CSE1222: Lecture 4The Ohio State University1

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 CSE1222: Lecture 4The Ohio State University2

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 CSE1222: Lecture 4The Ohio State University3

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??? CSE1222: Lecture 4The Ohio State University4

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 CSE1222: Lecture 4The Ohio State University5

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; CSE1222: Lecture 4The Ohio State University6

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? CSE1222: Lecture 4The Ohio State University7

Mathematical Functions (3)  What does the cmath library function pow do? Computes a b, 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 a b, so one answer  For example cout << pow(3.0, 4.0); would output 81 which is 3 to the 4 th power. Notice the two input parameters in the parenthesis (comma separated). The base must come BEFORE the power! CSE1222: Lecture 4The Ohio State University8

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 CSE1222: Lecture 4The Ohio State University9

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 CSE1222: Lecture 4The Ohio State University10

Common Math Functions FunctionReturned ValueData Type of Returned Value abs(a) Absolute value of a Data type of a pow(b,e) Value of b raised to the e th 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 a th power double CSE1222: Lecture 4The Ohio State University11

Example: sqrt() #include 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! CSE1222: Lecture 4The Ohio State University12

> distance.exe Enter x, y: 3 4 Distance of (3,4) to (0,0) = 5 > distance.exe Enter x, y: 5 8 Distance of (5,8) to (0,0) = CSE1222: Lecture 4The Ohio State University13 … 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 #include // math function library using 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)? CSE1222: Lecture 4The Ohio State University14

> gravity.exe Enter height: 100 It will take seconds to fall 100 feet. > gravity.exe Enter height (feet): 500 It will take seconds to fall 500 feet. CSE1222: Lecture 4The Ohio State University15 … 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 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; } // What are we computing here (in plain english)? CSE1222: Lecture 4The Ohio State University16

> interest.exe Enter annual interest rate (percentage): 7 Your money will double in years. > interest.exe Enter annual interest rate (percentage): 2 Your money will double in years. CSE1222: Lecture 4The Ohio State University17 … 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 CSE1222: Lecture 4The Ohio State University18

Example: sin(), cos() #include 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; } // What are we computing here (in plain english)? CSE1222: Lecture 4The Ohio State University19

> rotate_radians.exe Enter rotation angle (radians): 1.57 Point (1,0) rotates to point ( ,1) > rotate_radians.exe Enter rotation angle (radians): 0.78 Point (1,0) rotates to point ( , ) CSE1222: Lecture 4The Ohio State University20 … 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 ConstantValue M_PI π, … 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 CSE1222: Lecture 4The Ohio State University21

degrees2radians.cpp #include #include // cmath contains definitions of math constants using 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; } // What are we computing here (in plain english)? CSE1222: Lecture 4The Ohio State University22

rotate_degrees.cpp #include #include // cmath contains definitions of math constants using 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; } // What are we computing here (in plain english)? CSE1222: Lecture 4The Ohio State University23

> rotate_degrees.exe Enter rotation angle (degrees): 90 Point (1,0) rotates to point ( e-17,1) > rotate_degrees.exe Enter rotation angle (degrees): 45 Point (1,0) rotates to point ( , ) CSE1222: Lecture 4The Ohio State University24 … 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 #include // 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; } // What are we computing here (in plain english)? CSE1222: Lecture 4The Ohio State University25

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); CSE1222: Lecture 4The Ohio State University26

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; } CSE1222: Lecture 4The Ohio State University27

> g++ logError.cpp –o logError.exe Compiling 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) CSE1222: Lecture 4The Ohio State University28 … 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? CSE1222: Lecture 4The Ohio State University29

> logExample.cpp Enter value: 10 At 10% interest, it will take years for a $1 investment to be worth $10. > CSE1222: Lecture 4The Ohio State University30 … 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; …

CSE1222: Lecture 4The Ohio State University31

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 = ? CSE1222: Lecture 4The Ohio State University32

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 CSE1222: Lecture 4The Ohio State University33

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 CSE1222: Lecture 4The Ohio State University34

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); CSE1222: Lecture 4The Ohio State University35

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); CSE1222: Lecture 4The Ohio State University36

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; } CSE1222: Lecture 4The Ohio State University37

logExample2.cpp... int main() { int value(0); cout << "Enter value: "; cin >> value; // 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; } CSE1222: Lecture 4The Ohio State University38

CSE1222: Lecture 4The Ohio State University39

Evaluate  What does this express evaluate to? * 6 – 4 / 2 = ??? Try it out on a sheet of paper CSE1222: Lecture 4The Ohio State University40

Evaluate  What does this express evaluate to? 20 – 16 / * 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 CSE1222: Lecture 4The Ohio State University41

Operator Precedence and Associativity  The minus sign is overloaded Used as a binary operator for subtraction Used 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 CSE1222: Lecture 4The Ohio State University42

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 PrecedenceAssociativity () Unary - * / %Left to right + -Left to right CSE1222: Lecture 4The Ohio State University43

arithmetic3.cpp // Precedence of arithmetic operators #include 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; } CSE1222: Lecture 4The Ohio State University44

Math in C++ Review  Use #include 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 CSE1222: Lecture 4The Ohio State University45

Math in C++ Review  Arguments to functions should always be double  Mixed mode operations: ( ) or (3.0 * 5) have type double;  Operator precedence: Multiplication and division before addition and subtraction CSE1222: Lecture 4The Ohio State University46