Review. How to represent real numbers

Transcription

1 PCWrite PC IorD Review ALUSrcA emread Address Write data emory emwrite em Data IRWrite [3-26] [25-2] [2-6] [5-] [5-] RegDst Read register Read register 2 Write register Write data RegWrite Read data Read data 2 Registers A B ALU Zero Reslt ALUOp ALU Ot PCSorce Instrction register emory data register Sign etend Shift left 2 ALUSrcB emtoreg Label ALU Src Src2 Register emory PCWrite Net control control control Fetch Add PC 4 Read PC ALU Seq Add PC Etshift Read Dispatch How to represent real nmbers In decimal scientific notation: has 3 parts sign (i.e. +) fraction a/k/a mantissa (i.e. 5.28) base (i.e., ) to some power (i.e. 3) ost of the time, the sal representation has one digit to the left of decimal point Eample: A nmber is normalized if the leading digit is not Eample:

2 Real representation inside a compter Use a representation akin to scientific notation sign mantissa base eponent where sign is or - any variations in choice of representation for mantissa (cold be 2 s complement, sign and magnitde etc.) base (cold be 2, 8, 6 etc.; is a little difficlt) eponent (like mantissa) Arithmetic spport for real nmbers is called floating-point arithmetic What is the 52 bridge called? 3 Floating-point representation: IEEE Std Floating Point is necessarily an approimation Basic choices A single precision nmber mst fit into word (4 bytes, 32 bits) A doble precision nmber mst fit into 2 words The base for the eponent is 2 There shold be approimately as many positive and negative eponents Additional criteria The mantissa will be represented in sign and magnitde form Nmbers will be normalized This standard was significant milestone 4 2

3 E: IPS representation of IEEE Standard A nmber is represented as : (-) S. F. 2 E In single precision the representation is: 8 bits 23 bits s eponent mantissa IPS representation (contined) Bit 3 sign bit for mantissa ( pos, neg) Eponent 8 bits ( biased eponent, see net slide) mantissa 23 bits : always a fraction with an implied binary point at left of bit 22 Nmber is normalized (see implication net slides) is represented by all zero s woldn t have to be Note that having the most significant bit as sign bit makes it easier to test for positive and negative 6 3

4 Biased eponent Biased notation is yet another techniqe for representing signed nmbers The middle of the range will represent. For eight bits is 27 () All eps starting with a will be positive eponents Eample: is eponent 2 (- ) All eps starting with a will be negative eponents Eample is eponent - ( - ) The largest positive eponent will be, abot 38 The smallest negative eponent is abot Normalization Since nmbers mst be normalized, there is always a one at the left of the binary point:. No need to pt it in (improves precision by bit) Often referred to as the implied Bt need to reinstate it when performing operations In smmary, in IPS a floating-point nmber has the vale: (-) S. ( + mantissa). 2 (eponent - 27) 8 4

5 Doble precision Takes 2 words (64 bits) Eponent bits (instead of 8) what s the bias? antissa 52 bits (instead of 23) Still biased eponent and normalized nmbers Still is represented by all zeros We can still have overflow (the eponent cannot handle sper big nmbers) and nderflow (the eponent cannot handle sper small nmbers) 9 Floating-Point Addition Qite comple (more comple than fl mltiplication) Need to know which of the addends is larger (compare eponents) Need to shift smaller mantissa Need to know if mantissas have to be added or sbtracted (since it s a sign/magnitde representation) Need to normalize the reslt Correct rond-off procedres are not simple (not covered in detail here) 5

6 One of the 4 rond-off modes Rond to nearest even Eample : in base. Assme 2 digit accracy 3. * * -2 = 3.46 * clearly shold be ronded to 3. * Eample 2: 3. * + 5.* -2 = 3.5 * By convention, rond-off to nearest even nmber 3.2 * Other rond-off modes: towards, +, - F-P add (details for rond-off omitted). Compare eponents. If e < e2, swap the 2 operands sch that d = e - e2 >=. Tentatively set eponent of reslt to e 2. Insert s at left of mantissas. If the signs of operands differ, replace 2nd mantissa by its 2 s complement. 3. Shift 2nd mantissa d bits to the right (this is an arithmetic shift, i.e., insert either s or s depending on the sign of the second operand) 4. Add the (shifted) mantissas. (There is one case where the reslt cold be negative and yo have to take the 2 s complement; this can happen only when d = and the signs of the operands are different.) 2 6

7 F-P Add (contined) 5. Normalize (if there was a carry-ot in step 4, shift right once; else shift left ntil the first appears on msb) 6. odify eponent to reflect the nmber of bits shifted in previos step 3 Add decimal: Eample 3/ /2 2 =. +. = Now add: Align fractions: Add fractions:. 2 - Normalize:. 2 - =. 2 Rond: Not needed. 2 = + /2 3 = + /8 =.25 decimal In IEEE single precision.75 i.e.. is why? 4 7

8 Stage Eponent compare Stage 2 Shift and Add Using pipelining Stage 3 Rond-off, normalize and fi eponent ost of the time, done in 2 stages 5 Floating-point mltiplication Conceptally easier. Add eponents (carefl, sbtract one bias ) 2. ltiply mantissas (don t have to worry abot signs) 3. Normalize and rond-off and get the correct sign 6 8

9 Special Vales Allow comptation to contine in face of eceptional conditions For eample: divide by, overflow, nderflow Special vale: NaN (Not a Nmber; e.g., sqrt(-)) Operations sch as + NaN yield NaN Special vales: + and - (e.g, / is + ) Can also se denormal nmbers for nderflow and overflow allowing a wider range of vales 7 9