feat: remove truncations which can be seen to be noops using type inf…

…ormation (#3953)

…

# Description

## Problem\*

Resolves <!-- Link to GitHub Issue -->

## Summary\*

`Truncate`s which occur on outputs of `Binary(BinaryOp::Div)` can
sometimes be determined to be noops based on type information.

Put simply, if we have `(numerator / constant_denom) as T` then if
`type_of(numerator)::max / constant_denom` fits inside `T` then we don't
need to truncate the value.

## Additional Context

Testing this on noir-rsa takes us from

```
+---------+----------------------+--------------+----------------------+
| Package | Expression Width     | ACIR Opcodes | Backend Circuit Size |
+---------+----------------------+--------------+----------------------+
| dkim    | Bounded { width: 3 } | 1649234      | 3049613              |
+---------+----------------------+--------------+----------------------+
```

to

```
+---------+----------------------+--------------+----------------------+
| Package | Expression Width     | ACIR Opcodes | Backend Circuit Size |
+---------+----------------------+--------------+----------------------+
| dkim    | Bounded { width: 3 } | 1486175      | 2764246              |
+---------+----------------------+--------------+----------------------+
```


## Documentation\*

Check one:
- [x] No documentation needed.
- [ ] Documentation included in this PR.
- [ ] **[Exceptional Case]** 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.

compiler/noirc_evaluator/src/ssa/ir/instruction.rs

@@ -533,16 +533,43 @@ impl Instruction {
                     let truncated = numeric_constant.to_u128() % integer_modulus;
                     SimplifiedTo(dfg.make_constant(truncated.into(), typ))
                 } else if let Value::Instruction { instruction, .. } = &dfg[dfg.resolve(*value)] {
-                    if let Instruction::Truncate { bit_size: src_bit_size, .. } = &dfg[*instruction]
-                    {
-                        // If we're truncating the value to fit into the same or larger bit size then this is a noop.
-                        if src_bit_size <= bit_size && src_bit_size <= max_bit_size {
-                            SimplifiedTo(*value)
-                        } else {
-                            None
+                    match &dfg[*instruction] {
+                        Instruction::Truncate { bit_size: src_bit_size, .. } => {
+                            // If we're truncating the value to fit into the same or larger bit size then this is a noop.
+                            if src_bit_size <= bit_size && src_bit_size <= max_bit_size {
+                                SimplifiedTo(*value)
+                            } else {
+                                None
+                            }
                         }
-                    } else {
-                        None
+
+                        Instruction::Binary(Binary {
+                            lhs, rhs, operator: BinaryOp::Div, ..
+                        }) if dfg.is_constant(*rhs) => {
+                            // If we're truncating the result of a division by a constant denominator, we can
+                            // reason about the maximum bit size of the result and whether a truncation is necessary.
+
+                            let numerator_type = dfg.type_of_value(*lhs);
+                            let max_numerator_bits = numerator_type.bit_size();
+
+                            let divisor = dfg
+                                .get_numeric_constant(*rhs)
+                                .expect("rhs is checked to be constant.");
+                            let divisor_bits = divisor.num_bits();
+
+                            // 2^{max_quotient_bits} = 2^{max_numerator_bits} / 2^{divisor_bits}
+                            // => max_quotient_bits = max_numerator_bits - divisor_bits
+                            //
+                            // In order for the truncation to be a noop, we then require `max_quotient_bits < bit_size`.
+                            let max_quotient_bits = max_numerator_bits - divisor_bits;
+                            if max_quotient_bits < *bit_size {
+                                SimplifiedTo(*value)
+                            } else {
+                                None
+                            }
+                        }
+
+                        _ => None,
                     }
                 } else {
                     None

compiler/noirc_evaluator/src/ssa/opt/constant_folding.rs

@@ -184,10 +184,10 @@ mod test {
         function_builder::FunctionBuilder,
         ir::{
             function::RuntimeType,
-            instruction::{BinaryOp, Instruction, TerminatorInstruction},
+            instruction::{Binary, BinaryOp, Instruction, TerminatorInstruction},
             map::Id,
             types::Type,
-            value::Value,
+            value::{Value, ValueId},
         },
     };
 
@@ -247,6 +247,117 @@ mod test {
         }
     }
 
+    #[test]
+    fn redundant_truncation() {
+        // fn main f0 {
+        //   b0(v0: u16, v1: u16):
+        //     v2 = div v0, v1
+        //     v3 = truncate v2 to 8 bits, max_bit_size: 16
+        //     return v3
+        // }
+        //
+        // After constructing this IR, we set the value of v1 to 2^8.
+        // The expected return afterwards should be v2.
+        let main_id = Id::test_new(0);
+
+        // Compiling main
+        let mut builder = FunctionBuilder::new("main".into(), main_id, RuntimeType::Acir);
+        let v0 = builder.add_parameter(Type::unsigned(16));
+        let v1 = builder.add_parameter(Type::unsigned(16));
+
+        // Note that this constant guarantees that `v0/constant < 2^8`. We then do not need to truncate the result.
+        let constant = 2_u128.pow(8);
+        let constant = builder.numeric_constant(constant, Type::field());
+
+        let v2 = builder.insert_binary(v0, BinaryOp::Div, v1);
+        let v3 = builder.insert_truncate(v2, 8, 16);
+        builder.terminate_with_return(vec![v3]);
+
+        let mut ssa = builder.finish();
+        let main = ssa.main_mut();
+        let instructions = main.dfg[main.entry_block()].instructions();
+        assert_eq!(instructions.len(), 2); // The final return is not counted
+
+        // Expected output:
+        //
+        // fn main f0 {
+        //   b0(Field 2: Field):
+        //     return Field 9
+        // }
+        main.dfg.set_value_from_id(v1, constant);
+
+        let ssa = ssa.fold_constants();
+        let main = ssa.main();
+
+        println!("{ssa}");
+
+        let instructions = main.dfg[main.entry_block()].instructions();
+        assert_eq!(instructions.len(), 1);
+        let instruction = &main.dfg[instructions[0]];
+
+        assert_eq!(
+            instruction,
+            &Instruction::Binary(Binary { lhs: v0, operator: BinaryOp::Div, rhs: constant })
+        );
+    }
+
+    #[test]
+    fn non_redundant_truncation() {
+        // fn main f0 {
+        //   b0(v0: u16, v1: u16):
+        //     v2 = div v0, v1
+        //     v3 = truncate v2 to 8 bits, max_bit_size: 16
+        //     return v3
+        // }
+        //
+        // After constructing this IR, we set the value of v1 to 2^8 - 1.
+        // This should not result in the truncation being removed.
+        let main_id = Id::test_new(0);
+
+        // Compiling main
+        let mut builder = FunctionBuilder::new("main".into(), main_id, RuntimeType::Acir);
+        let v0 = builder.add_parameter(Type::unsigned(16));
+        let v1 = builder.add_parameter(Type::unsigned(16));
+
+        // Note that this constant does not guarantee that `v0/constant < 2^8`. We must then truncate the result.
+        let constant = 2_u128.pow(8) - 1;
+        let constant = builder.numeric_constant(constant, Type::field());
+
+        let v2 = builder.insert_binary(v0, BinaryOp::Div, v1);
+        let v3 = builder.insert_truncate(v2, 8, 16);
+        builder.terminate_with_return(vec![v3]);
+
+        let mut ssa = builder.finish();
+        let main = ssa.main_mut();
+        let instructions = main.dfg[main.entry_block()].instructions();
+        assert_eq!(instructions.len(), 2); // The final return is not counted
+
+        // Expected output:
+        //
+        // fn main f0 {
+        //   b0(v0: u16, Field 255: Field):
+        //      v5 = div v0, Field 255
+        //      v6 = truncate v5 to 8 bits, max_bit_size: 16
+        //      return v6
+        // }
+        main.dfg.set_value_from_id(v1, constant);
+
+        let ssa = ssa.fold_constants();
+        let main = ssa.main();
+
+        let instructions = main.dfg[main.entry_block()].instructions();
+        assert_eq!(instructions.len(), 2);
+
+        assert_eq!(
+            &main.dfg[instructions[0]],
+            &Instruction::Binary(Binary { lhs: v0, operator: BinaryOp::Div, rhs: constant })
+        );
+        assert_eq!(
+            &main.dfg[instructions[1]],
+            &Instruction::Truncate { value: ValueId::test_new(5), bit_size: 8, max_bit_size: 16 }
+        );
+    }
+
     #[test]
     fn arrays_elements_are_updated() {
         // fn main f0 {