CSE 246: Computer Arithmetic Algorithms and Hardware Design Instructor: Prof. Chung-Kuan Cheng Fall 2006 Lecture 9: Floating Point Numbers

CSE 2462 Motivation  Maximal information with given bit numbers.  Arithmetic with proper precision.  Fairness of rounding.  Features at the expenses of the complexity of the operations.

CSE 2463 Topics:  Floating Point Numbers (IEEE P754)  Standard  Operations  Exceptional Situations  Rounding Modes  Numerical Computing with IEEE Floating Point Arithmetic, Michael L. Overton, SIAM

CSE 2464 Standard 2 32  Typically  Goal: Dynamic Range: largest #/ smallest #  If too large, holes between # ’ s

CSE 2465 Standard  ulp (unit in the last place)  Difference between two consecutive values of the significand. 3 Parts  x = ~s b e :sign, significand, exponent Sign Bit 8-bit exponent 23-bit Significand

CSE 2466 Standard  ~ e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 s 1 s 2 s 3 … s 22 s s 1 s 2 s 3 … s 22 s 23 normalized number 0. s 1 s 2 s 3 … s 22 s 23 denormalized number e 1 e 2 e 3 e 4 e 5 e 6 e 7 e x= 0.s 1 s 2 s 3 … s 22 s x= 1.s 1 s 2 s 3 … s 22 s x= 1.s 1 s 2 s 3 … s 22 s x= 1.s 1 s 2 s 3 … s 22 s x= 1.s 1 s 2 s 3 … s 22 s x= 1.s 1 s 2 s 3 … s 22 s x= Inf if (s 1 … s 23 )= 0, NaN otherwise. NaN  Not a Number

CSE 2467 Standard 0.01x2 -3 = 0.001x2 -2  Same number, so normalize to remove redundancy  Use a default 1 in front for one more bit precision.  Smallest Number 0.00 … 01x = 1.0x2 -23 x = 1x2 -149

CSE 2468 Standard - Example ~ eeeeeeee sssss sssss sssss sssss sss = … 0x = … 0x = … 1x = … 0x normalized minimum = … 1x = … 0x = … 1x = … 1x2 1

CSE 2469 Standard – Example Cont = … 0x = … 1x = … 1x Normalized Maximum = Inf N min = 1.0 x N max = (2 – )2 127

CSE Double Floating Point ~ e 1 e 2 … e 11 s 1 s 2 … s … 000 s 1 s 2 … s 52 x=0.s 1 s 2 … s … 001 s 1 s 2 … s 52 x=1.s 1 s 2 … s … 111 s 1 s 2 … s 52 x=1.s 1 s 2 … s … 000 s 1 s 2 … s 52 x=1.s 1 s 2 … s … 110 s 1 s 2 … s 52 x=1.s 1 s 2 … s … 111 s 1 s 2 … s 52 x=Inf if (s 1 … s 52 )=0

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

CSE Rounding Mode  Adder Output = C out 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 normalize Guard bit

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

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