ufunc signature for true_divide and sqrt seem a little odd

[tdimitri]

I tried to make a signature of two inputs:int32, int32 returning output; float64 and this failed.
I notice some inputs like int16 return float32.

On sqrt, some inputs, like int8, return float16.
I thought it was common practice for these operations to return float64 unless the input is a float16, float32 or float128.

Second issue: I believe these are examples of internal upcast loops. That is, no need to convert int32 to float64 (in a first pass creating a temp array), then divide. Rather the loop can upcast ints to floats and then divide.

[mattip]

I tried to make a signature of two inputs:int32, int32 returning output; float64 and this failed

I am pretty sure you cannot do this via PyUFunc_ReplaceLoopBySignature . The only way (unfortunately) to add a signature is at UFunc creation time. The way NumPy does this is that a generator function spits out __umath_generated.c, which has static arrays of functions and signatures, here is the part for arccosh:

static PyUFuncGenericFunction arccosh_functions[] = { \
   NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL};

static void * arccosh_data[] = {(void *)NULL, (void *)NULL, \
    (void *)NULL, (void *)NULL, (void *)NULL, (void *)NULL, \
    (void *)NULL, (void *)"arccosh"};

static char arccosh_signatures[] = {NPY_HALF, NPY_HALF, 
    NPY_FLOAT, NPY_FLOAT, NPY_DOUBLE, NPY_DOUBLE, 
    NPY_LONGDOUBLE, NPY_LONGDOUBLE, NPY_CFLOAT, NPY_CFLOAT,
    NPY_CDOUBLE, NPY_CDOUBLE, NPY_CLONGDOUBLE, NPY_CLONGDOUBLE,
    NPY_OBJECT, NPY_OBJECT};

and here is the part of the generator code](https://github.com/numpy/numpy/blob/v1.19.4/numpy/core/code_generators/generate_umath.py#L618) that creates that

'arccosh':
    Ufunc(1, 1, None,
          docstrings.get('numpy.core.umath.arccosh'),
          None,
          TD(inexact, f='acosh', astype={'e':'f'}),
          TD(P, f='arccosh'),
          ),