ECE232: Hardware Organization and Design


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1 ECE232: Hardware Organization and Design Lecture 11: Floating Point & Floating Point Addition Adapted from Computer Organization and Design, Patterson & Hennessy, UCB
2 Last time: Single Precision Format Note that the exponent has no explicit sign bit Base? 32 bits M: Mantissa (23 bits) E: Exponent (8 bits) S: Sign of mantissa (1 bit) ECE331: Floating Point 2
3 Last time: Normalization The mantissa M is a normalized fraction Has an implied decimal place on left Has an implied (hidden) 1 on left of the decimal place E.g., Fraction Represents = The significand=1.f is in the range [1, 2ulp] ulp unit in the last position (what remains to reach a whole number when all bits are set to one) F ECE331: Floating Point 3 S = ( 1) 1. f 2 E Bias Value of exponent (unsigned integer) Bias value (known; set by convention)
4 Binary Fractions To convert binary fractions to floating point = 1*(0.5) + 1*(0.25) + 1*(0.125) + 0*(0.0625) + 0*( ) + 0*( ) + 1*( ) + 0*( ) = ECE331: Floating Point 4
5 Binary Fractions To convert floating point to binary whole number 9 à fraction * 2 = 1.25 = * 2 = * 2 = à = x 2 4 = x 2 3 Note that we can shift positions left and right of the decimal point by multiplying by different powers of 2 ECE331: Floating Point 5
6 FloatingPoint Example To convert floating point number to binary Represent = = ( 1) S = 1 Fraction = Exponent = 1 + Bias Single: = 126 = Double: = 1022 = Single: fraction sign exponent and bias Double: ECE331: Floating Point 6
7 FloatingPoint Example To convert from binary to floating point What number is represented by the singleprecision float? Identify the components S = 1 Fraction = Exponent = = 129 Calculate the value x = ( 1) 1 ( ) 2 ( ) = ( 1) = 5.0 ECE331: Floating Point 7
8 FloatingPoint Addition in Decimal Consider a 4digit decimal example In scientific notation: Align decimal points Shift number with smaller exponent Add significands = Normalize result & check for over/underflow Round and renormalize if necessary Optionally convert back to nonexponential form In this example, accurate up to 4 decimal places ECE331: Floating Point 8
9 FloatingPoint Addition in binary Now consider a 4digit binary example ( ) ( ) 1. Align binary points Shift number with smaller exponent Add significands = Normalize result & check for over/underflow , with no over/underflow 4. Round and renormalize if necessary (no change) = When converted to floating point representation: Mantissa, f = (23 bits of zeros) Exponent: E Bias = 4; if Bias = 127, then E = E = Sign bit, S = 0 The entire 32bit floating point representation in binary ECE331: Floating Point 9
10 An0ther Single precision example = 3 0 = positive mantissa x 2 3 = = ECE331: Floating Point 10
11 Converting to IEEE format Example  decimal number: X 10 0 What is the sign? What is the exponent? What is the mantissa? Converting Mixed Numbers Decimal to Binary How we do it in decimal = 4 x x x x x 102 How it is done in binary = 1 x x x x x x 22 = /2 + ¼ = = ECE331: Floating Point 11
12 How to convert whole Decimal to Binary Successive division by = value remainder Start dividing by 2 here and move upward Binary value read downwards ECE331: Floating Point 12
13 Converting fractional Decimal to Binary Successive multiplication by Decimal = ECE331: Floating Point 13
14 Floating Point Special Representations S E 127 F = ( 1) 1. f f < 2 There are two Zeroes, ±0, and two Infinities ± NaN (NotaNumber) may have a sign and have a nonzero fraction  used for program diagnostics NaNs and Infinities have all 1s in the Exp field, E=255. F+ =, F/ =0 ECE331: Floating Point 14 Source: I. Koren, Computer Arithmetic Algorithms, 2nd Edition, 2002
15 Floating Point Special Representations S E 127 F = ( 1) 1. f < 2 f 1 E 254 (single precision) Single Precision Double Precision Object represented Exponent Fraction Exponent Fraction nonzero 0 nonzero ± denormalized number Anything Anything ± floating point number ± infinity 255 nonzero 2047 nonzero NaN (not a number) ECE331: Floating Point 15
16 Smallest & Largest Numbers The smallest nonzero positive and largest nonzero negative normalized numbers (represented by 1 in the Exp field and 0 0 in the Fraction field) are ±2 126 ± The smallest nonzero positive and largest nonzero negative denormalized numbers (represented by all 0s in the Exp field and 0 01 in the Fraction field) are ±2 149 ± The largest finite positive and smallest finite negative numbers (represented by 254 in the Exp field and 1 1 in the Fraction field) are ±(2)(2 127 ) ± ECE331: Floating Point 16
17 FP Adder Hardware Step 1 Step 2 Step 3 Step 4 ECE331: Floating Point 17
18 Single Precision Summary Type Exponent Mantissa Value Zero One Denormalized number Largest normalized number Smallest normalized number Infinity Infinity NaN NaN ECE331: Floating Point 18
19 Summary Floating point numbers represent large numbers with fractions Number formats are different than 2 s complement. Requires some memorization Addition requires aligning, adding, and then realigning Do examples! The best way to learn floating point operations ECE331: Floating Point 19
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