Design and Implementation of Floating-Point Addition and Floating-Point Multiplication

Abstract

In this paper, we present the design and implementation of Floating point addition and Floating point Multiplication. There are many multipliers in existence in which Floating point Multiplication and Floating point addition offers a high precision and more accuracy for the data representation of the image. This project is designed and simulated on Xilinx ISE 14.7 version software using verilog. Simulation results show area reduction and delay reduction as compared to the conventional method.

Introduction

I. INTRODUCTION

In computing, floating-point arithmetic (FP) is arithmetic using formulaic representation of real numbers as an approximation to support a trade-off between range and precision. For this reason, floating-point computation is often used in systems with very small and very large real numbers that require fast processing times. In general, a floating-point number is represented approximately with a fixed number of significant digits (the significand) and scaled using an exponent in some fixed base; the base for the scaling is normally two, ten, or sixteen. A number that can be represented exactly is of the following form:

Here significand is an integer, base is an integer greater than or equal to two, and exponent is also an integer. For example:

II. FLOATING POINT ADDITION

Contrasted with a fixed point addition , a floating  point addition is more complicated and equipment consuming. This is on the grounds that type field is absent if there should be an occurrence of fixed point arithmetic. A floating point addition of two numbers and can be communicated as

Sa.Ma.2Ea + Sb.Mb.2Eb =S.2Eb(Ma+Mb*)

A. Flow Chart

B. Procedure

Floating point addition algorithm

                                                            X3=X1+X2

                                                X3 = (M1 x 2E1) + (M2 x 2E2)

1. X1 and X2 must be added if the exponents  are the equivalent i.e E1=E2.
2. We accept that X1 has the bigger absolute value of the 2 numbers. Absolute value of    X1 must be equal to o X2,otherwise swap those values .

                                                          Abs(X1)>Abs(X2).

3. Initial value  of the exponent must  to be the bigger of the 2 numbers, since we know exponent  of X1 will be bigger,hence Initial exponent result E3 = E1.

4. Calculate  the difference of those exponents i.e.Exp_diff=(E1-E2).

5. After calculating the exponent difference left shift the decimal point of mantissa (M2) .now we having both exponents of  X1 and X2 are same.

6. Depending on the sign bit S1 and S2 now add the mantissas.

if both signs are equal then add mantissa (S1==S2)

if not equal then subtract (S1!=S2)

7. Normalize the resultant mantissa (M3) if necessary. (1.m3 arrangement) and the initial exponent result E3=E1 should be changed by the normalization  of mantissa.

8. if (E3>Emax) then overflow occurred ,if (E3<Emin) then underflow occurred, then output is to be zero

9. Nan's are not upheld.

III. FLOATING POINT MULTIPLICATION

Floating point multiplication is comparatively easy than the floating point addition algorithm but off course consumes more hardware than fixed point multiplier circuit. Major hardware block is the multiplier which is same as fixed point multiplier. This multiplier is used to multiply the mantissas of the two numbers. A floating point multiplication between two numbers a  and b can be expressed as

A. Flow Chart

1. Procedure

Explanation of floating point algorithm has explained

Result X3 = X1 * X2

1. Check in the event that one/the two operands = 0 or infinity. Set the output to 0 or infinty. for example exponent = all 0 or all 1
2. S1 is XOR with the S2 where S1 is sign bit  multiplicand and S2 is the sign bit of multiplier.
3. The mantissa of M1 and M2 are multiplicated where.M1 is the multiplier and M2 is multiplicand and the output is placed in resultant field of mantissa.

                    =M1 * M2

4. The exponents of the M1 is (E1) and M2 is (E2) bits are added and the base value is subtracted from the output. That result is placed in the exponential field of the output block.

                      =E1+E2-bias

5. Now normalize the sum, By shifting right and increment the exponent or shifting left and decrement the exponent.

6. Now check the underflow/overflow, if it is underflow set output is zero and if it is overflow output is infinity. 

7. If (E1 + E2 -bias) >= to Emax then, at that point, set the product to infinity.

8. If E1 + E2 - bias) is lesser than/equivalent to Emin then, at that point, set output to  zero.

IV. RESULTS

This project is implemented on Xilinx software 14.7 version using veilog language, Spartan 5 Vertex device family, XC6SLX100T Device and the package used is FGG900 with a speed grade of “-3”.

Conclusion

In this brief, by the design and implementation of Floating point addition and Multiplication , it has been proved that the speed of multiplication addition can be increased and also it requires less area compared to the conventional method. here the Carry Look Ahead Adder is used to increase the speed. Thus this proposed method can be used for less area and high-speed digital image processing applications.

References

[1] Grover N, Soni M. Design of FPGA based 32-bit floating point arithmetic unit and verification of its VHDL code using matlab. Modern Education and Computer Science Press; 2014. p. 1–14. [23] Iqbal F. Wavelet transform based image compression on FPGA. Master of Science Thesis, Florida State University; 2004. diginole.lib.fsu.edu [2] Dhobale R, Chaturvedi S. Implementation of 32 bit binary floating point adder using IEEE-754 single precision format. IOSR J VLSI Signal Process (IOSR-JVSP) 2015;5(1):50–3. [3] Dahiya S, Kumar R. Performance Analysis of different bit carry look ahead adder using VHDL environment. Int J Eng Sci Innov Technol (IJESIT) 2013;2:80–8. [4] Hassan H, Ismail S. CLA based Floating-point adder suitable for chaotic generators on FPGA. In: 30th international conference on microelectronics (ICM); 2018.

Copyright

Copyright © 2022 Nagireddy Kavya. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.