x_mul(): This failed to normalize its result.

tim-one · tim-one · commit 44121a6bc982 · 2002-08-12T06:17:58.000Z
k_mul():  This didn't allocate enough result space when one input had
more than twice as many bits as the other.  This was partly hidden by
that x_mul() didn't normalize its result.

The Karatsuba recurrence is pretty much hosed if the inputs aren't
roughly the same size.  If one has at least twice as many bits as the
other, we get a degenerate case where the "high half" of the smaller
input is 0.  Added a special case for that, for speed, but despite that
it helped, this can still be much slower than the "grade school" method.
It seems to take a really wild imbalance to trigger that; e.g., a
2**22-bit input times a 1000-bit input on my box runs about twice as slow
under k_mul than under x_mul.  This still needs to be addressed.

I'm also not sure that allocating a-&gt;ob_size + b-&gt;ob_size digits is
enough, given that this is computing k = (ah+al)*(bh+bl) instead of
k = (ah-al)*(bl-bh); i.e., it's certainly enough for the final result,
but it's vaguely possible that adding in the "artificially" large k may
overflow that temporarily.  If so, an assert will trigger in the debug
build, but we'll probably compute the right result anyway(!).
diff --git a/Objects/longobject.c b/Objects/longobject.c
@@ -1556,7 +1556,7 @@ x_mul(PyLongObject *a, PyLongObject *b)
 			carry >>= SHIFT;
 		}
 	}
-	return z;
+	return long_normalize(z);
 }
 
 /* A helper for Karatsuba multiplication (k_mul).
@@ -1630,8 +1630,15 @@ k_mul(PyLongObject *a, PyLongObject *b)
 	}
 
 	/* Use gradeschool math when either number is too small. */
-	if (ABS(a->ob_size) <= KARATSUBA_CUTOFF)
-		return x_mul(a, b);
+	if (ABS(a->ob_size) <= KARATSUBA_CUTOFF) {
+		/* 0 is inevitable if one kmul arg has more than twice
+		 * the digits of another, so it's worth special-casing.
+		 */
+		if (a->ob_size == 0)
+			return _PyLong_New(0);
+		else
+			return x_mul(a, b);
+	}
 
 	shift = ABS(b->ob_size) >> 1;
 	if (kmul_split(a, shift, &ah, &al) < 0) goto fail;
@@ -1641,16 +1648,21 @@ k_mul(PyLongObject *a, PyLongObject *b)
 	assert(ahbh->ob_size >= 0);
 
 	/* Allocate result space, and copy ahbh into the high digits. */
-	ret = _PyLong_New(ahbh->ob_size + 2*shift + 1);
+	ret = _PyLong_New(ABS(a->ob_size) + ABS(b->ob_size));
 	if (ret == NULL) goto fail;
 #ifdef Py_DEBUG
 	/* Fill with trash, to catch reference to uninitialized digits. */
 	memset(ret->ob_digit, 0xDF, ret->ob_size * sizeof(digit));
 #endif
+	assert(2*shift + ahbh->ob_size <= ret->ob_size);
 	memcpy(ret->ob_digit + 2*shift, ahbh->ob_digit,
 	       ahbh->ob_size * sizeof(digit));
-	/* That didn't copy into the most-significant (overflow) digit. */
-	ret->ob_digit[ret->ob_size - 1] = 0;
+
+	/* Zero-out the digits higher than the ahbh copy. */
+	i = ret->ob_size - 2*shift - ahbh->ob_size;
+	if (i)
+		memset(ret->ob_digit + 2*shift + ahbh->ob_size, 0,
+		       i * sizeof(digit));
 
 	/* Compute al*bl, and copy into the low digits. */
 	if ((albl = k_mul(al, bl)) == NULL) goto fail;
