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Numerical Precision

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Numerical Precision

by

John Nelems

Professor Lu

CIS512

Strayer University

January 26th, 2014

Floating Point Format for storing floating numbers in memory
Numeric precision and representation are very familiar discussions in computer science. The way that numbers are stored and represented in a computer system has to be represented correctly in order to avoid costly mistakes. Numbers are stored as binary numbers in computerized system.
Floating point numbers are expressed as the product of two parts: the mantissa and a power of two. For example: ± mantissa x 2 exponent
The mantissa represents the binary digits of the floating number. The power of two is represented by the exponent. The stored form of the exponent is an 8 –bit value ranging from 0-255.
The mantissa is a 24-bit value (represents about seven decimal digits) that has a most significant bit (MSB) that is always 1 and not stored. Also, it has a sign bit that indicates if the floating point number is positive or negative. Floating point arithmetic is one of the most widely used ways of representing real numbers in today’s computer systems. The use of floating point arithmetic provides the following benefits: speed, accuracy, range, portability and ease of implementation and use for numerical applications. Floating point arithmetic uses IEEE 754 standard that is used by most computer manufacturers which makes it easy to use and manipulate floating numbers.

Is the floating point format more efficient in memory or not?
I believe that the floating point format is definitely efficient for memory. Memory can be thought of as an array of bytes. Each memory location has an address …the address of the first byte is 0, followed by 1, 2, 3, and so on. Floating point format is just like memory because it is assigned a location for a floating point value. Whenever the program is ran, the computer reserves a space for that variable in its memory.
Analyze the format used to store numbers using binary coded decimal format
Numbers are stored on the computer are in binary form. In other words, the information is encoded as a sequence of 1’s and 0’s. The memory is organized into 8-bit bytes on most computers. The 8- bit byte will be stored in memory and will have a separate address. Precision is what is used to express values which are represented in alternatives, decimal digits, bytes, or binary bits.
Determine if the BCD format is more or less memory efficient than the floating point format
Binary Coded Decimal format is a type of format that is used to simplify devices that use decimal numbers. The numbers are coded in a binary format. The BCD value is an unassigned 8- bit integer, with a range of 0 to 9. In contrast, the floating point format is a method used to store and calculate numbers in which the decimal points do not line up as in fixed numbers. Therefore, I think that in regards to which format is memory efficient would be Binary Coded Decimal because it can accurately represent numbers assuming that there are enough bytes to store the number. Floating Point formats can’t accurately represent that number but only provide an approximation.
Compare and contrast the BCD format to the floating point format in terms of: precision, performance in calculation, and memory usage. The binary coded decimal format precision uses the IEEE 754 and how they are in a format of single, double, double extended and quadruple formats that can accurately represent numbers. In contrast, the precision of the floating point format is accurate because of they are implemented into hardware which cannot accurately represent all decimal fractions. Now, in regards to the way that performance in calculation and memory usage will be precise in calculations and properly stored in memory because of having an assigned variable once the calculation is performed. However, the floating point format will not provide that same level of accuracy in calculation and memory usage.
Which format would best represent processing efficiency? The format that I think will best represent processing efficiency will be the binary coded decimal. Binary Coded Decimal is the format that represents decimal numbers (integers) in which each digit is represented by four bits (nibble). There are several advantages of BCD but one key advantage is that there is no limit to the size of a number. If there is another digit added, you would simply add a new 4-bit sequence. In contrast, the numbers that are represented in binary format are limited to the largest number that can be represented by 8, 16, 32 or 64 bits. Also, binary coded decimal represents other numbers as well, such as exponents and hexadecimal to help with the numeric representations in computer systems.

Rounding errors with floating point numbers in banking industries
Rounding errors occur because of approximating a number with fewer digits from an original number with more digits. Rounding errors are not an issue in manual computation but can become a problem when it comes to mathematical operations, especially in the banking industry. The way that rounding errors became an issue with floating point numbers have a limited number of digits which can’t accurately represent real numbers. The numbers that exceed the format are rounded. The way the numbers are rounded will be favorable towards the banking institution.
The rounding errors with the floating numbers have caused some ethical issues with banks. Banks knowingly profit from this error in which customers are unaware of what is happening. Now, some of the ethical issues that are concerns regarding the banking industry are the following: * Usurious practices - charging excessive interest rates, abusive commissions, and profitable credit charges on loans and etc. * Speculative banking- speculative investments and irresponsible credit lending practices.
This is just a couple of the ethical issues that are listed, in which will be favorable for the bank and a loss for the customer.

Differentiate between the exponential format and the floating point format
There is definitely a difference in the way that the exponential format and floating point format differs in the way that they’re represented in terms of format, processing speed and accuracy. Exponential notation is a method that is used to represent floating point numbers. Exponential notation consists of 6 components: * Mantissa * Sign of mantissa * Exponent * Sign of exponent * Base of exponent * Location of the fraction point
Example: -50.5 x 10 ̵ (20)
Floating point representation is very similar to exponential notation. Floating point representation uses the IEEE 754 Standards. * Binary is used instead a decimal * Storage size consists of 32, 64, and 128 bits * 32-bit floating point format
Left most bits is the mantissa sign
Followed by 8 bits exponent
Followed by 23 bits mantissa
The fraction point is implied to be at the beginning of the mantissa
Exponent is stored in excess- 128 notation
Base of the exponent is implied as base 2 Example Floating Point Format SEEMMMMM format Excess – 50 Base 10 exponent Base 10 mantissa Implied decimal point at the beginning of the number
The use of IEEE 754 Standard uses both exponential and floating point formats which process with great speed and accuracy.

References
Cody, W (1973). Static and dynamic numerical characteristics of floating- point arithmetic. IEEE Transactions on Computers
Jones, C.B. (1984). A Significance Rule for Multiple-Precision Arithmetic, ACM Transactions on Mathematical Software
Wegener, A. (2012). HPC and “Big Data” Apps Tap Floating-Point Number Compression. Electronic Design…...

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