ECE232: Hardware Organization and Design

Transcription

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, 2-ulp] 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 Floating-Point 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 Floating-Point Example To convert from binary to floating point What number is represented by the single-precision float? Identify the components S = 1 Fraction = Exponent = = 129 Calculate the value x = ( 1) 1 ( ) 2 ( ) = ( 1) = 5.0 ECE331: Floating Point 7

8 Floating-Point Addition in Decimal Consider a 4-digit 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 non-exponential form In this example, accurate up to 4 decimal places ECE331: Floating Point 8

9 Floating-Point Addition in binary Now consider a 4-digit 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 32-bit 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 10-2 How it is done in binary = 1 x x x x x x 2-2 = /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 (Not-a-Number) may have a sign and have a non-zero 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 non-zero positive and largest non-zero negative normalized numbers (represented by 1 in the Exp field and 0 0 in the Fraction field) are ±2 126 ± The smallest non-zero positive and largest non-zero 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