A TABLE LOOKUP-LESS METHOD FOR CORRECTLY ROUNDED IEEE-754 ELEMENTARY FUNCTIONS
George Kamal, Kamel Badawi · 2015
This thesis presents a hardware-oriented method for computing correctly rounded IEEE-754 single-precision floating-point elementary functions. The method presents a table “lookup-less” approach using a minimax approximation polynomial of a high degree within a single wide range as opposed to the currently dominant table-based designs that have to use large look-up tables while maintaining low-degree piecewise polynomials to achieve the accuracy required for correct rounding. The method is firstly applied to trigonometric functions for argument values |x| < 253. Range reduction and polynomial evaluation steps are carried out using a standard double-precision fused-multiply add (FMA) with the help of a simple control unit and combinational logic to support the range reduction, the implicit reconstruction and the final rounding steps. The method uses a single low-cost range reduction algorithm for all arguments without penalizing small and intermediate argument values. This approach makes computing correctly rounded trigonometric functions for a very large range of arguments possible entirely at the hardware level with reasonable logic resources and low memory requirements that can be less than 700 bits. This memory is 3.5 to 5 times smaller than the state-of-the-art faithful quadratic minimax table-based designs that cover a reduced range. Although the thesis addresses the round-to-nearest mode, the method can be easily applied to the directed rounding modes. Moreover, the method is applied for 2x and log2(x) within reduced intervals. We also argue that the method can be extended to involve the double-precision case for trigonometric functions.