Low Latency and Low Error Floating-Point Sine/Cosine Function Based TCORDIC Algorithm

Abstract

CORDIC algorithm is suitable to implement sine/cosine function, but the large number of iterations lead to great delay and overhead. Moreover, due to finite bit-width of operands and number of iterations, the relative error of floating-point sine or cosine is terrible when the input angle is close to 0 or $\pi /2$ , respectively. To overcome these shortcomings, TCORDIC algorithm, which combines low latency CORDIC and Taylor algorithm, is presented. After analyzing the latency of traditional CORDIC, low latency CORDIC is proposed, which adopts the technique of sign prediction, compressive iterations, and parallel iterations. Besides, the calculating boundary ( $N$ ), which is used for determining whether Taylor algorithm is selected or not in TCORDIC algorithm, is evaluated to achieve a trade-off between area and delay. Truncated multipliers are used to reduce the area further. Finally, Using TCORDIC algorithm, pipelined and iterative structures are implemented for IEEE-754 double precision floating-point sine/cosine with the input $Z\epsilon [0,\pi /2]$ . Under typical condition (1V, 25 °C), our designs are synthesized with 40 nm standard cell library. For a pipelined structure, the frequency is up to 1.70 GHz and area $194049.64~\mu {\mathrm { m}}^{2}$ . Frequency decreases to 1.45 GHz for iterative structure, but the area requires only $110590.81~\mu {\mathrm { m}}^{2}$ . TCORDIC is efficient in controlling relative error, and achieves the accuracy within one ulp (unit in the last place) for floating-point sine/cosine function.