CSE1222: Lecture 4The Ohio State University1. Mathematical Functions (1) The math library file...

CSE1222: Lecture 4 The Ohio State University 1

Mathematical Functions (1) The math library file cmath

Yes, this is a file with definitions for common and often used mathematical functions including:

sine, cosine, log, and square root

Provided by our C++ geniuses (Be thankful!)

Using the math library

Include the following at the top of your program

#include <cmath>

Mathematical Functions (1) How do you use or invoke a function in math?

Specify …

Its name Input parameters

Retrieve its …

Output or answer

Remember y = f(x) from math?

f is the name x is the input parameter y will hold the output

A function has three parts: Name, input, and output

Mathematical Functions (1) The cmath library defines a function called sqrt … What do you

think it does?

Function names are descriptive Function names must be spelled exactly how they are defined in their

defining library (cmath here, so don’t guess! Look it up) You must provide exactly how many input parameters a function is

defined to take A function can output at most one answer only

The square root function : Spelled sqrt Requires exactly one input parameter (a number) Outputs the square root of the input parameter

Question: Where does the output or answer go???

Mathematical Functions (2) This function invocation:

double a = 81;double b = sqrt(a);

Takes the value of the variable a as a single input parameter Executes the computation for a square root “Outputs” its answer in the assignment statement

Rule (Memorize this!): The output value of a function invocation will relace the syntax of the function call in your program during program execution

During execution of this code the second line will execute in the following way:

double b = sqrt(a); <---- Before execution

double b = sqrt(81); <---- Evaluates the input parameter

double b = 9.0; <---- This line uses the rule above

Mathematical Functions (2) Is this program snippet nonsensical … Why or why not?

double a = 81;sqrt(a);

Is there a syntax error, i.e. will it compile?

- No syntax error! But, avoid invoking a function in this way if it outputs a value. You are throwing away the output!

Is there a syntax error? Is it nonsensical?

double a = 81;cout << sqrt(a);

No syntax error! If a function invocation outputs a value, then ensure that the syntax of the function call is used with other code that performs useful work

This statement would output 11

cout << sqrt(121.0) << endl;

Mathematical Functions (2) Is there a syntax error?

double a = 81;double b = sqrt(a);double c = sqrt(sqrt(a)) * b;

No syntax error! A function can be used like a variable of the same data type

Remember our rule regarding the output of a function invocation How is it used here to evaluate the third line?

Here, a is 81, b is 9, and c is 27. Why? The sqrt function is called three separate times in this code snippet Which function is invoked first, which is invoked second, and then which

is invoked third?

Mathematical Functions (3) What does the cmath library function pow do?

Computes ab, where a and b are two numbers Its name tries to tell you what operation the function performs for you

How many input parameters does this function require? We need a base and an exponent, so two input parameters

Does this function output a single answer or no answer? It must return the value for ab, so one answer

For examplecout << pow(3.0, 4.0);

would output 81 which is 3 to the 4th power.

Notice the two input parameters in the parenthesis (comma separated). The base must come BEFORE the power!

Mathematical Functions (4) Arguments passed into any function can be a

constant, variable, or an expression that evaluates to the appropriate data type

For Example,

double j = 2.0;

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

Remember, the sqrt function requires exactly one argument

Mathematical Functions (5) Functions can themselves be part of an

expression

double x = 10;

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

Remember our rule regarding function output and where it goes?

Remember, the pow function requires exactly two arguments

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;}

// What are we computing here (in plain english)?// Compile and execute after class!

> 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;}

// What are we computing here (in plain english)?

> 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 Read your documentation on functions!

Trigonometric math functions (such as sin, cos, and tan) require input parameters to be 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 Besides functions, the cmath library file also defines some

commonly used math constants

Use these like variables Though, being “constants” you cannot assign a new value to them

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; // Convert degrees to radians

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;}

Arguments to Math Functions Input parameters (i.e., arguments) to math

functions should have type double

For example,

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 << "." <<

18. endl;

…// What is the problem here? What is the data type for variable “value”?

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;}

// Will this do the trick?

> 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;

Modulus Operator: % In math a mod b is the remainder after integer a is

divided by integer b

You can think of this operator as a function: 1) name (%), two operands (two integer input parameters), one output (an integer)

In C++ a mod b is written as:a % b

Examples:25 % 3 = ?137 % 10 = ?2751 % 2 = ?

Expression Types An expression consists of constants, variables,

and function invocations that output single values and evaluates to a single answer

For example,5 * pow( (x - 3), 2.0 )

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);

Exercisesint 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 expression "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;}

Evaluate

What does this express evaluate to?

1 + 3 * 6 – 4 / 2 = ???

Try it out on a sheet of paper

Evaluate What does this express evaluate to?

20 – 16 / 2 + 2 * 3 = ???

What if we insert parenthesis in different places?

(((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 is overloaded

Used as a binary operator for subtractionUsed as a unary operator to negate the sign

How do you know which one you are using?

int a = b – c; <---- Is this negation?int a = - c; <---- Is this negation?

Depends on the context, operator precedence, and associativity rules

Operator Precedence and Associativity Expressions are evaluated from left to right

If there is an ambiguity, then operator precedence determines which operators is evaluated first

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++ Review 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++ Review 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

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