cordicacos

CORDIC-based approximation of inverse cosine

Syntax

theta = cordicacos(x)

theta = cordicacos(x,niters)

Description

theta = cordicacos(x) returns the inverse cosine of x based on a double CORDIC (DCORDIC) algorithm approximation.

theta = cordicacos(x,niters) performs niters iterations of the DCORDIC algorithm.

Examples

Calculate CORDIC Inverse Cosine

Compute the inverse cosine of a fixed-point fi object using a CORDIC implementation.

a = fi(-1:.1:1,1,16);
b = cordicacos(a);
plot(a,b);
title('Inverse CORDIC Cosine');

Compare the output of the cordicacos function and the acos function.

c = acos(double(a));
error = double(b)-c;
plot(a,error);
title('Error');

Calculate CORDIC Inverse Cosine with Specified Number of Iterations

Find the inverse cosine of a fi object using a CORDIC implementation and specify the number of iterations the CORDIC kernel should perform. Plot the CORDIC approximation of the inverse cosine with varying numbers of iterations.

a = fi(-1:.1:1,1,16);
for i = 5:5:20
    b = cordicacos(a,i);
    plot(a,b);
    hold on;
end
legend('5 iterations','10 iterations',...
    '15 iterations','20 iterations')

Input Arguments

x — Cosine of angle

scalar | vector | matrix | multidimensional array

Cosine of angle, specified as a scalar, vector, matrix, or multidimensional array. x must be a real, finite value in the range [-1,1].

When x is specified as a fi object, the property 'DataTypeMode' must equal Fixed-point: binary point scaling', 'Scaled double: binary point scaling', 'double', or 'single'.

Data Types: single | double | fi
Complex Number Support: Yes

niters — Number of iterations CORDIC algorithm performs

positive integer-valued scalar

Number of iterations the CORDIC algorithm performs, specified as a positive integer-valued scalar.

If you do not specify niters, the algorithm uses a default value. For fixed-point inputs, the default value of niters is one less than the word length of the input array x. For double-precision inputs, the default value of niters is 54. For single-precision inputs, the default value is 25.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fi

Output Arguments

theta — Inverse cosine angle values

scalar | vector | matrix | multidimensional array

Inverse cosine angle values in radians, returned as a scalar, vector, matrix, or multidimensional array. The cordicacos function returns values in the interval [0,π].

Algorithms

CORDIC

CORDIC is an acronym for COordinate Rotation DIgital Computer. The Givens rotation-based CORDIC algorithm is one of the most hardware-efficient algorithms available because it requires only iterative shift-add operations (see References). The CORDIC algorithm eliminates the need for explicit multipliers. Using CORDIC, you can calculate various functions such as sine, cosine, arcsine, arccosine, arctangent, and vector magnitude. You can also use this algorithm for divide, square root, hyperbolic, and logarithmic functions.

The precision of the CORDIC algorithm is a function of the data type used and the maximum shift value or number of iterations of the CORDIC kernel. Using a data type with a larger word length and performing more iterations of the CORDIC algorithm can reduce the numeric error of the result. However, doing so also increases the latency of the computation and the utilizes more hardware resources. For more information, see How to Set CORDIC Input Word Length and Maximum Shift Value to Achieve Desired Precision.

References

[1] Volder, Jack E. “The CORDIC Trigonometric Computing Technique.” IRE Transactions on Electronic Computers. EC-8, no. 3 (Sept. 1959): 330–334.

[2] Andraka, Ray. “A Survey of CORDIC Algorithm for FPGA Based Computers.” In Proceedings of the 1998 ACM/SIGDA Sixth International Symposium on Field Programmable Gate Arrays, 191–200. https://dl.acm.org/doi/10.1145/275107.275139.

[3] Walther, J.S. “A Unified Algorithm for Elementary Functions.” In Proceedings of the May 18-20, 1971 Spring Joint Computer Conference, 379–386. https://dl.acm.org/doi/10.1145/1478786.1478840.

[4] Schelin, Charles W. “Calculator Function Approximation.” The American Mathematical Monthly, no. 5 (May 1983): 317–325. https://doi.org/10.2307/2975781.

Extended Capabilities

HDL Code Generation

Generate VHDL, Verilog and SystemVerilog code for FPGA and ASIC designs using HDL Coder™.

The cordicacos function also supports MATLAB^® to High-Level Synthesis (HLS) code generation.

Version History

Introduced in R2018b