A data processing apparatus has an instruction memory system arranged to output an instruction word, capable of containing a plurality of instructions, respective instruction words being output in response to respective instruction addresses. An instruction execution unit contains a plurality of functional units, each capable of executing a respective instruction from the instruction word in parallel with execution of other instructions from the instruction word by other ones of the functional units. A power saving circuit is provided to switch a selectable subset of the functional units and/or parts of the instruction memory to a power saving state, while other functional units and parts of the instruction memory continue processing instructions in a normal power consuming state. The power saving circuit selects the functional units and/or parts of the instruction memory dependent on program execution.