An Error Estimation Model for Fixed-Point Number Representations Hwanyong Lee Solution Division HUONE Inc. Daegu 702-250, Korea hylee@hu1.com Nakhoon Baek * School of EECS Kyungpook Nat’l Univ. Daegu 702-701, Korea oceancru@gmail.com Youngsul Shin School of EECS Kyungpook Nat’l Univ. Daegu 702-701, Korea youngsulshin@msn.com Kee Hyun Park Dept. of Comp. Eng. Keimyung Univ. Daegu 704-701, Korea khp@kmu.ac.kr ABSTRACT Recently, we need to display multimedia contents on rel- atively low-powered devices such as mobile phones and PDA’s. To support the multimedia capability on these de- vices, we fundamentally need efficient way of perform- ing various mathematical operations. Although most de- vices use floating-point number representation formats such as IEEE754, recent application programs often use fixed-point number representation formats for more effi- cient real number calculations. Currently, mobile devices without floating-point processing units and/or having rel- atively low computing powers tend to use it mostly due to its efficiency. In contrast, we cannot avoid greater numer- ical errors in comparison to the widely used floating-point number representation. In this paper, we analyze the nu- merical errors in the fixed-point number calculations and investigate the suitability of fixed-point number formats and its operations. We also show experimental results for various choices of fixed-point location in the real number representation. Keywords: Numerical Error, Fixed-point Number Rep- resentation, Error Propagation. 1 Introduction Nowadays, multimedia contents can be displayed on the more and more small devices. Although small electron- ics devices such as PDA’s and mobile phones can display various multimedia contents, they still need to reduce re- quired computing power. In this paper, we focused on a fundamental data representation model for real num- bers. Most applications need to handle real number data. In contrast, as we know, real number computation usu- ally requires relatively heavy computing power, and thus, most mobile devices are hard to use generic computa- tional models for the real numbers. * corresponding author In the area of recent mobile phone applications, for ex- ample, the vector graphics facility such as SVG(scalable vector graphics)[1] and Flash Lite[2] become widely used due to their remarkable graphics features for variety of applications, as shown in Figure 1. This vector graphics facility basically requires geometric operations based on the floating-point number calculations. Since we should meet the problem of numerical errors and their accumulations for almost all types of geomet- ric operations, they have tried to overcome these difficul- ties for several decades. As the results of various previ- ous works, we already have some solutions such as sym- bolic computations, rational number computations, etc. We also know the epsilon geometry[3] to derive some nu- merical algorithms to pre-calculate the error ranges. All these theoretically excellent methods, however, cannot be efficiently used on the small-scale computing devices such as mobile phones, PDA’s, etc. In this pa- per, we show the difficulties in the numerical and geomet- ric operations in the small-scale computing devices, and show that fixed-point number representations are more suitable for this purpose. And then, our theoretical cal- culations and experimental results show the errors in the typical geometric operations and their propagation mod- els. 2 Background Along to the advances in the computer technology, the computing powers of mobile and small-scale devices have been rapidly upgraded. In these days, typical mobile phones have the following computing powers[4]: • 200 to 500 MHz 32bit RISC CPU, • without floating-point number co-processors, • with more than 64MB RAM and • with more than 512MB flash memory. As an example of commercial products, ARM11 core- based CPU’s are now available, and they will include