1Floating Point NumbersSummer 2008CMPE12 – Summer 2008 – Slides by ADB 2Fractional numbers Fractional numbers – fixed point Floating point numbers – the IEEE 754 floating point standard Floating point operations Rounding modes2CMPE12 – Summer 2008 – Slides by ADB 3Positional representation of fractional numbers In base 102102Decimal pointNumber65431Position-4-3-2-1013Multiplier10-410-310-210-1100101103CMPE12 – Summer 2008 – Slides by ADB 4Positional representation of fractional numbers In base 22122Binary pointNumber10110Position-4-3-2-1013Multiplier2-42-32-22-12021233CMPE12 – Summer 2008 – Slides by ADB 5Fractional numbers – fixed point Fixed-point representation How much information is necessary to store? How do you choose a format for the bits? Fixed-point operations Addition Align binary points, and add straight down Multiplication ???CMPE12 – Summer 2008 – Slides by ADB 6Decimal to binary conversion Convert A = 3.141510to base 24CMPE12 – Summer 2008 – Slides by ADB 7Fixed-point number densityCMPE12 – Summer 2008 – Slides by ADB 8Scientific notation In base 10 Example: 3.0 × 108In base 2 Example: –1.00101 × 24(= –18.510) The general form r = Sign × signiFicand × baseExponent5CMPE12 – Summer 2008 – Slides by ADB 9Single-precision IEEE 754 floating-point numberstnenopxe30 29 28 27 26 25Sign31 23significand0123456789011121314151617181920212224 One-bit sign Eight-bit exponent 23-bit significand That’s the fractional partCMPE12 – Summer 2008 – Slides by ADB 10Single-precision IEEE 754 floating-point numbers Normalized numbers: only one non-zero bit to the left of the binary point Adjust the exponent as needed r = (–2)S× F ×2EImplicit leading 1 in the significand (the “hidden bit”) r = (–2)S×(1 + F)×2EBias notation to represent the exponent With the bias B = 127 r = (–2)S×(1 + F)×2E-B6CMPE12 – Summer 2008 – Slides by ADB 11How to convert a base-10 number into IEEE 754 single-precision floating point Convert the number to binary The big part  And the fractional part Normalize Isolate the hidden one Remove the significand’s hidden one Add bias to the exponent Represent the numberCMPE12 – Summer 2008 – Slides by ADB 12Example: 12.62530 29 28 27 26 25Sign31 23significand0123456789011121314151617181920212224 Convert to binary Normalize Remove hidden one Add bias exponent The end7CMPE12 – Summer 2008 – Slides by ADB 13Double-precision IEEE 754 floating-point numberstnenopxe63 62 61 60 59 58Sign64 56significand333435363738394041424344454647484950515253545557 One-bit sign Eleven-bit exponent 52-bit significand That’s the fractional part30 29 28 27 26 2531 23significand, continued0123456789011121314151617181920212224CMPE12 – Summer 2008 – Slides by ADB 14Summary of IEEE 754 formats≥ 7964≥ 4332Total bits≤ –16382–1022≤ –1022–126Emin≥ +16383+1023≥ +1023+127Emax≥ 1511≥ 118Bits for E≥ 6453≥ 3224Bits for FDouble Ext.DoubleSingle Ext.SingleParameterPrecision For every precision, there are reserved exponents, used for special quantities: Emin– 1 (i.e., E=0) is used for zero and denorms Emax+ 1 (i.e., E=255 or E=2047, with bias) is used for NaN and infinity8CMPE12 – Summer 2008 – Slides by ADB 15Special quantities: Infinity This special quantity avoids halt on overflow Much safer than returning the largest possible number Representation: E = Emax+ 1 E = 255 in single precision with bias E = 2047 in double precision with bias F = 0 Sign (+∞ or –∞)30 29 28 27 26 25Sign31 23significand0123456789011121314151617181920212224CMPE12 – Summer 2008 – Slides by ADB 16Special quantities: Infinity Examples of operations that return ±Inf1 / InfSqrt (+Inf)4 – Inf–1/01/0ResultOperation9CMPE12 – Summer 2008 – Slides by ADB 17Special quantities: NaN (Not a Number) This special quantity avoids halt on invalid operations Representation: E = Emax+ 1 E = 255 in single precision with bias E = 2047 in double precision with bias F ≠ 030 29 28 27 26 25Sign31 23significand0123456789011121314151617181920212224CMPE12 – Summer 2008 – Slides by ADB 18Special quantities: NaN (Not a Number) Examples of operations that return NaNlog(–0)log(+0)1/–Inf3/–03/+0–0/3+0/3ResultOperation10CMPE12 – Summer 2008 – Slides by ADB 19Special quantities: Zero Representation: E = Emin– 1 (i.e., E = 0) F ≠ 0 Sign: +0 or –030 29 28 27 26 25Sign31 23significand0123456789011121314151617181920212224CMPE12 – Summer 2008 – Slides by ADB 20Special quantities: Zero Examples of operations that involve ±0Sqrt/×+NaN produced byOperation11CMPE12 – Summer 2008 – Slides by ADB 21Floating point numbers range What is the largest number we can represent in IEEE 754 single-precision floating point? What is the smallest number?30 29 28 27 26 25Sign31 23significand0123456789011121314151617181920212224CMPE12 – Summer 2008 – Slides by ADB 22Floating point numbers range What is the largest number we can represent in IEEE 754 double-precision floating point? What is the smallest number?12CMPE12 – Summer 2008 – Slides by ADB 23Floating point numbers: density Fact 1: Floats are not reals E.g., 2/3 Fact 2: Floats are not decimals E.g., 0.1 (base 10) = 1.1001100… × 2–4(base 2) Fact 3: Not even all the integers in the range are represented E.g., 100,000,001 (base 10) =1011 1110 1011 1100 0010 0000 0001 (base 2)CMPE12 – Summer 2008 – Slides by ADB 24Floating point numbers: density Close to 0: high density Far from 0: high density13CMPE12 – Summer 2008 – Slides by ADB 25Special quantities: Denormals These are numbers smaller than 2^(Emin) Fill the gap between 2^(Emin) and 0 (gradual underflow) Representation: E = Emin– 1 (i.e., E = 0) F ≠ 0 The number represented is 0.f-1f-2…f-23×2^(Emin)30 29 28 27 26 25Sign31 23significand0123456789011121314151617181920212224CMPE12 – Summer 2008 – Slides by ADB 26Special quantities: Denormals From 0 to 2^(Emin)14CMPE12 – Summer 2008 – Slides by ADB 27Summary of IEEE 754 numbers≠ 02047≠ 0255020470255anything1 – 2046anything1 – 254≠ 00≠ 000000SignificandExponentSignificandExponentObjectDouble precisionSingle precisionCMPE12 – Summer 2008 – Slides