Table 9 Computational flow of the R4HR-CORDIC.

From: Radix-4 CORDIC algorithm based low-latency and hardware efficient VLSI architecture for Nth root and Nth power computations

Iteration index

Datapath

Operation

X

Y

Z

Prescaler

–

–

–

Pre-scale the input angle add selection function

\(j=0\)

\(X_1=\frac{1}{K_h}\)

\(Y_1=\frac{\sigma _0}{K_h}\left( 2^{-1}+2^{-3}\right)\)

\(Z_1=Z_0-\tanh _4^-1(0.625\sigma _0)\)

Compute \(X_1\), \(Y_1\), and \(Z_{1}\) based on \(\sigma _0\).

\(1\le j \le 3\)

\(X_{j+1}=X_j+\sigma _j4^{-j}Y_j\)

\(Y_{j+1}=Y_j+\sigma _j4^{-j}X_j\)

\(Z_{j+1}=Z_j-\tanh _4^{-1}(\sigma _j4^{-j})\)

Compute conventional radix-4 hyperbolic rotation

\(4\le j \le 6\)

\(X_{j+1}=X_j \left( 1+\frac{\sigma _j^24^{-2j}}{2} \right)\) \(+Y_j\left( \sigma _j4^{-j}+\frac{\sigma _j^34^{-3j}}{8}\right)\)

\(Y_{j+1}=Y_j \left( 1+\frac{\sigma _j^24^{-2j}}{2} \right)\) \(+X_j\left( \sigma _j4^{-j}+\frac{\sigma _j^34^{-3j}}{8}\right)\)

\(Z_{j+1}=Z_j -\tanh _4^{-1}\left( \frac{\sigma _j4^{-j}+\frac{\sigma _j^34^{-3j}}{8}}{ 1+\frac{\sigma _j^24^{-2j}}{2} } \right)\)

Compute scaling-free hyperbolic rotation with two terms of hyperbolic sine and cosine.

\(7\le j \le 12\)

\(X_{j+1}=X_j \left( 1+\frac{\sigma _j^24^{-2j}}{2} \right)\) \(Y_j\left( \sigma _j4^{-j}\right)\)

\(Y_{j+1}=Y_j \left( 1+\frac{\sigma _j^24^{-2j}}{2} \right)\) \(X_j\left( \sigma _j4^{-j}\right)\)

\(Z_{j+1}=Z_j -\tanh _4^{-1}\left( \frac{\sigma _j4^{-j}}{1+\frac{\sigma _j^24^{-2j}}{2}} \right)\)

Compute scaling-free hyperbolic rotation with two terms of hyperbolic cosine and one term of hyperbolic sine.

\(13\le j \le \frac{n}{2}\)

\(X_{j+1}=X_j+ Y_j\left( \sigma _j4^{-j} \right)\)

\(Y_{j+1}=X_j+ X_j\left( \sigma _j4^{-j} \right)\)

\(Z_{j+1}=Z_j -\tanh _4^{-1}\left( \sigma _j4^{-j}\right)\)

Compute scaling-free hyperbolic rotation with one term of hyperbolic sine and cosine.

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