Incorrect partial fraction expansion with floats about sympy

[oscarbenjamin]

The call to cancel here fails:

sympy/sympy/polys/partfrac.py

Lines 371 to 382 in 7dc74d8

	f, x, U = P/Q, P.gen, []
	u = Function('u')(x)
	a = Dummy('a')
	partial = []
	for d, n in Q.sqf_list_include(all=True):
	b = d.as_expr()
	U += [ u.diff(x, n - 1) ]
	h = cancel(f*b**n) / u**n

It is assumed that since f = P/Q and b**n is a factor of Q then cancel(f*b**n) will cancel the factor of b**n out of the denominator of f. This fails with floats though since b**n might not exactly divide Q.

We know that it is supposed to divide Q so we can just compute the quotient and discard the remainder:

diff --git a/sympy/polys/partfrac.py b/sympy/polys/partfrac.py
index 000c5d7354..b9a5d32609 100644
--- a/sympy/polys/partfrac.py
+++ b/sympy/polys/partfrac.py
@@ -379,7 +379,7 @@ def apart_list_full_decomposition(P, Q, dummygen):
         b = d.as_expr()
         U += [ u.diff(x, n - 1) ]
 
-        h = cancel(f*b**n) / u**n
+        h = cancel(P/Q.quo(d**n)) / u**n
 
         H, subs = [h], []

That gives the correct answer albeit with some nontrivial rounding error:

In [1]: e = (6.43369157032015e-9*x**3 + 1.35203404799555e-5*x**2 + 0.00357538393743079*x + 0.085)/(4.
   ...: 74334912634438e-11*x**4 + 4.09576274286244e-6*x**3 + 0.00334241812250921*x**2 + 0.15406018058
   ...: 983*x + 1.0)

In [2]: apart(nsimplify(e), full=True).evalf()
Out[2]: 
  133.599202650992      1.07757928431867      0.395006955518971      0.564264854137341  
──────────────────── + ─────────────────── + ──────────────────── + ────────────────────
x + 85524.0054884464   x + 774.88576677949   x + 40.7977016133126   x + 7.79746609204661

In [3]: apart(e, full=True).evalf()
Out[3]: 
  133.594113472755      1.07757928431773      0.395006955518863      0.564264854137269  
──────────────────── + ─────────────────── + ──────────────────── + ────────────────────
x + 85524.0054884464   x + 774.88576677949   x + 40.7977016133126   x + 7.79746609204661

[oscarbenjamin]

A little more is needed:

diff --git a/sympy/polys/partfrac.py b/sympy/polys/partfrac.py
index 000c5d7354..1ee1f653e0 100644
--- a/sympy/polys/partfrac.py
+++ b/sympy/polys/partfrac.py
@@ -368,6 +368,8 @@ def apart_list_full_decomposition(P, Q, dummygen):
     .. [1] [Bronstein93]_
 
     """
+    P_orig, Q_orig = P, Q
+
     f, x, U = P/Q, P.gen, []
 
     u = Function('u')(x)
@@ -379,7 +381,7 @@ def apart_list_full_decomposition(P, Q, dummygen):
         b = d.as_expr()
         U += [ u.diff(x, n - 1) ]
 
-        h = cancel(f*b**n) / u**n
+        h = cancel(P_orig/Q_orig.quo(d**n)) / u**n
 
         H, subs = [h], []

[oscarbenjamin]

I've opened gh-26649 with a fix.