Slide 1Quote of the dayReview of NumbersOther NumbersScientific Notation (in Decimal)Scientific Notation (in Binary)Floating Point Representation (1/2)Floating Point Representation (2/2)Double Precision Fl. Pt. RepresentationQUAD Precision Fl. Pt. RepresentationIEEE 754 Floating Point Standard (1/4)IEEE 754 Floating Point Standard (2/4)IEEE 754 Floating Point Standard (3/4)IEEE 754 Floating Point Standard (4/4)?Understanding the Significand (1/2)Understanding the Significand (2/2)Example: Converting Binary FP to DecimalPeer Instruction #1AnswerConverting Decimal to FP (1/3)Converting Decimal to FP (2/3)Converting Decimal to FP (3/3)Example: Representing 1/3 in MIPSAdministrivia“Father” of the Floating point standardRepresentation for ± ∞Representation for 0Special NumbersRepresentation for Not a NumberRepresentation for Denorms (1/2)Representation for Denorms (2/2)Peer Instruction 2“And in conclusion…”CS 61C L10 Floating Point (1)A Carle, Summer 2005 © UCBinst.eecs.berkeley.edu/~cs61c/su05CS61C : Machine StructuresLecture #10: Floating Point2005-07-06Andy CarleCS 61C L10 Floating Point (2)A Carle, Summer 2005 © UCBQuote of the day“95% of thefolks out there arecompletely clueless about floating-point.”James GoslingSun FellowJava Inventor1998-02-28CS 61C L10 Floating Point (3)A Carle, Summer 2005 © UCBReview of Numbers•Computers are made to deal with numbers•What can we represent in N bits?•Unsigned integers:0to 2N - 1•Signed Integers (Two’s Complement)-2(N-1)to 2(N-1) - 1CS 61C L10 Floating Point (4)A Carle, Summer 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...), (3.141...)•All represented in scientific notationCS 61C L10 Floating Point (5)A Carle, Summer 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 L10 Floating Point (6)A Carle, Summer 2005 © UCBScientific Notation (in Binary)1.0two x 2-1radix (base)“binary point”exponent•Normalized mantissa always has exactly one “1” before the point.•Computer arithmetic that supports it called floating point, because it represents numbers where binary point is not fixed, as it is for integers•Declare such variable in C as floatmantissaCS 61C L10 Floating Point (7)A Carle, Summer 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 Sign•Exponent represents y’s•Significand represents x’sRepresent numbers as small as 2.0 x 10-38 to as large as 2.0 x 1038CS 61C L10 Floating Point (8)A Carle, Summer 2005 © UCBFloating Point Representation (2/2)•What if result too large? (> 2.0x1038 )•Overflow!•Overflow Exponent larger than represented in 8-bit Exponent field•What if result too small? (>0, < 2.0x10-38 )•Underflow!•Underflow Negative exponent larger than represented in 8-bit Exponent field•How to reduce chances of overflow or underflow?CS 61C L10 Floating Point (9)A Carle, Summer 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 as 2.0 x 10-308 to almost as large as 2.0 x 10308 •But primary advantage is greater accuracy due to larger significand031S Exponent30 20 19Significand1 bit 11 bits 20 bitsSignificand (cont’d)32 bitsCS 61C L10 Floating Point (10)A Carle, Summer 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 version in progress has 15 bits for the exponent and 112 bits for the significandCS 61C L10 Floating Point (11)A Carle, Summer 2005 © UCBIEEE 754 Floating Point Standard (1/4)•Single Precision, DP similar•Sign bit:1 means negative 0 means positive•Significand:•To pack more bits, leading 1 implicit for normalized numbers•1 + 23 bits single, 1 + 52 bits double•Note: 0 has no leading 1, so reserve exponent value 0 just for number 0CS 61C L10 Floating Point (12)A Carle, Summer 2005 © UCBIEEE 754 Floating Point Standard (2/4)•Kahan wanted FP numbers to be used even if no FP hardware; e.g., sort records with 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 if possible, 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 L10 Floating Point (13)A Carle, Summer 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 of 1/2 v. 2 makes 1/2 > 2!•Instead, pick notation 0000 0001 is most negative, 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 L10 Floating Point (14)A Carle, Summer 2005 © UCBIEEE 754 Floating Point Standard (4/4)•Called Biased Notation, where bias is number subtracted to get real number•IEEE 754 uses bias of 127 for single prec.•Subtract 127 from Exponent field to get actual value for exponent•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 with exponent bias of 1023CS 61C L10 Floating Point (15)A Carle, Summer 2005 © UCB?0 0111 1101 0000 0000 0000 0000 0000 000Is this floating point number:> 0?= 0?< 0?CS 61C L10 Floating Point (16)A Carle, Summer 2005 © UCBUnderstanding the Significand (1/2)•Method 1 (Fractions):•In decimal: 0.34010 => 34010/100010 => 3410/10010•In binary: 0.1102 =>
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