Programming I Lecture 7

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Madeleine Higgins
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1 1. Deinition: eal types simulate real numbers. eal types are discrete whereas the set o real numbers is continuous. eal types are called loating-point numbers. The density o loating-point numbers is shown on a real number line in Figure 1. -4n -2n -n n 2n 4n Figure 1. Density o loating-point numbers. Sets: Each set is dependent on its representation. Sign( S) 1 bit 8 bits 23 bits Characteristic ( C) Fraction ( F) Little endian bit position = Figure 2. IEEE-754 single binary loating-point representation used to implement type loat. 23 s c b { r F, s {,1, c = 1 c < 254, b = 127, F = 2, {,1 Figure 3. Set contains the numbers that can be produced by the IEEE-754 single binary loating-point representation Set 24 e = r s 2 = is equivalent to set. Set 2, s Figure 4. Set = 1 { 1,1, 126 e 127, {,1 is abstracted rom set. 22 1

2 Sign( S) 1 bit 11 bits 52 bits = Characteristic ( C) Fraction ( F) Little endian bit position Figure 5. IEEE-754 double binary loating-point representation used to implement type double. 52 s c b { r F, s {,1, c = 1 c < 247b = 123, F = 2, {,1 Figure 6. Set contains the numbers that can be produced by the IEEE-754 single binary loating-point representation Set d 53 e = r s 2 = d is equivalent to set. Set d 2, s Figure 7. Set = 1 { 1,1, 122 e 123, {,1 d is abstracted rom set. Sign ( S) 1 bit 15 bits 8 bits Characteristic ( C) Fraction( F) Little endian bit position Figure 8. IEEE-754 double extended binary loating-point representation used to implement type long double = 8 s c b { r F, s {,1, c = 1 c < 32,767b = 16,383, F = 2, {,1 Figure 9. Set contains the numbers that can be produced by the IEEE-754 single binary loating-point representation Set l 53 e = r s 2 = l is equivalent to set. Set 2, s Figure 1. Set = 1 { 1,1, 122 e 123, {,1 d is abstracted rom set. l 2

3 2. Declarations: declarations: real-declaration-list ; real-declaration-list: real-declaration real-declaration-list, real-declaration real-declaration: real-declaration-speciier-sequence real-variable-name real-initialization opt real-declaration-speciier-sequence: real-declaration-speciier real-declaration-speciier-sequence real-declaration-speciier real-declaration-speciier: storage-class-speciier real-type-speciier storage-class-speciier: auto register static extern real type-speciier: loat double long double real-variable-name: identiier real-initialization: = assignment-expression ( assignment-expression ) Examples: loat ; double d; long double ld; 3. Constants: Floating-point constants may be written with a decimal point, a signed exponent, or both. A loatingpoint constant is always interpreted to be in decimal radix. C++ allows a suix letter (loating-suix) to designate constants o types loat, and long double. Without a suix, the type o the constant is double. loating-constant: digit-sequence exponent loating-suix op t dotted-digits exponent opt loating-suix op t loating-suix: 3

4 F l L exponent: E sign-part opt digit-sequence e sign-part opt digit-sequence sign-part: + - dotted-digits: digit-sequence. digit-sequence. digit-sequence. digit-sequence digit: Examples: Constant Type Description. double 3e1 double double π. double 1.E-3 double.1 1e-3 double.1 1. double 1.34 double e+9 double 2,,, 1. loat 1 1.e67L long double E1L long double Operations: Operations on real types consist o the standard arithmetic operations o addition, subtraction, multiplication, and division. The <cmath> library also provides a rich set o useul operations primarily on real types Standard arithmetic operations. Operation Operator Multiplication * Division / Addition + Subtraction - Less than < Less than or equal <= Greater than > Greater than or equal >= Equality == Inequality!= Table 1. eal operations 4

5 4.2. <cmath> library. Selected unctions rom the <cmath> library. Declaration Description Example int abs(int x); Function abs(x) returns the int x=-5; absolute value o its integer argument x. cout << abs(x); 5 long labs(long int x); Function labs(x) returns the absolute value o its integer argument x. double ceil(double x); Function ceil(x) returns the smallest loating-point number not less than x whose value is an exact mathematical integer. double loor(double x); Function loor(x) returns the largest loating-point number not greater than x whose value is an exact mathematical integer. double pow(double b,double e); e Function pow(b,e) returns double sqrt(double x); int srand(unsigned seed); int rand(void); Function sqrt(x) returns Function srand may be used to initialize the pseudo-random number generator that is used to generate successive values or calls to rand. Successive calls to unction rand return integer values in the range to the largest possible value o type int that are the results o a pseudo-random-number generator. x b long int x=-5; cout << labs(x); 5 double x=5.5; cout << ceil(x); 6 double x=5.5; cout << loor(x); 5 double b=2.,e=5.; cout << pow(b,e); 32 double x=81.; cout << sqrt(x); 9 Program p7 in Figure 9 illustrates how samples rom the uniorm distribution can be generated. Functions srand and rand are employed to initialize and produce the uniorm distribution. Program p7 in Figure 9 illustrates how samples rom the uniorm distribution can be generated. Functions srand and rand are employed to initialize and produce the uniorm distribution. 5

6 #include <iostream> #include <iomanip> #include <cmath> #include <ctime> using namespace std; int main() { time_t t; srand((unsigned)time(&t)); //Seed rand using the time o day or (int a=;a<1;a++) { i (a%5==) // //Print random samples rom the uniorm distribution // cout << " " << ixed << setprecision(4) << (double)rand()/and_max; return ; Figure 9. Program p7 illustrates the use o unctions srand and rand Figure 1. Program p7 output 1 1 Program p7 produces dierent output each time it is invoed because the pseudo-random-number generator seed is dierent. The pseudo-random-number generator seed is dierent because it is an unsigned integer representing the time o day. 6

7 5. Example programs Program p8 prints the amount by which the dollar is devalued or inlation rates o 3%, 5%, 7%, and 9%. A ten-year period is printed. #include <iostream> #include <iomanip> using namespace std; int main() { double w3=1.,w5=1.,w7=1.,w9=1.; cout << "Year"; cout << " " << setw(6) << "3%"; cout << " " << setw(6) << "5%"; cout << " " << setw(6) << "7%"; cout << " " << setw(6) << "9%"; or (int y=1;y<=1;y++) { cout << setw(4) << y; cout << " " << ixed << setprecision(4) << w3; cout << " " << ixed << setprecision(4) << w5; cout << " " << ixed << setprecision(4) << w7; cout << " " << ixed << setprecision(4) << w9; w3*=1.3; w5*=1.5; w7*=1.7; w9*=1.9; return ; Figure 11. Program p8. Year 3% 5% 7% 9% Figure 12. Program p8 output 7

8 5.2. Program p9 computes the uture value o a sequence o ixed deposit in an interest bearing account. The user is prompted or the monthly deposit, annual percentage on the account and the term. #include <iostream> #include <iomanip> #include <cmath> using namespace std; int main() { cout << "Enter the monthly deposit. "; double ; cin >> ; cout << "Enter the Annual Percentage ate (AP) on the account. "; double AP; cin >> AP; double i=ap/12; cout << "i=" << ixed << setprecision(6) << i; cout << "Enter the number o years in the term. "; double y; cin >> y; int n=(int)loor(y*12+.5); cout << "n=" << n << endl; double S=*(pow(1+i,n)-1)/i; cout << "The balance on the account ater " << y << " years will be " << "$" << ixed << setprecision(2) << S << "."; return ; Figure 13. Program p9. Enter the monthly deposit. 1 Enter the Annual Percentage ate (AP) on the account. 9 i=.75 Enter the number o years in the term. 2 n=24 The balance on the account ater 2. years will be $ Figure 14. Program p9 output. 8

9 eerences: 1. Horstman and Budd; Big C++; Section 2.1, 2.2, 2.3, Stroustrup; The C++ Programming Language, 3 rd Ed. Section 4.5 Exercises: 1. Horstman and Budd; Big C++; p 7, Horstman and Budd; Big C++; p 7, Horstman and Budd; Big C++; p 7, Write a program that given an initial distance, s, and initial velocity, v, a rate o acceleration, a, and the amount o time a body was accelerated, t, will compute the distance rom the origin. 1. Write a program that will ind the roots o a second order polynomial. Horstman and Budd; Big C++; p 7, 2.1 9

The cin Object. cout << "Enter the length and the width of the rectangle? "; cin >> length >> width;

The cin Object. cout << Enter the length and the width of the rectangle? ; cin >> length >> width; The cin Object Short for console input. It is used to read data typed at the keyboard. Must include the iostream library. When this instruction is executed, it waits for the user to type, it reads the