Comparative study of single precision floating point division using different computational algorithms

International Journal of Reconfigurable and Embedded Systems (IJRES)
Vol. 12, No. 3, November 2023, pp. 336~344
ISSN: 2089-4864, DOI: 10.11591/ijres.v12.i3pp336-344  336
Journal homepage: http://ijres.iaescore.com
Comparative study of single precision floating point division
using different computational algorithms
Naginder Singh, Kapil Parihar
Department of Electronics and Communication Engineering, M.B.M. University, Jodhpur, India
Article Info ABSTRACT
Article history:
Received Jun 25, 2022
Revised Dec 10, 2022
Accepted Mar 20, 2023
This paper presents different computational algorithms to implement single
precision floating point division on field programmable gate arrays (FPGA).
Fast division computation algorithms can apply to all division cases by
which an efficient result will be obtained in terms of delay time and power
consumption. 24-bit Vedic multiplication (Urdhva-Triyakbhyam-sutra)
technique enhances the computational speed of the mantissa module and this
module is used to design a 32-bit floating point multiplier which is the
crucial feature of this proposed design, which yields a higher computational
speed and reduced delay time. The proposed design of floating-point divider
using fast computational algorithms synthesized using Verilog hardware
description language has a 32-bit floating point multiplier module unit and a
32-bit floating point subtractor module unit. Xilinx Spartan 6 SP605
evaluation platform is used to verify this proposed design on FPGA.
Synthesis results provide the device utilization and propagation delay
parameters for the proposed design and a comparative study is done with
previous work. Input to the divider is provided in IEEE 754 32-bit formats.
Keywords:
Field programmable gate array
Floating point division
Goldschmidt algorithm
Newton-Raphson algorithm
Vedic-multiplication
This is an open access article under the CC BY-SA license.
Corresponding Author:
Naginder Singh
Department of Electronics and Communication Engineering, M.B.M. University
Jodhpur, Rajasthan, India
Email: naginder.singh0110@gmail.com
1. INTRODUCTION
Field programmable gate array (FPGA) provides a very versatile high computational environment
and currently plays an important role due to the advancement of technology process or architectures modified
instantaneously. For the proposed design the cost per logic cell is reduced voluntarily because of the
enhanced feature of Spartan 6. In many digital processing systems for many arithmetic and logical units,
floating-point dividers play a crucial role. To design a divider power consumption and throughput pays major
factors for processing the information. This paper is based on computational algorithms to design a single
precision floating point divider; in which multiplication and subtraction are the basic mathematical
operations as described earlier in [1]-[3]. The arithmetic operations such as multiplication and subtraction
modules are used to implement and design Goldschmidt and Newton-Raphson algorithm for the proposed 32-
bit floating point division [4]. For multiplication Urdhva-Tiryagbhyam-sutra is an ancient Vedic mathematics
method, this algorithm reduces the number of steps required for multiplication and can increase the speed of
multiplication [5]. In this proposed design of divider, the module to compute mantissa in the multiplication
unit is designed using medic mathematics (Urdhva-Tiryagbhyam-sutra). A module of 16*16 Vedic multiplier
implemented by using the concept Urdhva-Tiryagbhyam-sutra and Nikhilam-sutra multiplication techniques
[6]-[9]. In this paper the exponent module used in the floating-point multiplier uses a ripple carry adder to
provide a low area and simple layout [10]. Binary floating point multiplier implementation and blocks

Int J Reconfigurable & Embedded Syst ISSN: 2089-4864 
Comparative study of single precision floating point division using … (Naginder Singh)
337
described and Urdhva-Tiryagbhyam-sutra used to achieve high speed [11]. In designing of processor,
arithmetic-logic unit (ALU) unit arithmetic operations are done by addition, subtraction, and multiplication
Vedic algorithm plays an important role [12]. Nikhilam-sutra based multiplier is designed to get high speed
as the computational speed gained by Vedic mathematics described [13]. High speed squaring circuit for
binary numbers [14] is proposed based on Vedic mathematics. To perform signal processing Goldschmidt
computational algorithm-based division provides the high computational speed and throughput described
[15] and it’s suitable for high computational demanding applications. Power consumption plays an important
role and is reduced efficiently by designing a multiplier using Vedic multiplication which reduces the
required look up tables and slices resulting in reduced power consumption [16]. A floating-point multiplier is
implemented which is designed for efficient power consumption and device utilization is reduced [17].
In digital signal processing application encryption and decryption algorithms in cryptography; a
divider is one of the most important hardware blocks which is applicable in most applications [18] and also in
various mathematical computational algorithms used to design floating-point divide-add fused (DAF)
architecture [19]. The floating-point DAF architecture has a dedicated unit for floating point division
followed by addition or subtraction so that a combined operation is performed [20]. The delay time and better
device utilization which is an important factor and have to optimize are done by a division architecture based
on the Vedic mathematics algorithm Parvartya sutra [21]. The restoring division is described for sixteen-bit
division and well explained its implementation of restoring division algorithm in encryption system [22].
Floating point architectures have low frequency, larger area and high latency in nature described [23]. A
reversible Vedic multiplier is designed using reversible logic gates and results in less delay and less power
consumption [24]. High computational speed and throughput provide by Newton-Raphson and Goldschmidt
computational division algorithm methods both converge much faster within one iteration to produce the
final quotient, so these computational algorithms in a single iteration generate twice as many digits to find the
final quotient as they start with a close approximation to find the final quotient [25]. To represent 32-bit and
64-bit floating point numbers IEEE 754 standard format is used [26]. Booth-based multiplication is used to
pre-calculate the carry and the presented design improved the speed and dissipated less power with respect to
the existing multipliers [27]. A fast-floating point unit is designed and implemented for FPGAs [28]. As
floating-point number is separated into three parts where a fixed number of bits is used to represent these
three parts that are sign, exponent and mantissa. To describe single and double precision the structure of the
IEE 754 format is shown in Table 1. For single precision IEEE 754 format, 23 bits are used to represent
mantissa, exponent part is represented by 8-bits and the sign bit is represented by the most significant bit
(MSB) part where the sign of the floating-point number depends on the MSB bit, floating point number is
negative when MSB is 1 else the number is positive when MSB bit is 0. For division purposes in many
signals processing algorithms complex iterative process is used so, not only is the precision to be maintained
but it has to be maintained for very large data intervals and is achieved by using Newton-Raphson and
Goldschmidt computational algorithms.
This paper presents implementation using Goldschmidt and Newton-Raphson computational
algorithm. To design of 32-bit floating point division which uses arithmetic operations such as 32-bit floating
point multiplication module and floating-point subtraction modules and a 24-bit Vedic multiplication used
for computing mantissa in 32-floating point multiplier module in the proposed designed. The paper describes
the implementation and design of Newton-Raphson and Goldschmidt computational algorithms single
precision floating point divider on FPGA.
The basics of IEEE 754 floating-point representation are discussed: in section 2 and the division
algorithms which are i) Newton Raphson and ii) Goldschmidt computational algorithm methods are
discussed in section 3. The implementation of Vedic multiplication technique base 32-bit floating point
multiplier is discussed in section 4. Floating point sub-tractor describes in section 5. The simulation results in
6 and conclusion describes is in section 7.
2. FLOATING POINT STANDARD
IEEE 754 standard representation for binary floating-point numbers [26]. The binary number
represents the single and double precision formats, 32-bit represented by single precision and 64-bit
represented by double precision. Single and double precision formats of IEEE 754 structure are shown in
Table 1 sign, exponent and mantissa are 3 fields’ that present a fixed number of bits represented in IEEE 754
format. In single precision, the mantissa part represents by 23-bits and for normalization 1-bit is added to the
MSB, the exponent part represents by 8-bits, and it is biased to 127, and the exponent part is represented in
excess of 127-bit format and the sign bit represents by 1 bit or MSB. The sign bit or MSB represents the sign
of the floating-point number, hence the floating-point number is negative when MSB is 1 else the number is
positive when the MSB bit is 0.

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Table 1. IEEE 754 format representation of 32-bit and 64-bit
Precision Sign Exponent Mantissa
Single precision (32 bit) 1-bit 8-bit 23-bit
Double precision (64 bit) 1-bit 11-bit 52-bit
3. DIVISION ALGORITHMS
The division operation requires a very high number of iterations and is implemented as an inverse
process of the multiplication operation. When combined with shift operation the number of iterations is
reduced to the number of result valid bits. The fast algorithm was developed to produce a more precise result
that requires more iteration and for this purpose algorithms that were developed are Goldschmidt and
Newton-Raphson division algorithms. These algorithms within a single iteration converge much faster
however; several additions or subtractions and multiplications operations are needed. The Goldschmidt
division computational algorithm goal to converge the denominator to 1 is achieved by continual
multiplication of the numerator and denominator by the same factor Fi. The quotient produced as a result
directly converges the numerator at the same time. By calculating the multiplicative inverse of denominator
which is produced by multiplied numerator to produce the result and this process is based on Newton-
Raphson division computational algorithm.
3.1. Goldschmidt algorithm
A complex initialization is used to compute continual iterations for the Goldschmidt computational
division algorithm. To get the resultant quotient this algorithm used a common factor 𝐺𝑖, this common factor
continually multiplying both the dividend and divisor that converge the divisor (𝐷) to one is an iterative
process used by this algorithm [25]. In this proposed floating point divider design to perform arithmetic
operations, a 32-floating point multiplier module and subtractor module are used in Goldschmidt
computational division algorithm. In this division algorithm, the divisor should be in the interval (0, 1) by
scaling the numerator and denominator, shifting operation is used for scaling the divisor. The fast division
algorithms are developed to produce a precise result for which more iteration is required. For a single
iteration, the Goldschmidt algorithm converges much faster for computing the result to perform the continual
iteration process thus; more multiplication and subtraction operations are needed. This computational
algorithm produces an optimized result by computing iterations in one cycle. As shown in (1) where N is the
numerator, 𝑄 is a quotient, 𝐷 is the denominator and 𝐺𝑖 is factoring at iteration 𝑖.
𝑄 =
𝑁
𝐷
∏ 𝐺𝑖
𝑛
𝑖
∏ 𝐺𝑖
𝑛
𝑖
(1)
The 𝐺𝑖+1 is the factor for the next iteration calculated using (2) and that is used to substitute in (1)
for calculating the further, iterations. For the next iteration, the numerator and denominator have to be
calculated and it’s done by (3).
𝐺𝑖+1 = 2 − 𝐷𝑖 (2)
𝑁𝑖+1
𝐷𝑖+1
=
𝑁𝑖
𝐷𝑖
𝐺𝑖+1
𝐺𝑖+1
(3)
Floating point division is based on the Goldschmidt computational algorithm where inputs 𝑁 and 𝐷
are the numerator and the denominator given as input for division. To calculate one iteration Goldschmidt's
computational algorithm uses one multiplier and one subtraction module. This computational algorithm
scales the numerator and denominator by the common factor 𝐺𝑖, in four iterations this algorithm computes
directly and produces the quotient in the result. Hence, the numerator converges directly to the quotient as the
denominator converges to one. For more refined results, more iterations are performed. F1, F2 and F3 are the
three floating point input numbers the resultant division of F1 by F2 is followed by the addition or
subtraction of F3 this operation performed by the Goldschmidt computational algorithm based proposed
floating-point divider is carried out as shown in the simulation results.
3.2. Newton-Raphson algorithm
Newton-Raphson division algorithm, this computational algorithm starts with the computation of the
multiplicative inverse; the multiplicative inverse is calculated by an iterative process and the resultant
quotient is found by l multiply dividend and the calculated multiplicative inverse. The shifting operation is
used for scaling the divisor. To get more precise results more iterations are required and for this purpose, fast

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division algorithms are used. Newton-Raphson computational algorithm-based floating point division
flowchart, using this algorithm to compute iteration converges much faster. Thus, for getting precise results
number of iterations required and to perform continual iterations several multiplication and subtraction
operations are required.
In this computational algorithm, an optimized result can be achieved in one cycle by computing
three iterations, for which two multipliers and a subtraction module are for iteration. More iteration is
performed to refine the multiplicative inverse. The Newton-Raphson division algorithm is based on
calculating of reciprocal of the denominator and producing the result reciprocal of the denominator
multiplied by the numerator. The formula to calculate the multiplicative inverse of the denominator) where 𝑍𝑖
is the multiplicative inverse at iteration 𝑖 is given by (4) in which D is the denominator.
𝑍𝑖+1 = 𝑍𝑖 + 𝑍𝑖(1 − 𝐷𝑍𝑖) = 𝑍𝑖(2 − 𝐷𝑍𝑖) (4)
In Newton-Raphson division by scaling the denominator, minimization of maximal relative error can
be achieved in the interval [0.5, 1]. The Newton-Raphson division uses complex initialization which requires
addition, multiplication and subtraction operations and therefore this initialization has been classified as the first
iteration cycle. 𝑍0 represents the initial iteration and its initialization is carried out by (5).
𝑍0 =
48
17
−
32
17
𝐷 (5)
4. FLOATING POINT MULTIPLIER
Figure 1 shows the architecture of the proposed multiplier module which used Vedic multiplication,
so (24x24) Vedic multiplier is used for calculating the mantissa part which enhances the overall performance
of the proposed 32-bit multiplier. The multiplication module unit is divided into three parts for two inputs,
input A [31-0] and B [31-0] which are in 32-bit floating point number IEEE 754 format for multiplication,
this format describes a fixed number of bits i.e.; sign (s1 and s2), exponent (e1, e2) & mantissa (m1, m2).
Figure 1. The proposed design for 32-bit floating point multiplier (FPM)
4.1. Mantissa unit
For getting high computation speed in this module for multiplication of the mantissa unit, lower bits
of inputs m1 and m2 that is A [22-0] and B [22-0]. Calculation is done using (24x24)-bit Vedic multiplier
and produces an output which is 24-bit normalized output which has leading 1 as their MSB. This multiplier
module efficiently used Vedic mathematics which also provides higher throughput.

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4.2. Exponent unit
A ripple carry adder is used in the exponent unit for the exponent calculation, e1 and e2 that is A
[30-23]) and B [30–23]. A [30-23]) and B [30–23] are the inputs given to the 8-bit ripple carry adder unit
which computes and the result is biased to 127. The result by this exponent unit the two cases overflow and
underflow are carefully handled so that to get the optimized result.
4.3. Sign unit
The two inputs s1 and s2 that A [31] and (B [31] is the 31st bit of the 32-bit floating point numbers
are given to the sign unit and computed by XORing s1 and s2 and the output of the exclusively-OR (XOR)
gate is the sign floating point number. Both Goldschmidt and Newton-Raphson computational algorithms
were designed using this multiplier module which is implemented using Vedic mathematics for getting
efficient results. If 32-bit single precision IEEE-754 format is considered then a 24-bit Vedic multiplier is
used to calculate the mantissa. To calculate the exponent, an 8-bit ripple carries adder is used, this
implementation results in a low–area, ease of implementation and simple layout. The input to the multiplier
for multiplication is provided in standard format for single precision which is in IEEE 754 format.
5. FLOATING POINT SUBTRACTOR
In the substractor module X [31-0] and Y [31-0] are the inputs given to the floating point
substractor. The floating-point inputs to the substractor module are separated as a sign, exponent and
mantissa and they are represented in IEEE 754 format further, the substraction operation is done in a stepwise
manner. Initially, to perform the substraction operation the floating-point number is unpacked so that the sign
(b1, b2), exponent (e1, e2) and mantissa (s1, s2) are identified for both the inputs. Then the alignment is done
to equalize the exponent and normalization is performed in the mantissa part. By comparing the mantissa m1
and m2 the relation between inputs exponent e1 and e2 which are (X [30-23]) and (Y [30-23]) is determined
if neither of the operands is infinite. After this the shifting operation is done on mantissa until the exponents
e1 and e2 that is becomes equal that is X [30-23]) = Y [30-23]. Further, the mantissa m1 and m2 are
substracted that is X [23-0] and (Y [23-0] after alignment and normalization process and after this resultant
value of mantissa are rounded off. Finally, the concatenation is done to the sign, exponent and mantissa parts.
The result of this concatenation is the single precision floating point number. The complete architecture of
the 32-bit floating point substractor module is shown in Figure 2 where X and Y are the inputs given to
substractor and this is used to calculate the iterative process computes in the fast computational division
algorithms that are the Goldschmidt algorithm and Newton-Raphson algorithm for getting the optimized
division result.
Figure 2. Architecture of floating point substractor (FPS)

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6. SIMULATION RESULT
ISim simulator is used for performing simulations and the Xilinx Spartan 6 SP605 evaluation
platform is used to implement and synthesize the proposed floating point divider design which is based on
Vedic mathematics. The two numbers (N, D) are given as input for the division to 32-bit floating point
divider. Table 2 shows the sample inputs (N, D) and the resultant output quotient (Q) of the sample inputs for
simulation for 32-bit floating point division.
Table 2. The sample inputs (N, D) and the resultant output (Q) of the sample inputs
Parameter Decimal Sign Exponent Mantissa
N 3.14 0 10000000 10010001111010111000011
D 8.56 0 10000010 00010001111010111000011
Q 0.366822 0 01111101 01110111101000000100110
The device utilization parameters of the Xilinx power estimator (XPE)-14.3 are shown in Table 3
and Table 4 for the Goldschmidt and Newton-Raphson computational algorithm-based 32-bit floating point
division implemented on the Xilinx Spartan 6 SP605 valuation platform FPGA. The device utilization
parameters such as the number of slice registers, look-up tables (LUTs), occupied slices and bounded input
and bounded output utilization are efficiently less utilized. By using Goldschmidt-based single floating-point
divider with respect to Newton-Raphson based single precision floating point divider as shown in device
utilization parameter Table 3 and Table 4.
Table 3. Device utilization parameters Goldschmidt computational algorithm
Logic utilization Used Available Utilization
Slice Registers Nos. 408 54,576 0.7%
Slice LUTs Nos. 9,185 27,288 33%
Occupied Slices Nos. 2,913 6,822 42%
Bonded IOBs NOs. 193 296 65%
Table 4. Device utilization parameters for Newton- Raphson computational algorithm
Logic utilization Used Available Utilization
Slice Registers Nos. 556 54,576 1%
Slice LUTs Nos. 11,584 27,288 42%
Occupied Slices Nos. 3,719 6,822 54%
Bonded IOBs NOs. 257 296 86%
The Table 5 compares the performance of the proposed DAF units using two computational
algorithms, Newton-Raphson and Goldschmidt, in terms of power consumption and latency time. The table
clearly indicates that the Goldschmidt-based DAF performs better in both aspects compared to the Newton-
Raphson-based DAF. Therefore, the statement correctly concludes that the Goldschmidt-based DAF gives
better performance in terms of latency and power with respect to the others.
Table 5. Comparative analysis of DAF unit in terms of power and latency time
S.No. Jeevan et al. [9] Nagendra et al. [10]
Using DAF
Newton-Raphson
Using DAF
Goldschmidt based
Latency time 175.49 ns 130.8 ns 110 ns 75 ns
Power - 0.050 W 0.057 W 0.031 W
The device summary reports of XPE-14.3 for the proposed 32-bit floating point division using
Goldschmidt and Newton-Raphson computational algorithm is shown in Table 6 where Goldschmidt-based
division parameters such as junction temperature, on-chip power, thermal margin and ambient temperature
(ϴJA) are less as compared to the Newton-Raphson based division. The synthesis and implementation is
done on Spartan-6 (SP605) evaluation platform field programmable gate array of the proposed 32-bit floating
point divider. Figures 3 and 4 show the simulation results where Q is used to represent the quotient and the
two numbers N and D represent the numerator and denominator given as input in IEEE-754 format to 32-bit
floating-point divider given as input to the proposed 32-bit floating point divider which is designed using
Goldschmidt and Newton-Raphson computational algorithms.

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Table 6. Device summary report of XPE-14.3 on Spartan-6 SP605 evaluation platform FPGA of floating
point divider
Using Goldschmidth algorithm Using Newton-Raphson algoroitm
Specifications Values Values
Junction temperature (ºC) 22.5 25.5
On-Chip power (Total) 0.031W 0.037 W
Thermal margin
58.5 ºC
(3.8 W)
59.5 ºC
(4.0 W)
Effective (ϴJA) 13.3 ºC/W 14.3 ºC/W
Figure 3. Goldschmidt computational algorithm based floating point division simulation result
In Figure 3 for Goldschmidt-based single precision division computations G1, G2, and G3 are used
to represent the first, second, and third iteration results and the final iteration result is represented by G4. In
Figure 4 for Newton-Raphson-based single precision division computations. The initial value represented by
Z0, the Z1, and Z2 are used to represent the first and second iteration results, and the final iteration result is
represented by Z3.
Figure 4. Newton-Raphson computational algorithm based floating point division simulation result

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7. CONCLUSION
Goldschmidt and Newton–Raphson computational algorithm-based single precision floating point
division were synthesized and implemented using Xilinx Spartan 6 SP605 evaluation platform on FPGA and
simulations of the proposed module were done on ISim simulator. For high computational arithmetic
operations design presented in this paper is useful. Floating point divider performance is improved by
designing multiplier module used in it using Vedic mathematics so, high computational speed and throughput
are achieved. The device utilization parameters are optimized which results in an improvement in power
consumption which is reduced by 26% and latency time reduced by 42.6% respectively with respect to
existing DAF using Goldschmidt-based single precision floating point division.
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BIOGRAPHIES OF AUTHORS
Naginder Singh completed his Master of Technology from the National Institute
of Technology, Kurukshetra (Haryana) and Bachelor of Technology from College of
Engineering and Technology, Bikaner. He is currently serving as an Assistant Professor in the
Department of Electronics and Communication Engineering, M.B.M. University, Jodhpur,
Rajasthan (India). His research interests are in FPGA-based design, embedded systems and
sensors, embedded system design, and VLSI. He can be contacted at email:
naginder.ece@mbm.ac.in.
Kapil Parihar received his B.Tech. degree in Electronics and Communication
Engineering at Bikaner Engineering College, Bikaner and M.E. in Digital Communication at
M.B.M. Engineering College, Jodhpur, Rajasthan (India). He is full-time Assistant professor at
M.B.M. University, Jodhpur, India. His research lines are optical communication and photonic
crystal fiber. He doing his Ph.D. in Electronics and Communication Engineering Department
from M.B.M. University Jodhpur and his field interest is terahertz antennas, electromagnetic
engineering and antennas/sensors for communication, radar, imaging and sensing systems,
UWB wireless communication, antennas for portable devices, and antennas for base stations in
wireless communications. He can be contacted at email: kapil.ece@mbm.ac.in.

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