Introduction to Computer Systems Recitation 2 May 29, Marjorie Carlson Aditya Gupta Shailin Desai
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1 Introduction to Computer Systems Recitation 2 May 29, 2014 Marjorie Carlson Aditya Gupta Shailin Desai 1
2 Agenda! Goal: translate any real number (plus some!) into and out of machine representation.! Integers! Biasing division! Floats! Binary fractions! IEEE standard! Normalized numbers! Denormalized numbers! Special values 2
3 Integer * and /! You can multiply and divide by powers of 2 with bitshifting alone.! Your computer does a lot of math this way!! Multiply:! To multiply by 2 k, simply left shift by k.! What s * 2?! What s * 4? 3
4 Integer * and /! You can multiply and divide by powers of 2 with bitshifting alone.! Your computer does a lot of math this way!! Divide:! To multiply by 2 k, right shift by k.! Let s try this with 01111/2?! How about (signed) 10001/2?! Uh- oh!! Shifting rounds down, but we want to round toward zero.! Solution: biasing when the number is negative 4
5 Biasing Division Only bias when the k bits dividend is negative! Dividend: x 1 Divisor: +2 k / 2 k Before we discard the last k bits, we add a biasing term consisting of k 1 s. 1 This bit will increment if appropriate. k bits of 1 s Then we >> k. x / 2 k And voila! The number now rounds toward zero instead of down. 5
6 Agenda! Goal: translate any real number (plus some!) into and out of machine representation.! Integers! Biasing division! Floats! Binary fractions! IEEE standard! Normalized numbers! Denormalized numbers! Special values 6
7 FloaCng Point FracCons in Binary bi bi- 1 b2 b1 b0 b- 1 b- 2 b- 3 b- j 1/2 1/4 1/8 2 - j 2 i 2 i
8 FloaCng Point FracCons in Binary! Convert binary to decimal:! 1.1! ! ! Convert decimal to binary:! 3 3/4! 2 3/32!
9 How to Represent #s Efficiently?! Forget about binary for a minute.! What do we do if we want to convey in only ten digits? 9
10 How to Represent #s Efficiently?! Forget about binary for a minute.! What do we do if we want to convey in only ten digits?! Hint: - _. _ * 10
11 How to Represent #s Efficiently?! Forget about binary for a minute.! What do we do if we want to convey in only ten digits? * sign mancssa exponent 11
12 FloaCng Point ScienCfic NotaCon! How can we put binary numbers into scientific notation? * 2 2 sign (S) mancssa (M) exponent (E)! Numerical form: ( 1) S M 2 E 12
13 Agenda! Goal: translate any real number (plus some!) into and out of machine representation.! Integers! Biasing division! Floats! Binary fractions! IEEE standard! Normalized numbers! Denormalized numbers! Special values 13
14 FloaCng Point IEEE Standard! Floating points encode binary scientific notation. s exp frac 1 8 bits 23 bits! exp encodes E (the exponent).! frac encodes M (the mantissa).! But, due to optimizations, exp E and frac M.! (Note: for the next five slides, forget denormalized floats exist. We ll get back to them, I promise.) 14
15 FloaCng Point First OpCmizaCon! In decimal scientific notation, the digit before the decimal place can be any number from 1 to 9: ! But in binary scientific notation, that digit will always be 1! ! So, encoding the 1 is unnecessary.! Instead of representing all of M in the frac, we discard the leading 1 and only encode the part after the decimal ! ! !.1111! frac = M 1 M = 1 + frac 15
16 FloaCng Point Second OpCmizaCon! exp is an unsigned 8- bit number, so it can represent the numbers 0 (0x ) to 255 (0x ).! Do we want floats be able to represent number from around 2 0 to around 2 255?! It would actually be much more useful to represent numbers from, say, to ! So, to get a more useful range of possible exponents, we subtract a bias of 127 from E to get the exp.! exp = E + bias bias = 2 k- 1-1 E = exp - bias 16
17 ConverCng a Number to a Float * 2 2 sign (S) mancssa (M) exponent (E)! S = + so s = 0! E = 2 so exp = 129! M = so frac = Remember! M = 1 + frac E = exp bias bias =
18 ConverCng a Number to a Float * 2 2 sign (S) mancssa (M) exponent (E) converted to a 32- bit float looks like: s exp frac
19 FloaCng Point Example! Consider the following 5- bit floating point representation based on the IEEE floating point format. This format does not have a sign bit; it can only represent nonnegative numbers.! There are k=3 exponent bits.! There are n=2 fraction bits ! What s the bias?! What does represent?! What does represent?! How would you represent 6?! How would you represent ¼? exp frac 19
20 FloaCng Point Denormalized Range! Given what we ve just discussed, the smallest representable number would be 1.0 * 2 - bias, which is not really that small.! It d be represented as all zeros. There d be no way to represent zero as a float!! IEEE uses a trick to give us numbers closer to 0, and 0 itself: for really small numbers (i.e., exp = 0), drop the implied leading 1.! This basically sacrifices a little precision for a wider range. 20
21 FloaCng Point Denormalized Range Normalized Denormalized exp 0 exp = 0 implied leading 1 no implied leading 1 E = exp - bias denser near origin represents most numbers E = 1 bias evenly spaced represents tiny numbers Why not 0 bias? Because we have to increment the exponent to counteract the missing leading 1. 21
22 FloaCng Point Examples! Back to our mini- floats:! There are k=3 exponent bits.! There are n=2 fraction bits.! Bias = exp frac! What does represent?! What s the smallest representable nonzero value?! What s the largest representable finite number?! What s the smallest normalized number?! What s the largest denormalized number? 22
23 FloaCng Point Special Cases! OK, denormalizing got us our zero. Now how about infinity? How about NaN (not a number)? 23
24 Last Two Tips to Convert Anything! 1. The tricky part about dec! float conversion is figuring out whether your number should be encoded as normalized or denormalized.! Strategy 1: compute the smallest possible noramlized number, then compare your number to it.! Strategy 2: try to encode it as normalized; if your exponent doesn t fit in exp, change exp to 0 and shift your decimal point accordingly. 2. You need to know how to round! 24
25 FloaCng Point Rounding Floats round to even if precisely between two options. (Avoids statistical bias from always rounding the same way.) truncate below half; round down interescng case; round to even above half; round up truncate below half; round down interescng case; round to even above half; round up truncate
26 FloaCng Point Examples! Back to our mini- floats:! There are k=3 exponent bits.! There are n=2 fraction bits.! Bias = exp frac Value Floating Point Bits Rounded Value 9/
27 FloaCng Point Examples! Back to our mini- floats:! There are k=3 exponent bits.! There are n=2 fraction bits.! Bias = exp frac Value Floating Point Bits Rounded Value 9/ /
28 CongratulaCons! You can now translate any real number (plus infinity and NaN) into and out of machine representation! 28
29 QuesCons? 29
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