CS 61C L15 Floating Point I (1)Krause, Spring 2005 © UCBThis day in history…TA Danny Krauseinst.eecs.berkeley.edu/~cs61c-tdinst.eecs.berkeley.edu/~cs61cCS61C : Machine Structures Lecture 15 – Floating Point I !!!!!!!!!!!!!!!!!!!!!! !!2004-02-231455 - Publication of the Gutenberg Bible1998 - Netscape founds Mozilla.orgCS 61C L15 Floating Point I (2)Krause, Spring 2005 © UCBQuote of the day“95% of thefolks out there arecompletely cluelessabout floating-point.”James GoslingSun FellowJava Inventor1998-02-28CS 61C L15 Floating Point I (3)Krause, Spring 2005 © UCBReview of Numbers• Computers are made to deal withnumbers• What can we represent in N bits?• Unsigned integers:0 to 2N - 1• Signed Integers (Two’s Complement)-2(N-1)to 2(N-1) - 1CS 61C L15 Floating Point I (4)Krause, Spring 2005 © UCBOther Numbers• What about other numbers?• Very large numbers? (seconds/century)3,155,760,00010 (3.1557610 x 109)• Very small numbers? (atomic diameter)0.0000000110 (1.010 x 10-8)• Rationals (repeating pattern) 2/3 (0.666666666. . .)• Irrationals21/2 (1.414213562373. . .)• Transcendentalse (2.718...), π (3.141...)• All represented in scientific notationCS 61C L15 Floating Point I (5)Krause, Spring 2005 © UCBScientific Notation (in Decimal)6.0210 x 1023radix (base)decimal pointmantissaexponent• Normalized form: no leadings 0s(exactly one digit to left of decimal point)• Alternatives to representing 1/1,000,000,000• Normalized: 1.0 x 10-9• Not normalized: 0.1 x 10-8,10.0 x 10-10CS 61C L15 Floating Point I (6)Krause, Spring 2005 © UCBScientific Notation (in Binary)1.0two x 2-1radix (base)“binary point”exponent• Computer arithmetic that supports itcalled floating point, because itrepresents numbers where the binarypoint is not fixed, as it is for integers• Declare such variable in C as floatmantissaCS 61C L15 Floating Point I (7)Krause, Spring 2005 © UCBFloating Point Representation (1/2)• Normal format: +1.xxxxxxxxxxtwo*2yyyytwo• Multiple of Word Size (32 bits)031S Exponent30 23 22Significand1 bit 8 bits 23 bits• S represents SignExponent represents y’sSignificand represents x’s• Represent numbers as small as2.0 x 10-38 to as large as 2.0 x 1038CS 61C L15 Floating Point I (8)Krause, Spring 2005 © UCBFloating Point Representation (2/2)• What if result too large? (> 2.0x1038 )• Overflow!• Overflow ⇒ Exponent larger thanrepresented in 8-bit Exponent field• What if result too small? (>0, < 2.0x10-38 )• Underflow!• Underflow ⇒ Negative exponent larger thanrepresented in 8-bit Exponent field• How to reduce chances of overflow orunderflow?CS 61C L15 Floating Point I (9)Krause, Spring 2005 © UCBDouble Precision Fl. Pt. Representation• Next Multiple of Word Size (64 bits)• Double Precision (vs. Single Precision)• C variable declared as double• Represent numbers almost as small as2.0 x 10-308 to almost as large as 2.0 x 10308• But primary advantage is greater accuracydue to larger significand031S Exponent30 20 19Significand1 bit 11 bits 20 bitsSignificand (cont’d)32 bitsCS 61C L15 Floating Point I (10)Krause, Spring 2005 © UCBQUAD Precision Fl. Pt. Representation• Next Multiple of Word Size (128 bits)• Unbelievable range of numbers• Unbelievable precision (accuracy)• This is currently being worked on• The current version has 15 bits for theexponent and 112 bits for thesignificand• Oct-Precision? That’s just silly! It’sbeen implemented before…CS 61C L15 Floating Point I (11)Krause, Spring 2005 © UCBIEEE 754 Floating Point Standard (1/4)• Single Precision, DP similar• Sign bit: 1 means negative0 means positive• Significand:• To pack more bits, leading 1 implicit fornormalized numbers• 1 + 23 bits single, 1 + 52 bits double• always true: Significand < 1 (for normalized numbers)• Note: 0 has no leading 1, so reserveexponent value 0 just for number 0CS 61C L15 Floating Point I (12)Krause, Spring 2005 © UCBIEEE 754 Floating Point Standard (2/4)• Kahan wanted FP numbers to be usedeven if no FP hardware; e.g., sort recordswith FP numbers using integer compares• Could break FP number into 3 parts:compare signs, then compare exponents,then compare significands• Wanted it to be faster, single compare ifpossible, especially if positive numbers• Then want order:• Highest order bit is sign ( negative < positive)• Exponent next, so big exponent => bigger #• Significand last: exponents same => bigger #CS 61C L15 Floating Point I (13)Krause, Spring 2005 © UCBIEEE 754 Floating Point Standard (3/4)• Negative Exponent?• 2’s comp? 1.0 x 2-1 v. 1.0 x2+1 (1/2 v. 2)0 1111 1111 000 0000 0000 0000 0000 00001/20 0000 0001 000 0000 0000 0000 0000 00002• This notation using integer compare of1/2 v. 2 makes 1/2 > 2!• Instead, pick notation 0000 0001 is mostnegative, and 1111 1111 is most positive• 1.0 x 2-1 v. 1.0 x2+1 (1/2 v. 2)1/20 0111 1110 000 0000 0000 0000 0000 00000 1000 0000 000 0000 0000 0000 0000 00002CS 61C L15 Floating Point I (14)Krause, Spring 2005 © UCBIEEE 754 Floating Point Standard (4/4)• Called Biased Notation, where bias isnumber subtract to get real number• IEEE 754 uses bias of 127 for single prec.• Subtract 127 from Exponent field to getactual value for exponent• 1023 is bias for double precision• Summary (single precision):031S Exponent30 23 22Significand1 bit 8 bits 23 bits• (-1)S x (1 + Significand) x 2(Exponent-127)• Double precision identical, except withexponent bias of 1023CS 61C L15 Floating Point I (15)Krause, Spring 2005 © UCB“Father” of the Floating point standardIEEE Standard754 for BinaryFloating-PointArithmetic.www.cs.berkeley.edu/~wkahan/…/ieee754status/754story.htmlProf. Kahan1989ACM TuringAward Winner!CS 61C L15 Floating Point I (16)Krause, Spring 2005 © UCBAdministrivia…Midterm in 2 weeks!• Midterm 1 LeConte Mon 2004-03-07 @ 7-10pm• Conflicts/DSP? Email Head TA Andy, cc Dan• How should we study for the midterm?• Form study groups -- don’t prepare in isolation!• Attend the review session(2004-03-06 @ 2pm in 10 Evans)• Look over HW, Labs, Projects• Write up your 1-page study sheet--handwritten• Go over old exams – HKN office has put themonline (link from 61C home page)CS 61C L15 Floating Point I (17)Krause, Spring 2005 © UCBUpcoming CalendarStateElementsRunningProgramFloatingPt IWedComb.LogicMidtermgradesoutFiniteStateMachinesDigitalSystemsMidterm@
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