MAINT: stats: Improve precision of argus.sf. (scipy#21326)

Replace (1-x**2) with (1 - x)*(1 + x) to avoid loss of
precision when x is near 1.

Author: WarrenWeckesser

scipy/stats/_continuous_distns.py

@@ -11467,7 +11467,7 @@ def _cdf(self, x, chi):
         return 1.0 - self._sf(x, chi)
 
     def _sf(self, x, chi):
-        return _argus_phi(chi * np.sqrt(1 - x**2)) / _argus_phi(chi)
+        return _argus_phi(chi * np.sqrt((1 - x)*(1 + x))) / _argus_phi(chi)
 
     def _rvs(self, chi, size=None, random_state=None):
         chi = np.asarray(chi)

scipy/stats/tests/test_distributions.py

@@ -9242,6 +9242,17 @@ def test_sf_small_chi(self, chi, expected):
         x = np.array([0.1, 0.5, 0.9])
         assert_allclose(stats.argus.sf(x, chi), expected, rtol=1e-14)
 
+    # Expected values were computed with mpmath.
+    @pytest.mark.parametrize(
+        'x, chi, expected',
+        [(0.9999999, 0.25, 9.113252974162428e-11),
+         (0.9999999, 3.0, 6.616650419714568e-10),
+         (0.999999999, 2.5, 4.130195911418939e-13),
+         (0.999999999, 10.0, 2.3788319094393724e-11)])
+    def test_sf_near_1(self, x, chi, expected):
+        sf = stats.argus.sf(x, chi)
+        assert_allclose(sf, expected, rtol=5e-15)
+
     # Expected values were computed with mpmath (code: see gh-13370).
     @pytest.mark.parametrize(
         'chi, expected',