L15 - CS61C Machine Structures Lecture 15 - Floating Point...

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CS 61C L15 Floating Point I (1) Wawrzynek Fall 2007 © UCB 10/1/2007 John Wawrzynek (www.cs.berkeley.edu/~johnw) www-inst.eecs.berkeley.edu/~cs61c/ CS61C – Machine Structures Lecture 15 - Floating Point Numbers I CS 61C L15 Floating Point I (2) Wawrzynek Fall 2007 © UCB Introduction ° What does the above bit pattern mean? “4UCB” ori $21, $2, 17218 878,003,010 ° Data can be anything. Its meaning is determined by how it is interpreted. (As an instruction, as an operand for a arithmetic instruction, …) ° And as we shall see today … as a floating point number. 0011 0100 0101 0101 0100 0011 0100 0010
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CS 61C L15 Floating Point I (3) Wawrzynek Fall 2007 © UCB Review of Numbers ° Computers are made to deal with numbers ° What can we represent in N bits? 2 N things, and no more! They could be… Unsigned integers: 0 to 2 N - 1 (for N=32, 2 N –1 = 4,294,967,295) Signed Integers (Two s Complement) -2 (N-1) to 2 (N-1) - 1 (for N=32, 2 (N-1) = 2,147,483,648) CS 61C L15 Floating Point I (4) Wawrzynek Fall 2007 © UCB What about other numbers? 1. Very large numbers? (seconds/millennium) 31,556,926,000 10 (3.1556926 10 x 10 10 ) 2. Very small numbers? (Bohr radius) 0.0000000000529177 10 m (5.29177 10 x 10 -11 ) 3. Numbers with both 1.5 First consider #3. …our solution will also help with 1 and 2.
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CS 61C L15 Floating Point I (5) Wawrzynek Fall 2007 © UCB Representation of Fractions “Binary Point” like decimal point signifies boundary between integer and fractional parts: xx . yyyy 2 1 2 0 2 -1 2 -2 2 -3 2 -4 Example 5-bit representation: 10.101 2 = 1x2 1 + 1x2 -1 + 1x2 -3 = 2.625 10 If we assume “fixed binary point”, range of 5-bit representations with this format: 0 to 3.9375 (almost 4) CS 61C L15 Floating Point I (6) Wawrzynek Fall 2007 © UCB Fractional Powers of 2 0 1.0 1 1 0.5 1/2 2 0.25 1/4 3 0.125 1/8 4 0.0625 1/16 5 0.03125 1/32 6 0.015625 7 0.0078125 8 0.00390625 9 0.001953125 10 0.0009765625 11 0.00048828125 12 0.000244140625 13 0.0001220703125 14 0.00006103515625 15 0.000030517578125 i 2 -i
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L15 - CS61C Machine Structures Lecture 15 - Floating Point...

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