Fix multiplication and division of complex numbers

Multiplication and division of complex numbers are not just pointwise
applications of those operations.

Fixes: #8375

Author: tautschnig

regression/cbmc/complex2/main.c

@@ -0,0 +1,15 @@
+#include <complex.h>
+
+int main()
+{
+  char choice;
+  float re = choice ? 1.3f : 2.1f; // a non-constant well-behaved float
+  _Complex float z1 = I + re;
+  _Complex float z2 = z1 * z1;
+  _Complex float expected = 2 * I * re + re * re - 1; // (a+i)^2 = 2ai + a^2 - 1
+  _Complex float actual =
+    re * re + I; // (a1 + b1*i)*(a2 + b2*i) = (a1*a2 + b1*b2*i)
+  __CPROVER_assert(z2 == expected, "right");
+  __CPROVER_assert(z2 == actual, "wrong");
+  return 0;
+}

regression/cbmc/complex2/test.desc

@@ -0,0 +1,13 @@
+CORE no-new-smt gcc-only
+main.c
+
+^\[main.assertion.1\] line 12 right: SUCCESS$
+^\[main.assertion.2\] line 13 wrong: FAILURE$
+^VERIFICATION FAILED$
+^EXIT=10$
+^SIGNAL=0$
+--
+^warning: ignoring
+--
+Visual Studio does not directly support `complex` or `_Complex`, or using
+standard arithmetic operators over complex numbers.

src/goto-programs/remove_complex.cpp

@@ -127,13 +127,10 @@ static void remove_complex(exprt &expr)
 
   if(expr.type().id()==ID_complex)
   {
-    if(expr.id()==ID_plus || expr.id()==ID_minus ||
-       expr.id()==ID_mult || expr.id()==ID_div)
+    if(expr.id() == ID_plus || expr.id() == ID_minus)
     {
-      // FIXME plus and mult are defined as n-ary operations
-      //      rather than binary. This code assumes that they
-      //      can only have exactly 2 operands, and it is not clear
-      //      that it is safe to do so in this context
+      // plus and mult are n-ary expressions, but front-ends currently ensure
+      // that we only see them as binary ones
       PRECONDITION(expr.operands().size() == 2);
       // do component-wise:
       // x+y -> complex(x.r+y.r,x.i+y.i)
@@ -153,6 +150,53 @@ static void remove_complex(exprt &expr)
 
       expr=struct_expr;
     }
+    else if(expr.id() == ID_mult)
+    {
+      // plus and mult are n-ary expressions, but front-ends currently ensure
+      // that we only see them as binary ones
+      PRECONDITION(expr.operands().size() == 2);
+      exprt lhs_real = complex_member(to_binary_expr(expr).op0(), ID_real);
+      exprt lhs_imag = complex_member(to_binary_expr(expr).op0(), ID_imag);
+      exprt rhs_real = complex_member(to_binary_expr(expr).op1(), ID_real);
+      exprt rhs_imag = complex_member(to_binary_expr(expr).op1(), ID_imag);
+
+      struct_exprt struct_expr{
+        {minus_exprt{
+           mult_exprt{lhs_real, rhs_real}, mult_exprt{lhs_imag, rhs_imag}},
+         plus_exprt{
+           mult_exprt{lhs_imag, rhs_real}, mult_exprt{lhs_real, rhs_imag}}},
+        expr.type()};
+
+      struct_expr.op0().add_source_location() = expr.source_location();
+      struct_expr.op1().add_source_location() = expr.source_location();
+
+      expr = struct_expr;
+    }
+    else if(expr.id() == ID_div)
+    {
+      exprt lhs_real = complex_member(to_binary_expr(expr).op0(), ID_real);
+      exprt lhs_imag = complex_member(to_binary_expr(expr).op0(), ID_imag);
+      exprt rhs_real = complex_member(to_binary_expr(expr).op1(), ID_real);
+      exprt rhs_imag = complex_member(to_binary_expr(expr).op1(), ID_imag);
+
+      plus_exprt numerator_real{
+        mult_exprt{lhs_real, rhs_real}, mult_exprt{lhs_imag, rhs_imag}};
+      minus_exprt numerator_imag{
+        mult_exprt{lhs_imag, rhs_real}, mult_exprt{lhs_real, rhs_imag}};
+
+      plus_exprt denominator{
+        mult_exprt{rhs_real, rhs_real}, mult_exprt{rhs_imag, rhs_imag}};
+
+      struct_exprt struct_expr{
+        {div_exprt{numerator_real, denominator},
+         div_exprt{numerator_imag, denominator}},
+        expr.type()};
+
+      struct_expr.op0().add_source_location() = expr.source_location();
+      struct_expr.op1().add_source_location() = expr.source_location();
+
+      expr = struct_expr;
+    }
     else if(expr.id()==ID_unary_minus)
     {
       auto const &unary_minus_expr = to_unary_minus_expr(expr);

src/solvers/flattening/boolbv_floatbv_op.cpp

@@ -6,16 +6,14 @@ Author: Daniel Kroening, [email protected]
 
 \*******************************************************************/
 
-#include "boolbv.h"
-
-#include <algorithm>
-
 #include <util/bitvector_types.h>
 #include <util/c_types.h>
 #include <util/floatbv_expr.h>
 
 #include <solvers/floatbv/float_utils.h>
 
+#include "boolbv.h"
+
 bvt boolbvt::convert_floatbv_typecast(const floatbv_typecast_exprt &expr)
 {
   const exprt &op0=expr.op(); // number to convert
@@ -131,44 +129,66 @@ bvt boolbvt::convert_floatbv_op(const ieee_float_op_exprt &expr)
         sub_width > 0 && width % sub_width == 0,
         "width of a complex subtype must be positive and evenly divide the "
         "width of the complex expression");
+      DATA_INVARIANT(
+        sub_width * 2 == width, "a complex type consists of exactly two parts");
 
-      std::size_t size=width/sub_width;
-      bvt result_bv;
-      result_bv.resize(width);
+      bvt lhs_real{lhs_as_bv.begin(), lhs_as_bv.begin() + sub_width};
+      bvt rhs_real{rhs_as_bv.begin(), rhs_as_bv.begin() + sub_width};
 
-      for(std::size_t i=0; i<size; i++)
+      bvt lhs_imag{lhs_as_bv.begin() + sub_width, lhs_as_bv.end()};
+      bvt rhs_imag{rhs_as_bv.begin() + sub_width, rhs_as_bv.end()};
+
+      bvt result_real, result_imag;
+
+      if(expr.id() == ID_floatbv_plus || expr.id() == ID_floatbv_minus)
+      {
+        result_real = float_utils.add_sub(
+          lhs_real, rhs_real, expr.id() == ID_floatbv_minus);
+        result_imag = float_utils.add_sub(
+          lhs_imag, rhs_imag, expr.id() == ID_floatbv_minus);
+      }
+      else if(expr.id() == ID_floatbv_mult)
+      {
+        // Could be optimised to just three multiplications with more additions
+        // instead, but then we'd have to worry about the impact of possible
+        // overflows. So we use the naive approach for now:
+        result_real = float_utils.add_sub(
+          float_utils.mul(lhs_real, rhs_real),
+          float_utils.mul(lhs_imag, rhs_imag),
+          true);
+        result_imag = float_utils.add_sub(
+          float_utils.mul(lhs_real, rhs_imag),
+          float_utils.mul(lhs_imag, rhs_real),
+          false);
+      }
+      else if(expr.id() == ID_floatbv_div)
       {
-        bvt lhs_sub_bv, rhs_sub_bv, sub_result_bv;
-
-        lhs_sub_bv.assign(
-          lhs_as_bv.begin() + i * sub_width,
-          lhs_as_bv.begin() + (i + 1) * sub_width);
-        rhs_sub_bv.assign(
-          rhs_as_bv.begin() + i * sub_width,
-          rhs_as_bv.begin() + (i + 1) * sub_width);
-
-        if(expr.id()==ID_floatbv_plus)
-          sub_result_bv = float_utils.add_sub(lhs_sub_bv, rhs_sub_bv, false);
-        else if(expr.id()==ID_floatbv_minus)
-          sub_result_bv = float_utils.add_sub(lhs_sub_bv, rhs_sub_bv, true);
-        else if(expr.id()==ID_floatbv_mult)
-          sub_result_bv = float_utils.mul(lhs_sub_bv, rhs_sub_bv);
-        else if(expr.id()==ID_floatbv_div)
-          sub_result_bv = float_utils.div(lhs_sub_bv, rhs_sub_bv);
-        else
-          UNREACHABLE;
-
-        INVARIANT(
-          sub_result_bv.size() == sub_width,
-          "we constructed a new complex of the right size");
-        INVARIANT(
-          i * sub_width + sub_width - 1 < result_bv.size(),
-          "the sub-bitvector fits into the result bitvector");
-        std::copy(
-          sub_result_bv.begin(),
-          sub_result_bv.end(),
-          result_bv.begin() + i * sub_width);
+        bvt numerator_real = float_utils.add_sub(
+          float_utils.mul(lhs_real, rhs_real),
+          float_utils.mul(lhs_imag, rhs_imag),
+          false);
+        bvt numerator_imag = float_utils.add_sub(
+          float_utils.mul(lhs_imag, rhs_real),
+          float_utils.mul(lhs_real, rhs_imag),
+          true);
+
+        bvt denominator = float_utils.add_sub(
+          float_utils.mul(rhs_real, rhs_real),
+          float_utils.mul(rhs_imag, rhs_imag),
+          false);
+
+        result_real = float_utils.div(numerator_real, denominator);
+        result_imag = float_utils.div(numerator_imag, denominator);
       }
+      else
+        UNREACHABLE;
+
+      bvt result_bv = std::move(result_real);
+      result_bv.reserve(width);
+      result_bv.insert(
+        result_bv.end(),
+        std::make_move_iterator(result_imag.begin()),
+        std::make_move_iterator(result_imag.end()));
 
       return result_bv;
     }