diff --git a/sql/parser/col_types.go b/sql/parser/col_types.go
index 9c9d6fe71d7e..a5ca0d01a214 100644
--- a/sql/parser/col_types.go
+++ b/sql/parser/col_types.go
@@ -19,6 +19,8 @@ package parser
 import (
 	"bytes"
 	"fmt"
+
+	"github.com/pkg/errors"
 )
 
 // ColumnType represents a type in a column definition.
@@ -247,3 +249,30 @@ func (node *TimestampTZColType) String() string { return AsString(node) }
 func (node *IntervalColType) String() string    { return AsString(node) }
 func (node *StringColType) String() string      { return AsString(node) }
 func (node *BytesColType) String() string       { return AsString(node) }
+
+// DatumTypeToColumnType produces a SQL column type equivalent to the
+// given Datum type. Used to generate CastExpr nodes during
+// normalization.
+func DatumTypeToColumnType(d Datum) (ColumnType, error) {
+	switch d.(type) {
+	case *DInt:
+		return intColTypeInt, nil
+	case *DFloat:
+		return floatColTypeFloat, nil
+	case *DDecimal:
+		return decimalColTypeDecimal, nil
+	case *DTimestamp:
+		return timestampColTypeTimestamp, nil
+	case *DTimestampTZ:
+		return timestampTzColTypeTimestampWithTZ, nil
+	case *DInterval:
+		return intervalColTypeInterval, nil
+	case *DDate:
+		return dateColTypeDate, nil
+	case *DString:
+		return stringColTypeString, nil
+	case *DBytes:
+		return bytesColTypeBytes, nil
+	}
+	return nil, errors.Errorf("internal error: unknown Datum type %T", d)
+}
diff --git a/sql/parser/expr.go b/sql/parser/expr.go
index d9641a74d201..cfd8860dda6c 100644
--- a/sql/parser/expr.go
+++ b/sql/parser/expr.go
@@ -460,6 +460,21 @@ type IfExpr struct {
 	typeAnnotation
 }
 
+// TypedTrueExpr returns the IfExpr's True expression as a TypedExpr.
+func (node *IfExpr) TypedTrueExpr() TypedExpr {
+	return node.True.(TypedExpr)
+}
+
+// TypedCondExpr returns the IfExpr's Cond expression as a TypedExpr.
+func (node *IfExpr) TypedCondExpr() TypedExpr {
+	return node.Cond.(TypedExpr)
+}
+
+// TypedElseExpr returns the IfExpr's Else expression as a TypedExpr.
+func (node *IfExpr) TypedElseExpr() TypedExpr {
+	return node.Else.(TypedExpr)
+}
+
 // Format implements the NodeFormatter interface.
 func (node *IfExpr) Format(buf *bytes.Buffer, f FmtFlags) {
 	buf.WriteString("IF(")
@@ -496,6 +511,11 @@ type CoalesceExpr struct {
 	typeAnnotation
 }
 
+// TypedExprAt returns the expression at the specified index as a TypedExpr.
+func (node *CoalesceExpr) TypedExprAt(idx int) TypedExpr {
+	return node.Exprs[idx].(TypedExpr)
+}
+
 // Format implements the NodeFormatter interface.
 func (node *CoalesceExpr) Format(buf *bytes.Buffer, f FmtFlags) {
 	buf.WriteString(node.Name)
@@ -971,6 +991,25 @@ func (node *BinaryExpr) memoizeFn() {
 	node.fn = fn
 }
 
+// newBinExprIfValidOverload constructs a new BinaryExpr if and only
+// if the pair of arguments have a valid implementation for the given
+// BinaryOperator.
+func newBinExprIfValidOverload(op BinaryOperator, left TypedExpr, right TypedExpr) *BinaryExpr {
+	leftRet, rightRet := left.ReturnType(), right.ReturnType()
+	fn, ok := BinOps[op].lookupImpl(leftRet, rightRet)
+	if ok {
+		expr := &BinaryExpr{
+			Operator: op,
+			Left:     left,
+			Right:    right,
+			fn:       fn,
+		}
+		expr.typ = fn.returnType()
+		return expr
+	}
+	return nil
+}
+
 // Format implements the NodeFormatter interface.
 func (node *BinaryExpr) Format(buf *bytes.Buffer, f FmtFlags) {
 	binExprFmtWithParen(buf, f, node.Left, node.Operator.String(), node.Right)
@@ -1017,6 +1056,11 @@ func (node *UnaryExpr) Format(buf *bytes.Buffer, f FmtFlags) {
 	exprFmtWithParen(buf, f, node.Expr)
 }
 
+// TypedInnerExpr returns the UnaryExpr's inner expression as a TypedExpr.
+func (node *UnaryExpr) TypedInnerExpr() TypedExpr {
+	return node.Expr.(TypedExpr)
+}
+
 // FuncExpr represents a function call.
 type FuncExpr struct {
 	Name  *QualifiedName
diff --git a/sql/parser/normalize.go b/sql/parser/normalize.go
index f0393c2c4081..e41ec871065b 100644
--- a/sql/parser/normalize.go
+++ b/sql/parser/normalize.go
@@ -16,27 +16,194 @@
 
 package parser
 
-import "fmt"
+import "github.com/pkg/errors"
 
 type normalizableExpr interface {
 	Expr
 	normalize(*normalizeVisitor) TypedExpr
 }
 
+func (expr *CastExpr) normalize(v *normalizeVisitor) TypedExpr {
+	if expr.Expr == DNull {
+		return DNull
+	}
+	return expr
+}
+
+func (expr *CoalesceExpr) normalize(v *normalizeVisitor) TypedExpr {
+	// This normalization checks whether COALESCE can be simplified
+	// based on constant expressions at the start of the COALESCE
+	// argument list. All known-null constant arguments are simply
+	// removed, and any known-nonnull constant argument before
+	// non-constant argument cause the entire COALESCE expression to
+	// collapse to that argument.
+	last := len(expr.Exprs) - 1
+	for i := range expr.Exprs {
+		subExpr := expr.TypedExprAt(i)
+
+		if i == last {
+			return subExpr
+		}
+
+		if !v.isConst(subExpr) {
+			exprCopy := *expr
+			exprCopy.Exprs = expr.Exprs[i:]
+			return &exprCopy
+		}
+
+		val, err := subExpr.Eval(v.ctx)
+		if err != nil {
+			v.err = err
+			return expr
+		}
+
+		if val != DNull {
+			return subExpr
+		}
+	}
+	return expr
+}
+
+func (expr *IfExpr) normalize(v *normalizeVisitor) TypedExpr {
+	if v.isConst(expr.Cond) {
+		cond, err := expr.TypedCondExpr().Eval(v.ctx)
+		if err != nil {
+			v.err = err
+			return expr
+		}
+		if d, err := GetBool(cond); err == nil {
+			if d {
+				return expr.TypedTrueExpr()
+			}
+			return expr.TypedElseExpr()
+		}
+		return DNull
+	}
+	return expr
+}
+
+func (expr *UnaryExpr) normalize(v *normalizeVisitor) TypedExpr {
+	val := expr.TypedInnerExpr()
+
+	if val == DNull {
+		return val
+	}
+
+	switch expr.Operator {
+	case UnaryPlus:
+		// +a -> a
+		return val
+	case UnaryMinus:
+		// -0 -> 0 (except for float which has negative zero)
+		if !val.ReturnType().TypeEqual(TypeFloat) && IsNumericZero(val) {
+			return val
+		}
+		switch b := val.(type) {
+		// -(a - b) -> (b - a)
+		case *BinaryExpr:
+			if b.Operator == Minus {
+				newBinExpr := newBinExprIfValidOverload(Minus,
+					b.TypedRight(), b.TypedLeft())
+				if newBinExpr != nil {
+					newBinExpr.memoizeFn()
+					b = newBinExpr
+				}
+				return b
+			}
+		// - (- a) -> a
+		case *UnaryExpr:
+			if b.Operator == UnaryMinus {
+				return b.TypedInnerExpr()
+			}
+		}
+	}
+
+	return expr
+}
+
 func (expr *BinaryExpr) normalize(v *normalizeVisitor) TypedExpr {
 	left := expr.TypedLeft()
 	right := expr.TypedRight()
+	expectedType := expr.ReturnType()
 
 	if left == DNull || right == DNull {
 		return DNull
 	}
 
-	return expr
+	var final TypedExpr
+
+	switch expr.Operator {
+	case Plus:
+		if IsNumericZero(right) {
+			final, v.err = ReType(left, expectedType)
+			break
+		}
+		if IsNumericZero(left) {
+			final, v.err = ReType(right, expectedType)
+			break
+		}
+	case Minus:
+		if IsNumericZero(right) {
+			final, v.err = ReType(left, expectedType)
+			break
+		}
+	case Mult:
+		if IsNumericOne(right) {
+			final, v.err = ReType(left, expectedType)
+			break
+		}
+		if IsNumericOne(left) {
+			final, v.err = ReType(right, expectedType)
+			break
+		}
+		if IsNumericZero(left) || IsNumericZero(right) {
+			final, v.err = SameTypeZero(expectedType)
+			break
+		}
+	case Div, FloorDiv:
+		if IsNumericOne(right) {
+			final, v.err = ReType(left, expectedType)
+			break
+		}
+		if IsNumericZero(left) {
+			cbz, err := CanBeZeroDivider(right)
+			if err != nil {
+				v.err = err
+				break
+			}
+			if !cbz {
+				final, v.err = ReType(left, expectedType)
+				break
+			}
+		}
+	case Mod:
+		if IsNumericOne(right) {
+			final, v.err = SameTypeZero(expectedType)
+			break
+		}
+		if IsNumericZero(left) {
+			cbz, err := CanBeZeroDivider(right)
+			if err != nil {
+				v.err = err
+				break
+			}
+			if !cbz {
+				final, v.err = ReType(left, expectedType)
+				break
+			}
+		}
+	}
+
+	if final == nil {
+		return expr
+	}
+	return final
 }
 
 func (expr *AndExpr) normalize(v *normalizeVisitor) TypedExpr {
 	left := expr.TypedLeft()
 	right := expr.TypedRight()
+	var dleft, dright Datum
 
 	if left == DNull && right == DNull {
 		return DNull
@@ -44,31 +211,31 @@ func (expr *AndExpr) normalize(v *normalizeVisitor) TypedExpr {
 
 	// Use short-circuit evaluation to simplify AND expressions.
 	if v.isConst(left) {
-		left, v.err = left.Eval(v.ctx)
+		dleft, v.err = left.Eval(v.ctx)
 		if v.err != nil {
 			return expr
 		}
-		if left != DNull {
-			if d, err := GetBool(left.(Datum)); err == nil {
+		if dleft != DNull {
+			if d, err := GetBool(dleft); err == nil {
 				if !d {
-					return left
+					return dleft
 				}
 				return right
 			}
 			return DNull
 		}
 		return NewTypedAndExpr(
-			left,
+			dleft,
 			right,
 		)
 	}
 	if v.isConst(right) {
-		right, v.err = right.Eval(v.ctx)
+		dright, v.err = right.Eval(v.ctx)
 		if v.err != nil {
 			return expr
 		}
-		if right != DNull {
-			if d, err := GetBool(expr.Right.(Datum)); err == nil {
+		if dright != DNull {
+			if d, err := GetBool(dright); err == nil {
 				if !d {
 					return right
 				}
@@ -78,7 +245,7 @@ func (expr *AndExpr) normalize(v *normalizeVisitor) TypedExpr {
 		}
 		return NewTypedAndExpr(
 			left,
-			right,
+			dright,
 		)
 	}
 	return expr
@@ -87,10 +254,6 @@ func (expr *AndExpr) normalize(v *normalizeVisitor) TypedExpr {
 func (expr *ComparisonExpr) normalize(v *normalizeVisitor) TypedExpr {
 	switch expr.Operator {
 	case EQ, GE, GT, LE, LT:
-		if expr.TypedLeft() == DNull || expr.TypedRight() == DNull {
-			return DNull
-		}
-
 		// We want var nodes (VariableExpr, QualifiedName, etc) to be immediate
 		// children of the comparison expression and not second or third
 		// children. That is, we want trees that look like:
@@ -114,6 +277,10 @@ func (expr *ComparisonExpr) normalize(v *normalizeVisitor) TypedExpr {
 		// tree or we would not have entered this code path.
 		exprCopied := false
 		for {
+			if expr.TypedLeft() == DNull || expr.TypedRight() == DNull {
+				return DNull
+			}
+
 			if v.isConst(expr.Left) {
 				switch expr.Right.(type) {
 				case *BinaryExpr, VariableExpr:
@@ -121,6 +288,13 @@ func (expr *ComparisonExpr) normalize(v *normalizeVisitor) TypedExpr {
 				default:
 					return expr
 				}
+
+				invertedOp, err := invertComparisonOp(expr.Operator)
+				if err != nil {
+					v.err = nil
+					return expr
+				}
+
 				// The left side is const and the right side is a binary expression or a
 				// variable. Flip the comparison op so that the right side is const and
 				// the left side is a binary expression or variable.
@@ -130,11 +304,8 @@ func (expr *ComparisonExpr) normalize(v *normalizeVisitor) TypedExpr {
 					expr = &exprCopy
 					exprCopied = true
 				}
-				expr = NewTypedComparisonExpr(
-					invertComparisonOp(expr.Operator),
-					expr.TypedRight(),
-					expr.TypedLeft(),
-				)
+
+				expr = NewTypedComparisonExpr(invertedOp, expr.TypedRight(), expr.TypedLeft())
 			} else if !v.isConst(expr.Right) {
 				return expr
 			}
@@ -156,6 +327,22 @@ func (expr *ComparisonExpr) normalize(v *normalizeVisitor) TypedExpr {
 				//    [+-/]   2  ->  a   [-+*]
 				//   /     \            /     \
 				//  a       1          2       1
+				var op BinaryOperator
+				switch left.Operator {
+				case Plus:
+					op = Minus
+				case Minus:
+					op = Plus
+				case Div:
+					op = Mult
+				}
+
+				newBinExpr := newBinExprIfValidOverload(op,
+					expr.TypedRight(), left.TypedRight())
+				if newBinExpr == nil {
+					// Substitution is not possible type-wise. Nothing else to do.
+					break
+				}
 
 				if !exprCopied {
 					exprCopy := *expr
@@ -163,26 +350,12 @@ func (expr *ComparisonExpr) normalize(v *normalizeVisitor) TypedExpr {
 					exprCopied = true
 				}
 
-				leftCopy := *left
-				left = &leftCopy
-
-				switch left.Operator {
-				case Plus:
-					left.Operator = Minus
-				case Minus:
-					left.Operator = Plus
-				case Div:
-					left.Operator = Mult
-				}
-
 				expr.Left = left.Left
-				left.Left = expr.Right
-
-				left.memoizeFn()
-				expr.Right, v.err = left.Eval(v.ctx)
+				expr.Right, v.err = newBinExpr.Eval(v.ctx)
 				if v.err != nil {
-					return nil
+					return expr
 				}
+
 				expr.memoizeFn()
 				if !isVar(expr.Left) {
 					// Continue as long as the left side of the comparison is not a
@@ -197,30 +370,45 @@ func (expr *ComparisonExpr) normalize(v *normalizeVisitor) TypedExpr {
 				//   /    \           /    \
 				//  1      a         1      2
 
+				op := expr.Operator
+				var newBinExpr *BinaryExpr
+
+				switch left.Operator {
+				case Plus:
+					//
+					// (A + X) cmp B => X cmp (B - C)
+					//
+					newBinExpr = newBinExprIfValidOverload(Minus,
+						expr.TypedRight(), left.TypedLeft())
+				case Minus:
+					//
+					// (A - X) cmp B => X cmp' (A - B)
+					//
+					newBinExpr = newBinExprIfValidOverload(Minus,
+						left.TypedLeft(), expr.TypedRight())
+					op, v.err = invertComparisonOp(op)
+					if v.err != nil {
+						return expr
+					}
+				}
+
+				if newBinExpr == nil {
+					break
+				}
+
 				if !exprCopied {
 					exprCopy := *expr
 					expr = &exprCopy
 					exprCopied = true
 				}
 
-				leftCopy := *left
-				left = &leftCopy
-
-				// Clear the function caches; we're about to change stuff.
-				left.Right, expr.Right = expr.Right, left.Right
-				if left.Operator == Plus {
-					left.Operator = Minus
-					left.Left, left.Right = left.Right, left.Left
-				} else {
-					expr.Operator = invertComparisonOp(expr.Operator)
-				}
-
-				left.memoizeFn()
-				expr.Left, v.err = left.Eval(v.ctx)
+				expr.Operator = op
+				expr.Left = left.Right
+				expr.Right, v.err = newBinExpr.Eval(v.ctx)
 				if v.err != nil {
 					return nil
 				}
-				expr.Left, expr.Right = expr.Right, expr.Left
+
 				expr.memoizeFn()
 				if !isVar(expr.Left) {
 					// Continue as long as the left side of the comparison is not a
@@ -259,6 +447,7 @@ func (expr *ComparisonExpr) normalize(v *normalizeVisitor) TypedExpr {
 func (expr *OrExpr) normalize(v *normalizeVisitor) TypedExpr {
 	left := expr.TypedLeft()
 	right := expr.TypedRight()
+	var dleft, dright Datum
 
 	if left == DNull && right == DNull {
 		return DNull
@@ -266,31 +455,31 @@ func (expr *OrExpr) normalize(v *normalizeVisitor) TypedExpr {
 
 	// Use short-circuit evaluation to simplify OR expressions.
 	if v.isConst(left) {
-		left, v.err = left.Eval(v.ctx)
+		dleft, v.err = left.Eval(v.ctx)
 		if v.err != nil {
 			return expr
 		}
-		if left != DNull {
-			if d, err := GetBool(left.(Datum)); err == nil {
+		if dleft != DNull {
+			if d, err := GetBool(dleft); err == nil {
 				if d {
-					return left
+					return dleft
 				}
 				return right
 			}
 			return DNull
 		}
 		return NewTypedOrExpr(
-			left,
+			dleft,
 			right,
 		)
 	}
 	if v.isConst(right) {
-		right, v.err = right.Eval(v.ctx)
+		dright, v.err = right.Eval(v.ctx)
 		if v.err != nil {
 			return expr
 		}
-		if right != DNull {
-			if d, err := GetBool(right.(Datum)); err == nil {
+		if dright != DNull {
+			if d, err := GetBool(dright); err == nil {
 				if d {
 					return right
 				}
@@ -300,14 +489,21 @@ func (expr *OrExpr) normalize(v *normalizeVisitor) TypedExpr {
 		}
 		return NewTypedOrExpr(
 			left,
-			right,
+			dright,
 		)
 	}
 	return expr
 }
 
 func (expr *ParenExpr) normalize(v *normalizeVisitor) TypedExpr {
-	return expr.TypedInnerExpr()
+	newExpr := expr.TypedInnerExpr()
+	if normalizeable, ok := newExpr.(normalizableExpr); ok {
+		newExpr = normalizeable.normalize(v)
+		if v.err != nil {
+			return expr
+		}
+	}
+	return newExpr
 }
 
 func (expr *AnnotateTypeExpr) normalize(v *normalizeVisitor) TypedExpr {
@@ -324,17 +520,27 @@ func (expr *RangeCond) normalize(v *normalizeVisitor) TypedExpr {
 
 	if expr.Not {
 		// "a NOT BETWEEN b AND c" -> "a < b OR a > c"
-		return NewTypedOrExpr(
-			NewTypedComparisonExpr(LT, left, from),
-			NewTypedComparisonExpr(GT, left, to),
-		)
+		newLeft := NewTypedComparisonExpr(LT, left, from).normalize(v)
+		if v.err != nil {
+			return expr
+		}
+		newRight := NewTypedComparisonExpr(GT, left, to).normalize(v)
+		if v.err != nil {
+			return expr
+		}
+		return NewTypedOrExpr(newLeft, newRight).normalize(v)
 	}
 
 	// "a BETWEEN b AND c" -> "a >= b AND a <= c"
-	return NewTypedAndExpr(
-		NewTypedComparisonExpr(GE, left, from),
-		NewTypedComparisonExpr(LE, left, to),
-	)
+	newLeft := NewTypedComparisonExpr(GE, left, from).normalize(v)
+	if v.err != nil {
+		return expr
+	}
+	newRight := NewTypedComparisonExpr(LE, left, to).normalize(v)
+	if v.err != nil {
+		return expr
+	}
+	return NewTypedAndExpr(newLeft, newRight).normalize(v)
 }
 
 // NormalizeExpr normalizes a typed expression, simplifying where possible,
@@ -370,25 +576,7 @@ func (v *normalizeVisitor) VisitPre(expr Expr) (recurse bool, newExpr Expr) {
 		return false, expr
 	}
 
-	// Normalize expressions that know how to normalize themselves.
-	if normalizeable, ok := expr.(normalizableExpr); ok {
-		expr = normalizeable.normalize(v)
-		if v.err != nil {
-			return false, expr
-		}
-	}
-
 	switch expr.(type) {
-	case *CaseExpr, *IfExpr, *NullIfExpr, *CoalesceExpr:
-		// Conditional expressions need to be evaluated during the downward
-		// traversal in order to avoid evaluating sub-expressions which should
-		// not be evaluated due to the case/conditional.
-		if v.isConst(expr) {
-			expr, v.err = expr.(TypedExpr).Eval(v.ctx)
-			if v.err != nil {
-				return false, expr
-			}
-		}
 	case *Subquery:
 		// Avoid normalizing subqueries. We need the subquery to be expanded in
 		// order to do so properly.
@@ -415,7 +603,11 @@ func (v *normalizeVisitor) VisitPost(expr Expr) Expr {
 
 	// Evaluate all constant expressions.
 	if v.isConst(expr) {
-		expr, v.err = expr.(TypedExpr).Eval(v.ctx)
+		newExpr, err := expr.(TypedExpr).Eval(v.ctx)
+		if err != nil {
+			return expr
+		}
+		expr = newExpr
 	}
 	return expr
 }
@@ -424,20 +616,20 @@ func (v *normalizeVisitor) isConst(expr Expr) bool {
 	return v.isConstVisitor.run(expr)
 }
 
-func invertComparisonOp(op ComparisonOperator) ComparisonOperator {
+func invertComparisonOp(op ComparisonOperator) (ComparisonOperator, error) {
 	switch op {
 	case EQ:
-		return EQ
+		return EQ, nil
 	case GE:
-		return LE
+		return LE, nil
 	case GT:
-		return LT
+		return LT, nil
 	case LE:
-		return GE
+		return GE, nil
 	case LT:
-		return GT
+		return GT, nil
 	default:
-		panic(fmt.Sprintf("unable to invert: %s", op))
+		return op, errors.Errorf("internal error: unable to invert: %s", op)
 	}
 }
 
@@ -502,3 +694,98 @@ func ContainsVars(expr Expr) bool {
 	WalkExprConst(&v, expr)
 	return v.containsVars
 }
+
+// DecimalZero represents the constant 0 as DECIMAL.
+var DecimalZero DDecimal
+
+// DecimalOne represents the constant 1 as DECIMAL.
+var DecimalOne DDecimal
+
+func init() {
+	DecimalOne.Dec.SetUnscaled(1).SetScale(0)
+	DecimalZero.Dec.SetUnscaled(0).SetScale(0)
+}
+
+// IsNumericZero returns true if the datum is a number and equal to
+// zero.
+func IsNumericZero(expr TypedExpr) bool {
+	if d, ok := expr.(Datum); ok {
+		switch t := d.(type) {
+		case *DDecimal:
+			return t.Dec.Cmp(&DecimalZero.Dec) == 0
+		case *DFloat:
+			return *t == 0
+		case *DInt:
+			return *t == 0
+		}
+	}
+	return false
+}
+
+// CanBeZeroDivider returns true if the expr may be a number and equal
+// to zero. It also returns true if it is not known yet whether the
+// expr is a number (e.g. it is a more complex sub-expression that has
+// resisted normalization).
+func CanBeZeroDivider(expr TypedExpr) (bool, error) {
+	if d, ok := expr.(Datum); ok {
+		switch t := d.(type) {
+		case *DDecimal:
+			return t.Dec.Cmp(&DecimalZero.Dec) == 0, nil
+		case *DFloat:
+			return *t == 0, nil
+		case *DInt:
+			return *t == 0, nil
+		case dividerDatum:
+			return true, errors.Errorf("internal error: unknown dividerDatum in CanBeZeroDivider(): %T", d)
+		}
+		// All other Datums are non-numeric and thus cannot be zero.
+		return false, nil
+	}
+	// Other type of expression; may evaluate to zero.
+	return true, nil
+}
+
+// IsNumericOne returns true if the datum is a number and equal to
+// one.
+func IsNumericOne(expr TypedExpr) bool {
+	if d, ok := expr.(Datum); ok {
+		switch t := d.(type) {
+		case *DDecimal:
+			return t.Dec.Cmp(&DecimalOne.Dec) == 0
+		case *DFloat:
+			return *t == 1.0
+		case *DInt:
+			return *t == 1
+		}
+	}
+	return false
+}
+
+// SameTypeZero returns a datum of equivalent type with value zero.
+// The argument must be a datum of a numeric type.
+func SameTypeZero(d Datum) (TypedExpr, error) {
+	switch d.(type) {
+	case *DDecimal:
+		return &DecimalZero, nil
+	case *DFloat:
+		return NewDFloat(0.0), nil
+	case *DInt:
+		return NewDInt(0), nil
+	}
+	return nil, errors.Errorf("internal error: zero not defined for Datum type %T", d)
+}
+
+// ReType ensures that the given numeric expression evaluates
+// to the requested type, inserting a cast if necessary.
+func ReType(expr TypedExpr, wantedType Datum) (TypedExpr, error) {
+	if expr.ReturnType().TypeEqual(wantedType) {
+		return expr, nil
+	}
+	reqType, err := DatumTypeToColumnType(wantedType)
+	if err != nil {
+		return nil, err
+	}
+	res := &CastExpr{Expr: expr, Type: reqType}
+	res.typ = wantedType
+	return res, nil
+}
diff --git a/sql/parser/normalize_test.go b/sql/parser/normalize_test.go
index 433f50175df0..53de6be4fc97 100644
--- a/sql/parser/normalize_test.go
+++ b/sql/parser/normalize_test.go
@@ -32,6 +32,34 @@ func TestNormalizeExpr(t *testing.T) {
 	}{
 		{`(a)`, `a`},
 		{`((((a))))`, `a`},
+		{`CAST(NULL AS INTEGER)`, `NULL`},
+		{`+a`, `a`},
+		{`-(-a)`, `a`},
+		{`-+-a`, `a`},
+		{`-(a-b)`, `b - a`},
+		{`-0`, `0`},
+		{`-NULL`, `NULL`},
+		{`-1`, `-1`},
+		{`a+0`, `a`},
+		{`0+a`, `a`},
+		{`a+(2-2)`, `a`},
+		{`a-0`, `a`},
+		{`a*1`, `a`},
+		{`1*a`, `a`},
+		{`0*a`, `0`},
+		{`a+NULL`, `NULL`},
+		{`a/1`, `CAST(a AS DECIMAL)`},
+		{`0/a`, `0 / a`},
+		{`0/1`, `0`},
+		{`0%a`, `0 % a`},
+		{`0%1`, `0`},
+		{`1%1`, `0`},
+		{`0%2`, `0`},
+		{`1%2`, `1`},
+		{`1%-2`, `1`},
+		{`12 BETWEEN 24 AND 36`, `false`},
+		{`12 BETWEEN 10 AND 20`, `true`},
+		{`10 BETWEEN a AND 20`, `a <= 10`},
 		{`a BETWEEN b AND c`, `(a >= b) AND (a <= c)`},
 		{`a NOT BETWEEN b AND c`, `(a < b) OR (a > c)`},
 		{`a BETWEEN NULL AND c`, `NULL`},
@@ -96,6 +124,10 @@ func TestNormalizeExpr(t *testing.T) {
 		{`ANNOTATE_TYPE(1, float)`, `1.0`},
 		// TODO(nvanbenschoten) introduce a shorthand type annotation notation.
 		// {`1!float`, `1.0`},
+		{`IF((true AND a < 0), (0 + a)::decimal, 2 / (1 - 1))`, `IF(a < 0, CAST(a AS DECIMAL), 2 / 0)`},
+		{`IF((true OR a < 0), (0 + a)::decimal, 2 / (1 - 1))`, `CAST(a AS DECIMAL)`},
+		{`COALESCE(NULL, (NULL < 3), a = 2 - 1, d)`, `COALESCE(a = 1, d)`},
+		{`COALESCE(NULL, a)`, `a`},
 	}
 	for _, d := range testData {
 		expr, err := ParseExprTraditional(d.expr)
diff --git a/sql/parser/type_check.go b/sql/parser/type_check.go
index 159713c21508..4c5525510d6d 100644
--- a/sql/parser/type_check.go
+++ b/sql/parser/type_check.go
@@ -49,6 +49,23 @@ var (
 	TypeTuple Datum = &DTuple{}
 )
 
+// dividerDatum is used during normalization to determine which Datums
+// are possible valid second operands to an arithmetic divide. See
+// CanBeZeroDivider() for an example.
+// Ensure this is implemented appropriately if/when new numeric types
+// are added.
+type dividerDatum interface {
+	dividerDatum()
+}
+
+var _ dividerDatum = NewDInt(0)
+var _ dividerDatum = NewDFloat(0)
+var _ dividerDatum = &DDecimal{}
+
+func (*DInt) dividerDatum()     {}
+func (*DFloat) dividerDatum()   {}
+func (*DDecimal) dividerDatum() {}
+
 // SemaContext defines the context in which to perform semantic analysis on an
 // expression syntax tree.
 type SemaContext struct {