Symbolic Toom-Cook multiplication

Implements the algorithm of Section 4 of "Further Steps Down The Wrong
Path : Improving the Bit-Blasting of Multiplication" (see
https://ceur-ws.org/Vol-2908/short16.pdf).

Author: tautschnig

src/solvers/flattening/bv_utils.cpp

@@ -8,6 +8,8 @@ Author: Daniel Kroening, kroening@kroening.com
 
 #include "bv_utils.h"
 
+#include <util/arith_tools.h>
+
 #include <list>
 #include <utility>
 
@@ -783,7 +785,7 @@ bvt bv_utilst::dadda_tree(const std::vector<bvt> &pps)
 // trees (and also the default addition scheme), but isn't consistently more
 // performant with simple partial-product generation. Only when using
 // higher-radix multipliers the combination appears to perform better.
-#define DADDA_TREE
+// #define DADDA_TREE
 
 // The following examples demonstrate the performance differences (with a
 // time-out of 7200 seconds):
@@ -926,7 +928,8 @@ bvt bv_utilst::dadda_tree(const std::vector<bvt> &pps)
 // while not regressing substantially in the matrix of different benchmarks and
 // CaDiCaL and MiniSat2 as solvers.
 // #define RADIX_MULTIPLIER 8
-#define USE_KARATSUBA
+// #define USE_KARATSUBA
+#define USE_TOOM_COOK
 #ifdef RADIX_MULTIPLIER
 #  define DADDA_TREE
 #endif
@@ -1869,6 +1872,155 @@ bvt bv_utilst::unsigned_karatsuba_multiplier(const bvt &_op0, const bvt &_op1)
   return add(z0, z1);
 }
 
+bvt bv_utilst::unsigned_toom_cook_multiplier(const bvt &_op0, const bvt &_op1)
+{
+  PRECONDITION(!_op0.empty());
+  PRECONDITION(!_op1.empty());
+
+  if(_op1.size() == 1)
+    return unsigned_multiplier(_op0, _op1);
+
+  // break up _op0, _op1 in groups of at most GROUP_SIZE bits
+  PRECONDITION(_op0.size() == _op1.size());
+#define GROUP_SIZE 8
+  const std::size_t d_bits =
+    2 * GROUP_SIZE +
+    2 * address_bits((_op0.size() + GROUP_SIZE - 1) / GROUP_SIZE);
+  std::vector<bvt> a, b, c_ops, d;
+  for(std::size_t i = 0; i < _op0.size(); i += GROUP_SIZE)
+  {
+    std::size_t u = std::min(i + GROUP_SIZE, _op0.size());
+    a.emplace_back(_op0.begin() + i, _op0.begin() + u);
+    b.emplace_back(_op1.begin() + i, _op1.begin() + u);
+
+    c_ops.push_back(zeros(i));
+    d.push_back(prop.new_variables(d_bits));
+    c_ops.back().insert(c_ops.back().end(), d.back().begin(), d.back().end());
+    c_ops.back() = zero_extension(c_ops.back(), _op0.size());
+  }
+  for(std::size_t i = a.size(); i < 2 * a.size() - 1; ++i)
+  {
+    d.push_back(prop.new_variables(d_bits));
+  }
+
+  // r(0)
+  bvt r_0 = d[0];
+  prop.l_set_to_true(equal(
+    r_0,
+    unsigned_multiplier(
+      zero_extension(a[0], r_0.size()), zero_extension(b[0], r_0.size()))));
+
+  for(std::size_t j = 1; j < a.size(); ++j)
+  {
+    // r(2^(j-1))
+    bvt r_j = zero_extension(
+      d[0], std::min(_op0.size(), d[0].size() + (j - 1) * (d.size() - 1)));
+    for(std::size_t i = 1; i < d.size(); ++i)
+    {
+      r_j = add(
+        r_j,
+        shift(
+          zero_extension(d[i], r_j.size()), shiftt::SHIFT_LEFT, (j - 1) * i));
+    }
+
+    bvt a_even = zero_extension(a[0], r_j.size());
+    for(std::size_t i = 2; i < a.size(); i += 2)
+    {
+      a_even = add(
+        a_even,
+        shift(
+          zero_extension(a[i], a_even.size()),
+          shiftt::SHIFT_LEFT,
+          (j - 1) * i));
+    }
+    bvt a_odd = zero_extension(a[1], r_j.size());
+    for(std::size_t i = 3; i < a.size(); i += 2)
+    {
+      a_odd = add(
+        a_odd,
+        shift(
+          zero_extension(a[i], a_odd.size()),
+          shiftt::SHIFT_LEFT,
+          (j - 1) * (i - 1)));
+    }
+    bvt b_even = zero_extension(b[0], r_j.size());
+    for(std::size_t i = 2; i < b.size(); i += 2)
+    {
+      b_even = add(
+        b_even,
+        shift(
+          zero_extension(b[i], b_even.size()),
+          shiftt::SHIFT_LEFT,
+          (j - 1) * i));
+    }
+    bvt b_odd = zero_extension(b[1], r_j.size());
+    for(std::size_t i = 3; i < b.size(); i += 2)
+    {
+      b_odd = add(
+        b_odd,
+        shift(
+          zero_extension(b[i], b_odd.size()),
+          shiftt::SHIFT_LEFT,
+          (j - 1) * (i - 1)));
+    }
+
+    prop.l_set_to_true(equal(
+      r_j,
+      unsigned_multiplier(
+        add(a_even, shift(a_odd, shiftt::SHIFT_LEFT, j - 1)),
+        add(b_even, shift(b_odd, shiftt::SHIFT_LEFT, j - 1)))));
+
+    // r(-2^(j-1))
+    bvt r_minus_j = zero_extension(
+      d[0], std::min(_op0.size(), d[0].size() + (j - 1) * (d.size() - 1)));
+    for(std::size_t i = 1; i < d.size(); ++i)
+    {
+      if(i % 2 == 1)
+      {
+        r_minus_j = sub(
+          r_minus_j,
+          shift(
+            zero_extension(d[i], r_minus_j.size()),
+            shiftt::SHIFT_LEFT,
+            (j - 1) * i));
+      }
+      else
+      {
+        r_minus_j = add(
+          r_minus_j,
+          shift(
+            zero_extension(d[i], r_minus_j.size()),
+            shiftt::SHIFT_LEFT,
+            (j - 1) * i));
+      }
+    }
+
+    prop.l_set_to_true(equal(
+      r_minus_j,
+      unsigned_multiplier(
+        sub(a_even, shift(a_odd, shiftt::SHIFT_LEFT, j - 1)),
+        sub(b_even, shift(b_odd, shiftt::SHIFT_LEFT, j - 1)))));
+  }
+
+  if(c_ops.empty())
+    return zeros(_op0.size());
+  else
+  {
+#ifdef WALLACE_TREE
+    return wallace_tree(c_ops);
+#elif defined(DADDA_TREE)
+    return dadda_tree(c_ops);
+#else
+    bvt product = c_ops.front();
+
+    for(auto it = std::next(c_ops.begin()); it != c_ops.end(); ++it)
+      product = add(product, *it);
+
+    return product;
+#endif
+  }
+}
+
 bvt bv_utilst::unsigned_multiplier_no_overflow(
   const bvt &op0,
   const bvt &op1)
@@ -1921,6 +2073,8 @@ bvt bv_utilst::signed_multiplier(const bvt &op0, const bvt &op1)
 
 #ifdef USE_KARATSUBA
   bvt result = unsigned_karatsuba_multiplier(neg0, neg1);
+#elif defined(USE_TOOM_COOK)
+  bvt result = unsigned_toom_cook_multiplier(neg0, neg1);
 #else
   bvt result=unsigned_multiplier(neg0, neg1);
 #endif
@@ -1994,6 +2148,9 @@ bvt bv_utilst::multiplier(
 #ifdef USE_KARATSUBA
   case representationt::UNSIGNED:
     return unsigned_karatsuba_multiplier(op0, op1);
+#elif defined(USE_TOOM_COOK)
+  case representationt::UNSIGNED:
+    return unsigned_toom_cook_multiplier(op0, op1);
 #else
   case representationt::UNSIGNED: return unsigned_multiplier(op0, op1);
 #endif

src/solvers/flattening/bv_utils.h

@@ -80,6 +80,7 @@ class bv_utilst
 
   bvt unsigned_multiplier(const bvt &op0, const bvt &op1);
   bvt unsigned_karatsuba_multiplier(const bvt &op0, const bvt &op1);
+  bvt unsigned_toom_cook_multiplier(const bvt &op0, const bvt &op1);
   bvt signed_multiplier(const bvt &op0, const bvt &op1);
   bvt multiplier(const bvt &op0, const bvt &op1, representationt rep);
   bvt multiplier_no_overflow(