CSE 246: Computer Arithmetic Algorithms and Hardware Design Instructor: Prof. Chung-Kuan Cheng Winter 2004 Lecture 9

CSE 2462 Topics:  Floating Point Numbers (IEEE P754)  Standard  Operations  Exceptional Situations  Rounding Modes

CSE 2463 Standard 2 32  Typically  Goal: Dynamic Range: largest #/ smallest #  If too large, holes between # ’ s

CSE 2464 Standard  ulp (unit in the last place)  Difference between two consecutive values of the significand. 3 Parts  x =  s  b e Sign Bit 8-bit exponent Significand

CSE 2465 Standard   a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 b 1 b 2 b 3  b 22 b 23 1.* normalized number 0.* denormalized number 0 0.b 1 b 2 b 3  b 22 b 23  b 1 b 2 b 3  b 22 b 23  b 1 b 2 b 3  b 22 b 23    if b i = 0 for all i = 1,2, …,23, NaN otherwise NaN  Not a Number

CSE 2466 Standard 0.01x2 -3 = 0.00x2 -2  Same number, so normalize to remove redundancy  Smallest Number 0.00 … 01x = 1.0x2 -23 x = 1x   Difference between 2 # ’ s small for normalized times compared to magnitudes

CSE 2467 Standard - Example s. eeeeeeee nnnnnnnnnnnnnnnnnnnnnnn = … 0x = = … 0x minimal normalized # = … 1x = … 1x = … 0x2 1

CSE 2468 Standard – Example Cont = … 1x = … 1x Normalized Maximum =  N min = 1.0 x N max = (2 – )2 127

CSE 2469 Double Floating Point  a 1 a 2 … a 11 b 1 b 2 … b … b 1 b 2 … b 52 x … b 1 b 2 … b 52 x … b 1 b 2 … b 52 x … b 1 b 2 … b 52 x … b 1 b 2 … b 52 x … 11 =  if b i = 0 for all i = 1,2, …,52

CSE Overflow/Underflow N max N min SparserDenser Overflow Underflow

CSE Addition/Multiplication  s 1 xb e1 + (  s 2 xb e2 ) =  sxb e =  s 1 xb e1 +  s 2 /b e1-e2 x b e1 = (  s 1  s 2 /b e1-e2 ) x b e1  (  s 1 xb e1 ) x (  s 2 xb e2 ) =  (s 1 xs 2 )b e1+e2

CSE Exceptions a/0 =  if a > 0 a/  = 0if a != 0 a · 0 = 0 a ·  =  if a > 0 0 ·  = invalid operation (NaN) 0/0 = invalid operation (NaN) NaP op a = NaN a +  =   -  = NaN

CSE Rounding Mode  Adder Output = Cout z 1 z 0.z -1 z -2 … z - l GRS Guard Bit Round Bit Sticky Bit, OR of all bits below bit R x x x x2 4 Normalize – need to round or

CSE Rouding x x x x 2 0 normalize x x x x x x 2 2 Guard bit

CSE Rounding  Round to the nearest even toward Toward +  Toward - 

CSE Conventional Rounding Error Rounding Error  1.101=  1.101=  =  = Average Error = 0.5/4 = 0.125