cse202: lecture 4the ohio state university1 mathematical functions

CSE202: Lecture 4 The Ohio State University 1

Mathematical Functions


Mathematical Functions (1)

• The math library file cmath contains common and often used mathematical functions such as sine, cosine, log, square root, etc. To use, your program must have the following line in the header:

#include <cmath>

• As in math, functions take input and produce an output. For example, the square root function, sqrt(), needs a number input to product an output.



• These math functions can be used like a variable of the same data type:double a = 81;double b = sqrt(a);double c = sqrt(sqrt(a)) * b;

• Here, a is 81, b is 9, and c is 27

• This statement would output 11: cout << sqrt(121.0) << endl;



• When data needs to be passed into the function, you place it inside the parentheses (). The passed data are called arguments or parameters.

• A function can have more than one input. For instance, the power function needs a base and an exponent to produce an output.– For example, cout << pow(3.0, 4.0); would

output 81 which is 3 to the 4th power.



• Arguments passed into any function can be a constant, variable, an expression that evaluates to the appropriate data type such as the following:

double j = 2.0;

cout << sqrt(j * 32.0); //outputs 8



• Functions can themselves be part of an expression.

double x = 10;

cout << 5 * pow( (x - 3), 2.0 ); //output?


Common Math FunctionsFunction Returned Value Data Type of

Returned Value

abs(a) Absolute value of a Data type of a

pow(b,e) Value of b raised to the eth power float or double

Data type of b

sqrt(a) Square root of a double

sin(a) Sine of a in radians double

cos(a) Cosine of a in radians double

tan(a) Tangent of a in radians double

log(a) Natural log of a double

log10(a) Common log (base 10) of a double

exp(a) e raised to the ath power double


Example: sqrt()#include <iostream>#include <cmath>using namespace std;

int main(){ double x(0.0), y(0.0); double dist(0.0);

cout << "Enter x, y: "; cin >> x >> y;

dist = sqrt(x*x + y*y); cout << "Distance of (" << x << "," << y << ") to (0,0) =

" << dist << endl; return 0;}


> distance.exeEnter x, y: 3 4Distance of (3,4) to (0,0) = 5

> distance.exeEnter x, y: 5 8Distance of (5,8) to (0,0) = 9.43398

… cout << "Enter x, y: "; cin >> x >> y;

dist = sqrt(x*x + y*y); cout << "Distance of (" << x << "," << y << ") to (0,0) =

" << dist << endl;

…


Example 2: sqrt()#include <iostream>#include <cmath> // math function libraryusing namespace std;

int main (){

double height(0), time(0.0);

cout << "Enter height (feet): "; cin >> height;

time = sqrt(2.0 * height / 32.2);cout << "It will take " << time << " seconds to fall "

<< height << " feet. " << endl;

return 0;}


> gravity.exeEnter height: 100It will take 2.49222 seconds to fall 100 feet.

> gravity.exeEnter height (feet): 500It will take 5.57278 seconds to fall 500 feet.

… cout << "Enter height (feet): "; cin >> height;

time = sqrt(2.0 * height / 32.2); cout << "It will take " << time << " seconds to fall " << height << " feet. " << endl;

…


Example: log()#include <iostream>#include <cmath>using namespace std;

int main(){ double rate(0.0), years(0.0);

cout << "Enter annual interest rate (percentage): "; cin >> rate;

years = log(2.0)/log(1+(rate/100.0)); cout << "Your money will double in " << years << " years." << endl; return 0;}


> interest.exeEnter annual interest rate (percentage): 7Your money will double in 10.2448 years.

> interest.exeEnter annual interest rate (percentage): 2Your money will double in 35.0028 years.

… cout << "Enter annual interest rate (percentage): "; cin >> rate;

years = log(2.0)/log(1+(rate/100.0)); cout << "Your money will double in " << years << " years." << endl;…

Trigonometric Math Functions

• Trigonometric math functions (sin, cos, tan) require input in radians.



Example: sin(), cos()#include <iostream>#include <cmath>using namespace std;

int main(){ double angle(0.0), x(0.0), y(0.0);

cout << "Enter rotation angle (radians): "; cin >> angle;

x = cos(angle); y = sin(angle); cout << "Point (1,0) rotates to point" << "(" << x << "," << y << ")" << endl; return 0;}


> rotate_radians.exeEnter rotation angle (radians): 1.57Point (1,0) rotates to point (0.000796327,1)

> rotate_radians.exeEnter rotation angle (radians): 0.78Point (1,0) rotates to point (0.710914,0.703279)

… cout << "Enter rotation angle (radians): "; cin >> angle;

x = cos(angle); y = sin(angle); cout << "Point (1,0) rotates to point" << "(" << x << "," << y << ")" << endl;…


Common Math Constants

• In addition to functions, cmath also defines some commonly used math constants.

• These are used like variables, except that they cannot be assigned a new value (hence “constants”).

Constant ValueM_PI π, 3.14159…

M_E e, the base of natural logarithms

M_LOG2E Base-2 logarithm of e

M_LOG10E Base-10 logarithm of e

M_LN2 Natural log of 2

M_LN10 Natural log of 10


degrees2radians.cpp#include <iostream>#include <cmath> // cmath contains definitions of math constantsusing namespace std;

int main(){ double degrees(0.0), radians(0.0);

cout << "Enter angle in degrees: "; cin >> degrees;

radians = (degrees * M_PI) / 180.0;

cout << "Angle in radians = " << radians << endl;

return 0;}


rotate_degrees.cpp#include <iostream>#include <cmath> // cmath contains definitions of math constantsusing namespace std;int main(){ double degrees(0.0), radians(0.0), x(0.0), y(0.0);

cout << "Enter rotation angle (degrees): "; cin >> degrees;

radians = (degrees * M_PI)/180.0; x = cos(radians); y = sin(radians); cout << "Point (1,0) rotates to point (" << x << "," << y << ")" << endl;

return 0;}


> rotate_degrees.exeEnter rotation angle (degrees): 90Point (1,0) rotates to point (6.12323e-17,1)

> rotate_degrees.exeEnter rotation angle (degrees): 45Point (1,0) rotates to point (0.707107,0.707107)

… cout << "Enter rotation angle (degrees): "; cin >> degrees; radians = (degrees * M_PI)/180.0; x = cos(radians); y = sin(radians); cout << "Point (1,0) rotates to point" << "(" << x << "," << y << ")" << endl;…


log_2.cpp#include <iostream>#include <cmath> // cmath contains definitions of math constants

using namespace std;

int main(){ double x(0.0), y(0.0);

cout << "Enter number: "; cin >> x;

y = log(x) / M_LN2; cout << "log_e(" << x << ") = " << log(x) << endl; cout << "log_2(" << x << ") = " << y << endl;

return 0;}

Modulus Operator: %

• a mod b is the remainder after integer a is divided by integer b;

• In C++ a mod b is written as:

a%b• Examples:

25 % 3 = ?

137 % 10 = ?

2751 % 2 = ?


Arguments to Math Functions

• Arguments to math functions should have type double, e.g.:

double x(3.6), y(0.3), z(0.0);

z = sin(1.2);

z = sqrt(x);

z = log(3.2 * x);

z = pow(x / 0.5, 1.2 * y);



logError.cpp. . .int main(){ int value(0);

cout << "Enter value: "; cin >> value;

// log(value) generates a compiler error cout << "At 10% interest, it will take " << log(value)/log(1.1) << " years for a $1 investment to be worth $" << value << "." <<

endl; return 0;}


> g++ logError.cpp –o logError.exeCompiling logError.cpp into logError.exe.logError.cpp: In function `int main()':logError.cpp:16: call of overloaded `log(int&)' is ambiguous/usr/include/iso/math_iso.h:52: candidates are: double log(double)/usr/local/include/g++-v3/bits/std_cmath.h:333: long double std::log(long double)/usr/local/include/g++-v3/bits/std_cmath.h:323: float std::log(float)

…15. // log(value) generates a compiler error

16. cout << "At 10% interest, it will take " << log(value) / log(1.1)

17. << " years for a $1 investment to be worth $" << value << "." << endl;

…


logExample.cpp. . .int main(){ int value(0); double x(0.0);

cout << "Enter value: "; cin >> value; x = value; // implicit conversion to double

cout << "At 10% interest, it will take " << log(x)/log(1.1) << " years for a $1 investment to be worth $" << value << "." << endl; return 0;}


> logExample.cpp

Enter value: 10

At 10% interest, it will take 24.1589 years for a $1 investment to be worth $10.

>

…

double x(0.0);

...

x = value; // implicit conversion to double

cout << "At 10% interest, it will take " << log(x)/log(1.1)

<< " years for a $1 investment to be worth $" << value << "." << endl;

…

Mixed Mode Expressions



Expression Types

• An expression that contains only integer operands is an integer expression.

• An expression that contains only floating point operands is a floating-point expression.


Mixing Expression Types

• A mixed-mode expression has both floating-point and integer data types.

• The rules governing the data type of the result are:

1. If both operands are integers, the result is an integer

2. If one operand is a floating-point number, then the result is a double

Mixed Mode Expressions

int a(3);

double x(3.5), y(5), z(0.0);

z = 3.0 * 25;

z = a * x;

z = a * y;

What about:

z = x + (a/2);


Exercises

int a(3), b(2);

double y(5), z(0.0);

After each operation, what is z?

z = (y + a) / b;

z = (y * a) / b;

z = y * (a / b);

z = (y / b) * (a / b);

z = (a + b) / (b*y);



logError.cpp. . .int main(){ int value(0);


// log(value) generates a compiler error cout << "At 10% interest, it will take " << log(value)/log(1.1) << " years for a $1 investment to be worth $" << value << "." <<

endl; return 0;}


logExample2.cpp. . .int main(){ int value(0);


// Mixed mode operation "value*1.0" returns double.

cout << "At 10% interest, it will take " << log(value*1.0)/log(1.1) << " years for a $1 investment to be worth $" << value << "." << endl; return 0;}

Operator Precedence


Evaluate

• Evaluate:

1 + 3*6 – 4/2 = ???


Evaluate

• Evaluate:

20 – 16 / 2 + 2 * 3 = ???

• (((20 – 16) /2) +2) *3 = 12.

• ((20 – 16) / (2 + 2)) * 3 = 3.

• 20 – ((16/2) + (2*3)) = -4.



Operator Precedence and Associativity

• The minus sign can also be used as a unary operator that serves to negate the sign of a number

Precedence Associativity

Unary -

* / % Left to right

+ - Left to right


arithmetic3.cpp// Precedence of arithmetic operators

#include <iostream>using namespace std;

int main(){ cout << "-3+5*2 = " << -3+5*2 << endl << endl;

// Is this? cout << "((-3)+5)*2 = " << ((-3)+5)*2 << endl; cout << "(-(3+5))*2 = " << (-(3+5))*2 << endl; cout << "(-3)+(5*2) = " << (-3)+(5*2) << endl; cout << "-(3+(5*2)) = " << -(3+(5*2)) << endl; return 0;}

Math in C++• Use #include<cmath> for math functions;

• Common math functions:

abs(a), pow(b,e), sqrt(a), sin(a), cos(a), tan(a), log(a), log10(a), exp(a);

• Common math constants:

M_PI, M_E, M_LN2, M_LN10;


Math in C++• Arguments to functions should always be

double;

• Mixed mode operations:

(3.0 + 5) or (3.0 * 5) have type double;

• Operator precedence:

Multiplication and division before addition and subtraction.


cse202: lecture 4the ohio state university1 mathematical functions

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