Numerical Precision, Research Paper Example

The arithmetic application of floating points is a subject which is unknown to many computer consumers. Notwithstanding, all of the computing systems apply floating point information types. All of the computer operating systems reference the floating point system as being an overflow. In computer systems architecture, the system builders frequently require additional information with regards to arithmetic expression in the decimal floating point system. The IEEE floating point system is acknowledged as being the core concepts for computer architecture. The floating point system has the aspect of rounding uncertainties. This essay will include the perspectives of Goldberg (1991), Duale et al. (2007), Droogendyk (2011) and Brinkman and Sanders (2012)

Floating point numbers are stored in binary. It is loess challenging to apply the fixed position binary numbers. Some of the binary bits would be used to express the segment which is to the right of the decimal point. The factional part or the decimal part of the number must be viewed as an aggregate of the binary fractions. This is viewed in the same manner as any other type of number base (Goldberg, 1991). A fixed position decimal point format with one number before the decimal point and three numbers after the decimal point would be expressed as the following example: 0.001 would be expressed in extended binary form as 0 * 2?(0) + 0 * 2?(-1) + 0 * 2?(-2)+ 0 * 2?(-3)+ 1 * 2?(-4 ).  In fractional form this number would be 1/16 and in base ten numbers it would be 0. 0625.

The application of decimal arithmetic functions are implemented in business and economic models. Notwithstanding, the correlating computations are administrated by software. Consequently, the software applications which utilize decimal information are plagued with performance challenges. The application of the innovative decimal point floating matrix is anticpated to enhance the production of software arithmetic. IBM is the inventor of the system z9th which applies the decimal floating point instruction to computer algorithms (Duale et al., 2007).

The operand system for the decimal floating system is composed of a coefficient, exponent and sign bit. The arithmetic value of the operand is delineated as coefficient x 10^operand  x (-1) ^sign . The numbers which are applied in the decimal floating system are not standard. There are various methods of expressing a value. The number 100 may be expressed as 1 x 10² or 100 x 1⁰. In order to provide criteria, the IEEE Standard P754 provides a definition of an exponent which is preferred for data input of all operations. The decimal floating numbers are stored in a format which is designated as a densely packed decimal (DPD). The DPD matrix is composed of four distinct fields (Duale et al., 2007).

These fields are s which represents the sign, CF which represents a combination field, BNXCF which is used to describe the integers which are to the left of the decimal point. The BXCF is composed of between six to twelve bits which are applied for expressions which vary from a short aspect to an extended aspect. The CCF which represents the coefficient continuing field applies data values which vary from twenty bits to one hundred and ten bits. These data values are used to express values which vary from short aspects to extensive aspects. The DPD blocks each represent one declet. There are twenty four repetitive strings. In the three digit blocks, the encoding is repetitive (Duale et al., 2007). The BCD binary coding decimal offers a larger range of storage. The BCD numbers are evaluated in base ten. The floating point numbers are expressed in binary or base 2. There are an unlimited number of digits which may be applied in the BCD system. The floating point numbers have the availability of a non-changing numbers of bytes in the memory. The floating point system can only accurately calculate factions which are divisible by two. The arithmetic operation becomes tedious when the fractions are demonstrated in base ten. Consequently, the BCD system is more efficient than the floating point matrix (Goldberg, 1991; Droogendyk, 2011).

The SAS formats are the most efficient and effective numerical systems to apply. The SAS format technique enhances coding efficiency and program effectiveness. The SAS formats are the most effective formats for data summarization, data presentation and data conversion. The SAS formats can be adapted to the needs of the user. In the SAS formats, the formatting requisites are requires a prefix to the expression. The format may reach six characters in length and may in the case of numeric expressions, may be extended to eight characters. The formal expressions in SAS do not terminate in numerical expressions. The letter w is used to detail the breadth of the value which is being interpreted or expressed. The dollar sign, decimal places and commas are included. Periods are also included in order to differentiate the variable designation from the formats. The application of decimal places is considered to be optional and may be used in numerical values (Droogendyk, 2011).

In the case which involved Intel Computer Corporation, a defect had been discovered in the Intel Pentium P5 chip. This defect pertained to the division function in the floating decimal points. The rounding errors had gone unnoticed by the majority of the users. Notwithstanding, the majority of the personal computers used the Intel Pentium P5 chip in 1994. Intel Computers replaced all of the chips for the users who became aware of the rounding problems. The expense for the correction of the rounding uncertainties was estimated at a cost of $500,000,000. The ethical dilemma caused many to mistrust the IntelPentium brand (Brinkman & Sanders, 29012).

The benefits which are derived from the decimal floating point system overt the exponential fixed point numerical values are that the decimal floating point values can demonstrate a greater scale of values. The decimal floating point system is composed of seven decimal places whereas the fixed decimal point system is composed of two decimal places. This aspect causes the decimal floating point system to have greater precision. The decimal floating point matrix has the requisite of greater storage space in comparison to the fixed point systems with regards to computer memory storage. The velocity of the floating point function is a significant measure of commuter production. The computer production is established in FLOPS (Goldberg, 1991).

Conclusion

The usage of floating points in arithmetic calculations is a subject which many are unaware. The IEEE floating point system is recognized in the design of computer systems. The floating point system has the aspect of rounding errors. At times, the errors may cause ethical problems for the users, as demonstrated in the case of Intel Computers.

References

Brinkman, W. & Sanders, A. (2012). Ethics in a computing culture. Cengage Learning.

Droogendyk, H. (2011). SAS formats: effective and efficient, Paper CC-06. SESUG, 2011: 1- 16.

Duale, A.Y., Decker, M. H., Zipperer, H. – G., Aharoni, M., & Bohzic, T. J. (2007). Decimal floating point in z9th: an implementation and testing perspective. Journal of Research and Development, 51(1/2): 217- 227.

Goldberg, D. (1991). What every computer scientists should know about floating point arithmetic. Xerox.