An apparatus and method are provided for processing instructions. The apparatus has execution circuitry for executing instructions, where each instruction requires an associated operation to be performed using one or more source operand values in order to produce a result value. Issue circuitry is used to maintain a record of pending instructions awaiting execution by the execution circuitry, and prediction circuitry is used to produce a predicted source operand value for a chosen pending instruction. Optimisation circuitry is then arranged to detect an optimisation condition for the chosen pending instruction when the predicted source operand value is such that, having regard to the associated operation for the chosen pending instruction, the result value is known without performing the associated operation. In response to detection of the optimisation condition, an optimisation operation is implemented instead of causing the execution circuitry to perform the associated operation in order to execute the chosen pending instruction. This can lead to significant performance and/or power consumption improvements.