@@ -238,9 +238,11 @@ reducedEvals := f.gl.ReduceWithPowers(evals, friAlpha)
 		numerator := f.gl.SubExtensionNoReduce(reducedEvals, reducedOpenings)
 		denominator := f.gl.SubExtension(subgroupX_QE, point)
 		sum = f.gl.MulExtension(f.gl.ExpExtension(friAlpha, uint64(len(evals))), sum)
+		inv, hasInv := f.gl.InverseExtension(denominator)
+		f.api.AssertIsEqual(hasInv, frontend.Variable(1))
 		sum = f.gl.MulAddExtension(
 			numerator,
-			f.gl.InverseExtension(denominator),
+			inv,
 			sum,
 		)
 	}
@@ -272,17 +274,23 @@ lX = f.gl.SubMulExtension(x, xPoints[i], lX)
 	}
 
 	sum := gl.ZeroExtension()
+
+	lookupFromPoints := frontend.Variable(1)
 	for i := 0; i < len(xPoints); i++ {
+		quotient, hasQuotient := f.gl.DivExtension(
+			barycentricWeights[i],
+			f.gl.SubExtension(
+				x,
+				xPoints[i],
+			),
+		)
+
+		lookupFromPoints = f.api.Mul(hasQuotient, lookupFromPoints)
+
 		sum = f.gl.AddExtension(
 			f.gl.MulExtension(
-				f.gl.DivExtension(
-					barycentricWeights[i],
-					f.gl.SubExtension(
-						x,
-						xPoints[i],
-					),
-				),
 				yPoints[i],
+				quotient,
 			),
 			sum,
 		)
@@ -290,17 +298,17 @@ }
 
 	interpolation := f.gl.MulExtension(lX, sum)
 
-	returnField := interpolation
+	lookupVal := gl.ZeroExtension()
 	// Now check if x is already within the xPoints
 	for i := 0; i < len(xPoints); i++ {
-		returnField = f.gl.Lookup(
+		lookupVal = f.gl.Lookup(
 			f.gl.IsZero(f.gl.SubExtension(x, xPoints[i])),
-			returnField,
+			lookupVal,
 			yPoints[i],
 		)
 	}
 
-	return returnField
+	return f.gl.Lookup(lookupFromPoints, lookupVal, interpolation)
 }
 
 func (f *Chip) computeEvaluation(
@@ -367,7 +375,9 @@ }
 		}
 		// Take the inverse of the barycentric weights
 		// OPTIMIZE: Can provide a witness to this value
-		barycentricWeights[i] = f.gl.InverseExtension(barycentricWeights[i])
+		inv, hasInv := f.gl.InverseExtension(barycentricWeights[i])
+		f.api.AssertIsEqual(hasInv, frontend.Variable(1))
+		barycentricWeights[i] = inv
 	}
 
 	return f.interpolate(beta, xPoints, yPoints, barycentricWeights)

@@ -237,18 +237,21 @@ return nil
 }
 
 // Computes the inverse of a field element x such that x * x^-1 = 1.
-func (p *Chip) Inverse(x Variable) Variable {
-	result, err := p.api.Compiler().NewHint(InverseHint, 1, x.Limb)
+func (p *Chip) Inverse(x Variable) (Variable, frontend.Variable) {
+	result, err := p.api.Compiler().NewHint(InverseHint, 2, x.Limb)
 	if err != nil {
 		panic(err)
 	}
 
 	inverse := NewVariable(result[0])
+	hasInv := frontend.Variable(result[1])
 	p.RangeCheck(inverse)
 
 	product := p.Mul(inverse, x)
-	p.api.AssertIsEqual(product.Limb, frontend.Variable(1))
-	return inverse
+	productToCheck := p.api.Select(hasInv, product.Limb, frontend.Variable(1))
+	p.api.AssertIsEqual(productToCheck, frontend.Variable(1))
+
+	return inverse, hasInv
 }
 
 // The hint used to compute Inverse.
@@ -264,10 +267,18 @@ }
 
 	inputGl := goldilocks.NewElement(input.Uint64())
 	resultGl := goldilocks.NewElement(0)
+
+	// Will set resultGL if inputGL == 0
 	resultGl.Inverse(&inputGl)
 
 	result := big.NewInt(0)
 	results[0] = resultGl.BigInt(result)
+
+	hasInvInt64 := int64(0)
+	if !inputGl.IsZero() {
+		hasInvInt64 = 1
+	}
+	results[1] = big.NewInt(hasInvInt64)
 
 	return nil
 }

@@ -126,7 +126,7 @@ return p.ReduceExtension(acc)
 }
 
 // Computes the inverse of a quadratic extension variable in the Goldilocks field.
-func (p *Chip) InverseExtension(a QuadraticExtensionVariable) QuadraticExtensionVariable {
+func (p *Chip) InverseExtension(a QuadraticExtensionVariable) (QuadraticExtensionVariable, frontend.Variable) {
 	a0IsZero := p.api.IsZero(a[0].Limb)
 	a1IsZero := p.api.IsZero(a[1].Limb)
 	p.api.AssertIsEqual(p.api.Mul(a0IsZero, a1IsZero), frontend.Variable(0))
@@ -135,12 +135,15 @@ a[0],
 		p.Mul(a[1], NewVariable(DTH_ROOT)),
 	}
 	aPowR := p.MulExtension(aPowRMinus1, a)
-	return p.ScalarMulExtension(aPowRMinus1, p.Inverse(aPowR[0]))
+
+	aPowRInv, hasInv := p.Inverse(aPowR[0])
+	return p.ScalarMulExtension(aPowRMinus1, aPowRInv), hasInv
 }
 
 // Divides two quadratic extension variables in the Goldilocks field.
-func (p *Chip) DivExtension(a, b QuadraticExtensionVariable) QuadraticExtensionVariable {
-	return p.MulExtension(a, p.InverseExtension(b))
+func (p *Chip) DivExtension(a, b QuadraticExtensionVariable) (QuadraticExtensionVariable, frontend.Variable) {
+	bInv, hasInv := p.InverseExtension(b)
+	return p.MulExtension(a, bInv), hasInv
 }
 
 // Exponentiates a quadratic extension variable to some exponent in the Golidlocks field.

@@ -59,7 +59,7 @@ }
 
 func (c *TestQuadraticExtensionDivCircuit) Define(api frontend.API) error {
 	glAPI := New(api)
-	actualRes := glAPI.DivExtension(c.Operand1, c.Operand2)
+	actualRes, _ := glAPI.DivExtension(c.Operand1, c.Operand2)
 	glAPI.AssertIsEqual(actualRes[0], c.ExpectedResult[0])
 	glAPI.AssertIsEqual(actualRes[1], c.ExpectedResult[1])
 	return nil

@@ -71,10 +71,15 @@ denominator := glApi.SubExtension(
 		glApi.ScalarMulExtension(x, p.DEGREE),
 		p.DEGREE_QE,
 	)
-	return glApi.DivExtension(
+
+	quotient, hasQuotient := glApi.DivExtension(
 		evalZeroPoly,
 		denominator,
 	)
+
+	p.api.AssertIsEqual(hasQuotient, frontend.Variable(1))
+
+	return quotient
 }
 
 func (p *PlonkChip) checkPartialProducts(