NumPy-Discussion at python.org
> > > https://mail.python.org/mailman/listinfo/numpy-discussion
> >
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