Floating point operators and floating point expressions

1
Floating point operators and floating point
expressions
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Floating point computation

• Humans can do floating point computations
• So can a computer
• Computer are very picky about the correctness in
  the syntax of a computer program
• That includes syntax for describing arithmetic
  expressions

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Floating point computation (cont.)

• In this webnote, we will learn about the details
  of arithmetic operators and arithmetic expression
  
• I will take advantage of your knowledge in
  arithmetic expressions from your elementary
  school education.

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Floating point operators

• Floating point operators are arithmetic operators
  that manipulate (operate on) floating point
  values
• All of the floating point (arithmetic) operators
  should look familiar to you...

• A floating point operator only operates on 2
  floating point values
• The result of an integer operator is always a
  floating point value

• Specifically A floating point operator cannot
  operate on integer values

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Floating point operators (cont.)

• Floating point arithmetic operators

  Operator symbol Operation Note  
addition Binary operator, e.g. 4.0 5.0 9.0
- subtraction Binary operator, e.g. 4.0 - 5.0 -1.0
multiplication Binary operator, e.g. 4.0 5.0 20.0
/ division Binary operator, e.g. 4.0 / 5.0 0.8
( ... ) brackets Changes order of computation
- negation Changes the sign of the value - 4.0 (-4.0) (Read as - 4.0 negative 4.0)
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Floating point operators (cont.)

• Example

Result of the integer operation 5.0 3.0 is 8.0
Result of the integer operation 5.0 - 3.0 is 2.0
Result of the integer operation 5.0 3.0 is
15.0 Result of the integer operation 5.0 / 3.0
is 1.666666666666...
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Floating point expressions

• An Floating point expression is a (legal)
  combination of

• Floating point values
• Floating point variables and
• Floating point operators

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Floating point expressions (cont.)

• Example compute the area of a circle for a given
  radius

public class AreaOfCircle public static void
main(String args) double r //
variable containing the radius double
area // variable containing the area
r 4 // Give the radius area 3.1415
r r // Compute the area of the circle
System.out.print("The radius ")
System.out.println(r)
System.out.print("The area ")
System.out.println(area)
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Floating point expressions (cont.)

• Notes

• The expression "3.1415 r r" computes the
  area of a circle using the formula pr2
• You must use the operator to indicate a
  multiplication This is result in a syntax error
• The print statement   System.out.println("...")
  will print the string (text) between the quotes
  and then starts a new line
• The print statement   System.out.print("...")
  will print the string (text) between the quotes
  without starting a new line

area 3.1415 r r // Syntax error !!
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Priority of the floating point arithmetic
operators

• Each arithmetic operator is given a priority
• Priority assignment of the floating point
  arithmetic operators

Operator Priority   Note  
(   ....   ) Highest
- (negation) Higher Unary operator, e.g. -3.0
  / High   Binary operator, e.g. 4.0 5.0
  - Lowest   Binary operator, e.g. 4.0 5.0
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Priority of the floating point arithmetic
operators (cont.)

• When operators of different priority appear in
  one single arithmetic expression, then the
  operator with the highest priority is executed
  first.

• It's the same as what you have learned in
  Elementary School...

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Priority of the floating point arithmetic
operators (cont.)

• Example 1

Floating point expression 22.0 - 3.0 4.0
Evaluated as follows 22.0 - 3.0 4.0
22.0 - 12.0 10.0
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Priority of the floating point arithmetic
operators (cont.)

• Example 2 using brackets

Floating point expression (22.0 - 3.0) 4.0
Evaluated as follows (22.0 - 3.0) 4.0
19.0 4.0 76.0
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Priority of the floating point arithmetic
operators (cont.)

• Example 3 a negation operator

Floating point expression 22.0 - - 3.0 4.0
Evaluated as follows 22.0 - - 3.0 4.0
22.0 - (-3.0) 4.0 22.0 - (-12.0)
34.0
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Priority of the floating point arithmetic
operators (cont.)

• Example Program (Demo above code)        
• Prog file
• http//mathcs.emory.edu/cheung/Courses/170/Sylla
  bus/04/Progs/Priority01.java
• How to run the program

• Right click on link and save in a scratch
  directory
• To compile   javac Priority01.java
• To run          java Priority01

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A simple Java program demonstrating arithmetic
expressions

• Example computing the average of 3 numbers

public class Average public static
void main(String args) double
a, b, c, avg // Define 4 variables
a 3.0 b 4.0 c 6.0
avg (a b c)/3.0
System.out.print("The average ")
System.out.println(avg)
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A simple Java program demonstrating arithmetic
expressions (cont.)

• Example Program (Demo above code)  
• Prog file
• http//mathcs.emory.edu/cheung/Courses/170/Sylla
  bus/04/Progs/Average.java           
• How to run the program
•          

• Right click on link and save in a scratch
  directory
• To compile   javac Average.java
• To run          java Average

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Left and right associative operators

• Left-associative operators

• A left-associative operator is an operator that
  groups its operands from left to right

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Left and right associative operators (cont.)

• Example of left-associative operators

The binary arithmetic operators are
left-associative     

•  
• /
• -

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Left and right associative operators (cont.)

• Example

Floating point expression 7.0 - 2.0 - 1.0
Evaluated as follows 7.0 - 2.0 - 1.0 (group
from left to right) 5.0 - 1.0 4.0
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Left and right associative operators (cont.)

• Right-associative operators
• Example of right-associative operators

• A right-associative operator is an operator that
  groups its operands from right to left

• The unary negation operator
•        
• is right-associative       

• -

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Left and right associative operators (cont.)

• Example

Floating point expression - - - - 1.0 Evaluated
as follows - - - - 1.0 - - -
(-1.0) (Note (-1.0) denotes negative
1.0) - - (1.0) -
(-1.0) 1.0
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Left and right associative operators (cont.)

• Example Program (Demo above code)  
• Prog file http//mathcs.emory.edu/cheung/Courses
  /170/Syllabus/04/Progs/RightAssoc.java
• How to run the program                      

• Right click on link and save in a scratch
  directory
• To compile   javac RightAssoc.java
• To run          java RightAssoc

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Applying priority and associativity rules

• Rules for evaluation an arithmetic expression

• First, apply the operator priority rules     
• I.e. perform the operation of the highest
  priority first
• 2. Then, apply the operator associativity rules

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Applying priority and associatively rules

• Example

Floating point expression 7.0 / 2.0 - 6.0
2.0 / 4.0 - 4.0 / 2.0 / 2.0 Evaluated as
follows 1. Make groups containing only
operators of the highest priority (7.0 / 2.0) -
(6.0 2.0 / 4.0) - (4.0 / 2.0 / 2.0) 2. Within
each group, apply the operator associativity
rule (7.0 / 2.0) - (6.0 2.0 / 4.0) - (4.0 /
2.0 / 2.0) (3.5) - (12.0 / 4.0) - (2.0 /
2.0) (3.5) - (3.0) - (1.0) 3.5 - 3.0 - 1.0
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Applying priority and associatively rules (cont.)

• Repeat
• 1. Make groups containing only operators of the
  highest priority
• (3.5 - 3.0 - 1.0) (only one group !)
• 2. Within each group, apply the operator
  associativity rule
• (3.5 - 3.0 - 1.0)
• (0.5 - 1.0)
• -0.5