Integers and Floating Point
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1 CMPE12 More about Numbers Integers and Floating Point (Rest of Textbook Chapter 2 plus more)"
2 Review: Unsigned Integer A string of 0s and 1s that represent a positive integer." String is X n-1, X n-2, X 1, X 0, where X k is either a 0 or a 1 and has a weight of 2 k." The represented number is the sum of all the weights for each 1 in the string." 14-2"
3 Signed Integers" Allow us to represent positive and negative integers." 4 important types:" "Sign and Magnitude -- Leftmost bit is the sign and the remaining bits are the unsigned magnitude." "1ʼs complement -- The additive inverse of a number is the bit-wise complement of the number." "2ʼs complement -- The additive inverse of a number is the bit-wise complement plus one to the number." "Bias or excess notation -- a bias is subtracted from the unsigned value to get the bias value." 14-3"
4 Quick Review of Signed Integers" Decimal 1 s complement 2 s complement Sign-and- Magnitude "
5 Biased notation" How does it work? The signed integer is biased so that the bias value is represented by " Advantages" Preserves lexical order" Single zero" Most versatile " Disadvantages:" - Add and sub require one additional operation to adjust the bias" Rep Value Bias "
6 Conversion: Decimal/BiasX" Rep D Value Decimal -> BiasX" Add X, then Convert to Binary" BiasX -> Decimal" Convert Binary to Decimal, then Subtract X" "
7 Addition of Bias representations?" unsigned" x! +y! z! (x+b)! +(y+b)! (z+b)! What you want" What you get" (x+ b)! +(y+ b)! (z+2b)! z+2b! - b! z+ b! How you convert" So you must subtract out the additional Bias when you are finished!" 14-7"
8 unsigned" x -y z (x+b) -(y+b) (z+b) What you want Subtraction?" What you get" (x+b) -(y+b) (z-0) z+0 + b z+b How you convert" So you must add back the Bias when you are finished!" 14-8"
9 Biased notation mapping" Number" represented" -3! -2! -1! 0! 1! 2! 3! 4! Number" 000! 001! 010! 011! 100! 101! 110! 111! encoded" Range on n bits:" -(2 n-1-1) to 2 n-1" if Bias is " 14-9"
10 32-bit word" 32 bit word can represent ~ 4.3 billion values" Integers: 0 -> ~4.3 billion" Signed Integers: ~ -2.15B -> 2.15B" Fractional numbers?" Very large numbers?" Numbers with very small magnitude?" 14-10"
11 Scientific Notation" Example: 6.023*10 23" Of form A.xxx *(BASE) exponent" In Binary: 1.xxx * 2 exponent" Or maybe Y 16.xx 16 * 16 exponent" Standard: IEEE standard for floating point arithmetic " 14-11"
12 IEEE standard for floating point" 1.xxx * 2 exponent in a 32-bit word " The 1. and the 2 can be assumed." xxx xx and exponent (and sign) is all that must be specified." 14-12"
13 Floating Point Numbers" 8 bits 23 bits" S Exponent Fraction (xx xx)" 1 means" negative" (In Bias 127)" How do we convert to Decimal?" If < Exponent < " N = (-1) S * 1.Fraction * 2 Exponent-127" 14-13"
14 Converting from Decimal to Float" 1. Convert to Binary (eg )" 2. Normalize (form = 1.xxxxxx *2 EXP )" 3. Convert EXP to bias127 (add 127 to it)" 4. MSB [31] gets sign" 5. [23:30] gets EXP (bias127)" 6. [0:22] gets xxxxxxxxxxxxxxxxxxxxxxx" 14-14"
15 Convert to IEEE FP" 56.5" " (do to 5 binary places)" 14-15"
16 Your hard work has not gone unnoticed!" From: The Chronicle of Higher Education " The average full-time undergraduate student studies about 15 hours a week but the duration varies by major, according to this year's National Survey of Student Engagement." Engineering majors spend the most time studying, 19 hours a week, but even among those who exceed 20 hours, nearly a quarter still often show up for class without assignments completed." 14-16"
17 Misconceptions about floats" Floats are not reals. Ex. 2/3" Floats are not decimals = " Not all integers < 2 31 can be represented = " 13-17"
18 More on IEEE 754 FP Standard" Distribution of floats on number line" Denormalized floats" Double precision floats" Arithmetic on floates" 14-18"
19 How FP numbers distributed" A 32 bit number can represent at most 2 32 values" IEEE 754 FP can represent numbers larger than so many integers between 0 and are not represented." High density close to 0." Low density far from 0 " 13-19"
20 Specifically: 2 23 values for each value of exponent (23 bits)" Between 1/2048 and 1/1024 there are 2 23 floats." Between 1 and 2 there are 2 23 floats." Between 2 30 and 2 31 there are 2 23 floats." Between 2 x to 2 x+1 for -127 < x < 128 there are 2 23 floats." 13-20"
21 Number Line" "
22 Denormalized Floating Point Numbers" 8 bits 23 bits" S Fraction (xx xx)" 1 means" negative" (Shows as Denormalized)" How do we convert to Decimal?" N = (-1) S * 0.Fraction * 2-126" * is smallest Normalized number; * is largest Denormalized number. " 16-22"
23 What if Exponent is ?" If FRAC is 0, the 32 bits represent + or infinity." If FRAC is nonzero, the 32 bits represent NaN (Not a Number)" Ex: 0/0" 15-23"
24 Infinity: EXP = ; FRAC=0" Infinity avoids exception on overflow. (overflow definition: result exceeds value that can be represented)" Examples of operations that return infinity: 1/0, -1/0, 3 inf, sqrt(+inf) 13-24"
25 Double Precision IEEE 754 Floating Point Numbers" 11 bits 52 bits" S Exponent Fraction (xx xx)" 1 means" (In Bias 1023)" negative" To convert to Decimal" If < Exponent < " N = (-1) S * 1.Fraction * 2 Exponent-1023" 16-25"
26 Double precision floating-point" 11 bits 52 bits" S Exponent Fraction (xx xx)" 1 means" negative" (In Bias 1023)" -( ) <= exp <= " is about 2*10 308" 15-26"
27 Double Precision Float" 52 significant figures base 2 is approximately 16 significant figures in base 10." 14-27"
28 Single vs. Double FP" Range:" SP: ~2-126 to approximately: to 10 38" DP approximately: 2* to 2*10 308" Significant figures: "" SP: 23 significant bits, 2 23 = 8,388,608" almost 9 significant decimal digits" DP: 52 significant bits, 2 52 = 4*2 20 *2 30 " > 15 significant decimal digits " 13-28"
29 What is this single-precision floating-point number?" A. 2-5 B. 0 C D. 1 * 2exp( ) E. None of the above 15-29"
30 What is this floating-point number?" A B. 1.01*2-127 C D E. None of the above 15-30"
31 Adding two scientific notation numbers" 5.345* *10 25" 1. Make their exponent the same ( * *10 25 )" 2. Add the non-exponents ( *10 25 )" 3. Normalize (already done)" 15-31"
32 Adding two floats" * * " 2. Make their exponent the same (1.011* * )" 3. Add nonexponents ( * )" 4. Normalize (already done)" "
33 Multiplying two scientific notation numbers" *10 23 * 8.1*10 25" 2. Multiply the non-exponents and add the exponents (42.93*10 48 )" 3. Normalize (4.293*10 49 )" 15-33"
34 Multiplying two floats" *2 4 * 1.11*2 2" 2. Multiply the non-exponents and add the exponents ( *2 6 )" 3. Normalize ( *2 7 )" "
35 Add these two floats" Write each in normalized form " 2. Make their exponent the same " 3. Add nonexponents" 4. Normalize" 15-35"
36 Multiplying these two floats" Write normal form of numbers " 2. Multiply the non-exponents and add the exponents" 3. Normalize" 15-36"
37 How is FP arithmetic done?" Software: very, very slow." Hardware floating-point: expensive, but usually worth it." Two measures of performance:" "1. MIPS: millions of instructions executed per second." "2. MFLOP: millions of floating point operations per second." 15-37"
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