Real Reciprocal HDL Optimized

Compute reciprocal using CORDIC algorithm and generate optimized HDL code

Since R2021a

• 

Libraries:
Fixed-Point Designer HDL Support / Math Operations

Description

The Real Reciprocal HDL Optimized block computes 1/u, where u is a real scalar.

Examples

Implement Hardware-Efficient Real Divide HDL Optimized

How to use the Real Divide HDL Optimized block.

• Open Live Script

Customize Output Value of Real Divide HDL Optimized Block When Denominator Is Zero

Use the divideByZero port to customize the value of the block output when division by zero occurs.

• Open Live Script

How to Set CORDIC Input Word Length and Maximum Shift Value to Achieve Desired Precision

Provides a starting point for the input data type and number of iterations or maximum shift value required for the CORDIC algorithm to achieve a desired accuracy.

• Open Live Script

Limitations

Data type override is not supported for the Real Reciprocal HDL Optimized block.

Ports

Input

u — Value to take reciprocal of

scalar | vector | matrix

Value to take the reciprocal of, specified as a real-valued scalar, vector, or matrix.

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

Data Types: single | double | fixed point

validIn — Whether input is valid

Boolean scalar

Whether input is valid, specified as a Boolean scalar. This control signal indicates when the data from the u input port is valid. When this value is 1 (true), the block captures the value at the u input port. When this value is 0 (false), the block ignores the input samples.

Data Types: Boolean

Output

y — Reciprocal

real scalar

Reciprocal, returned as a real scalar with the data type specified by the Output datatype parameter.

Data Types: single | double | fixed point

divideByZero — Whether the value at output is the result of division by zero

Boolean scalar

Since R2024b

Whether the value at the y output port is the result of a division by zero operation, returned as a Boolean scalar. When the value of this signal is 1 (true), the corresponding output value at the y port is the result of division by zero. When the value of this signal is 0 (false), the corresponding output value at the y port is the result of division by a nonzero value.

See Division by Zero Behavior for a description of the default divide by zero behavior.

Dependencies

To enable this port, select the Show divide by zero port parameter.

Data Types: Boolean

validOut — Whether output data is valid

Boolean scalar

Whether output data is valid, returned as a Boolean scalar. When the value of this control signal is 1 (true), the block has successfully computed the output at port y. When this value is 0 (false), the output data is not valid.

Data Types: Boolean

Parameters

Output datatype — Data type of output

fixdt(1,18,10) (default) | single | double | fixdt(1,16,0) | <data type expression>

Data type of output y, specified as fixdt(1,18,10), single, double, fixdt(1,16,0), or as a user-specified data type expression. The type can be specified directly or expressed as a data type object, such as Simulink.NumericType.

In R2024b: Unsigned data types are supported for this parameter.

Programmatic Use

To set the block parameter value programmatically, use the set_param function.

To get the block parameter value programmatically, use the get_param function.

Parameter:	OutputType
Values:	fixdt(1,18,10) (default) | single | double | fixdt(1,16,0) | <data type expression>
Data Types:	char | string

Example: set_param(gcb,"OutputType","fixdt(1,16,0)")

Show divide by zero port — Whether to show the divideByZero port

off (default) | on

Since R2024b

Select this parameter to show the divideByZero port.

Programmatic Use

To set the block parameter value programmatically, use the set_param function.

To get the block parameter value programmatically, use the get_param function.

Parameter:	dbzPort
Values:	0 (false) (default) | 1 (true)
Data Types:	logical

Example: set_param(gcb,"dbzPort",1)

Automatically select CORDIC maximum shift value based on input word length — Automatically select CORDIC maximum shift value based on input word length

on (default) | off

Since R2024b

Automatically select the CORDIC maximum shift value based on input word length. When this parameter is selected, the default CORDIC maximumShiftValue is equal to wl - 1, where wl = u.WordLength + ~issigned(u).

Programmatic Use

To set the block parameter value programmatically, use the set_param function.

To get the block parameter value programmatically, use the get_param function.

Parameter:	autoMaximumShiftVal
Values:	on (default) | off
Data Types:	char | string

Example: set_param(gcb,"autoMaximumShiftVal","off")

CORDIC maximum shift value — Maximum shift value of hyperbolic vectoring CORDIC

wl - 1 (default) | 10 | positive integer-valued scalar

Since R2024b

Maximum shift value of hyperbolic vectoring CORDIC, specified as a positive integer-valued scalar. The default value for this parameter is wl - 1, where wl = u.WordLength + ~issigned(u).

Dependencies

To enable this parameter, clear the Automatically select CORDIC maximum shift value based on input word length parameter.

Tips

See Customizable Pipelining for more information.

Programmatic Use

To set the block parameter value programmatically, use the set_param function.

To get the block parameter value programmatically, use the get_param function.

Parameter:	maximumShiftValue
Values:	10 (default) | positive integer-valued scalar
Data Types:	char | string

Example: set_param(gcb,"maximumShiftValue","10")

Number of iterations per pipeline register — Number of CORDIC iterations to perform per pipeline stage

1 (default) | positive integer-valued scalar

Since R2024b

Number of CORDIC iterations to perform per pipeline stage, specified as a positive integer-valued scalar.

See Customizable Pipelining for more information. See How to Interface with the Real Reciprocal HDL Optimized Block and Hardware Resource Utilization for more information and examples showing how this parameter impacts latency and hardware resource utilization.

Programmatic Use

To set the block parameter value programmatically, use the set_param function.

To get the block parameter value programmatically, use the get_param function.

Parameter:	nIterPerReg
Values:	1 (default) | positive integer-valued scalar
Data Types:	char | string

Example: set_param(gcb,"nIterPerReg","2")

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

When the input u is zero and has a fixed-point data type, then the output y is equal to upperbound(Output datatype).

For floating-point inputs, the Real Divide HDL Optimized block follows IEEE^® Standard 754.

How to Interface with the Real Reciprocal HDL Optimized Block

Because of its fully pipelined nature, the Real Reciprocal HDL Optimized block is able to accept input data on any cycle, including consecutive clock cycles. To send input data to the block, the validIn signal must be true. When the block has finished the computation and is ready to send the output, it will change validOut to true for one clock cycle. For inputs set of consecutive cycles, validOut will also be set to true on consecutive cycles.

The latency is defined from the input to the corresponding output. For example in the figure below, from In1 to Out1, In2 to Out2, In3 to Out3.

The latency depends on the input data type, as summarized in the table.

Input Type	Latency
Fixed point or scaled double fi	ceil((nextpow2(wl) + maximumShiftValue)/nIterPerReg) + 1 where wl = u.WordLength + ~issigned(u) and maximumShiftValue = wl - 1 or user-specified value.
Floating point	0

Customizable Pipelining

The Real Reciprocal HDL Optimized block uses fully pipelined architecture that implements iterative normalization and a CORDIC-based division algorithm. If the input u is a fixed-point or scaled double data type, the block uses multiple pipeline stages for computation. If the input is a signed data type, the normalization requires nextpow2(u.WordLength) iterations. The number of CORDIC iterations depends on the value of the CORDIC maximum shift value parameter. A larger word length can provide higher resolution, but requires more iterations to process. The Real Reciprocal HDL Optimized block can perform multiple iterations per pipeline stage, which results in lower latency at the cost of a longer critical path in the generated HDL code.

For example, if the word length of the input u is 18, then normalization requires 5 iterations. If the Automatically select CORDIC maximum shift value based on input word length parameter is selected, the CORDIC maximum shift value is 18 - 1 = 17 and requires 17 iterations. The total number of iterations is 5 + 17 = 22 and the latency of the block is ceil((total number of iterations)/nIterPerReg) + 1. If the number of iterations per pipeline register is set to 1, then the block latency is 23; if the number of iterations per pipeline register is set to 2, then the block latency is 12. If the number of iterations per pipeline register is greater than the total number of required iterations, the block performs all iterations in one pipeline stage and the total latency is minimized to 2.

Hardware Resource Utilization

This block supports HDL code generation using the Simulink^® HDL Workflow Advisor. For an example, see HDL Code Generation and FPGA Synthesis from Simulink Model (HDL Coder) and Implement Digital Downconverter for FPGA (DSP HDL Toolbox).

This example data was generated by synthesizing the block on a Xilinx^® Zynq^®-7000 xc7z045 SoC. The synthesis tool was Vivado^® v2023.1.2.

The following synthesis results show the effect of the Number of iterations per pipeline register parameter on the latency and hardware resource utilization.

nIterPerReg = 1

These parameters were used for synthesis:

• Input data type: sfix18_en10

• Output data type: sfix18_en10

• Input dimension: scalar

• Automatically select CORDIC maximum shift value based on input word length: on

• Number of iterations per pipeline register: 1

• Target frequency: 500 MHz

• Latency for this configuration: 23

Resource	Usage	Available	Utilization (%)
Slice LUTs	815	218600	0.37
Slice Registers	703	437200	0.16
DSPs	0	900	0.00
Block RAM Tile	0	545	0.00
URAM	0	0

	Value
Requirement	2 ns (500 MHz)
Data Path Delay	1.74 ns
Slack	0.176 ns
Clock Frequency	548.25 MHz

nIterPerReg = 2

These parameters were used for synthesis:

• Input data type: sfix18_en10

• Output data type: sfix18_en10

• Input dimension: scalar

• Automatically select CORDIC maximum shift value based on input word length: on

• Number of iterations per pipeline register: 2

• Target frequency: 300 MHz

• Latency for this configuration: 12

Resource	Usage	Available	Utilization (%)
Slice LUTs	705	218600	0.32
Slice Registers	374	437200	0.09
DSPs	0	900	0.00
Block RAM Tile	0	545	0.00
URAM	0	0

	Value
Requirement	3.3333 ns (300 MHz)
Data Path Delay	2.65 ns
Slack	0.676 ns
Clock Frequency	376.32 MHz

nIterPerReg = 3

These parameters were used for synthesis:

• Input data type: sfix18_en10

• Output data type: sfix18_en10

• Input dimension: scalar

• Automatically select CORDIC maximum shift value based on input word length: on

• Number of iterations per pipeline register: 3

• Target frequency: 200 MHz

• Latency for this configuration: 9

Resource	Usage	Available	Utilization (%)
Slice LUTs	680	218600	0.31
Slice Registers	281	437200	0.06
DSPs	0	900	0.00
Block RAM Tile	0	545	0.00
URAM	0	0

	Value
Requirement	5 ns (200 MHz)
Data Path Delay	3.889 ns
Slack	1.125 ns
Clock Frequency	258.06 MHz

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 Simulink® Coder™.

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

HDL Code Generation

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

HDL Coder™ provides additional configuration options that affect HDL implementation and synthesized logic.

HDL Architecture

This block has one default HDL architecture.

HDL Block Properties

General
ConstrainedOutputPipeline	Number of registers to place at the outputs by moving existing delays within your design. Distributed pipelining does not redistribute these registers. The default is 0. For more details, see ConstrainedOutputPipeline (HDL Coder).
In R2024b: FlattenHierarchy	Remove PWM Reference Generator block hierarchy from generated HDL code. The default is inherit. See also FlattenHierarchy (HDL Coder).
InputPipeline	Number of input pipeline stages to insert in the generated code. Distributed pipelining and constrained output pipelining can move these registers. The default is 0. For more details, see InputPipeline (HDL Coder).
OutputPipeline	Number of output pipeline stages to insert in the generated code. Distributed pipelining and constrained output pipelining can move these registers. The default is 0. For more details, see OutputPipeline (HDL Coder).

Restrictions

Supports fixed-point data types only.

Version History

Introduced in R2021a

R2024b: Custom pipelining, improved latency and resource utilization, optional divide by zero port

Several improvements have been made to the Real Reciprocal HDL Optimized block:

• Custom pipelining is supported via the new CORDIC maximum shift value and Number of iterations per pipeline register parameters.

• The latency of this block has been reduced. Latency depends on the specified data type and pipeline configuration. See How to Interface with the Real Reciprocal HDL Optimized Block for more information.

• HDL resource utilization has been further optimized to require fewer hardware resources. See Hardware Resource Utilization for example synthesis results.

• An optional divideByZero port has been added to output a flag when the corresponding output is a result of division by zero.

R2024b: Support for unsigned output

The Real Reciprocal HDL Optimized block now supports unsigned data types for the Output datatype parameter.