(Value Based Promotion) Current Behaviour
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) ```
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. All the best, Marten On Tue, Jun 11, 2019 at 8:46 PM Sebastian Berg <sebastian@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) ``` _______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@python.org https://mail.python.org/mailman/listinfo/numpy-discussion
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. 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@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) ``` _______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@python.org https://mail.python.org/mailman/listinfo/numpy-discussion
_______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@python.org https://mail.python.org/mailman/listinfo/numpy-discussion
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@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) ``` _______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@python.org https://mail.python.org/mailman/listinfo/numpy-discussion
_______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@python.org https://mail.python.org/mailman/listinfo/numpy-discussion
_______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@python.org https://mail.python.org/mailman/listinfo/numpy-discussion
Hi Sebastian, One way to avoid an ugly lookup table and special cases is to store the amount of sign bits, the amount of integer/mantissa bits and the amount of exponent bits for each numeric style. A safe cast can only happen if all three are exceeded or equal. Just a thought. Best Regards, Hameer Abbasi
On Wednesday, Jun 12, 2019 at 9:50 PM, Sebastian Berg <sebastian@sipsolutions.net (mailto:sebastian@sipsolutions.net)> wrote: 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@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) ``` _______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@python.org https://mail.python.org/mailman/listinfo/numpy-discussion
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(this may be a bit thinking out loudly...) On Thu, 2019-06-13 at 07:30 +0200, Hameer Abbasi wrote:
Hi Sebastian,
One way to avoid an ugly lookup table and special cases is to store the amount of sign bits, the amount of integer/mantissa bits and the amount of exponent bits for each numeric style. A safe cast can only happen if all three are exceeded or equal. Just a thought.
True, although I am not sure I like it as a general solution. Within numpy, we do not have much problems in any case, and I do not mind the table as such. What I am worrying more about right now is what happens with user dtypes and the "minimal dtype" logic for them. For example, if a user creates an int24, an python integer could probably not get cast to it automatically (since it is a user defined type). But for the sake of "minimal dtype" should we try to support it? I.e. should: `user_library.int24(1) - 2**20` have machinery to convert 2**20 to an int24 instead of an int32 as "minimal type"? But if a second user dtype does not know about int24 (say rational), it may be an invalid "minimal dtype" for that (at least unless numpy tries to automagically fill holes in the casting table, and that seems like too much magic to me)? (Another example is a masked_int8 which uses -128 to mean NA) Similarly, fixing the hierarchy by representing 0-127 with a uint7 is problematic since current dtypes cannot register with it or have not done so (instead it really means "uint8 or int8"). Of course you could do that for them, but its just another complexity hoop to jump through. A similar thing is Marten's thought about intermediate values of int16 casting safely to float16, requiring ever more fine grained value based logic. Just to be clear, I think we _can_ very much live with any hypothetical inconsistencies when it comes to "minimal dtypes" for the time being; they seem very much irrelevant! My issue is that I would like to figure out a good final solution that covers them (since I doubt we will get rid of them completely). And if that means that we cannot cache them very well (because any value _could_ behave differently) then maybe so be it. Even if that means that `arr + python_int` is slower then `arr + int32(python_int)` then that is maybe fine, right now for all we know the difference, may even be small. For such a scalar object instead what would seem necessary is to call a `dtype.__coerce_pyvalue__(scalar, casting="safe")`, or a `__can_coerce_pyvalue__` method/slot. It would replace the current `PyArray_CanCastArrayTo`, which can only handle the current hardcoded special "minimum value" rules. And of course it also means that resolvers need to handle such scalar(?) objects, but in many cases they do not need more than "can cast" anyway. Best, Sebastian
Best Regards, Hameer Abbasi
On Wednesday, Jun 12, 2019 at 9:50 PM, Sebastian Berg < sebastian@sipsolutions.net> wrote: 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@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) ``` _______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@python.org https://mail.python.org/mailman/listinfo/numpy-discussion
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On 14/6/19 1:37 am, Sebastian Berg wrote:
For such a scalar object instead what would seem necessary is to call a `dtype.__coerce_pyvalue__(scalar, casting="safe")`, or a `__can_coerce_pyvalue__` method/slot. It would replace the current `PyArray_CanCastArrayTo`, which can only handle the current hardcoded special "minimum value" rules.
This makes sense to me since it makes the problem explicit, rather than trying to generalize for some properties. I would suggest changing the first argument from "scalar" to "obj" to indicate it is not necessarily a np.scalar but could be any non-ndarray object, although I would exclude sequences. Matti
participants (4)
-
Hameer Abbasi -
Marten van Kerkwijk -
Matti Picus -
Sebastian Berg