Partial specialization specifically. Match some patterns and covert it to something else. For example:
struct F { double x; };
enum Op { Add, Mul };
auto eval(F x) { return x.x; }
template<class L, class R, Op op> struct Expr;
template<class L, class R> struct Expr<L,R,Add>{ L l; R r;
friend auto eval(Expr self) { return eval(self.l) + eval(self.r); } };
template<class L, class R> struct Expr<L,R,Mul>{ L l; R r;
friend auto eval(Expr self) { return eval(self.l) * eval(self.r); } };
template<class L, class R, class R2> struct Expr<Expr<L, R, Mul>, R2, Add>{ Expr<L,R, Mul> l; R2 r;
friend auto eval(Expr self) { return fma(eval(self.l.l), eval(self.l.r), eval(self.r));}};
template<class L, class R>
auto operator +(L l, R r) { return Expr<L, R, Add>{l, r}; }
template<class L, class R>
auto operator *(L l, R r) { return Expr<L, R, Mul>{l, r}; }
double optimized(F x, F y, F z) { return eval(x * y + z); }
double non_optimized(F x, F y, F z) { return eval(x + y * z); }
Optimized always generates a call to fma, non-optimized does not. Use -O1 to see the difference (will inline trivial functions, but will not do other optimizations). -O0 also generates the fma, but it is lost in the noise.
The magic happens by specifically matching the pattern Expr<Expr<L, R, Mul>, R2, Add>; try to add a rule to optimize x+y*z as well.