Total functional computable real numbers

Question

Is there any computable real number which can not be computed by a higher order primitive recursive algorithm?

For computable real number I mean those that can be computed by a Turing machine to any desired precision in finite time. For higher order primitive recursive algorithm I mean common primitive recursive functions theory extended with first-class functions (as in Ackermann function).

Turing machines are more powerful than higher order primitive recursive functions so there exists the possibility that some computable reals numbers are not expressible by them.