diff --git a/qpms/polynomials.c b/qpms/polynomials.c
index 7e16bda..b3c7eb4 100644
--- a/qpms/polynomials.c
+++ b/qpms/polynomials.c
@@ -59,6 +59,62 @@ void mpzs_hh_set(mpzs_hh_t x, const mpzs_hh_t y) {
   mpz_set(x->_2, y->_2);
 }
 
+/* Append an mpzs element to a mpzs sum that must not contain a matching key
+ * in advance.
+ */
+static inline void mpzs_hash_append(struct _qp_mpzs_hashed **hash, 
+    const struct _qp_mpzs_hashed *newelem) {
+  struct _qp_mpzs_hashed *n;
+  QPMS_CRASHING_MALLOC(n, sizeof(mpzs_hh_t));
+  mpzs_hh_init(n);
+  mpzs_hh_set(n, newelem);
+  HASH_ADD_KEYPTR(hh, *hash, mpz_limbs_read(n->_2),
+      mpz_size(n->_2) * sizeof(mp_limb_t), n);
+}
+
+// Arithmetically add an mpzs element to a mpzs sum/hash.
+static inline void mpzs_hash_addelem(struct _qp_mpzs_hashed **hash,
+    const struct _qp_mpzs_hashed *addend) {
+  struct _qp_mpzs_hashed *s;
+  HASH_FIND(hh, *hash, mpz_limbs_read(addend->_2),
+      mpz_size(addend->_2), s);
+  if (!s) mpzs_hash_append(hash, addend); // if not found
+  else {
+    mpz_add(s->_1, s->_1, addend->_1);
+    if(!mpz_sgn(s->_1)) { // If zero, annihilate
+      HASH_DEL(*hash, s);
+      mpzs_hh_clear(s);
+      free(s);
+    }
+  }
+}
+
+// Multiplies two mpzs hashes, adding them to *prod; 
+// *prod must differ from x and y (this is not checked)
+static inline void  mpzs_hash_mul(struct _qp_mpzs_hashed **prod, 
+    const struct _qp_mpzs_hashed *x, const struct _qp_mpzs_hashed *y) {
+  mpz_t gcd; mpz_init(gcd); // common denominator of the sqrt to factor out
+  mpz_t mx, my; mpz_init(mx); mpz_init(my);
+  mpzs_hh_t addend; mpzs_hh_init(addend);
+
+  for (const struct _qp_mpzs_hashed *nx = x; 
+      nx != NULL; nx = nx->hh.next) {
+    for (const struct _qp_mpzs_hashed *ny = y; 
+        ny != NULL; ny = ny->hh.next) {
+      mpz_gcd(gcd, nx->_2, ny->_2);
+      mpz_divexact(mx, nx->_2, gcd);
+      mpz_divexact(my, ny->_2, gcd);
+      mpz_mul(addend->_2, mx, my);
+      mpz_mul(addend->_1, ny->_1, nx->_1);
+      mpz_mul(addend->_1, addend->_1, gcd);
+      mpzs_hash_addelem(prod, addend);
+    }
+  }
+  mpzs_hh_clear(addend);
+  mpz_clear(mx); mpz_clear(my); mpz_clear(gcd);
+}
+
+
 //===== mpqs_t =====
 
 void mpqs_init(mpqs_t x) {
@@ -84,13 +140,9 @@ void mpqs_clear(mpqs_t x) {
   x->nt = 0;
 }
 
+// Append an element to mpqs_t's numerator hash without any checks.
 void mpqs_nt_append(mpqs_t x, const struct _qp_mpzs_hashed *numelem) {
-  struct _qp_mpzs_hashed *n;
-  QPMS_CRASHING_MALLOC(n, sizeof(mpzs_hh_t));
-  mpzs_hh_init(n);
-  mpzs_hh_set(n, numelem);
-  HASH_ADD_KEYPTR(hh, x->nt, mpz_limbs_read(n->_2),
-      mpz_size(n->_2) * sizeof(mp_limb_t), n);
+  mpzs_hash_append(&(x->nt), numelem);
 }
 
 void mpqs_set_z(mpqs_t x, const mpz_t num1, const mpz_t num2,
@@ -121,19 +173,9 @@ void mpqs_set_si(mpqs_t x, long num1, unsigned long num2,
   mpzs_hh_clear(tmp);
 }
 
-void mpqs_nt_add(mpqs_t x, const struct _qp_mpzs_hashed *addend) {
-  struct _qp_mpzs_hashed *s;
-  HASH_FIND(hh, x->nt, mpz_limbs_read(addend->_2),
-      mpz_size(addend->_2), s);
-  if (!s) mpqs_nt_append(x, addend); // if not found
-  else {
-    mpz_add(s->_1, s->_1, addend->_1);
-    if(!mpz_sgn(s->_1)) { // If zero, annihilate
-      HASH_DEL(x->nt, s);
-      mpzs_hh_clear(s);
-      free(s);
-    }
-  }
+// Arithmetically adds an element to mpqs's numerator hash.
+void mpqs_nt_addelem(mpqs_t x, const struct _qp_mpzs_hashed *addend) {
+  mpzs_hash_addelem(&(x->nt), addend);
 }
 
 void mpqs_init_set(mpqs_t dest, const mpqs_t src) {
@@ -175,6 +217,9 @@ void mpqs_canonicalise(mpqs_t x) {
   mpz_t gcd; mpz_init(gcd);
   mpqs_nt_gcd(gcd, x);
   mpz_mul(mpq_numref(x->f), mpq_numref(x->f), gcd);
+  for(struct _qp_mpzs_hashed *n = (x->nt); n != NULL; n = n->hh.next) {
+    mpz_divexact(n->_1, n->_1, gcd);
+  }
   mpz_clear(gcd);
   mpq_canonicalize(x->f);
 }
@@ -213,7 +258,7 @@ void mpqs_add(mpqs_t sum_final, const mpqs_t x, const mpqs_t y) {
   for(const struct _qp_mpzs_hashed *n = y->nt; n != NULL; n = n->hh.next) {
     mpz_mul(addend->_1, tmp, n->_1);
     mpz_set(addend->_2, n->_2);
-    mpqs_nt_add(sum, addend);
+    mpqs_nt_addelem(sum, addend);
   }
   mpzs_hh_clear(addend);
   mpz_clear(tmp);
@@ -232,31 +277,15 @@ void mpqs_sub(mpqs_t dif, const mpqs_t x, const mpqs_t y) {
   mpqs_clear(tmp);
 }
 
+
 void mpqs_mul(mpqs_t product_final, const mpqs_t x, const mpqs_t y) {
   mpqs_t prod; mpqs_init(prod);
 
   mpz_mul(mpq_numref(prod->f), mpq_numref(x->f), mpq_numref(y->f));
   mpz_mul(mpq_denref(prod->f), mpq_denref(x->f), mpq_denref(y->f));
 
-  mpz_t gcd; mpz_init(gcd); // common denominator of the sqrt to factor out
-  mpz_t mx, my; mpz_init(mx); mpz_init(my);
-  mpzs_hh_t addend; mpzs_hh_init(addend);
+  mpzs_hash_mul(&(prod->nt), x->nt, y->nt);
 
-  for (const struct _qp_mpzs_hashed *nx = x->nt->hh.next; 
-      nx != NULL; nx = nx->hh.next) {
-    for (const struct _qp_mpzs_hashed *ny = x->nt->hh.next; 
-        ny != NULL; ny = ny->hh.next) {
-      mpz_gcd(gcd, nx->_2, ny->_2);
-      mpz_divexact(mx, nx->_2, gcd);
-      mpz_divexact(my, ny->_2, gcd);
-      mpz_mul(addend->_2, mx, my);
-      mpz_mul(addend->_1, ny->_1, nx->_1);
-      mpz_mul(addend->_1, addend->_1, gcd);
-      mpqs_nt_add(prod, addend);
-    }
-  }
-  mpzs_hh_clear(addend);
-  mpz_clear(mx); mpz_clear(my); mpz_clear(gcd);
   mpqs_canonicalise(prod);
   mpqs_set(product_final, prod);
   mpqs_clear(prod);