chore: Avoid creating witness for simple multiplications (#5100)

# Description

## Problem\*

Related to  #4629

## Summary\*
Do not create a witness when multiplying a witness (or an affine
transformation of a witness) with an expression of degree 1.


## Additional Context
This PR does NOT fix the issue #4629, but it should help reducing the
overall gates number.
The arithmetic expressions resulting from the multiplication should not
explode in width since I only consider multiplication with a witness (or
'affine witness').

## Documentation\*

Check one:
- [X] No documentation needed.
- [ ] Documentation included in this PR.
- [ ] **[For Experimental Features]** Documentation to be submitted in a
separate PR.

# PR Checklist\*

- [X] I have tested the changes locally.
- [X] I have formatted the changes with [Prettier](https://prettier.io/)
and/or `cargo fmt` on default settings.

acvm-repo/acvm/src/compiler/transformers/csat.rs

@@ -210,16 +210,16 @@ impl CSatTransformer {
                             }
                         } else {
                             // No more usable elements left in the old opcode
-                            opcode.linear_combinations = remaining_linear_terms;
                             break;
                         }
                     }
+                    opcode.linear_combinations.extend(remaining_linear_terms);
+
                     // Constraint this intermediate_opcode to be equal to the temp variable by adding it into the IndexMap
                     // We need a unique name for our intermediate variable
                     // XXX: Another optimization, which could be applied in another algorithm
                     // If two opcodes have a large fan-in/out and they share a few common terms, then we should create intermediate variables for them
                     // Do some sort of subset matching algorithm for this on the terms of the polynomial
-
                     let inter_var = Self::get_or_create_intermediate_vars(
                         intermediate_variables,
                         intermediate_opcode,

compiler/noirc_evaluator/src/ssa/acir_gen/acir_ir/acir_variable.rs

@@ -614,13 +614,44 @@ impl AcirContext {
                 expr.push_multiplication_term(FieldElement::one(), lhs_witness, rhs_witness);
                 self.add_data(AcirVarData::Expr(expr))
             }
-            (
-                AcirVarData::Expr(_) | AcirVarData::Witness(_),
-                AcirVarData::Expr(_) | AcirVarData::Witness(_),
-            ) => {
+            (AcirVarData::Expr(expression), AcirVarData::Witness(witness))
+            | (AcirVarData::Witness(witness), AcirVarData::Expr(expression))
+                if expression.is_linear() =>
+            {
+                let mut expr = Expression::default();
+                for term in expression.linear_combinations.iter() {
+                    expr.push_multiplication_term(term.0, term.1, witness);
+                }
+                expr.push_addition_term(expression.q_c, witness);
+                self.add_data(AcirVarData::Expr(expr))
+            }
+            (AcirVarData::Expr(lhs_expr), AcirVarData::Expr(rhs_expr)) => {
+                let degree_one = if lhs_expr.is_linear() && rhs_expr.is_degree_one_univariate() {
+                    Some((lhs_expr, rhs_expr))
+                } else if rhs_expr.is_linear() && lhs_expr.is_degree_one_univariate() {
+                    Some((rhs_expr, lhs_expr))
+                } else {
+                    None
+                };
+                if let Some((lin, univariate)) = degree_one {
+                    let mut expr = Expression::default();
+                    let rhs_term = univariate.linear_combinations[0];
+                    for term in lin.linear_combinations.iter() {
+                        expr.push_multiplication_term(term.0 * rhs_term.0, term.1, rhs_term.1);
+                    }
+                    expr.push_addition_term(lin.q_c * rhs_term.0, rhs_term.1);
+                    expr.sort();
+                    expr = expr.add_mul(univariate.q_c, &lin);
+                    self.add_data(AcirVarData::Expr(expr))
+                } else {
+                    let lhs = self.get_or_create_witness_var(lhs)?;
+                    let rhs = self.get_or_create_witness_var(rhs)?;
+                    self.mul_var(lhs, rhs)?
+                }
+            }
+            _ => {
                 let lhs = self.get_or_create_witness_var(lhs)?;
                 let rhs = self.get_or_create_witness_var(rhs)?;
-
                 self.mul_var(lhs, rhs)?
             }
         };