Programming I Lecture 7
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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
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