Trac #24667: Clean up partitions_c.cc

1. Remove some unused stuff.

2. Compute the `double` and `long double` constants using MPFR. This
makes a lot of sense since we already computed MPFR versions of these
constants anyway. For extra safety, we also increase the precision used
to compute these constants.

3. No longer use the header file `partitions_c.h`. Instead, include the
`.cc` file in Cython.

Points 2. and 3. above allow to remove some platform-specific hacks.

4. Replace `T(int(x))` by `T(x)` to convert `x` to type `T`.

5. Only one temporary MPFR variable is really needed, call it `mptemp`.

6. Merge `initialize_mpz_and_mpq_variables`, `initialize_mpfr_variables`
and `initialize_constants` in one function `initialize_globals`.

7. Use some more specialized MPFR functions like `mpfr_set_prec()` where
applicable.

8. Define all internal functions as `static`.

9. Use `MPFR_RNDF` rounding mode where exact rounding is not required.

10. Ensure that the file compiles without compiler warnings.

11. Various style improvements.

URL: https://trac.sagemath.org/24667
Reported by: jdemeyer
Ticket author(s): Jeroen Demeyer
Reviewer(s): Dima Pasechnik

src/sage/combinat/partitions.pyx

@@ -1,7 +1,6 @@
 # The actual algorithm is implemented in the C++ file partitions_c.cc
 # which requires the gmp and mpfr libraries.
 #
-# distutils: sources = sage/combinat/partitions_c.cc
 # distutils: libraries = gmp mpfr
 # distutils: language = c++
 """
@@ -32,8 +31,8 @@ from cysignals.signals cimport sig_on, sig_off
 
 from sage.libs.gmp.types cimport mpz_t
 
-cdef extern from "partitions_c.h":
-    int part(mpz_t answer, unsigned int n)
+cdef extern from "partitions_c.cc":
+    void part(mpz_t answer, unsigned int n)
     int test(bint longtest, bint forever)
 
 from sage.rings.integer cimport Integer
@@ -45,6 +44,12 @@ def number_of_partitions(n):
     EXAMPLES::
 
         sage: from sage.combinat.partitions import number_of_partitions
+        sage: number_of_partitions(0)
+        1
+        sage: number_of_partitions(1)
+        1
+        sage: number_of_partitions(2)
+        2
         sage: number_of_partitions(3)
         3
         sage: number_of_partitions(10)
@@ -90,14 +95,11 @@ def number_of_partitions(n):
 
         sage: len([n for n in [1..500] if number_of_partitions(n) != Partitions(n).cardinality(algorithm='pari')])
         0
-
     """
     n = Integer(n)
     if n < 0:
         raise ValueError("n (=%s) must be a nonnegative integer"%n)
-    elif n <= 1:
-        return Integer(1)  # part hangs on n=1 as input.
-    if n >= Integer('4294967296'):
+    if n >= Integer(4294967296):
         raise ValueError("input must be a nonnegative integer less than 4294967296.")
     cdef unsigned int nn = n