Dokumen - IKKP0098 - STMIK EL RAHMA 09_Arithmetic

(1)

William Stallings

Computer Organization and Architecture

7th Edition

Chapter 9

(2)

Arithmetic & Logic Unit • _{Does the calculations}

• _{Everything else in the computer is there to} service this unit

• Handles integers

(3)

(4)

Integer Representation

• _{Only have 0 & 1 to represent everything} • _{Positive numbers stored in binary}

—e.g. 41=00101001

• No minus sign • No period

(5)

Sign-Magnitude

• _{Left most bit is sign bit} • _{0 means positive}

• _{1 means negative} • _{+18 = 00010010} • _{-18 = 10010010} • _Problems

—Need to consider both sign and magnitude in

arithmetic

(6)

(7)

Benefits

• _{One representation of zero}

• _{Arithmetic works easily (see later)} • _{Negating is fairly easy}

—3 = 00000011

—Boolean complement gives 11111100

(8)

(9)

(10)

Negation Special Case 2

• _{Monitor MSB (sign bit)}

(11)

Range of Numbers

• _{8 bit 2s compliment}

—+127 = 01111111 = 27 -1

— -128 = 10000000 = -27

• 16 bit 2s compliment

—_{+32767 = 011111111 11111111 = 2}15 - 1

(12)

Conversion Between Lengths

• _{Positive number pack with leading zeros} • _{+18 = 00010010}

• _{+18 = 00000000 00010010}

• _{Negative numbers pack with leading ones} • _{-18 = 10010010}

(13)

Addition and Subtraction • _{Normal binary addition}

• _{Monitor sign bit for overflow}

• _{Take twos compliment of substahend and} add to minuend

—i.e. a - b = a + (-b)

(14)

(15)

Multiplication • _Complex

(16)

(17)

(18)

(19)

(20)

Multiplying Negative Numbers • _{This does not work!}

• _{Solution 1}

—Convert to positive if required

—Multiply as above

—_{If signs were different, negate answer} • _{Solution 2}

(21)

(22)

(23)

Division

(24)

001111

Division of Unsigned Binary Integers

(25)

(26)

Real Numbers

• _{Numbers with fractions}

• _{Could be done in pure binary}

—1001.1010 = 24 + 20 +2-1 + 2-3 =9.625

• Where is the binary point? • Fixed?

—Very limited

• Moving?

(27)

Floating Point

• +/- .significand x 2exponent

• Misnomer

• Point is actually fixed between sign bit and body of mantissa

(28)

(29)

Signs for Floating Point

• _{Mantissa is stored in 2s compliment}

• _{Exponent is in excess or biased notation}

—e.g. Excess (bias) 128 means

—8 bit exponent field

—_{Pure value range 0-255}

—Subtract 128 to get correct value

(30)

Normalization

• _{FP numbers are usually normalized}

• _{i.e. exponent is adjusted so that leading} bit (MSB) of mantissa is 1

• Since it is always 1 there is no need to store it

• _{(c.f. Scientific notation where numbers are} normalized to give a single digit before the decimal point

(31)

FP Ranges

• _{For a 32 bit number} —8 bit exponent

—+/- 2256  1.5 x 1077

• Accuracy

—_{The effect of changing lsb of mantissa} —23 bit mantissa 2-23  1.2 x 10-7

(32)

(33)

(34)

IEEE 754

• _{Standard for floating point storage} • _{32 and 64 bit standards}

• _{8 and 11 bit exponent respectively}

(35)

(36)

FP Arithmetic +/-• _{Check for zeros}

• _{Align significands (adjusting exponents)} • _{Add or subtract significands}

(37)

(38)

FP Arithmetic x/ • _{Check for zero}

• _{Add/subtract exponents}

• _{Multiply/divide significands (watch sign)} • _Normalize

• _Round

(39)

(40)

(41)

Required Reading

• _{Stallings Chapter 9}