Computation for Physics計算物理概論數位資料表示法

Numeral systems計數系統Base基底 Name名稱 Representation

2 Binary 二進位 0,1

8 Octal八進位 0,1,2,3,4,5,6,7

10 Decimal十進位 0,1,2,3,4,5,6,7,8,9

16 Hexadecimal十六進位 0,1,2,3,4,5,6,7,8,9,A=10,B=11,C=12,D=13,E=14,F=15

(𝑎𝑛𝑎𝑛−1⋯𝑎1𝑎0 .𝑐1𝑐2𝑐3⋯ )𝑏=∑𝑘=0

𝑛

𝑎𝑘𝑏𝑘+∑

𝑘=1

𝑛

𝑐𝑘𝑏−𝑘

(001000101111)2=(1057)8=(22𝐹 )16=(559)10

Bit=Binary digit• Bit=0,1

• b, bit

• Byte=Eight bits• B

• K=Kilo=1000=103 or 1024=210

• M=Mega=106 or 1024=220

• G=Giga=109 or 1024=230

• T=Tera=1012 or 1024=240

• KB=kilobyte, MB=kilobyte, GB=gigabyte

Integer

Unsigned integer• Map N-bits to 0,1,…,2N-1=2N possible numbers• Unsigned byte=8 bits

• 0,…,255

• Unsigned short integer=16 bits• 0,…,65535

• Unsigned integer=32 bits• 0,…,4,294,967,295

• Unsigned long integer=64 bits• 0,…, 18,446,744,073,709,551,615

Most and leastsignificant bit

1 0 0 1 1 0 1 1

a 8-bits number

Most significant bit=msbLeast significant bit=lsb

Signed integerHow to represent negative numbers in N-bits without “-” ?• Sign-and-magnitude method

• Use msb as “sign”, N-1 bits as “magnitude”• Two representations for zero

• One’s complement• Use “bit complement” as the arithmetic negative• Two representations for zero

• Two’s complement• Use “two’s complement of the absolute value”• One representation for zero

Sign-and-magnitude

0 1 1 1 1 1 1 1=

0 0 0 0 0 0 0 0=

1 0 0 0 0 0 0 0=

1 1 1 1 1 1 1 1=

sign magnitude

Sign-and-magnitude• Addition of x and y

• If x and y have the same sign sgn(x)(x+y)• If x and y have different sign

• If |x|>|y| sgn(x)*(x-y)• If |y|>|x| sgn(y)*(y-x)

• Subtraction x and y=Addition of x and (-y)• If overflow return error message

One’s complement

0 1 1 1 1 1 1 1=

0 0 0 0 0 0 0 0=

1 1 1 1 1 1 1 1=

1 0 0 0 0 0 0 0=

𝑥=(𝑏7𝑏6𝑏5𝑏4𝑏3𝑏2𝑏1𝑏0)2❑⇔ −𝑥=(𝑏7𝑏6𝑏5𝑏4𝑏3𝑏2𝑏1𝑏0)2𝑏=1❑

⇔𝑏=0 ,𝑏=0❑

⇔𝑏=1

One’s complementDecimal Binary Decimal Binary

+0 0000 -0 1111+1 0001 -1 1110+2 0010 -2 1101+3 0011 -3 1100+4 0100 -4 1011+5 0101 -5 1010+6 0110 -6 1001+7 0111 -7 1000+8 1000 -8 0111

One’s complementAddition

1 1

0 0 0 1 0 1 1 0 +22

+ 0 0 0 0 0 0 1 1 +3

0 0 0 1 1 0 0 1 +25

One’s complementend-around carry

1 1 1 1

0 1 0 0 0 1 1 0 +70

+ 1 1 1 0 1 1 0 0 -19

1 0 0 1 1 0 0 1 0 +50

+ 0 0 0 0 0 0 0 1 +1

0 0 1 1 0 0 1 1 +51

One’s complementnegative zero

1 1 1 1 1 1 1

0 0 0 1 0 1 1 0 +22

+ 1 1 1 1 1 1 1 1 -0

1 0 0 0 1 0 1 0 1 +21

+ 0 0 0 0 0 0 0 1 +1

0 0 0 1 0 1 1 0 +22

TWO’s complement

0 1 1 1 1 1 1 1=

0 0 0 0 0 0 0 0=

0 0 0 0 0 0 0 0=

1 0 0 0 0 0 0 1=

𝑥=(𝑏7𝑏6𝑏5𝑏4𝑏3120100)2❑⇔ −𝑥=(𝑏7𝑏6𝑏5𝑏4𝑏3120100)2𝑏=1❑

⇔𝑏=0 ,𝑏=0❑

⇔𝑏=1

1 0 0 0 0 0 0 0=

two’s complementDecimal Binary Decimal Binary

+0 0000 -0 0000+1 0001 -1 1111+2 0010 -2 1110+3 0011 -3 1101+4 0100 -4 1100+5 0101 -5 1011+6 0110 -6 1010+7 0111 -7 1001+8 1000 -8 1000

Two’s ComplementAddition

1

0 0 0 1 0 0 0 0 +16

+ 0 0 0 1 1 0 0 0 +24

0 0 1 0 1 0 0 0 +40

Two’s ComplementAddition

1 1 1 1

1 1 1 1 0 0 0 0 -16

+ 0 0 0 1 1 0 0 0 +24

1 0 0 0 0 1 0 0 0 +8

ignore

Two’s ComplementAddition

1 1 1 0 1 0 0 0 -24

+ 0 0 0 1 0 0 0 0 +16

1 1 1 1 1 0 0 0 -8

Two’s ComplementAddition

1 1 1 1

1 1 1 1 0 0 0 0 -16

+ 1 1 1 0 1 0 0 0 -24

1 1 1 0 1 1 0 0 0 -40

ignore

Two’s ComplementAddition

1 1 1 1

0 1 1 1 1 0 0 0 +120

+ 0 0 0 0 1 0 0 1 +9

1 0 0 0 0 0 0 1 -126

overflow error message

Positive+Positive=Negative!

Most and leastsignificant bit

1 0 0 1 1 0 1 1

a 8-bits number

Most significant bit=msbLeast significant bit=lsb

Logic gates

Logic gates邏輯閘AND NAND

OR NOR

NOT XOR

XNOR

And Gate

Input OutputA B A AND B0 0 00 1 01 0 01 1 1

OR Gate

Input OutputA B A OR B0 0 00 1 11 0 11 1 1

NOT Gate

Input OutputA B0 11 0

NAND Gate

Input OutputA B A NAND B0 0 10 1 11 0 11 1 0

NOR Gate

Input OutputA B A NOR B0 0 10 1 01 0 01 1 0

XOR Gate

Input OutputA B A XOR B0 0 00 1 11 0 11 1 0

XNOR Gate

Input OutputA B A XNOR B0 0 10 1 01 0 01 1 1

AdderHalf Adder Full Adder

Real number

Radix point小數點• Base 10 notation decimal point• Base 2 notation binary point

(𝑎𝑛𝑎𝑛−1⋯𝑎1𝑎0 .𝑐1𝑐2𝑐3⋯ )𝑏=∑𝑘=0

𝑛

𝑎𝑘𝑏𝑘+∑

𝑘=1

𝑛

𝑐𝑘𝑏−𝑘

Radix point

Scientific notation科學記號

𝑥=𝑎×10𝑏 ,1≤|𝑎|<10

Floating point representation浮點數表示法+152853.5047=+12345×10− 4

significand (mantissa)

exponent

IEEE 754• IEEE

• Institute of Electrical and Electronics Engineers

• IEEE 754• IEEE Standard for Floating-Point Arithmetic• Arithmetic formats: sets of binary and decimal floating-point

data, which consist of finite numbers (including signed zeros and subnormal numbers), infinities, and special "not a number" values (NaNs)

• Interchange formats: encodings (bit strings) that may be used to exchange floating-point data in an efficient and compact form

• Rounding rules: properties to be satisfied when rounding numbers during arithmetic and conversions

• Operations: arithmetic and other operations on arithmetic formats

• Exception handling: indications of exceptional conditions (such as division by zero, overflow, etc.)

IEEE 754 binary16Half precision

• Sign bit: 1 bit• Exponent width: 5 bits• Significand precision: 11 (10 explicitly stored)• Exponent encoding

• Offset=15, Emin=-14,Emax=15

• Minimum positive value=2^-14• Maximum positive value=(2-2^-10)2^15=65504• Minimum subnormal value=2^-24≈5.96 ×10^-5

IEEE 754 binary32 single precision

• Sign bit: 1 bit• Exponent width: 8 bits• Significand precision: 24 (23 explicitly stored)• Exponent encoding

• Offset=127, Emin=-126,Emax=127

• Minimum positive value≈2.2 ×10^-308• Maximum positive value≈1.8 ×10^308• Minimum subnormal value≈4.9 ×10^-324

IEEE 754 binary64Double precision

• Sign bit: 1 bit• Exponent width: 11 bits• Significand precision: 53 (52 explicitly stored)• Exponent encoding

• Offset=1023, Emin=-1022,Emax=1023

• Minimum positive value=2^-126≈1.18 ×10^-38• Maximum positive value=(2-2^-23)2^127≈3.4

×10^38• Minimum subnormal value=2^-149≈1.4 ×10^-45

Representation errorNotation

Represent Approximate Error

1/7 0.142857142857142857… 0.142587

Ln(2) 0.69314718055994530941...

0.693147

Log(2) 0.30102999566398119521...

0.3010

1.25992104989487316476...

1.25992

1.41421356237309504880...

1.41421

e 2.71828182845904523536...

2.718281828459045

π 3.14159265358979323846...

3.141592653589793

Rounding Errors• In base-10 system

• ½=0.5• 1/3=0.333333333333333333333333333333333

333333

• In base-2 system• Terminating iff denominators are powers of 2

(1/2, 3/16)

IEEE Rounding Modes• Truncation:

• Keep the desired number of digits unchanged, removing all less-significant digits; also called rounding toward zero.

• 0.142857 ≈ 0.142 (All digits less significant than the third removed).

• Round to Nearest: (Default)• Round to the nearest valid representation. Break ties by rounding to an even

digit• +23.524, +24.524; -23.5-24, -24.5 -24 (symmetry between +/- numbers)

• Round to Nearest: • Round to the nearest valid representation. Break ties by rounding away from

zero.• +23.524,+24,525; -23.5-24, -24.5 -25 (symmetry between +/- numbers)

• Round to −∞: • Round to a value less than or equal to the original number. If the original

number is positive, this is equivalent to truncation.

• Round to +∞: • Round to a value greater than or equal to the original number. If the original

number is negative, this is equivalent to truncation.

IEEE 754Special values

• Positive infinity• Negative infinity• (positive zero=ordinary zero)• Negative zero: -0• NaNs: “Not a number” values• Subnormal numbers

Signed zero

• Sign=0+ or 1-• Exponent=0• Significand(Mantissa)=0• Arithmetic

• , , • ,

• , • NaN, NaN

Signed infinity • Sign=0+ or 1-• Exponent=maximum value

• 111112

• FFH

• 7FF16

• Significand(Mantissa)=0

NaNs• Sign quiet, signaling• Exponent=maximum value

• 111112

• FFH

• 7FF16

• Significand(Mantissa)≠0• Creation

• NaN, NaN, NaN

• NaN, =NaN

• =1 or NaN

• Square root or logarithm of negative number

• Inverse sine or cosine of a number with absolute value greater than 1

Subnormal numbers• Sign=0+ or 1-• Exponent=0

• 000002

• 00H

• 00016

• Significand(Mantissa)≠0

Floating pointaddition

• x=123456.7 = 1.234567 × 10^5• y=101.7654 = 1.017654 × 10^2 =

0.001017654 × 10^5• x+y=(1.234567+ 0.001017654) × 10^5• x+y=1.235584654 × 10^5• x+y≈1.235585 × 10^5• Round-off error!

Floating point addition of a small number

• x=1.234567 × 10^5• y=9.876543 × 10^-3=0.00000009876543

× 10^5• x+y=(1.234567+ 0.00000009876543) ×

10^5• x+y=1.23456709876543 × 10^5• x+y≈1.234567 × 10^5=x• Round-off error!

Floating point subtraction of two similar numbers

• x=123457.1467 ≈ 1.234571 × 10^5• y=123456.659 ≈ 1.234567 × 10^5• x-y=0.4877=4.877000 × 10^1• (1.234571-

1.234567)×10^5=0.0000040×10^5=4×10^1

• 4.877000 × 10^1 v.s. 4.000000 × 10^1 • 20% round-off error!

4.877000 × 10^1

Floating pointmultiplication

• x=4.734612 × 10^3• y=5.417242 × 10^5• x*y=25.648538980104 × 10^8• x*y≈25.64854 × 10^8= 2.564854 × 10^9• Round-off error!

Accuracy problems• 0.1*0.1=0.01• 0.1≈ 1.10011001100110011001101 × 2^-4• x=0.100000001490116119384765625• x*x=0.01000000029802322609739917425031308

0847263336181640625≠0.01• x*x≈0.010000000707805156707763671875

(single)• 0.01≈ 0.009999999776482582092285156250

≠x*x• tan(π/2) ≈ 16331239353195370.0 or −22877332.0

Arithmetic overflow

Arithmetic underflow

(a+b)+c≠a+(b+c)• a = 1234.567, b = 45.67834, c = 0.0004• (a+b)= 1280.24534 ≈ 1280.245• (a+b)+c=1280.245+0.0004=1280.2454 ≈

1280.245

• (b+c)= 45.67874• a+(b+c)= 1280.24574 ≈ 1280.246

(a+b)*c ≠A*C+B*C• a= 1234.567, b= 1.234567, c= 3.333333• (a+b)=1235.802• (a+b)*c=1235.802*3.333333 ≈ 4119.340

• a*c= 1234.567 × 3.333333 ≈ 4115.223• b*c= 1.234567 × 3.333333 ≈ 4.115223• a*c+b*c=4115.223+4.115223

≈4110.338≠4119.340

Numerical derivative

𝑓 ′ (𝑥 )=limh❑→0

𝑓 (𝑥+h )− 𝑓 (𝑥 )h