fixed.cordicReciprocal

CORDIC-based fixed-point reciprocal

Since R2021b

Syntax

y = fixed.cordicReciprocal(u,OutputType)

Description

y = fixed.cordicReciprocal(u,OutputType) returns 1./u with the output cast to the data type specified by OutputType.

Examples

Fixed-Point Reciprocal Using CORDIC

u = fi(10);
outputType = fi([],1,32,24);
y = fixed.cordicReciprocal(u,outputType)

y = 
    0.1000

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 32
        FractionLength: 24

Input Arguments

u — Value to take reciprocal of

scalar | vector | matrix | multidimensional array

Value to take reciprocal of, specified as a scalar, vector, matrix, or multidimensional array.

• If u is a floating-point type, then OutputType must specify a floating-point data type.

• If u is a built-in integer type, then OutputType must specify a built-in integer data type.

• If u is a fixed-point type, then OutputType must specify a fixed-point data type.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fi
Complex Number Support: Yes

OutputType — Data type of output

fi object | numerictype object | Simulink.NumericType object

Data type of the output, specified as a fi object, numerictype, or Simulink.NumericType object.

• If num is a floating-point type, den must also be a floating-point type and OutputType must specify a floating-point data type.

• If num is a built-in integer type, den must also be a built-in integer type and OutputType must specify a built-in integer data type.

• If num is a fixed-point type, den must also be a fixed-point type and OutputType must specify a fixed-point data type.

Example: fi([],1,16,15)

Example: numerictype(1,16,15)

Example: fixdt(1,16,15)

Tips

• The behavior of the Real Reciprocal HDL Optimized block is equivalent to the fixed.cordicReciprocal function. When the input to the function is a real number with a binary-point scaled fixed-point data type, the function and block provide bit-exact results.

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.

Division by Zero Behavior

For fixed-point input u, fixed.cordicReciprocal wraps on overflow for division by zero. The behavior for fixed-point division by zero is summarized in the table below.

Wrap Overflow	Saturate Overflow
0/0 = 0	0/0 = 0
1/0 = 0	1/0 = upper bound
-1/0 = 0	-1/0 = lower bound

For floating-point inputs, fixed.cordicReciprocal follows IEEE^® Standard 754.

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

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Slope-bias representation is not supported for fixed-point data types.

Fixed-Point Conversion

Design and simulate fixed-point systems using Fixed-Point Designer™.

Slope-bias representation is not supported for fixed-point data types.

Version History

Introduced in R2021b

R2025a: Bit-exact result with Simulink block

The output of the fixed.cordicReciprocal function has changed to be bit-exact with the output of the Real Reciprocal HDL Optimized block when the input to the function is a real number and the data type is fixed point with binary-point scaling. Small numerical differences may exist compared to previous releases.