k_mul: Rearranged computation for better cache use. Ignored overflow

(it's possible, but should be harmless -- this requires more thought,
and allocating enough space in advance to prevent it requires exactly
as much thought, to know exactly how much that is -- the end result
certainly fits in the allocated space -- hmm, but that's really all
the thought it needs!  borrows/carries out of the high digits really
are harmless).

Objects/longobject.c

@@ -1598,20 +1598,17 @@ kmul_split(PyLongObject *n, int size, PyLongObject **high, PyLongObject **low)
 static PyLongObject *
 k_mul(PyLongObject *a, PyLongObject *b)
 {
+	int asize = ABS(a->ob_size);
+	int bsize = ABS(b->ob_size);
 	PyLongObject *ah = NULL;
 	PyLongObject *al = NULL;
 	PyLongObject *bh = NULL;
 	PyLongObject *bl = NULL;
-	PyLongObject *albl = NULL;
-	PyLongObject *ahbh = NULL;
-	PyLongObject *k = NULL;
 	PyLongObject *ret = NULL;
-	PyLongObject *t1, *t2;
+	PyLongObject *t1, *t2, *t3;
 	int shift;	/* the number of digits we split off */
 	int i;
-#ifdef Py_DEBUG
-	digit d;
-#endif
+
 	/* (ah*X+al)(bh*X+bl) = ah*bh*X*X + (ah*bl + al*bh)*X + al*bl
 	 * Let k = (ah+al)*(bh+bl) = ah*bl + al*bh  + ah*bh + al*bl
 	 * Then the original product is
@@ -1623,59 +1620,75 @@ k_mul(PyLongObject *a, PyLongObject *b)
 	/* We want to split based on the larger number; fiddle so that b
 	 * is largest.
 	 */
-	if (ABS(a->ob_size) > ABS(b->ob_size)) {
+	if (asize > bsize) {
 		t1 = a;
 		a = b;
 		b = t1;
+
+		i = asize;
+		asize = bsize;
+		bsize = i;
 	}
 
 	/* Use gradeschool math when either number is too small. */
-	if (ABS(a->ob_size) <= KARATSUBA_CUTOFF) {
+	if (asize <= 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)
+		if (asize == 0)
 			return _PyLong_New(0);
 		else
 			return x_mul(a, b);
 	}
 
-	shift = ABS(b->ob_size) >> 1;
+	shift = bsize >> 1;
 	if (kmul_split(a, shift, &ah, &al) < 0) goto fail;
 	if (kmul_split(b, shift, &bh, &bl) < 0) goto fail;
 
-	if ((ahbh = k_mul(ah, bh)) == NULL) goto fail;
-	assert(ahbh->ob_size >= 0);
-
-	/* Allocate result space, and copy ahbh into the high digits. */
-	ret = _PyLong_New(ABS(a->ob_size) + ABS(b->ob_size));
+	/* Allocate result space. */
+	ret = _PyLong_New(asize + bsize);
 	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));
 
-	/* Zero-out the digits higher than the ahbh copy. */
-	i = ret->ob_size - 2*shift - ahbh->ob_size;
+	/* 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);
+	memcpy(ret->ob_digit + 2*shift, t1->ob_digit,
+	       t1->ob_size * sizeof(digit));
+
+	/* Zero-out the digits higher than the ah*bh copy. */
+	i = ret->ob_size - 2*shift - t1->ob_size;
 	if (i)
-		memset(ret->ob_digit + 2*shift + ahbh->ob_size, 0,
+		memset(ret->ob_digit + 2*shift + t1->ob_size, 0,
 		       i * sizeof(digit));
 
-	/* Compute al*bl, and copy into the low digits. */
-	if ((albl = k_mul(al, bl)) == NULL) goto fail;
-	assert(albl->ob_size >= 0);
-	assert(albl->ob_size <= 2*shift); /* no overlap with high digits */
-	memcpy(ret->ob_digit, albl->ob_digit, albl->ob_size * sizeof(digit));
+	/* t2 <- al*bl, and copy into the low digits. */
+	if ((t2 = k_mul(al, bl)) == NULL) {
+		Py_DECREF(t1);
+		goto fail;
+	}
+	assert(t2->ob_size >= 0);
+	assert(t2->ob_size <= 2*shift); /* no overlap with high digits */
+	memcpy(ret->ob_digit, t2->ob_digit, t2->ob_size * sizeof(digit));
 
 	/* Zero out remaining digits. */
-	i = 2*shift - albl->ob_size;	/* number of uninitialized digits */
+	i = 2*shift - t2->ob_size;	/* number of uninitialized digits */
 	if (i)
-		memset(ret->ob_digit + albl->ob_size, 0, i * sizeof(digit));
+		memset(ret->ob_digit + t2->ob_size, 0, i * sizeof(digit));
+
+	/* Subtract ah*bh (t1) and al*bl (t2) from "the middle" digits. */
+	i = ret->ob_size - shift;  /* # digits after shift */
+	v_isub(ret->ob_digit + shift, i, t2->ob_digit, t2->ob_size);
+	Py_DECREF(t2);
 
-	/* k = (ah+al)(bh+bl) */
+	v_isub(ret->ob_digit + shift, i, t1->ob_digit, t1->ob_size);
+	Py_DECREF(t1);
+
+	/* t3 <- (ah+al)(bh+bl) */
 	if ((t1 = x_add(ah, al)) == NULL) goto fail;
 	Py_DECREF(ah);
 	Py_DECREF(al);
@@ -1689,36 +1702,16 @@ k_mul(PyLongObject *a, PyLongObject *b)
 	Py_DECREF(bl);
 	bh = bl = NULL;
 
-	k = k_mul(t1, t2);
+	t3 = k_mul(t1, t2);
+	assert(t3->ob_size >= 0);
 	Py_DECREF(t1);
 	Py_DECREF(t2);
-	if (k == NULL) goto fail;
-
-	/* Add k into the result, starting at the shift'th LSD. */
-	i = ret->ob_size - shift;  /* # digits after shift */
-#ifdef Py_DEBUG
-	d =
-#endif
-	v_iadd(ret->ob_digit + shift, i, k->ob_digit, k->ob_size);
-	assert(d == 0);
-	Py_DECREF(k);
+	if (t3 == NULL) goto fail;
 
-	/* Subtract ahbh and albl from the result.  Note that this can't
-	 * become negative, since k = ahbh + albl + other stuff.
-	 */
-#ifdef Py_DEBUG
-	d =
-#endif
-	v_isub(ret->ob_digit + shift, i, ahbh->ob_digit, ahbh->ob_size);
-	assert(d == 0);
-	Py_DECREF(ahbh);
-
-#ifdef Py_DEBUG
-	d =
-#endif
-	v_isub(ret->ob_digit + shift, i, albl->ob_digit, albl->ob_size);
-	assert(d == 0);
-	Py_DECREF(albl);
+	/* Add t3. */
+	v_iadd(ret->ob_digit + shift, ret->ob_size - shift,
+	       t3->ob_digit, t3->ob_size);
+	Py_DECREF(t3);
 
 	return long_normalize(ret);
 
@@ -1728,9 +1721,6 @@ k_mul(PyLongObject *a, PyLongObject *b)
 	Py_XDECREF(al);
 	Py_XDECREF(bh);
 	Py_XDECREF(bl);
-	Py_XDECREF(ahbh);
-	Py_XDECREF(albl);
-	Py_XDECREF(k);
 	return NULL;
 }