16-bit Adder Multiplier Hardware for Fixed Point and Floating Point Format (binary16)

Contents of Readme

1. About
2. Number Formats
3. Modules
4. Simulation
5. Test
6. Helper Script
7. Status
8. Issues
9. Licence

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About

This project was originated from a laboratory assignment and rewritten with Xilinx Vivado to work on Digilent Basys 3 FPGA. Two adder and two multiplier modules implemented, one working with fixed point numbers, other with floating point (binary16) numbers.

Number Formats

Fixed Point Format:

Most significant 8 bits represent integer part and least significant 8 bits represent fraction part.

i.e. IIIIIIIIFFFFFFFF = IIIIIIII.FFFFFFFF

Floating Point Format:

binary16 (IEEE 754-2008) is used. MSB is used as sign bit. 10 least significant bits are used as fraction and remaining bits are used as exponent.

For SEEEEEFFFFFFFFFF:

Exponent	Fraction is 0	Fraction is not 0
b00000	0	(-1)^S * 0.FFFFFFFFFF * 2^(-14)
b00001 to b11110	(-1)^S * 2^(EEEEE-15)	(-1)^S * 1.FFFFFFFFFF * 2^(EEEEE-15)
b11111	Infinity	NaN

Modules

Flags:

Flag	Description
overflow	Result does not fit or is infinite
zero	Result is zero
NaN	One or both of the operands are not a number
precisionLost	Result has errors due to precision lost

Fixed Point Adder

Module fixed_adder is a simple 16 bit adder with overflow signal. Overflow signal can also be used as 17th bit of result.

Ports:

Port	Type	Width	Description
num1	I	16	First operant
num2	I	16	Second operant
result	O	16	Result of the addition
overflow	O	1	Overflow flag or bit 17 of the result

I: Input O: Output

Fixed Point Multiplier

Module fixed_multi multiply two 16 bit fixed point numbers. Multiplication is in 32 bit, thus no precision lost during multiplication process. Result can be obtained either in 32 bit or in 16 bit. 32 bit format is similar to 16 bit format, 16 most significant bits represent integer part and 16 least significant bits represent fraction part. For 16 bit result overflow and precision lost flags implemented.

Ports:

Port	Type	Width	Description
num1	I	16	First operant
num2	I	16	Second operant
result	O	16	16 bit result of the multiplication
overflow	O	1	Overflow flag
precisionLost	O	1	Precision lost flag
result_full	O	32	32 bit result of the multiplication

I: Input O: Output

Floating Point Adder

Module float_adder is an adder module that can add two half-precision floating-point format (binary16) numbers.

Ports:

Port	Type	Width	Description
num1	I	16	First operant
num2	I	16	Second operant
result	O	16	Result of the addition
overflow	O	1	Overflow flag
zero	O	1	Zero flag
NaN	O	1	NaN flag
precisionLost	O	1	Precision lost flag

I: Input O: Output

Floating Point Multiplier

Module float_multi is an multiplier module that can multiply two half-precision floating-point format (binary16) numbers. Currently, multiplying a normal and a subnormal value does not work properly.

Ports:

Port	Type	Width	Description
num1	I	16	First operant
num2	I	16	Second operant
result	O	16	Result of the multiplication
overflow	O	1	Overflow flag
zero	O	1	Zero flag
NaN	O	1	NaN flag
precisionLost	O	1	Precision lost flag

I: Input O: Output

Simulation

Fixed point modules simulated using operatorCore_sim.v. It contains four test cases.

Floating point adder module simulated using float_add_sim.v. It contains ten test cases.

Floating point multiplier module simulated using float_multi_sim.v.

Test

Inputs

• Number switches: Used to enter operands
• Reset/New calculation (Center button): Used to reset system or begin new calculation
• Operator buttons: Switch operation, legend follows as:
  • Up button: Floating Format Addition
  • Left button: Floating Format Multiplication
  • Down button: Fixed Format Addition
  • Right button: Fixed Format Multiplication

Outputs

• Overflow LED (Leftmost LED): Indicates overflow during operation
• Zero LED (Second Leftmost LED): Indicateds zero result
• NaN LED (Third Leftmost LED): Indicates NaN error
• precisionLost LED (Forth Leftmost LED): Indicates precision lost during operation
• State LEDs (Rightmost two LEDs): Shows machine state, States follow as:
  1. IDLE: System does nothing, waits for user input
  2. WAIT1: System stores operand 1
  3. WAIT2: System stores operand 2
  4. RESULT: Results are shown, operation can be changed for provided numbers
• Seven Segment Displays: Shows operation result in hexadecimal format

System description

• This project provides a 16 bit adder multiplier hardware and interface for testing designed hardware
• System works on two number formats shown at number formats section
• Output is provided in the same format as the operands
• Flow:
  • Start with Reset/New calculation button.
  • Enter the first operand via Switches and press any one of the Operator buttons.
  • Enter the second operand via Switches and press any one of the Operator buttons.
  • System will calculate results for all of the modules. Shown output can be chosen via Operator buttons.
  • To initiate a new calculation press Reset/New calculation button.

Helper Scripts

Helper python 3 scripts are added to help verfication. They can be found at Scripts directory. Script can be used to calculate operation results for binary16 format or decode 16 bit binary16.

Status

Simulation:

• Fixed Point Modules: 9 January 2021, with Vivado Simulator.
• Floating Point Adder: 22 May 2021, with Vivado Simulator.
• Floating Point Multiplier: 23 May 2021, with Vivado Simulator.

Test:

• Fixed Point Modules: 22 May 2021, on Digilent Basys 3
• Floating Point Adder: 22 May 2021, on Digilent Basys 3
• Floating Point Multiplier: 22 May 2021, on Digilent Basys 3

Issues

• Minor bug in floating multiplier module. Some of the results incorrect. See issue #4.

Licence

CERN Open Hardware Licence Version 2 - Weakly Reciprocal