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Floating Point (a brief look)

Floating Point (a brief look). We need a way to represent numbers with fractions, e.g., 3.1416 very small numbers, e.g., .000000001 very large numbers, e.g., 3.15576 ´ 10 9 Representation: sign, exponent, significand: (–1) sign ´ significand ´ 2 exponent

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Floating Point (a brief look)

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  1. Floating Point (a brief look) • We need a way to represent • numbers with fractions, e.g., 3.1416 • very small numbers, e.g., .000000001 • very large numbers, e.g., 3.15576 ´ 109 • Representation: • sign, exponent, significand: (–1)sign´ significand ´ 2exponent • more bits for significand gives more accuracy • more bits for exponent increases range • IEEE 754 floating point standard: • single precision: 8 bit exponent, 23 bit significand • double precision: 11 bit exponent, 52 bit significand

  2. IEEE 754 floating-point standard • Leading “1” bit of significand is implicit • Exponent is “biased” to make sorting easier • all 0s is smallest exponent all 1s is largest • bias of 127 for single precision and 1023 for double precision • summary: (–1)sign´ (1+significand) ´ 2exponent – bias • Example: • decimal: -.75 = -3/4 = -3/22 • binary: -.11 = -1.1 x 2-1 • floating point: exponent = 126 = 01111110 • IEEE single precision: 10111111010000000000000000000000

  3. Floating Point Complexities • Operations are somewhat more complicated (see text) • In addition to overflow we can have “underflow” • Accuracy can be a big problem • IEEE 754 keeps two extra bits, guard and round • four rounding modes • positive divided by zero yields “infinity” • zero divide by zero yields “not a number” • other complexities • Implementing the standard can be tricky • Not using the standard can be even worse • see text for description of 80x86 and Pentium bug!

  4. S i g n E x p o n e n t S i g n i f i c a n d S i g n E x p o n e n t S i g n i f i c a n d S m a l l A L U E x p o n e n t d i f f e r e n c e 0 1 0 1 0 1 C o n t r o l S h i f t r i g h t B i g A L U S h i f t l e f t o r r i g h t R o u n d i n g h a r d w a r C o m p a r e e x p o n e n t s S h i f t s m a l l e r n u m b e r r i g h t A d d 0 1 0 1 I n c r e m e n t o r N o r m a l i z e d e c r e m e n t R o u n d e S i g n E x p o n e n t S i g n i f i c a n d

  5. Floating point Multiplication • Multiply mantissas and add exponents • Steps: • Add exponents • Multiply mantissas • Normalise result • Round results • Fix sign of product

  6. Floating Accuracy • Floating point Numbers are normally approximations for the numbers they represent • IEEE-754 keeps two extra bits on the right during intermediate calculations called guard and round respectively • Example: add 2.56ten * 100 to 2.34ten*102 assumming three significant digits • First with guard and round digits and • then without them

  7. Floating Point Accuracy • Four Rounding modes : • round to nearest (default) • round towards plus infinity • round towards minus infinity • round towards 0

  8. Examples

  9. Special Symbols • Special symbols used by IEEE-754 • + or -  (largest exponent with 0 mantissa) (result of divide by zero) • NaN (Not a number) (largest exponent with 0 mantissa) (0/0 or  - ) • Unnormalised Numbers (no explicit 1 in MSB) (0 exponent non-zero mantissa) • Zero (zero Mantissa and zero expeonent)

  10. Summary

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