diff --git a/ceno_zkvm/src/chip_handler/general.rs b/ceno_zkvm/src/chip_handler/general.rs
index 66e4a41f6..b03418f9f 100644
--- a/ceno_zkvm/src/chip_handler/general.rs
+++ b/ceno_zkvm/src/chip_handler/general.rs
@@ -225,10 +225,7 @@ impl<'a, E: ExtensionField> CircuitBuilder<'a, E> {
         NR: Into<String>,
         N: FnOnce() -> NR,
     {
-        self.namespace(
-            || "require_one",
-            |cb| cb.cs.require_zero(name_fn, Expression::from(1) - expr),
-        )
+        self.namespace(|| "require_one", |cb| cb.cs.require_zero(name_fn, 1 - expr))
     }
 
     pub fn condition_require_equal<NR, N>(
@@ -260,7 +257,7 @@ impl<'a, E: ExtensionField> CircuitBuilder<'a, E> {
         when_true: &Expression<E>,
         when_false: &Expression<E>,
     ) -> Expression<E> {
-        cond.clone() * when_true.clone() + (1 - cond.clone()) * when_false.clone()
+        cond * when_true + (1 - cond) * when_false
     }
 
     pub(crate) fn assert_ux<NR, N, const C: usize>(
@@ -346,10 +343,7 @@ impl<'a, E: ExtensionField> CircuitBuilder<'a, E> {
     {
         self.namespace(
             || "assert_bit",
-            |cb| {
-                cb.cs
-                    .require_zero(name_fn, expr.clone() * (Expression::ONE - expr))
-            },
+            |cb| cb.cs.require_zero(name_fn, &expr * (1 - &expr)),
         )
     }
 
@@ -417,14 +411,10 @@ impl<'a, E: ExtensionField> CircuitBuilder<'a, E> {
         let is_eq = self.create_witin(|| "is_eq");
         let diff_inverse = self.create_witin(|| "diff_inverse");
 
+        self.require_zero(|| "is equal", is_eq.expr() * &lhs - is_eq.expr() * &rhs)?;
         self.require_zero(
             || "is equal",
-            is_eq.expr().clone() * lhs.clone() - is_eq.expr() * rhs.clone(),
-        )?;
-        self.require_zero(
-            || "is equal",
-            Expression::from(1) - is_eq.expr().clone() - diff_inverse.expr() * lhs
-                + diff_inverse.expr() * rhs,
+            1 - is_eq.expr() - diff_inverse.expr() * lhs + diff_inverse.expr() * rhs,
         )?;
 
         Ok((is_eq, diff_inverse))
diff --git a/ceno_zkvm/src/expression.rs b/ceno_zkvm/src/expression.rs
index efbf08502..8e489f781 100644
--- a/ceno_zkvm/src/expression.rs
+++ b/ceno_zkvm/src/expression.rs
@@ -5,7 +5,7 @@ use std::{
     fmt::Display,
     iter::Sum,
     mem::MaybeUninit,
-    ops::{Add, Deref, Mul, Neg, Sub},
+    ops::{Add, AddAssign, Deref, Mul, MulAssign, Neg, Sub, SubAssign},
 };
 
 use ff::Field;
@@ -315,6 +315,24 @@ impl<E: ExtensionField> Add for Expression<E> {
     }
 }
 
+macro_rules! binop_assign_instances {
+    ($op_assign: ident, $fun_assign: ident, $op: ident, $fun: ident) => {
+        impl<E: ExtensionField, Rhs> $op_assign<Rhs> for Expression<E>
+        where
+            Expression<E>: $op<Rhs, Output = Expression<E>>,
+        {
+            fn $fun_assign(&mut self, rhs: Rhs) {
+                // TODO: consider in-place?
+                *self = self.clone().$fun(rhs);
+            }
+        }
+    };
+}
+
+binop_assign_instances!(AddAssign, add_assign, Add, add);
+binop_assign_instances!(SubAssign, sub_assign, Sub, sub);
+binop_assign_instances!(MulAssign, mul_assign, Mul, mul);
+
 impl<E: ExtensionField> Sum for Expression<E> {
     fn sum<I: Iterator<Item = Expression<E>>>(iter: I) -> Expression<E> {
         iter.fold(Expression::ZERO, |acc, x| acc + x)
@@ -442,7 +460,64 @@ impl<E: ExtensionField> Sub for Expression<E> {
     }
 }
 
-macro_rules! binop_instances {
+/// Instances for binary operations that mix Expression and &Expression
+macro_rules! ref_binop_instances {
+    ($op: ident, $fun: ident) => {
+        impl<E: ExtensionField> $op<&Expression<E>> for Expression<E> {
+            type Output = Expression<E>;
+
+            fn $fun(self, rhs: &Expression<E>) -> Expression<E> {
+                self.$fun(rhs.clone())
+            }
+        }
+
+        impl<E: ExtensionField> $op<Expression<E>> for &Expression<E> {
+            type Output = Expression<E>;
+
+            fn $fun(self, rhs: Expression<E>) -> Expression<E> {
+                self.clone().$fun(rhs)
+            }
+        }
+
+        impl<E: ExtensionField> $op<&Expression<E>> for &Expression<E> {
+            type Output = Expression<E>;
+
+            fn $fun(self, rhs: &Expression<E>) -> Expression<E> {
+                self.clone().$fun(rhs.clone())
+            }
+        }
+
+        // for mutable references
+        impl<E: ExtensionField> $op<&mut Expression<E>> for Expression<E> {
+            type Output = Expression<E>;
+
+            fn $fun(self, rhs: &mut Expression<E>) -> Expression<E> {
+                self.$fun(rhs.clone())
+            }
+        }
+
+        impl<E: ExtensionField> $op<Expression<E>> for &mut Expression<E> {
+            type Output = Expression<E>;
+
+            fn $fun(self, rhs: Expression<E>) -> Expression<E> {
+                self.clone().$fun(rhs)
+            }
+        }
+
+        impl<E: ExtensionField> $op<&mut Expression<E>> for &mut Expression<E> {
+            type Output = Expression<E>;
+
+            fn $fun(self, rhs: &mut Expression<E>) -> Expression<E> {
+                self.clone().$fun(rhs.clone())
+            }
+        }
+    };
+}
+ref_binop_instances!(Add, add);
+ref_binop_instances!(Sub, sub);
+ref_binop_instances!(Mul, mul);
+
+macro_rules! mixed_binop_instances {
     ($op: ident, $fun: ident, ($($t:ty),*)) => {
         $(impl<E: ExtensionField> $op<Expression<E>> for $t {
             type Output = Expression<E>;
@@ -458,21 +533,38 @@ macro_rules! binop_instances {
             fn $fun(self, rhs: $t) -> Expression<E> {
                 self.$fun(Expression::<E>::from(rhs))
             }
-        })*
+        }
+
+        impl<E: ExtensionField> $op<&Expression<E>> for $t {
+            type Output = Expression<E>;
+
+            fn $fun(self, rhs: &Expression<E>) -> Expression<E> {
+                Expression::<E>::from(self).$fun(rhs)
+            }
+        }
+
+        impl<E: ExtensionField> $op<$t> for &Expression<E> {
+            type Output = Expression<E>;
+
+            fn $fun(self, rhs: $t) -> Expression<E> {
+                self.$fun(Expression::<E>::from(rhs))
+            }
+        }
+    )*
     };
 }
 
-binop_instances!(
+mixed_binop_instances!(
     Add,
     add,
     (u8, u16, u32, u64, usize, i8, i16, i32, i64, i128, isize)
 );
-binop_instances!(
+mixed_binop_instances!(
     Sub,
     sub,
     (u8, u16, u32, u64, usize, i8, i16, i32, i64, i128, isize)
 );
-binop_instances!(
+mixed_binop_instances!(
     Mul,
     mul,
     (u8, u16, u32, u64, usize, i8, i16, i32, i64, i128, isize)
@@ -686,6 +778,20 @@ impl<F: SmallField, E: ExtensionField<BaseField = F>> ToExpr<E> for F {
     }
 }
 
+macro_rules! impl_from_via_ToExpr {
+    ($($t:ty),*) => {
+        $(
+            impl<E: ExtensionField> From<$t> for Expression<E> {
+                fn from(value: $t) -> Self {
+                    value.expr()
+                }
+            }
+        )*
+    };
+}
+impl_from_via_ToExpr!(WitIn, Fixed, Instance);
+impl_from_via_ToExpr!(&WitIn, &Fixed, &Instance);
+
 // Implement From trait for unsigned types of at most 64 bits
 macro_rules! impl_from_unsigned {
     ($($t:ty),*) => {
@@ -880,8 +986,7 @@ mod tests {
 
         // scaledsum * challenge
         // 3 * x + 2
-        let expr: Expression<E> =
-            Into::<Expression<E>>::into(3usize) * x.expr() + Into::<Expression<E>>::into(2usize);
+        let expr: Expression<E> = 3 * x.expr() + 2;
         // c^3 + 1
         let c = Expression::Challenge(0, 3, 1.into(), 1.into());
         // res
@@ -897,7 +1002,7 @@ mod tests {
 
         // constant * witin
         // 3 * x
-        let expr: Expression<E> = Into::<Expression<E>>::into(3usize) * x.expr();
+        let expr: Expression<E> = 3 * x.expr();
         assert_eq!(
             expr,
             Expression::ScaledSum(
@@ -947,35 +1052,30 @@ mod tests {
         let z = cb.create_witin(|| "z");
         // scaledsum * challenge
         // 3 * x + 2
-        let expr: Expression<E> =
-            Into::<Expression<E>>::into(3usize) * x.expr() + Into::<Expression<E>>::into(2usize);
+        let expr: Expression<E> = 3 * x.expr() + 2;
         assert!(expr.is_monomial_form());
 
         // 2 product term
-        let expr: Expression<E> = Into::<Expression<E>>::into(3usize) * x.expr() * y.expr()
-            + Into::<Expression<E>>::into(2usize) * x.expr();
+        let expr: Expression<E> = 3 * x.expr() * y.expr() + 2 * x.expr();
         assert!(expr.is_monomial_form());
 
         // complex linear operation
         // (2c + 3) * x * y - 6z
         let expr: Expression<E> =
-            Expression::Challenge(0, 1, 2.into(), 3.into()) * x.expr() * y.expr()
-                - Into::<Expression<E>>::into(6usize) * z.expr();
+            Expression::Challenge(0, 1, 2_u64.into(), 3_u64.into()) * x.expr() * y.expr()
+                - 6 * z.expr();
         assert!(expr.is_monomial_form());
 
         // complex linear operation
         // (2c + 3) * x * y - 6z
         let expr: Expression<E> =
-            Expression::Challenge(0, 1, 2.into(), 3.into()) * x.expr() * y.expr()
-                - Into::<Expression<E>>::into(6usize) * z.expr();
+            Expression::Challenge(0, 1, 2_u64.into(), 3_u64.into()) * x.expr() * y.expr()
+                - 6 * z.expr();
         assert!(expr.is_monomial_form());
 
         // complex linear operation
         // (2 * x + 3) * 3 + 6 * 8
-        let expr: Expression<E> = (Into::<Expression<E>>::into(2usize) * x.expr()
-            + Into::<Expression<E>>::into(3usize))
-            * Into::<Expression<E>>::into(3usize)
-            + Into::<Expression<E>>::into(6usize) * Into::<Expression<E>>::into(8usize);
+        let expr: Expression<E> = (2 * x.expr() + 3) * 3 + 6 * 8;
         assert!(expr.is_monomial_form());
     }
 
@@ -988,8 +1088,7 @@ mod tests {
         let y = cb.create_witin(|| "y");
         // scaledsum * challenge
         // (x + 1) * (y + 1)
-        let expr: Expression<E> = (Into::<Expression<E>>::into(1usize) + x.expr())
-            * (Into::<Expression<E>>::into(2usize) + y.expr());
+        let expr: Expression<E> = (1 + x.expr()) * (2 + y.expr());
         assert!(!expr.is_monomial_form());
     }
 
diff --git a/ceno_zkvm/src/expression/monomial.rs b/ceno_zkvm/src/expression/monomial.rs
index 4c73c557b..da16ef753 100644
--- a/ceno_zkvm/src/expression/monomial.rs
+++ b/ceno_zkvm/src/expression/monomial.rs
@@ -39,7 +39,7 @@ impl<E: ExtensionField> Expression<E> {
                 for a in a {
                     for b in &b {
                         res.push(Term {
-                            coeff: a.coeff.clone() * b.coeff.clone(),
+                            coeff: &a.coeff * &b.coeff,
                             vars: a.vars.iter().chain(b.vars.iter()).cloned().collect(),
                         });
                     }
@@ -54,7 +54,7 @@ impl<E: ExtensionField> Expression<E> {
                 for x in x {
                     for a in &a {
                         res.push(Term {
-                            coeff: x.coeff.clone() * a.coeff.clone(),
+                            coeff: &x.coeff * &a.coeff,
                             vars: x.vars.iter().chain(a.vars.iter()).cloned().collect(),
                         });
                     }
diff --git a/ceno_zkvm/src/instructions/riscv/insn_base.rs b/ceno_zkvm/src/instructions/riscv/insn_base.rs
index 64ac91b12..023e3b60e 100644
--- a/ceno_zkvm/src/instructions/riscv/insn_base.rs
+++ b/ceno_zkvm/src/instructions/riscv/insn_base.rs
@@ -381,13 +381,13 @@ impl<E: ExtensionField> MemAddr<E> {
 
     /// Represent the address aligned to 2 bytes.
     pub fn expr_align2(&self) -> AddressExpr<E> {
-        self.addr.address_expr() - self.low_bit_exprs()[0].clone()
+        self.addr.address_expr() - &self.low_bit_exprs()[0]
     }
 
     /// Represent the address aligned to 4 bytes.
     pub fn expr_align4(&self) -> AddressExpr<E> {
         let low_bits = self.low_bit_exprs();
-        self.addr.address_expr() - low_bits[1].clone() * 2 - low_bits[0].clone()
+        self.addr.address_expr() - &low_bits[1] * 2 - &low_bits[0]
     }
 
     /// Expressions of the low bits of the address, LSB-first: [bit_0, bit_1].
@@ -425,7 +425,7 @@ impl<E: ExtensionField> MemAddr<E> {
             .invert()
             .unwrap()
             .expr();
-        let mid_u14 = (limbs[0].clone() - low_sum) * shift_right;
+        let mid_u14 = (&limbs[0] - low_sum) * shift_right;
         cb.assert_ux::<_, _, 14>(|| "mid_u14", mid_u14)?;
 
         // Range check the high limb.
diff --git a/ceno_zkvm/src/instructions/riscv/memory/gadget.rs b/ceno_zkvm/src/instructions/riscv/memory/gadget.rs
index 8c6a36998..64f2a4c40 100644
--- a/ceno_zkvm/src/instructions/riscv/memory/gadget.rs
+++ b/ceno_zkvm/src/instructions/riscv/memory/gadget.rs
@@ -80,7 +80,7 @@ impl<const N_ZEROS: usize> MemWordChange<N_ZEROS> {
                 let u8_base_inv = E::BaseField::from(1 << 8).invert().unwrap();
                 cb.assert_ux::<_, _, 8>(
                     || "rs2_limb[0].le_bytes[1]",
-                    u8_base_inv.expr() * (rs2_limbs[0].clone() - rs2_limb_bytes[0].expr()),
+                    u8_base_inv.expr() * (&rs2_limbs[0] - rs2_limb_bytes[0].expr()),
                 )?;
 
                 // alloc a new witIn to cache degree 2 expression
@@ -125,8 +125,8 @@ impl<const N_ZEROS: usize> MemWordChange<N_ZEROS> {
                     // degree 2 expression
                     low_bits[1].clone(),
                     expected_change.expr(),
-                    (1 << 16) * (rs2_limbs[0].clone() - prev_limbs[1].clone()),
-                    rs2_limbs[0].clone() - prev_limbs[0].clone(),
+                    (1 << 16) * (&rs2_limbs[0] - &prev_limbs[1]),
+                    &rs2_limbs[0] - &prev_limbs[0],
                 )?;
 
                 Ok(MemWordChange {
diff --git a/ceno_zkvm/src/scheme/utils.rs b/ceno_zkvm/src/scheme/utils.rs
index 81ebfcf78..cc3766014 100644
--- a/ceno_zkvm/src/scheme/utils.rs
+++ b/ceno_zkvm/src/scheme/utils.rs
@@ -446,7 +446,7 @@ mod tests {
         let expected_final_product: E = last_layer
             .iter()
             .map(|f| match f.evaluations() {
-                FieldType::Ext(e) => e.iter().cloned().reduce(|a, b| a * b).unwrap(),
+                FieldType::Ext(e) => e.iter().copied().reduce(|a, b| a * b).unwrap(),
                 _ => unreachable!(""),
             })
             .product();
diff --git a/ceno_zkvm/src/uint.rs b/ceno_zkvm/src/uint.rs
index 23b55606c..7173af3bd 100644
--- a/ceno_zkvm/src/uint.rs
+++ b/ceno_zkvm/src/uint.rs
@@ -169,10 +169,7 @@ impl<const M: usize, const C: usize, E: ExtensionField> UIntLimbs<M, C, E> {
                     .assert_ux::<_, _, C>(|| "range check", w.expr())
                     .unwrap();
                 circuit_builder
-                    .require_zero(
-                        || "create_witin_from_expr",
-                        w.expr() - expr_limbs[i].clone(),
-                    )
+                    .require_zero(|| "create_witin_from_expr", w.expr() - &expr_limbs[i])
                     .unwrap();
                 w
             })
@@ -299,7 +296,7 @@ impl<const M: usize, const C: usize, E: ExtensionField> UIntLimbs<M, C, E> {
                 chunk
                     .iter()
                     .zip(shift_pows.iter())
-                    .map(|(limb, shift)| shift.clone() * limb.expr())
+                    .map(|(limb, shift)| shift * limb.expr())
                     .reduce(|a, b| a + b)
                     .unwrap()
             })
@@ -317,7 +314,7 @@ impl<const M: usize, const C: usize, E: ExtensionField> UIntLimbs<M, C, E> {
         let shift_pows = {
             let mut shift_pows = Vec::with_capacity(k);
             shift_pows.push(Expression::Constant(E::BaseField::ONE));
-            (0..k - 1).for_each(|_| shift_pows.push(shift_pows.last().unwrap().clone() * (1 << 8)));
+            (0..k - 1).for_each(|_| shift_pows.push(shift_pows.last().unwrap() * (1 << 8)));
             shift_pows
         };
         let split_limbs = x
@@ -334,7 +331,7 @@ impl<const M: usize, const C: usize, E: ExtensionField> UIntLimbs<M, C, E> {
                 let combined_limb = limbs
                     .iter()
                     .zip(shift_pows.iter())
-                    .map(|(limb, shift)| shift.clone() * limb.clone())
+                    .map(|(limb, shift)| shift * limb)
                     .reduce(|a, b| a + b)
                     .unwrap();
 
@@ -508,11 +505,10 @@ impl<const M: usize, const C: usize, E: ExtensionField> UIntLimbs<M, C, E> {
 
     /// Get an Expression<E> from the limbs, unsafe if Uint value exceeds field limit
     pub fn value(&self) -> Expression<E> {
-        let base = Expression::from(1 << C);
         self.expr()
             .into_iter()
             .rev()
-            .reduce(|sum, limb| sum * base.clone() + limb)
+            .reduce(|sum, limb| sum * (1 << C) + limb)
             .unwrap()
     }
 
@@ -626,7 +622,7 @@ impl<E: ExtensionField> UIntLimbs<32, 8, E> {
         let u16_limbs = u8_limbs
             .chunks(2)
             .map(|chunk| {
-                let (a, b) = (chunk[0].clone(), chunk[1].clone());
+                let (a, b) = (&chunk[0], &chunk[1]);
                 a + b * 256
             })
             .collect_vec();
diff --git a/ceno_zkvm/src/uint/arithmetic.rs b/ceno_zkvm/src/uint/arithmetic.rs
index 01413f6a3..2bba30371 100644
--- a/ceno_zkvm/src/uint/arithmetic.rs
+++ b/ceno_zkvm/src/uint/arithmetic.rs
@@ -179,9 +179,9 @@ impl<const M: usize, const C: usize, E: ExtensionField> UIntLimbs<M, C, E> {
                 let idx = i + j;
                 if idx < c_limbs.len() {
                     if result_c.get(idx).is_none() {
-                        result_c.push(a.clone() * b.clone());
+                        result_c.push(a * b);
                     } else {
-                        result_c[idx] = result_c[idx].clone() + a.clone() * b.clone();
+                        result_c[idx] += a * b;
                     }
                 }
             });
diff --git a/ceno_zkvm/src/virtual_polys.rs b/ceno_zkvm/src/virtual_polys.rs
index 9aaafdf4f..75c930191 100644
--- a/ceno_zkvm/src/virtual_polys.rs
+++ b/ceno_zkvm/src/virtual_polys.rs
@@ -202,8 +202,7 @@ mod tests {
         let mut virtual_polys = VirtualPolynomials::new(1, 0);
 
         // 3xy + 2y
-        let expr: Expression<E> =
-            Expression::from(3) * x.expr() * y.expr() + Expression::from(2) * y.expr();
+        let expr: Expression<E> = 3 * x.expr() * y.expr() + 2 * y.expr();
 
         let distrinct_zerocheck_terms_set = virtual_polys.add_mle_list_by_expr(
             None,
@@ -216,7 +215,7 @@ mod tests {
         assert!(virtual_polys.degree() == 2);
 
         // 3x^3
-        let expr: Expression<E> = Expression::from(3) * x.expr() * x.expr() * x.expr();
+        let expr: Expression<E> = 3 * x.expr() * x.expr() * x.expr();
         let distrinct_zerocheck_terms_set = virtual_polys.add_mle_list_by_expr(
             None,
             wits_in.iter().collect_vec(),