Trac #32775: Mostly fix slow down by #30022

We avoid multiple imports caused by #30022:

Before (and with #30022):

{{{
sage: a = 1/1000
sage: %timeit a.__pari__()
810 ns ± 2.33 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops
each)
sage: p = a.__pari__()
sage: %timeit QQ(p)
1.52 µs ± 1.37 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops
each)
sage: a = 12314
sage: %timeit a.__pari__()
723 ns ± 0.804 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops
each)
sage: p = a.__pari__()
sage: %timeit ZZ(p)
715 ns ± 1.32 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops
each)
sage: %timeit a.divisors(method='pari')
1.23 µs ± 1.51 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops
each)
sage: %timeit a.is_prime_power()
647 ns ± 0.164 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops
each)
sage: %timeit a.is_prime()
604 ns ± 0.486 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops
each)
sage: %timeit a.factor()
3.6 µs ± 13.4 ns per loop (mean ± std. dev. of 7 runs, 100000 loops
each)
}}}

Reverting #30022:

{{{
sage: a = 1/1000
sage: %timeit a.__pari__()
221 ns ± 0.289 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops
each)
sage: p = a.__pari__()
sage: %timeit QQ(p)
810 ns ± 0.852 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops
each)
sage: a = 12314
sage: %timeit a.__pari__()
176 ns ± 0.217 ns per loop (mean ± std. dev. of 7 runs, 10000000 loops
each)
sage: p = a.__pari__()
sage: %timeit ZZ(p)
150 ns ± 0.104 ns per loop (mean ± std. dev. of 7 runs, 10000000 loops
each)
sage: %timeit a.divisors(method='pari')
643 ns ± 1.38 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops
each)
sage: %timeit a.is_prime_power()
53.9 ns ± 0.336 ns per loop (mean ± std. dev. of 7 runs, 10000000 loops
each)
sage: %timeit a.is_prime()
58 ns ± 0.174 ns per loop (mean ± std. dev. of 7 runs, 10000000 loops
each)
sage: %timeit a.factor()
2.99 µs ± 4.62 ns per loop (mean ± std. dev. of 7 runs, 100000 loops
each)
}}}

With this ticket:

{{{
sage: a = 1/1000
sage: %timeit a.__pari__()
274 ns ± 0.344 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops
each)
sage: p = a.__pari__()
sage: %timeit QQ(p)
867 ns ± 0.999 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops
each)
sage: a = 12314
sage: %timeit a.__pari__()
200 ns ± 0.0656 ns per loop (mean ± std. dev. of 7 runs, 10000000 loops
each)
sage: p = a.__pari__()
sage: %timeit ZZ(p)
183 ns ± 0.179 ns per loop (mean ± std. dev. of 7 runs, 10000000 loops
each)
sage: %timeit a.divisors(method='pari')
673 ns ± 0.928 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops
each)
sage: %timeit a.is_prime_power()
88.4 ns ± 0.075 ns per loop (mean ± std. dev. of 7 runs, 10000000 loops
each)
sage: %timeit a.is_prime()
68.9 ns ± 0.0878 ns per loop (mean ± std. dev. of 7 runs, 10000000 loops
each)
sage: %timeit a.factor()
3.19 µs ± 10.2 ns per loop (mean ± std. dev. of 7 runs, 100000 loops
each)
}}}

URL: https://trac.sagemath.org/32775
Reported by: gh-kliem
Ticket author(s): Jonathan Kliem
Reviewer(s): Matthias Koeppe, Samuel Lelièvre

src/sage/rings/integer.pyx

@@ -198,6 +198,15 @@ except ImportError:
     pari_gen = ()
 
 
+set_integer_from_gen = None
+pari_divisors_small = None
+n_factor_to_list = None
+pari_is_prime_power = None
+pari_is_prime = None
+objtogen = None
+new_gen_from_integer = None
+
+
 cdef extern from *:
     int unlikely(int) nogil  # Defined by Cython
 
@@ -637,7 +646,9 @@ cdef class Integer(sage.structure.element.EuclideanDomainElement):
                     raise TypeError("Cannot convert non-integral float to integer")
 
             elif isinstance(x, pari_gen):
-                from sage.libs.pari.convert_sage import set_integer_from_gen
+                global set_integer_from_gen
+                if set_integer_from_gen is None:
+                    from sage.libs.pari.convert_sage import set_integer_from_gen
                 set_integer_from_gen(self, x)
 
             else:
@@ -3046,12 +3057,14 @@ cdef class Integer(sage.structure.element.EuclideanDomainElement):
             raise ValueError("n must be nonzero")
 
         if (method is None or method == 'pari') and mpz_fits_slong_p(self.value):
-            try:
-                from sage.libs.pari.convert_sage import pari_divisors_small
-            except ImportError:
-                if method == 'pari':
-                    raise ImportError("method `pari` requested, but cypari2 not present")
-            else:
+            global pari_divisors_small
+            if pari_divisors_small is None:
+                try:
+                    from sage.libs.pari.convert_sage import pari_divisors_small
+                except ImportError:
+                    if method == 'pari':
+                        raise ImportError("method `pari` requested, but cypari2 not present")
+            if pari_divisors_small is not None:
                 if mpz_sgn(self.value) > 0:
                     return pari_divisors_small(self)
                 else:
@@ -3909,11 +3922,13 @@ cdef class Integer(sage.structure.element.EuclideanDomainElement):
             return factor_trial_division(self, limit)
 
         if mpz_fits_slong_p(n.value):
-            try:
-                from sage.libs.flint.ulong_extras import n_factor_to_list
-            except ImportError:
-                pass
-            else:
+            global n_factor_to_list
+            if n_factor_to_list is None:
+                try:
+                    from sage.libs.flint.ulong_extras import n_factor_to_list
+                except ImportError:
+                    pass
+            if n_factor_to_list is not None:
                 if proof is None:
                     from sage.structure.proof.proof import get_flag
                     proof = get_flag(proof, "arithmetic")
@@ -5108,11 +5123,13 @@ cdef class Integer(sage.structure.element.EuclideanDomainElement):
             return (self, zero) if get_data else False
 
         if mpz_fits_slong_p(self.value):
-            try:
-                from sage.libs.pari.convert_sage import pari_is_prime_power
-            except ImportError:
-                pass
-            else:
+            global pari_is_prime_power
+            if pari_is_prime_power is None:
+                try:
+                    from sage.libs.pari.convert_sage import pari_is_prime_power
+                except ImportError:
+                    pass
+            if pari_is_prime_power is not None:
                 return pari_is_prime_power(self, get_data)
 
         cdef long n
@@ -5194,11 +5211,13 @@ cdef class Integer(sage.structure.element.EuclideanDomainElement):
             return False
 
         if mpz_fits_ulong_p(self.value):
-            try:
-                from sage.libs.pari.convert_sage import pari_is_prime
-            except ImportError:
-                pass
-            else:
+            global pari_is_prime
+            if pari_is_prime is None:
+                try:
+                    from sage.libs.pari.convert_sage import pari_is_prime
+                except ImportError:
+                    pass
+            if pari_is_prime is not None:
                 return pari_is_prime(self)
 
         if proof is None:
@@ -5550,7 +5569,9 @@ cdef class Integer(sage.structure.element.EuclideanDomainElement):
 
 
         """
-        from cypari2.gen import objtogen
+        global objtogen
+        if objtogen is None:
+            from cypari2.gen import objtogen
         if self.is_square():
             raise ValueError("class_number not defined for square integers")
         if not self%4 in [0,1]:
@@ -5958,7 +5979,9 @@ cdef class Integer(sage.structure.element.EuclideanDomainElement):
             1041334
 
         """
-        from sage.libs.pari.convert_sage import new_gen_from_integer
+        global new_gen_from_integer
+        if new_gen_from_integer is None:
+            from sage.libs.pari.convert_sage import new_gen_from_integer
         return new_gen_from_integer(self)
 
     def _interface_init_(self, I=None):

src/sage/rings/rational.pyx

@@ -110,6 +110,10 @@ except ImportError:
     pari_gen = ()
 
 
+set_rational_from_gen = None
+new_gen_from_rational = None
+
+
 cdef sage.rings.fast_arith.arith_int ai
 ai = sage.rings.fast_arith.arith_int()
 
@@ -650,7 +654,9 @@ cdef class Rational(sage.structure.element.FieldElement):
             mpq_canonicalize(self.value)
 
         elif isinstance(x, pari_gen):
-            from sage.libs.pari.convert_sage import set_rational_from_gen
+            global set_rational_from_gen
+            if set_rational_from_gen is None:
+                from sage.libs.pari.convert_sage import set_rational_from_gen
             set_rational_from_gen(self, x)
 
         elif isinstance(x, list) and len(x) == 1:
@@ -3777,7 +3783,9 @@ cdef class Rational(sage.structure.element.FieldElement):
             sage: m.type()
             't_FRAC'
         """
-        from sage.libs.pari.convert_sage import new_gen_from_rational
+        global new_gen_from_rational
+        if new_gen_from_rational is None:
+            from sage.libs.pari.convert_sage import new_gen_from_rational
         return new_gen_from_rational(self)
 
     def _interface_init_(self, I=None):