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Module 2:Floating-Point Representation

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Floating Point Numbers

■ Significant x base exponent

■ Example:

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Example1

Fixed point Floating point

Significant/fraction Base/Radix Exponent

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Normalized and Unnormalized■ A floating point number is said to be normalized if the

number after the radix point is a non-zero that is, it is not a ‘0’ value.

■ Unnormalized floating number is when the number after the radix point is ‘0’.

■ Example: normalized

unnormalized

unnormalized

normalized

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Normalization Process■ Normalization is the process of deleting the zeroes until

a non-zero value is detected.

■ Example :

■ A rule of thumb:– moving the radix point to the right subtract exponent – moving the radix point to the left add exponent

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Example 2

Decimal

Binary

-

-

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Floating Point Format for Binary Numbers

■ General form:

■ In binary:

sign signExponent

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Biased Exponent■ To eliminate the sign for the exponent value that is the

exponent will be positive.

sign Biased exponent

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Conversion to Floating Point Number

■ Normalized the number

■ Change the number to biased exponent

■ Form the word (3 fields)

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Example 3■ Transform –33.625 to floating point word using the following format (radix

2)

■ The biased constant

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Floating-Point Representation

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Overflow and Underflow

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Normalized Scientific Notation

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IEEE 754 Floating-Point Standard

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IEEE 754 Encoding ofFloating-Point Numbers

■ Purpose of NaNs is to allow programmers to postpone some tests and decision a later time in the program when it is convenient.

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Challenge of Negative Exponents■ Placing the exponent before the significand simplifies sorting of

floating-point numbers using integer comparison instructions.

■ However, using 2’s complement in the exponent field makes a negative exponent look like a big number.

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Biased Notation

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Convert 10.4ten to single precision floating point

IEEE 754 Conversion : Example 1

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IEEE 754 Conversion : Example 2

-0.75 = -0.11

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IEEE 754 Conversion : Example 2

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Converting Binary to Decimal Floating-Point

Fraction = 0.01b = 0.25

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Module 2:Floating-Point Operations

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Floating-Point Addition Flows

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Decimal Floating-Point AdditionAssume 4 decimal digit for significand and 2 decimal digits for exponent

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Binary Floating-Point Addition

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Floating-Point Multiplication Flows

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Decimal Floating-Point MultiplicationAssume 4 decimal digit for significand and 2 decimal digits for exponent

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Binary Floating-Point Multiplication

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Floating-Point ALU

010 1 0 1

Control

Small ALU

Big ALU

Sign Exponent Significand Sign Exponent Significand

Exponentdifference

Shift right

Shift left or right

Rounding hardware

Sign Exponent Significand

Increment ordecrement

0 10 1

Shift smallernumber right

Compareexponents

Add

Normalize

Round

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Accurate Arithmetic

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… Accurate Arithmetic