Difference between np.int, np.int_, int, and np.int_t in cython?

It's a bit complicated because the names have different meanings depending on the context.

int

1. In Python

   The int is normally just a Python type, it's of arbitrary precision, meaning that you can store any conceivable integer inside it (as long as you have enough memory).

   >>> int(10**50)
   100000000000000000000000000000000000000000000000000
   

2. However, when you use it as dtype for a NumPy array it will be interpreted as np.int_ ¹. Which is not of arbitrary precision, it will have the same size as C's long:

   >>> np.array(10**50, dtype=int)
   OverflowError: Python int too large to convert to C long
   

   That also means the following two are equivalent:

   np.array([1,2,3], dtype=int)
   np.array([1,2,3], dtype=np.int_)
   

3. As Cython type identifier it has another meaning, here it stands for the c type int. It's of limited precision (typically 32bits). You can use it as Cython type, for example when defining variables with cdef:

   cdef int value = 100    # variable
   cdef int[:] arr = ...   # memoryview
   

   As return value or argument value for cdef or cpdef functions:

   cdef int my_function(int argument1, int argument2):
       # ...
   

   As "generic" for ndarray:

   cimport numpy as cnp
   cdef cnp.ndarray[int, ndim=1] val = ...
   

   For type casting:

   avalue = <int>(another_value)
   

   And probably many more.

4. In Cython but as Python type. You can still call int and you'll get a "Python int" (of arbitrary precision), or use it for isinstance or as dtype argument for np.array. Here the context is important, so converting to a Python int is different from converting to a C int:

   cdef object val = int(10)  # Python int
   cdef int val = <int>(10)   # C int
   

np.int

Actually this is very easy. It's just an alias for int:

>>> int is np.int
True

So everything from above applies to np.int as well. However you can't use it as a type-identifier except when you use it on the cimported package. In that case it represents the Python integer type.

cimport numpy as cnp

cpdef func(cnp.int obj):
    return obj

This will expect obj to be a Python integer not a NumPy type:

>>> func(np.int_(10))
TypeError: Argument 'obj' has incorrect type (expected int, got numpy.int32)
>>> func(10)
10

My advise regarding np.int: Avoid it whenever possible. In Python code it's equivalent to int and in Cython code it's also equivalent to Pythons int but if used as type-identifier it will probably confuse you and everyone who reads the code! It certainly confused me...

np.int_

Actually it only has one meaning: It's a Python type that represents a scalar NumPy type. You use it like Pythons int:

>>> np.int_(10)        # looks like a normal Python integer
10
>>> type(np.int_(10))  # but isn't (output may vary depending on your system!)
numpy.int32

Or you use it to specify the dtype, for example with np.array:

>>> np.array([1,2,3], dtype=np.int_)
array([1, 2, 3])

But you cannot use it as type-identifier in Cython.

cnp.int_t

It's the type-identifier version for np.int_. That means you can't use it as dtype argument. But you can use it as type for cdef declarations:

cimport numpy as cnp
import numpy as np

cdef cnp.int_t[:] arr = np.array([1,2,3], dtype=np.int_)
     |---TYPE---|                         |---DTYPE---|

This example (hopefully) shows that the type-identifier with the trailing _t actually represents the type of an array using the dtype without the trailing t. You can't interchange them in Cython code!

Notes

There are several more numeric types in NumPy I'll include a list containing the NumPy dtype and Cython type-identifier and the C type identifier that could also be used in Cython here. But it's basically taken from the NumPy documentation and the Cython NumPy pxd file:

NumPy dtype          Numpy Cython type         C Cython type identifier

np.bool_             None                      None
np.int_              cnp.int_t                 long
np.intc              None                      int       
np.intp              cnp.intp_t                ssize_t
np.int8              cnp.int8_t                signed char
np.int16             cnp.int16_t               signed short
np.int32             cnp.int32_t               signed int
np.int64             cnp.int64_t               signed long long
np.uint8             cnp.uint8_t               unsigned char
np.uint16            cnp.uint16_t              unsigned short
np.uint32            cnp.uint32_t              unsigned int
np.uint64            cnp.uint64_t              unsigned long
np.float_            cnp.float64_t             double
np.float32           cnp.float32_t             float
np.float64           cnp.float64_t             double
np.complex_          cnp.complex128_t          double complex
np.complex64         cnp.complex64_t           float complex
np.complex128        cnp.complex128_t          double complex

Actually there are Cython types for np.bool_: cnp.npy_bool and bint but both they can't be used for NumPy arrays currently. For scalars cnp.npy_bool will just be an unsigned integer while bint will be a boolean. Not sure what's going on there...

¹ Taken From the NumPy documentation "Data type objects"

Built-in Python types

Several python types are equivalent to a corresponding array scalar when used to generate a dtype object:

int           np.int_
bool          np.bool_
float         np.float_
complex       np.cfloat
bytes         np.bytes_
str           np.bytes_ (Python2) or np.unicode_ (Python3)
unicode       np.unicode_
buffer        np.void
(all others)  np.object_

np.int_ is the default integer type (as defined in the NumPy docs), on a 64bit system this would be a C long. np.intc is the default C int either int32 or int64. np.int is an alias to the built-in int function

>>> np.int(2.4)
2
>>> np.int is int  # object id equality
True

The cython datatypes should reflect C datatypes, so cdef int a is a C int and so on.

As for np.int_t that is the Cython compile time equivalent of the NumPy np.int_ datatype, np.int64_t is the Cython compile time equivalent of np.int64