cse1222: lecture 4the ohio state university1. mathematical functions (1) the math library file...
TRANSCRIPT
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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>
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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
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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???
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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
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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;
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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?
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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!
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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
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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
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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
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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!
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> 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
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… cout << "Enter x, y: "; cin >> x >> y;
dist = sqrt(x * x + y * y); cout << "Distance of (" << x << "," << y << ") to (0,0) =
" << dist << endl;
…
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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)?
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> 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.
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… cout << "Enter height (feet): "; cin >> height;
time = sqrt(2.0 * height / 32.2); cout << "It will take " << time << " seconds to fall " << height << " feet. " << endl;
…
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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;}
// What are we computing here (in plain english)?
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> 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.
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… 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;…
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Trigonometric Math Functions Read your documentation on functions!
Trigonometric math functions (such as sin, cos, and tan) require input parameters to be in radians
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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;}
// What are we computing here (in plain english)?
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> 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)
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… cout << "Enter rotation angle (radians): "; cin >> angle;
x = cos(angle); y = sin(angle); cout << "Point (1,0) rotates to point" << "(" << x << "," << y << ")" << endl;…
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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
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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;}
// What are we computing here (in plain english)?
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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;}
// What are we computing here (in plain english)?
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> 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)
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… 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;…
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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;}
// What are we computing here (in plain english)?
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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);
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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;}
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> 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)
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…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”?
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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?
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> logExample.cpp
Enter value: 10
At 10% interest, it will take 24.1589 years for a $1 investment to be worth $10.
>
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…
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;
…
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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 = ?
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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
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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
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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);
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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);
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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;}
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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;}
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Evaluate
What does this express evaluate to?
1 + 3 * 6 – 4 / 2 = ???
Try it out on a sheet of paper
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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
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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
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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
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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;}
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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
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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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