Abstract

1. Research of a presentation of formats with floating-point numbers and their features

2. Research of methods of a presentation and a development of converter of mantissa number

3. The programming model of postbinary formats

4. Investigation of the temporal characteristics

References

Introduction

         At present, hardware of computers is able to produce all sorts of calculations with numbers, which can be represented only in two formats - integer and floating point numbers. Integer arithmetic uses a finite subset of the set of integers. If the hardware is working, but the programs do not contain errors, integer arithmetic and software adds work without errors. For example, on the base of the integer arithmetic it is possible to construct the arithmetic of systematic fractions of arbitrary digit capacity and common fractions.

         For an accurate representation of the results of arithmetic operations, a modern computer is able to save the required quantity of digits, for an accurate representation of an arbitrarily large integer. A floating point numbers in a computer are represented in floating point format. In general, the format of representation of numbers is the best cost of memory and CPU time to work with numbers and calculation accuracy. But as it contains initially some errors, then its error is often unacceptably high for the exact science and critical industrial computing.

         The purpose of this research is to study the shortcomings of modern methods and formats for floating point numbers in computer memory and to improve the accuracy of these standards by means of the development of modified formats, the development of software and hardware model number conversion to new formats.

1. Research of a presentation of formats with floating-point numbers and their features

         Nowadays  one of the most prospective research directions in  the computer science is to design specialized processor devices  with  a low power consumption. This research shows the instantiation of the 1st stage, during which it is suggested a departure from the conventional (binary) representation of real numbers in computers in the direction post binary. In this representation of floating-point format it is offered to use  a number of modified formats  [2, 3, 5], which are  able to represent not only real numbers as such, and their aggregate in the form of fractions (a couple of numbers - numerator and denominator) and intervals (a couple of numbers - the boundaries of the interval .)

         For the further test the algorithms of arithmetic and logical operations with numbers postbinary it was necessary to develop an appropriate programming model of formats. In this case the unit numbers  should be so flexible in order to have  possibility to represent  an arbitrarily accurate representation of the set of all real numbers, that is  the transformation of the input number x ? postbinary format (pbinary [x]) and pbinary [x] ? Output number y must be as accurate as possible and strive for equality x = y.

2. Research of methods of a presentation and a development of converter of mantissa number

         In [3, 5] it is suggested  a number of ways to overcome the problems which  associated with limited numbers of digits, because the use during  the calculation of digits, which is significantly exceeding the standard, is one way to get the correct results. All these  ways  in the process of computation can perform the following operations:

- Increase (or alignment) of a capacity in order to avoid an overflow of a result and the correct implementation of algorithms;

- Implementation of the so-called deferred division, when numerator and denominator separately computed, and division is the last step of the calculation;

- The use of interval computations.

The suggested modified format floating-point to 128-bit number (S, P, M - a sign, mantissa and the order number, MF, CF - modifier and format numbers, n - index)

Figure 2.1 - The suggested modified format floating-point to 128-bit number (S, P, M - a sign, mantissa and the order number, MF, CF - modifier and format numbers, n - index)

 

The suggested modified format floating-point to 128-bit number (S, P, M - a sign, mantissa and the order number, MF, CF - modifier and format numbers, n - index)

Figure 2.2 - The suggested modified format floating-point to 128-bit number (S, P, M - a sign, mantissa and the order number, MF, CF - modifier and format numbers, n - index)

 

The suggested modified format floating-point to 256-bit number (S, P, M - a sign, mantissa and the order number, MF, CF - modifier and format numbers, n - index)

Figure 2.3 - The suggested modified format floating-point to 256-bit number (S, P, M - a sign, mantissa and the order number, MF, CF - modifier and format numbers, n - index)

 

3. The programming model of postbinary formats

         In modern  computing devices formats ,that conform to IEEE754-2008, which consist of field sign, mantissa and the order number, are used to represent the actual (real) numbers. The same structure is saved in post binary   formats, but  a mantissa field has undergone a modification, selecting a small number of bits in the ID field [3, 5].

The program interface PBinary

Figure 3.1 - The program interface PBinary

         The programming model of  PBinary  is a source of  a converter of  decimal to binary, which is "packaged" in postbinary  formats. PBinary implements  in Java and takes about 300 KB of disk space. Creating of a model of PBinary as Java-based applications have been justified, first, by a cross-platform, and, secondly,by the increased speed of execution of elementary mathematical operations. The program interface of  PBinary  is shown in Fig. 3, the source code presented in Appendix B.

Presentation format fields pbinary128

Figure 3.2 - Presentation format fields pbinary128

The exact value of the number of 10 c / s, obtained from the format pbinary128, in the simplest (a) and exponential (b) a

Figure 3.3 - The exact value of the number of 10 c / s, obtained from the format pbinary128, in the simplest (a) and exponential (b) a

4. Investigation of the temporal characteristics

         For investigating of the temporal characteristics of the converter a series of numbers is selected, which was given to the input program. The principle of run-time checking of the program was based on samples of the system time when you click "start" until the end of all transformations. The results are presented in Fig. 4, where run time is indicated on   the vertical axis, in milliseconds, and exponential value for each number - on the horizontal axis. Testing was done on the computer with the following parameters: CPU: Celeron M 1.6 GHz, RAM: 1024 MB, HDD: SATA 80 026 MB.

         The research results show that for small values of the exponent, the time depends on the number of significant digits of the number itself, and for large values of the exponent the difference between the initial numbers is not significant. Analyzing this results it can be assumed that the negative values of the exponent, the dependence of time from the magnitude is linear, but with positive values - exponential. This phenomenon can be explained in the following way: for negative values of the exponent the plural operation of multiplication occurs (in fact, repeated addition), while for positive values of the exponent a division operation  runs (in fact, repeated subtraction).

Stages of the converter

Figure 4 - Stages of the converter (4 frames, lenght 2 s., 7 repeads, size — 62 КБ.)

This master's work is not completed yet. Final completion: December 2012. The full text of the work and materials on the topic can be obtained from the author or his head after this date.

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