Floating Point NumbersExpressions

Scanner InputAlgorithms to Programs

Shirley MooreCS 1401 Spring 2013

February 12, 2013

Learning Outcomes• Explain how floating point numbers are represented inside a

computer• Define what is meant by range and precision of a floating point

number representation• Define roundoff error and explain how it can accumulate to produce

results with significant errors• Evaluate expressions using operator precedence rules• Use the Java Scanner class to obtain user keyboard input• Solve simple IPO (Input-Processing-Output) programming problems

by– Writing a step-by-step algorithm– Implementing the algorithm in Java– Verifying correct output

Real vs. Floating Point Numbers

• How many real numbers are there between 0 and 1?• Floating point numbers can be represented in 32 bits

(single precision) or 64 bits (double precision).– How many floating point numbers can we represent in 32 bits?– How many floating point numbers can we represent in 64 bits?

• How can we represent 0.5 in binary?• How can we represent 1/3 as a decimal fraction?• How can we represent 0.1 in binary?• Converting decimal fractions to binary

– http://cs.furman.edu/digitaldomain/more/ch6/dec_frac_to_bin.htm

Roundoff Error Examples• Try the following in Dr. Java Interactions:

> 1.0 - .90.09999999999999998> 1.0 - 0.1 - 0.1 - 0.1 - 0.1 - 0.10.5000000000000001> int d = 2147483647> float e = (float) d> d2147483647> e2.14748365E9> e = e - 21474836460.0> d = d - 21474836461> d = (int) e0

Floating Point Number Representation

• IEEE Standard 754 Floating Point Numbers– http://steve.hollasch.net/cgindex/coding/ieeefloat.html

• What types of errors are the following?> float f = (float) Math.pow(10,38)> f1.0E38> int g = (int) f> g2147483647> float h = (float) Math.pow(10,39)> hInfinity> float p = (float) Math.pow(10,-50)> p0.0

• Decimal to floating point converter– http://www.binaryconvert.com/convert_double.html

Expression Evaluation• In the Dr Java Interactions window, try evaluating the following

expressions: 5/3 5 % 3 5./3. 5 / 0 5./0. 5 < 6 5. < 6. 3 – 2*5 (3 - 2)*5 3 + .100000000 * .10000000 - 3.

• Did you get the results you expected?

Scanner Class

• In the Dr. Java Interactions window type the followingScanner input = new Scanner(System.in);System.out.println(“Enter an integer: “);int n = input.nextInt();nSystem.out.println(“Enter a real number: “);double r = input.nextDouble();r

Programming Exercise

• Now let’s try a complete program: Problem 2.2, page 75. 2.2 (Compute the volume of a cylinder) Write a

program that reads in the radius and length of a cylinder and computes the area of the base and the volume of the cylinder using the following formulas:

area = π * radius * radius volume = area * length

Algorithm for Problem 2.2

Lab 3 – Projectile Motion

• By Thursday– Problem description– Algorithm– Set of test cases