Computer Organization and ASSEMBLY LANGUAGE

Computer Organization and ASSEMBLY LANGUAGE Lecture 7 & 8 Floating Point Representation Binary Coded Decimal Course Instructor: Engr. Aisha Danish

Real Numbers • Numbers with fractions • Could be done in pure binary • 1001.1010 = 24 + 20 +2-1 + 2-3 =9.625 • Where is the binary point? • Fixed? • Very limited • Moving? • How do you show where it is?

Floating point • In computing, floating point describes a method of representing an approximation of a real number in a way that can support a wide range of values • The numbers are, in general, represented approximately to a fixed number of significant digits (the significand) and scaled using an exponent • The term floating point refers to the fact that a number's radix point (decimal point, or, more commonly in computers, binary point) can "float"; that is, it can be placed anywhere relative to the significant digits of the number • Floating point can be thought of as a computer realization of scientific notation

How floating-point numbers work? The idea is to compose a number of two main parts: • A significand that contains the number’s digits. • Negative significands represent negative numbers. • An exponent that says where the decimal (or binary) point is placed relative to the beginning of the significand. • Negative exponents represent numbers that are very small (i.e. close to zero). Such a format satisfies all the requirements: • It can represent numbers at widely different magnitudes (limited by the length of the exponent) • It provides the same relative accuracy at all magnitudes (limited by the length of the significand)

Floating Point • A typical 32-bit floating-point format • The leftmost bit stores the sign of the number • The exponent value is stored in the next 8 bits • The representation used is known as a biased representation • A fixed value, called the bias, is subtracted from the field to get the true exponent value • Typically, the bias equals 2^(k-1)-1where k is the number of bits in the binary exponent. In this case, the 8-bit field yields the numbers 0 through 255 • With a bias of 127(2^7-1), the true exponent values are in the range -127 to +128 • In this example, the base is assumed to be 2

Floating Point Examples

Signs for Floating Point • Mantissa/Significand is stored in 2s compliment • Exponent is in excess or biased notation • e.g. Excess (bias) 128 means • 8 bit exponent field • Pure value range 0-255 • Subtract 127 to get correct value • Range -127 to +128

Normalization • A normalized number is one in which the most significant digit of the significand is nonzero • For base 2 representation, a normalized number is therefore one in which the most significant bit of the significand is one. • The typical convention is that there is one bit to the left of the radix point • i.e. exponent is adjusted so that leading bit (MSB) of mantissa is 1 • Since it is always 1 there is no need to store it • (Scientific notation where numbers are normalized to give a single digit before the decimal point e.g. 3.123 x 103)

IEEE 754 • Developed to facilitate the portability of programs from one processor to another and to encourage the development of sophisticated, numerically oriented programs • The standard has been widely adopted and is used on virtually all contemporary processors and arithmetic coprocessors • 32 and 64 bit standards • 8 and 11 bit exponent respectively

IEEE 754 Formats

Required Reading • Stallings Chapter 9 • IEEE 754 on IEEE Web site

The 8421 BCD Code • BCD stands for Binary-Coded Decimal. • A BCD number is a four-bit binary group that represents one of the ten decimal digits 0 through 9. Example: Decimal number 4926 4 9 2 6 8421 BCD coded number0100 10010010 0110

QUIZ Convert the BCD coded number 1000 0111 0001 into decimal. BCD Coded Number1000 0111 0001 Decimal Number 8 7 1

QUIZ Convert the decimal number 350 to its BCD equivalent. Decimal Number 3 5 0 BCD Coded Number0011 0101 0000

Binary Coded Decimal (BCD) • Would it be easy for you if you can replace a decimal number with an individual binary code? • Such as 00011001 = 1910 • The 8421 code is a type of BCD to do that. • BCD code provides an excellent interface to binary systems: • Keypad inputs • Digital readouts

ex2: BCD-to-dec 10000110 001101010001 1001010001110000 Binary Coded Decimal ex1: dec-to-BCD • 35 • 98 • 170 • 2469 Note: 1010, 1011, 1100, 1101, 1110, and 1111 are INVALID CODE!

BCD Addition • BCD is a numerical code and can be used in arithmetic operations. Here is how to add two BCD numbers: • Add the two BCD numbers, using the rules for basic binary addition. • If a 4-bit sum is equal to or less than 9, it is a valid BCD number. • If a 4-bit sum > 9, or if a carry out of the 4-bit group is generated it is an invalid result. Add 6 (0110) to a 4-bit sum in order to skip the six the invalid states and return the code to 8421. If a carry results when 6 is added, simply add the carry to the next 4-bit group.

BCD Addition • Add the following numbers (a) 0011+0100 (b) 00100011 + 00010101 (c) 10000110 + 00010011 (d) 010001010000 + 010000010111 (e) 1001 + 0100 (f) 1001 + 1001 (g) 00010110 + 00010101 (h) 01100111 + 01010011

Computer Organization and ASSEMBLY LANGUAGE