Multiplication/division Oku H&P sections 4.6-4.8 Bir kac integer multiplication algorithm Bir integer division algorithms Floating point math 10/22/2004 Bilgisayar Mimarisi 6.1 10/22/2004 Bilgisayar Mimarisi 6.2 1000 10 1001 10 ----------- 1000 0000 0000 1000 ----------- 100 10 A B ----- C multiplicand m bit multiplier n bit result (m+n) bit 111 Value = 7 in 3 bits 11 Value = 3 in 2 bits --------- 111 111 --------- 10101 Value = 21 in 5 bits 10/22/2004 Bilgisayar Mimarisi 6.3 10/22/2004 Bilgisayar Mimarisi 6.4
multiply version 1 2 3 ------ --- 0110 6 Shift left 64bits Start Multiplier0 = 1 1. Test M ultip lier0 = 0 Multiplier 1a. Add multiplicand to product and place the result in register 2. Shift the register left 1 bit 3. Shift the M ultiplier register right 1 bit First version worked Not surprising.worked in grade school too! Slow; averages 2.5 processing steps per bit Wide; 64-bit adder, 160 bits of data storage 64-bit ALU Multiplier Shift right 32bits N o: < 32 repetitions 32nd repetition? Yes: 32 repetitions Write 64bits Done Control test from H&P, pages 251-2 10/22/2004 Bilgisayar Mimarisi 6.5 10/22/2004 Bilgisayar Mimarisi 6.6 Multiply version 2 Start Shift product right Add only 32 bits Multiplier0 = 1 1. Test Multiplier0 Multiplier0 = 0 Second version better 32 bits 1a. Add multiplicand to the left half of the product and place the result in thelefthalfofthe register 2. Shift the register right 1bit Still slow; averages 2.5 processing steps per bit Narrower; 32-bit adder, 128 bits of data storage 32-bit ALU Multiplier Shift right 32 bits 3. Shift the Multiplier register right 1 bit 32nd repetition? No: < 32 repetitions Shift right Write Control test Yes: 32 repetitions 10/22/2004 64 bits Bilgisayar Mimarisi Done 6.7 10/22/2004 Bilgisayar Mimarisi 6.8
Multiply version 3 Start Baslangicta multiplier product in sag yari tarafinda 32-bit ALU 32 bits 0 =1 1. Test 0 =0 0 1a. Add multiplicand to the left half of the product and place the result in thelefthalfoftheregister 2. Shift the register right 1 bit No: < 32 repetitions 32nd repetition? Yes: 32 repetitions Done Iteration 0 1 2 3 4 Step Initial values 0000 0000 0000 0111 0111 0001 1011 0100 1011 0101 0101 0101 1010 Shift right Control Write test 10/22/2004 64 bits Bilgisayar Mimarisi 6.9 10/22/2004 Bilgisayar Mimarisi 6.10 Iteration 0 1 2 3 4 Step Initial values 0000 1111 0001 1101 0001 1000 1000 1111 0100 1101 0100 1010 Third version best yet Faster; averages 1.5 processing steps per bit Width same as version 2; 32-bit adder, 128 bits of data storage Simpler; only one register requires shifting 10/22/2004 Bilgisayar Mimarisi 6.11 10/22/2004 Bilgisayar Mimarisi 6.12
Booth s algorithm for int multiply Shift isleminin add isleminden hizli oldugu durumda degerlilik kazanir Gunumuzde adder lar hizli oldugu icin bu algorithm onemini yitirdi Booth s algorithm 3. Multiply versiyonuna benzer Multiplier result in sag 32 bitinde Multiplier yine shift right islemine tabi. 3. Version daki bazi durumlarda add islemlerinden kurtulundu 10/22/2004 Bilgisayar Mimarisi 6.13 10/22/2004 Bilgisayar Mimarisi 6.14 Booth algorithm Itera tion Multip licand Original algorithm Step Booth s algorithm Step product Multiplier in simdiki (current) ve bir onceki (previous) bit lerine bakilarak asagidaki islemlerin biri gerceklenir 00 arithmetic islem yok 01 multiplicand i product in sol yarisina ekle 10 multiplicand i product in sol yarisindan cikar 11 arithmetic islem yok register 1 bit right shift yapilir 0 1 2 3 4 Initial values Prod = Prod + Mcand Prod = Prod + Mcand 0000 0110 0000 0110 0000 0001 0001 0001 0001 1000 0001 1000 0000 1100 Initial values Prod = Prod Mcand Prod = Prod + Mcand 0000 0110 0 0000 0110 0 0000 0 0 1111 0001 1 1111 0001 1 1111 1000 1 0001 1000 1 0000 1100 0 10/22/2004 Bilgisayar Mimarisi 6.15 10/22/2004 Bilgisayar Mimarisi 6.16
Integer Division Quotient = dividend / divisor with remainder Try this example: 2619 / 32 Hardest step: determine the largest quotient digit that, when multiplied times divisor, is less than the dividend. Division Version 1 64 bit divisor 64 ALU 64 bit remainder 32 bit quotient To divide 7 / 2 (each 4 bits) Divisor reg starts with 0000 Rem reg starts with 0000 0111 Quotient starts with 0000 10/22/2004 Bilgisayar Mimarisi 6.17 10/22/2004 Bilgisayar Mimarisi 6.18 Start Rem = Rem - divisor Division Version 1 Floating point representation Shift Quotient left, set LSB = 1 Remainder Shift Divisor right Repeat 33 times Rem = Rem + Divisor Shift quotient left, set LSB = 0 10/22/2004 Bilgisayar Mimarisi 6.19 Bilgisayar lar fraction a sahip sayilar icin bir gosterime sahip olmali (math islemlerini yapabilmek icin) Fraction li sayilar 3.1416 Cok kucuk sayilar.000000001 Cok buyuk sayilar 6.02 x 10 23 10/22/2004 Bilgisayar Mimarisi 6.20
Exponent siz sayilarin bilgisayarlarda saklanmasi 32 bitle 4 milyar sayi saklanabilir 64 bitle 16x10 18 sayi saklanabilir Ancak scientific hesaplarda cok daha buyuk sayilarin saklanmasina ihtiyac duyulur Floating point representation Sign, exponent, significand (-1) sign x significand x 2 exponent Bit lerin cogu significand icin kullanilirsa More accuracy Bit lerin cogu exponent icin kullanilirsa more range IEEE 754 floating point standard single precision: 8 bit exp, 23 bit significand double: 11 bit exp, 52 bit significand 10/22/2004 Bilgisayar Mimarisi 6.21 10/22/2004 Bilgisayar Mimarisi 6.22 Normalization In scientific notation: 1234000 = 1234 x 10 3 = 1.234 x 10 6 in floating point: 1010000 = 1011 x 2 3 = 1.011 x 2 6 Ilk bit 1, bunu binary point izler Ilk bit in 1 oldugu icin bunu bit kullanarak temsil etmemize gerek yok IEEE 754 standard Leading 1 is implicit Buyuk sayilar icin positive exponent lere ihtiyac Kucuk sayilar icin negative exponentlere ihtiyac Biasing kullanilarak siralama daha kolay sekle sokulur En kucuk exponent de tum bit ler 0, en buyuk exponent de butun bit ler 1 Single prcision icin bias 127, double precision icin bias 1023 10/22/2004 Bilgisayar Mimarisi 6.23 10/22/2004 Bilgisayar Mimarisi 6.24
Convert 2.75 to floating point Sign bit 0 2.75 10 10.110 2 Normalize to 1.0110 x 2 +1 Stored exp: 127 + 1 = 128 = 10000000 Final value: 010000000000000 0 Convert -6.25 to IEEE floating point..? 10/22/2004 Bilgisayar Mimarisi 6.25 Convert 010001000 0 Sign is positive Exponent is 129-127 = 2 Binary value is 1.01101 x 2 2 or 101.101 Apply weights 4 2 1.5.25.125 Result is 5.625 Convert 101111000000000.0 =? 10/22/2004 Bilgisayar Mimarisi 6.26 Special values Addition in floating point S (1 bit) E (8 bit) F (23 bit) single precision 0 [ E [ 255 e = E 127 (Biased-127) S E F Number 0/1 0 0 0 0 255 0 + 1 255 0-0/1 255 g 0 NaN 10/22/2004 Bilgisayar Mimarisi 6.27 Kucuk exponent li sayiyi shift ederek exponent ler esitlenir Significand lari topla Sonucu normalize et 4.56 x 10 3 --> 4.56 x 10 3 2.34 x 10 2 -->.234 x 10 3 ------------------------------------- 4.794 x 10 3 Already normalized 10/22/2004 Bilgisayar Mimarisi 6.28
Multiplication in floating point Multiply the significands Add the exponents Normalize the result 1.03 x 10 3 0.6 x 10 2 ------------- 0.618 x 10 5 normalize to 6.18 x 10 4 10/22/2004 Bilgisayar Mimarisi 6.29 Gizli Tehlikeler Overflow (sonuc cok buyuk) Sonucun exponen ti 8 (11) bit alanina sigmiyor Underflow Sonucun exponenti cok kucuk, ornek., 10-6 Note: too small doesn t mean zero! 10/22/2004 Bilgisayar Mimarisi 6.30 Floating point is not natural 1 ile 3 * (1/3) ayni sey degil 1/3 is a forever repeating fraction Bolme islemi belli bir adimdan sonra sonlandirilir Carpma yapildiginda, sonuc 1 den farkli Floating point addition associative degil X = -1.5 x 10 38 Y = 1.5 x 10 38 Z = 1.0 X + (Y+Z) = -1.5 x 10 38 + (1.5 x 10 38 + 1.0) = -1.5 x 10 38 + 1.5 x 10 38 = 0 (X+Y) +Z = (-1.5 x 10 38 + 1.5 x 10 38 ) + 1.0 = 0 + 1.0 = 1.0 10/22/2004 Bilgisayar Mimarisi 6.31 10/22/2004 Bilgisayar Mimarisi 6.32
There are repeating fractions in binary 0.1 in decimal is 1/16 + 1/32 + 1/256 + 1/512 + 1/4096 =.0 etc Once bolme sonra carpma yapildiginda baslangictaki sayidan farkli bir sayi elde edilir Loss of accuracy with subtraction Assume 6 digits of accuracy with A and B A = 3.03492 B = 3.03471 A - B = 0.00021 Only two accurate digits left Restate: Bill Gates net worth yesterday was 3.61 B dollars, today it is still 3.61 B dollars. I would love to have the difference. 10/22/2004 Bilgisayar Mimarisi 6.33 10/22/2004 Bilgisayar Mimarisi 6.34