An apparatus and method are provided for performing arithmetic operations to accumulate floating-point numbers. The apparatus comprises execution circuitry to perform arithmetic operations, and decoder circuitry to decode a sequence of instructions in order to generate control signals to control the arithmetic operations performed by the execution circuitry. A convert and accumulate instruction is provided, and the decoder circuitry is responsive to decoding such a convert and accumulate instruction within the sequence of instructions to generate one or more control signals to control the execution circuitry. In particular, the execution circuitry is responsive to such control signals to convert at least one floating-point operand identified by the convert and accumulate instruction into a corresponding N-bit fixed-point operand having M fraction bits, where M is less than N and M is dependent on a format of the floating-point operand. In addition, the execution circuitry accumulates each corresponding N bit fixed-point operand and a P bit fixed-point operand identified by the convert and accumulate instruction in order to generate a P bit fixed-point result value, where P is greater than N and also has M fraction bits. This provides a fast and efficient mechanism for accumulating floating-point numbers in a manner that is associative, and hence enables reproducible and correct results to be generated irrespective of the order in which the floating-point numbers are accumulated.