Floating Point Sept 2, 2008Creating Floating Point NumberNormalizeRoundingPostnormalizeFloating PointSept 2, 2008TopicsTopicsIEEE Floating Point StandardRoundingFloating Point OperationsMathematical propertiesclass03.ppt15-213“The course that gives CMU its Zip!”15-213, F’08– 28 –15-213, F’08Creating Floating Point NumberCreating Floating Point NumberStepsStepsNormalize to have leading 1Round to fit within fractionPostnormalize to deal with effects of roundingCase StudyCase StudyConvert 8-bit unsigned numbers to tiny floating point formatExample Numbers 128 10000000 15 00001101 33 00010001 35 00010011 138 10001010 63 00111111s exp frac02367– 29 –15-213, F’08NormalizeNormalizeRequirementRequirementSet binary point so that numbers of form 1.xxxxxAdjust all to have leading oneDecrement exponent as shift leftValue Binary Fraction Exponent 128 10000000 1.0000000 7 15 00001101 1.1010000 3 17 00010001 1.0001000 5 19 00010011 1.0011000 5 138 10001010 1.0001010 7 63 00111111 1.1111100 5s exp frac02367– 30 –15-213, F’08RoundingRoundingRound up conditionsRound up conditionsRound = 1, Sticky = 1 > 0.5Guard = 1, Round = 1, Sticky = 0 Round to evenValue Fraction GRS Incr? Rounded 128 1.0000000 000 N 1.000 15 1.1010000 100 N 1.101 17 1.0001000 010 N 1.000 19 1.0011000 110 Y 1.010 138 1.0001010 011 Y 1.001 63 1.1111100 111 Y 10.0001.BBGRXXXGuard bit: LSB of resultRound bit: 1st bit removedSticky bit: OR of remaining bits– 31 –15-213, F’08PostnormalizePostnormalizeIssueIssueRounding may have caused overflowHandle by shifting right once & incrementing exponentValue Rounded Exp Adjusted Result 128 1.000 7 128 15 1.101 3 15 17 1.000 4 16 19 1.010 4 20 138 1.001 7 134 63 10.000 5 1.000/6
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