CS代考计算机代写 Floating Point

Floating Point

á Fixedpointrepresentations á Big and Small Numbers
á ScientificNotation
á IEEE754floatingpointstandard ± Special symbols
± Underflow overflow
á Floatingpointadditionandmultiplication á Material from section 3.5 of textbook
Agenda

How to Represent Real Numbers?

Real Numbers á Positionalnotationallowsforfractions
anan-1………a1a0 . a-1a-2……..a-m
á Let’s start with fixed point representation ± Choose n and m
± Radix point is always in the same position
± Easy to implement
± Limited range

á 152.310
á 1011.012
Real Numbers

Binary to Decimal
á Integersscaledbyanappropriatefactor á Directexpansionwithpositionalweights á 0.110012

Binary to Hexadecimal á Use the same trick as before
0.1101010012
0.2BE16

Decimal to Binary
á Multiply by 2 and note the integer part
á Subtractintegerpartandrepeatuntilnofractionleft
0.62510

Decimal to Binary
á Can all decimal fractions be expressed exactly in Binary? 0.110

How to Represent Small and Big Numbers in Decimal?

How big is Coronavirus?
Particle
Size (meter)
PM10
Red Blood Cell
0.00001 0.000007
PM2.5
0.0000025
Bacteria Coronavirus
0.0000005 0.0000001
0.000000007

What numbers do we need?
ÔXÌðÌÒıÓY ÓXÛÌÙÓÙY
1.0 × 10-9
3.15576 × 109 1.47 × 1013 2.99792458 × 1010 6.67300 × 10-11 1.98892 × 1030 2.08 × 1022
S
1.0 × 10-15
e
Seconds per nanosecond Seconds per century
US National Debt
Speed of light in cm/s Gravitational constant
Mass of sun in kilograms Distance to Andromeda in m Size of a proton in meters

Scientific Notation for Decimal
á Weusescientificnotationforbigandsmallnumbers ± Use a single digit to the left of the decimal point
± Multiplied by base (e.g., 10) raised to some exponent
± Use e or E to denote the exponent part
1.0 × 10-15 1.0e-15 1.0E-15
á Anormalizednumberhasnoleadingzero ± 1.010 x 10-9 normalized
± 0.110 x 10-8 not normalized
± 10.010 x 10-10 not normalized

Scientific Notation for Binary
á How do we represent very small and big numbers in Binary?
á Binary numbers can be written in scientific notation too
1.02 x 2-1 1.12 x 211

How to Represent Floating Points?

Floating Point
Normalized Binary number always has the form:
± x is the fraction / significand / coefficient / mantissa ± y is the exponent
± always has a one to the left of the binary point
1.xxxxxxx2 × 2yyyy

Floating Point Standards
á Manyoptionsforrepresentingfloatingpoint ± Number of bits for the fraction
± Number of bits for the exponent
± How to represent zero?
± How to represent negative numbers?
á Standardsareimportantforexchangingdata

Floating Point Standards á IEEE 754 used in nearly all computers today
± Defines two representations á single precision (32 bits)
á double precision (64 bits)
In high level languages, data of this type is called
á float (single precision)
á double (for double precision)

Single Precision
Sign Exponent Fraction 1 bit 8 bits 23 bits
SEF
A real number can be described as (-1)S x (1+F) x 2E
á IEEE 754 does not use 2’s complement
á Clarification:
± Fraction refers to the 23-bit number F
± Mantissa refers to the 24-bit number 1+F

á Numbers are in normalized form. Why? Base 2
S Exponent
Mantissa
Single Precision
Sign Exponent Fraction 1 bit 8 bits 23 bits
SEF
A real number can be described as (-1)S
x (1+F) x 2E
0.0011 ×20 0.011 × 2>Ì
0 0000000
001100…
0 1111111
0 1111110
011000… 110000…
0.11 × 2>Ó
All equivalent to the same real number. The encoding is wasteful

Biased Notation
Sign Exponent Fraction 1 bit 8 bits 23 bits
SEF
á In IEEE 754, actual representation is
(-1)S x (1 + Fraction) x 2(Exponent t Bias)
á In single-precision, bias = 127
á Represent negative exponents
á Wanteasyintegerstylecomparison/sorting

Single Precision Floating Point
Sign Exponent Fraction 1 bit 8 bits 23 bits
SEF
á Largest number?
á Smallestnumber?
á How many numbers can we represent?
(-1)S x (1+F) x 2E

Single Precision Floating Point
á Convert -0.75 from decimal to single precision

Single Precision Floating Point
á ConvertÌÏÏÏÏÏÏÌÏÌÏÏÏYÏÏÏÏfromsingleprecisionto decimal:

Double Precision
Sign Exponent Fraction 1 bit 11 bits 52 bits
SEF
á More bits!
á More precision
á Double precision uses a bias of 1023
á Can do more before underflow / overflow ± Approximately 1E-308 to 1E308

Double Precision
á Convert 3.25 from decimal to double precision

Tricky Questions
á What is the largest number that can be represented in single precision?
á What is the smallest number that can be represented in single precision?

Floating Point Arithmetic

± Make the smaller number to match the larger
á Normalize the result
± What if one number is positive and the other negative? ± May need to shift a lot!
± Check for overflow or underflow when shifting!
á Round so number fits in available digits/bits ± If bad luck when rounding, renormalize

9.999e1 + 1.610e-1 with 4 digits precision

Floating Point Multiplication
á Multiplythesignificands
á Normalize the result (check for overflow)
á Round to fit in available digits/bits ± Normalize again if necessary
á Compute sign of result
± Positive if signs of operands match, negative otherwise

Floating Point Multiplication
1.110e10 times 9.200e-5 with 4 digits precision

Special Cases?

Special symbols
Exponent Fraction Object represented
000
0 1-254 255 255
Nonzero Anything 0 Nonzero
± denormalized number ± floating point number ± infinity
NaN (Not a Number)

Denormalized Numbers
á The exponent 00000000 is used to represent a set of numbers in the tiny interval ( -2-126, 2-126 )
á This includes the number 0
á Calleddenormalizednumbers
± Smallest normalized is 1.0 x 2-126 = 2-126
± Smallest denormalized is 0.000 μ μ μ 01 x 2-126 = 2-149
á Allows us to squeeze more precision out of a floating point operation
á Tricky to implement. We will come back to this topic later

Unusual events
á Nonzerodividedbyzero
± Not the end of the world!
± Results in positive or negative infinity
á 0/0(invalid),orsubtractinginfinityfrominfinity ± Results in NaN
á Notes on NaN
± Using NaN in math always results in NaN
± Allows us to avoid tests or decisions until a later time in our program

What can go wrong?

Overflow / Underflow
á Largest number that can be represented in single precision:
Approximately ±2.0 x 2128 = 2.0 x 1038
á Smallest fraction that can be represented in single precision:
Approximately ±2.0 x 2-128 = 2.0 x 10-38
á Overflow: representing a number larger than the one above;
á Underflow: representing a number smaller than the one above

Loss of Precision
https://imgur.com/r/totallynotrobots/lsNcv

}
Compare these for loops
for ( int i = 0; i <= 10; i += 1 ) { System.out.println( i/10f ); for ( float y = 0; y <= 1; y += 0.1f ) { } System.out.println( y ); Same or different? Questions á Represent 0.110 in IEEE 754 single precision floating point á Represent 1.110 in IEEE 754 single precision floating point? Review and more information á Big and Small Numbers á Scientific Notation á IEEE754floatingpointstandard á Floating point addition and multiplication á Material from Section 3.5 of textbook