k_mul() and long_mul(): I'm confident that the Karatsuba algorithm is

correct now, so added some final comments, did some cleanup, and enabled
it for all long-int multiplies.  The KARAT envar no longer matters,
although I left some #if 0'ed code in there for my own use (temporary).
k_mul() is still much slower than x_mul() if the inputs have very
differenent sizes, and that still needs to be addressed.

Misc/NEWS

@@ -57,9 +57,16 @@ Type/class unification and new-style classes
 
 Core and builtins
 
-- XXX Karatsuba multiplication.  This is currently used if and only
+- When multiplying very large integers, a version of the so-called
-  if envar KARAT exists.  It needs more correctness and speed testing,
+  Karatsuba algorithm is now used.  This is most effective if the
-  the latter especially with unbalanced bit lengths.
+  inputs have roughly the same size.  If they both have about N digits,
+  Karatsuba multiplication has O(N**1.58) runtime (the exponent is
+  log_base_2(3)) instead of the previous O(N**2).  Measured results may
+  be better or worse than that, depending on platform quirks.  Note that
+  this is a simple implementation, and there's no intent here to compete
+  with, e.g., gmp.  It simply gives a very nice speedup when it applies.
+  XXX Karatsuba multiplication can be slower when the inputs have very
+  XXX different sizes.
 
 - u'%c' will now raise a ValueError in case the argument is an
   integer outside the valid range of Unicode code point ordinals.

Objects/longobject.c

@@ -1645,7 +1645,23 @@ k_mul(PyLongObject *a, PyLongObject *b)
 	if (kmul_split(a, shift, &ah, &al) < 0) goto fail;
 	if (kmul_split(b, shift, &bh, &bl) < 0) goto fail;
 
-	/* Allocate result space. */
+	/* The plan:
+	 * 1. Allocate result space (asize + bsize digits:  that's always
+	 *    enough).
+	 * 2. Compute ah*bh, and copy into result at 2*shift.
+	 * 3. Compute al*bl, and copy into result at 0.  Note that this
+	 *    can't overlap with #2.
+	 * 4. Subtract al*bl from the result, starting at shift.  This may
+	 *    underflow (borrow out of the high digit), but we don't care:
+	 *    we're effectively doing unsigned arithmetic mod
+	 *    BASE**(sizea + sizeb), and so long as the *final* result fits,
+	 *    borrows and carries out of the high digit can be ignored.
+	 * 5. Subtract ah*bh from the result, starting at shift.
+	 * 6. Compute (ah+al)*(bh+bl), and add it into the result starting
+	 *    at shift.
+	 */
+
+	/* 1. Allocate result space. */
 	ret = _PyLong_New(asize + bsize);
 	if (ret == NULL) goto fail;
 #ifdef Py_DEBUG
@@ -1653,7 +1669,7 @@ k_mul(PyLongObject *a, PyLongObject *b)
 	memset(ret->ob_digit, 0xDF, ret->ob_size * sizeof(digit));
 #endif
 
-	/* t1 <- ah*bh, and copy into high digits of result. */
+	/* 2. t1 <- ah*bh, and copy into high digits of result. */
 	if ((t1 = k_mul(ah, bh)) == NULL) goto fail;
 	assert(t1->ob_size >= 0);
 	assert(2*shift + t1->ob_size <= ret->ob_size);
@@ -1666,7 +1682,7 @@ k_mul(PyLongObject *a, PyLongObject *b)
 		memset(ret->ob_digit + 2*shift + t1->ob_size, 0,
 		       i * sizeof(digit));
 
-	/* t2 <- al*bl, and copy into the low digits. */
+	/* 3. t2 <- al*bl, and copy into the low digits. */
 	if ((t2 = k_mul(al, bl)) == NULL) {
 		Py_DECREF(t1);
 		goto fail;
@@ -1680,15 +1696,17 @@ k_mul(PyLongObject *a, PyLongObject *b)
 	if (i)
 		memset(ret->ob_digit + t2->ob_size, 0, i * sizeof(digit));
 
-	/* Subtract ah*bh (t1) and al*bl (t2) from "the middle" digits. */
+	/* 4 & 5. Subtract ah*bh (t1) and al*bl (t2).  We do al*bl first
+	 * because it's fresher in cache.
+	 */
 	i = ret->ob_size - shift;  /* # digits after shift */
-	v_isub(ret->ob_digit + shift, i, t2->ob_digit, t2->ob_size);
+	(void)v_isub(ret->ob_digit + shift, i, t2->ob_digit, t2->ob_size);
 	Py_DECREF(t2);
 
-	v_isub(ret->ob_digit + shift, i, t1->ob_digit, t1->ob_size);
+	(void)v_isub(ret->ob_digit + shift, i, t1->ob_digit, t1->ob_size);
 	Py_DECREF(t1);
 
-	/* t3 <- (ah+al)(bh+bl) */
+	/* 6. t3 <- (ah+al)(bh+bl), and add into result. */
 	if ((t1 = x_add(ah, al)) == NULL) goto fail;
 	Py_DECREF(ah);
 	Py_DECREF(al);
@@ -1709,8 +1727,7 @@ k_mul(PyLongObject *a, PyLongObject *b)
 	if (t3 == NULL) goto fail;
 
 	/* Add t3. */
-	v_iadd(ret->ob_digit + shift, ret->ob_size - shift,
+	(void)v_iadd(ret->ob_digit + shift, i, t3->ob_digit, t3->ob_size);
-	       t3->ob_digit, t3->ob_size);
 	Py_DECREF(t3);
 
 	return long_normalize(ret);
@@ -1743,10 +1760,14 @@ long_mul(PyLongObject *v, PyLongObject *w)
 		return Py_NotImplemented;
 	}
 
+#if 0
 	if (Py_GETENV("KARAT") != NULL)
 		z = k_mul(a, b);
 	else
 		z = x_mul(a, b);
+#else
+	z = k_mul(a, b);
+#endif
 	if(z == NULL) {
 		Py_DECREF(a);
 		Py_DECREF(b);