PowerPoint PresentationReview“Father” of the Floating point standardPrecision and AccuracyFractional Powers of 2Representation of FractionsUnderstanding the Significand (1/2)Understanding the Significand (2/2)Example: Converting Binary FP to DecimalConverting Decimal to FP (1/3)Converting Decimal to FP (2/3)Converting Decimal to FP (3/3)Example: Representing 1/3 in MIPSRepresentation for ± ∞Representation for 0Special NumbersRepresentation for Not a NumberRepresentation for Denorms (1/2)Representation for Denorms (2/2)OverviewPeer InstructionPeer Instruction AnswerAdministrivia…Midterm in 11 days!More AdministriviaRoundingIEEE Four Rounding ModesInteger Multiplication (1/3)Integer Multiplication (2/3)Integer Multiplication (3/3)Integer Division (1/2)Integer Division (2/2)Unsigned Instructions & OverflowFP Addition & SubtractionMIPS Floating Point Architecture (1/4)MIPS Floating Point Architecture (2/4)MIPS Floating Point Architecture (3/4)MIPS Floating Point Architecture (4/4)Slide 38Slide 39“And in conclusion…”CS61C L11 Floating Point II (1)Beamer, Summer 2007 © UCBScott Beamer, Instructorinst.eecs.berkeley.edu/~cs61c CS61C : Machine Structures Lecture #11 – Floating Point II 2007-7-12Sony & Nintendo make E3 Newswww.nytimes.comCS61C L11 Floating Point II (2)Beamer, Summer 2007 © UCBReview•Floating Point numbers approximate values that we want to use.•IEEE 754 Floating Point Standard is most widely accepted attempt to standardize interpretation of such numbers•Every desktop or server computer sold since ~1997 follows these conventions•Summary (single precision):031S Exponent30 23 22Significand1 bit 8 bits 23 bits•(-1)S x (1 + Significand) x 2(Exponent-127)•Double precision identical, bias of 1023CS61C L11 Floating Point II (3)Beamer, Summer 2007 © UCB“Father” of the Floating point standardIEEE Standard 754 for Binary Floating-Point Arithmetic.www.cs.berkeley.edu/~wkahan/…/ieee754status/754story.htmlProf. Kahan1989ACM TuringAward Winner!CS61C L11 Floating Point II (4)Beamer, Summer 2007 © UCBPrecision and AccuracyPrecision is a count of the number bits in a computer word used to represent a value.Accuracy is a measure of the difference between the actual value of a number and its computer representation.Don’t confuse these two terms!High precision permits high accuracy but doesn’t guarantee it. It is possible to have high precisionbut low accuracy. Example: float pi = 3.14;pi will be represented using all 24 bits of the significant (highly precise), but is only an approximation (not accurate).CS61C L11 Floating Point II (5)Beamer, Summer 2007 © UCBFractional Powers of 20 1.0 11 0.5 1/22 0.25 1/43 0.125 1/84 0.0625 1/165 0.03125 1/326 0.0156257 0.00781258 0.003906259 0.00195312510 0.000976562511 0.0004882812512 0.00024414062513 0.000122070312514 0.0000610351562515 0.000030517578125i 2-iCS61C L11 Floating Point II (6)Beamer, Summer 2007 © UCBRepresentation of Fractions“Binary Point” like decimal point signifies boundary between integer and fractional parts:xx.yyyy21202-12-22-32-4Example 6-bit representation:10.10102 = 1x21 + 1x2-1 + 1x2-3 = 2.62510 If we assume “fixed binary point”, range of 6-bit representations with this format: 0 to 3.9375 (almost 4)CS61C L11 Floating Point II (7)Beamer, Summer 2007 © UCBUnderstanding the Significand (1/2)•Method 1 (Fractions):•In decimal: 0.34010 => 34010/100010 => 3410/10010•In binary: 0.1102 => 1102/10002 = 610/810 => 112/1002 = 310/410•Advantage: less purely numerical, more thought oriented; this method usually helps people understand the meaning of the significand betterCS61C L11 Floating Point II (8)Beamer, Summer 2007 © UCBUnderstanding the Significand (2/2)•Method 2 (Place Values):•Convert from scientific notation•In decimal: 1.6732 = (1x100) + (6x10-1) + (7x10-2) + (3x10-3) + (2x10-4)•In binary: 1.1001 = (1x20) + (1x2-1) + (0x2-2) + (0x2-3) + (1x2-4)•Interpretation of value in each position extends beyond the decimal/binary point•Advantage: good for quickly calculating significand value; use this method for translating FP numbersCS61C L11 Floating Point II (9)Beamer, Summer 2007 © UCBExample: Converting Binary FP to Decimal•Sign: 0 => positive•Exponent: •0110 1000two = 104ten•Bias adjustment: 104 - 127 = -23•Significand:•1 + 1x2-1+ 0x2-2 + 1x2-3 + 0x2-4 + 1x2-5 +...=1+2-1+2-3 +2-5 +2-7 +2-9 +2-14 +2-15 +2-17 +2-22= 1.0ten + 0.666115ten0 0110 1000 101 0101 0100 0011 0100 0010•Represents: 1.666115ten*2-23 ~ 1.986*10-7 (about 2/10,000,000)CS61C L11 Floating Point II (10)Beamer, Summer 2007 © UCBConverting Decimal to FP (1/3)•Simple Case: If denominator is an exponent of 2 (2, 4, 8, 16, etc.), then it’s easy.•Show MIPS representation of -0.75•-0.75 = -3/4•-11two/100two = -0.11two•Normalized to -1.1two x 2-1•(-1)S x (1 + Significand) x 2(Exponent-127)• (-1)1 x (1 + .100 0000 ... 0000) x 2(126-127)1 0111 1110 100 0000 0000 0000 0000 0000CS61C L11 Floating Point II (11)Beamer, Summer 2007 © UCBConverting Decimal to FP (2/3)•Not So Simple Case: If denominator is not an exponent of 2.•Then we can’t represent number precisely, but that’s why we have so many bits in significand: for precision•Once we have significand, normalizing a number to get the exponent is easy.•So how do we get the significand of a neverending number?CS61C L11 Floating Point II (12)Beamer, Summer 2007 © UCBConverting Decimal to FP (3/3)•Fact: All rational numbers have a repeating pattern when written out in decimal.•Fact: This still applies in binary.•To finish conversion:•Write out binary number with repeating pattern.•Cut it off after correct number of bits (different for single v. double precision).•Derive Sign, Exponent and Significand fields.CS61C L11 Floating Point II (13)Beamer, Summer 2007 © UCBExample: Representing 1/3 in MIPS•1/3 = 0.33333…10= 0.25 + 0.0625 + 0.015625 + 0.00390625 + … = 1/4 + 1/16 + 1/64 + 1/256 + …= 2-2 + 2-4 + 2-6 + 2-8 + …= 0.0101010101… 2 * 20= 1.0101010101… 2 * 2-2•Sign: 0•Exponent = -2 + 127 = 125 = 01111101•Significand = 0101010101…0 0111 1101 0101 0101 0101 0101 0101 010CS61C L11 Floating Point II (14)Beamer, Summer 2007 © UCBRepresentation for ± ∞•In FP, divide by 0 should produce ± ∞, not overflow.•Why?•OK to do further computations with ∞ E.g., X/0 > Y may be a valid comparison•Ask math majors•IEEE 754 represents ± ∞•Most positive
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