CORDIC

CORDIC, short for COordinate Rotation DIgital Computer, is an iterative algorithm for computing trigonometric, hyperbolic, exponential, and logarithmic functions using only additions, subtractions, and bit shifts. It was developed to enable efficient hardware implementations without multipliers, and became widely used in calculators, digital signal processors, and embedded systems.

The method exploits a sequence of small rotations. In rotation mode, starting from a vector (x0, y0) and a target angle z, the algorithm performs a series of micro-rotations by angles arctan(2^-i). Each step uses only shifts and adds to update the coordinates, guided by the sign of the residual angle. After n iterations the result is approximately (K_n cos z, K_n sin z), where K_n is a known scaling factor equal to the product of 1/sqrt(1+2^-2i). A final scaling correction using K_n yields the true cosine and sine values. In vectoring mode, the process is reversed to compute the magnitude r = sqrt(x^2 + y^2) and the angle arctan(y/x) from a given vector.

CORDIC also has hyperbolic variants that compute hyperbolic functions, exponential, and logarithms by using hyperbolic rotations and a modified sequence of steps. These variants can require extra iterations or repeated indices to maintain convergence, but they retain the shift-add simplicity.

Implementation aspects include fixed-point arithmetic, a precomputed table of arctan(2^-i), and a fixed number of iterations to achieve desired precision. Its main advantages are low hardware complexity and reliability, while the need to compensate the scale factor and manage angular ranges can be drawbacks.

Applications span scientific calculators, image processing, communication systems, and FPGA/ASIC implementations that require efficient, deterministic trigonometric and related function calculations.