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Chapters 4 and 6

Chapters 4 and 6. Floating Point Numbers. Floating Point Numbers. Registers for real numbers usually contain 32 or 64 bits, allowing 2 32 or 2 64 numbers to be represented. Which reals to represent? There are an infinite number between 2 adjacent integers. (or two reals!!)

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Chapters 4 and 6

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  1. Chapters 4 and 6 Floating Point Numbers

  2. Floating Point Numbers • Registers for real numbers usually contain 32 or 64 bits,allowing 232 or 264 numbers to be represented. • Which reals to represent? There are an infinite numberbetween 2 adjacent integers. (or two reals!!) • Which bit patterns for reals selected? • Answer: use scientific notation • Consider: A x 10B, where A is one digit • Scientific notation in binary • Standard: IEEE 754 Floating-Point A B A x 10B 0 any 0 1 .. 9 0 1 .. 9 1 .. 9 1 10 .. 90 1 .. 9 2 100 .. 900 1 .. 9 -1 0.1 .. 0.9 1 .. 9 -2 0.01 .. 0.09

  3. Floating Point Representation: S E F • S is one bit representing the sign of the number • E is an 8 bit biased integer representing the exponent • F is an unsigned integer The true value represented is: (-1)S x f x 2e • e = E – bias • f = F/2n + 1 • for single precision numbers n=23, bias=127

  4. Floating Point • S, E, F all represent fields within a representation. Eachis just a bunch of bits. • S is the sign bit • (-1)S (-1)0 = +1 and (-1)1 = -1 • Just a sign bit for signed magnitude • E is the exponent field • The E field is a biased-127 representation. • True exponent is (E – bias) • The base (radix) is always 2 (implied). • Some early machines used radix 4 or 16 (IBM) • F (or M) is the fractional or mantissa field. • It is in a strange form. • There are 23 bits for F. • A normalized FP number always has a leading 1. • No need to store the one, just assume it. • This MSB is called the HIDDEN BIT.

  5. Floating Point: 64.2 into IEEE SP • Get a binary representation for 64.2 • Binary of left of radix point 64 is: • Binary of right of radix • .2 x 2 = 0.4 0 • .4 x 2 = 0.8 0 • .8 x 2 = 1.6 1 • .6 x 2 = 1.2 1 • Binary for .2: • 64.2 is: • Normalize binary form • Produces: • Turn true exponent into bias-127 • Put it together: • 23-bit F is: • S E F is: • In hex:

  6. Floating Point • Since floating point numbers are always stored innormal form, how do we represent 0? • 0x0000 0000 and 0x8000 0000 represent 0. • What numbers cannot be represented because of this?

  7. Floating Point • Other special values: • + 5 / 0 = + • + 0 11111111 00000… (0x7f80 0000) • -7/0 = - • - 1 11111111 00000… (0xff80 0000) • 0/0 or + + - =NaN (Not a number) • NaN ? 11111111 ?????… • (S is either 0 or 1, E=0xff, and F is anything but all zeroes) • Also de-normalized numbers (beyond scope)

  8. Floating Point What is 0x4228 0000 if in a SP FP notation?

  9. Floating Point What is 47.625 as a SP FP?

  10. Floating Point • What do floating-point numbers represent? • Rational numbers with non-repeating expansionsin the given base within the specified exponent range. • They do not represent repeating rational or irrationalnumbers, or any number too small (0 < |x|<Bemin) –underflow, or too large (|x| > Bemax) – overflow • IEEE Double Precision is similar to SP • 52-bit M • 53 bits of precision with hidden bit • 11-bit E, excess 1023, representing –1022:1023 • One sign bit • Always use DP unless memory/file size is important • SP ~ 10-38 … 1038 • DP ~ 10-308 … 10308 • Be very careful of these ranges in numeric computation

  11. Floating Point: Chapter 6 • Floating Point operations include • ? • ? • ? • ? • ? • They are complicated because…

  12. Floating Point Addition 9.997 x 102 + 4.631 x 10-1 • Align decimal points • Add • Normalize the result • Often already normalized • Otherwise move one digit • 1.0001631 x 103 • Round result • 1.000 x103 9.997 x 102 + 0.004631 x 102 10.001631 x 102

  13. Floating Point Addition • .25 = 0 01111101 00000000000000000000000 • 100 = 0 10000101 10010000000000000000000 • Or with hidden bit • .25 = 0 01111101 1 00000000000000000000000 • 100 = 0 10000101 1 10010000000000000000000 • Align radix points • Shifting F left by 1 bit, decreasing e by 1 • Shifting F right by 1 bit, increasing e by 1 • Shift mantissa right so least significant bits fall off • Which should we shift?

  14. Floating Point Addition • Shift the .25 to increase its exponent • 01111101 – 1111111 (127) = • 10000101 – 1111111 (127) = • Shift smaller by: • Bias cancels with subtraction, so • 10000101 • - 01111101 • 00001000 • Carefully shifting, • 0 01111101 1 00000000000000000000000 (original value) • 0 01111110 0 10000000000000000000000 (shifted by 1) • 0 01111111 0 01000000000000000000000 (shifted by 2) • 0 10000000 0 00100000000000000000000 (shifted by 3) • 0 10000001 0 00010000000000000000000 (shifted by 4) • 0 10000010 0 00001000000000000000000 (shifted by 5) • 0 10000011 0 00000100000000000000000 (shifted by 6) • 0 10000100 0 00000010000000000000000 (shifted by 7) • 0 10000101 0 00000001000000000000000 (shifted by 8)

  15. Floating Point Addition • 2. Add with hidden bit • 0 10000101 1 10010000000000000000000 (100) • + 0 10000101 0 00000001000000000000000 (.25) • 0 10000101 1 10010001000000000000000 • 3. Normalize the result • Get a ‘1’ back in hidden bit • Already normalized • 4. Round result • Renormalize if needed • 0 10000101 10010001000000000000000 • Post-normalization example: • s exp h frtn • 0 011 1 1100 • + 0 011 1 1011 • 0 011 11 0111 • 0 100 1 1011 1 discarded

  16. Floating Point Subtraction • Mantissa’s are sign-magnitude • Watch out when the numbers are close • 1.23456 x 102 • - 1.23455 x 102 • A many-digit normalization is possible • This is why FP addition is in many ways moredifficult than FP multiplication

  17. Floating Point Subtraction • Align radix points • Perform sign-magnitude operand swap • Compare magnitudes (with hidden bit) • Change sign bit if order of operands is changed. • Subtract • Normalize • Round • Renormalize if needed s exp h frtn 0 011 1 1011 smaller - 0 011 1 1101 bigger switch and make difference negative 0 011 1 1101 bigger - 0 011 1 1011 smaller 1 011 0 0010 1 000 1 0000 switch sign

  18. Floating Point Multiplication • Multiplication Example • 3.0 x 101 • x 5.0 x 102 • Multiply mantissas • 3.0 • x 5.0 • 15.00 • Add exponents • 1 + 2 = 3 • 15.00 x 103 • Normalize if needed • 1.50 x 104

  19. Floating Point Multiplication • Multiplication in binary (4-bit F) • 0 10000100 0100 • x 1 00111100 1100 • Multiply mantissas 1.0100 x 1.1100 00000 00000 10100 10100 10100 1000110000 10.00110000

  20. Floating Point Multiplication • Add exponents, subtract extra bias. • Biased: • 10000100 • + 00111100 • Check by converting to true exponents and adding • Renormalize, correcting exponent • 1 01000001 10.00110000 • Becomes • 1 01000010 1.000110000 • Drop the hidden bit • 1 01000010 000110000

  21. Floating Point Division • Division • True division • Unsigned, full-precision division on mantissas • This is much more costly (e.g. 4x) than mult. • Subtract exponents • Faster division • Newton’s method to find reciprocal • Multiply dividend by reciprocal of divisor • May not yield exact result without some work • Similar speed as multiplication

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