You can have a look at Richard Bird's book _Pearls of Functional Algorithm Design_. It's much more accessible than his earlier influential book _Algebra of Programming_.
In this book, many well-known algorithms, e.g. KMP pattern matching, are derived from naive implementations by step-by-step applications of algebraic rules of (functional) programs.
In this book, many well-known algorithms, e.g. KMP pattern matching, are derived from naive implementations by step-by-step applications of algebraic rules of (functional) programs.