1CS365 1Floating Point ArithmeticCS 365 Lecture 8Prof. Yih HuangCS365 2Scientific Notation6.02 x 10 23exponentradix (base)Mantissadecimal pointSign, magnitudeSign, magnitude2CS365 3Floating Point Numbers  We need a way to represent– numbers with fractions, e.g., 3.1416– very small numbers, e.g., .000000001– very large numbers, e.g., 3.15576 × 109 Representation:– sign, exponent, significand: (–1)sign×××× significand ×××× 2exponent– more bits for significand gives more accuracy– more bits for exponent increases rangeCS365 4Issues Arithmetic (+, -, ×, /) Representation, Normal form Range and Precision Rounding Exceptions (e.g., divide by zero, overflow, underflow) Errors3CS365 5IEEE 754 Standards single precision: 8 bit exponent, 23 bit significand double precision: 11 bit exponent, 52 bit significandCS365 6IEEE 754 Single Precision Exponent: biased 127 binary integer– Real exponent e = E − 127 Mantissa: Signed magnitude, normalized significant with a hidden 1. Value:S E (Exponent) M (Magnitude)1 8 bits23 bits1272).1()1(−××−ESM4CS365 7DiscussionsLeading “1” bit of significand is implicitExponent is “biased”–bias of 127 –all 0s is smallest exponent; all 1s is largest Magnitude of numbers that can be represented: 2-126to 2127(2−2-23) Or in decimal numbers: 1.8×10-38to 3.4×1038CS365 8Exercise Give the single precision representation of 1.0.  Give the single precision representation of -0.875  Give the single precision representation of 0.15CS365 9Exercise Calculate the value of 0 1 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0CS365 10Floating Point Arithmetic1.12+ 0.0000012 Step 1: alignment of decimal point Step 2: add significands Step 3: normalization6CS365 110 10 1 0 1ControlSmall ALUBig ALUSign Exponent Significand Sign Exponent SignificandExponentdifferenceShift rightShift left or rightRounding hardwareSign Exponent SignificandIncrement ordecrement0 10 1Floating Point