math Namespace Reference

for IO operations More...

Typedefs
using	Point
	structure of points containing two numbers, x and y, such that 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1.

Functions
uint64_t	aliquot_sum (const uint64_t num)
	to return the aliquot sum of a number
double	approximate_pi (const std::vector< Point > &pts)
	This function uses the points in a given vector 'pts' (drawn at random) to return an approximation of the number π.
template<typename T >
T	square_area (T length)
	area of a square (l * l)
template<typename T >
T	rect_area (T length, T width)
	area of a rectangle (l * w)
template<typename T >
T	triangle_area (T base, T height)
	area of a triangle (b * h / 2)
template<typename T >
T	circle_area (T radius)
	area of a circle (pi
template<typename T >
T	parallelogram_area (T base, T height)
	area of a parallelogram (b * h)
template<typename T >
T	cube_surface_area (T length)
	surface area of a cube ( 6 * (l
template<typename T >
T	sphere_surface_area (T radius)
	surface area of a sphere ( 4 * pi * r^2)
template<typename T >
T	cylinder_surface_area (T radius, T height)
	surface area of a cylinder (2 * pi * r * h + 2 * pi * r^2)
int	sum_of_divisor (int num)
	Function to calculate the sum of all the proper divisor of an integer.
bool	are_amicable (int x, int y)
	Function to check whether the pair is amicable or not.
bool	is_factorial (uint64_t n)
	Function to check if the given number is factorial of some number or not.
bool	is_prime (int64_t num)
	Function to check if the given number is prime or not.
void	sieve (std::vector< bool > *vec)
	Performs the sieve.
void	print_primes (std::vector< bool > const &primes)
	Prints all the indexes of true values in the passed std::vector.
uint64_t	phiFunction (uint64_t n)
	Function to calculate Euler's Totient.
uint64_t	factorial (uint8_t n)
	function to find factorial of given number
double	integral_approx (double lb, double ub, const std::function< double(double)> &func, double delta=.0001)
	Computes integral approximation.
void	test_eval (double approx, double expected, double threshold)
	Wrapper to evaluate if the approximated value is within .XX% threshold of the exact value.
uint64_t	iterativeFactorial (uint8_t n)
	Calculates the factorial iteratively.
uint64_t	largestPower (uint32_t n, const uint16_t &p)
	Function to calculate largest power.
uint64_t	lcmSum (const uint16_t &num)
bool	magic_number (const uint64_t &n)
uint64_t	power (uint64_t a, uint64_t b, uint64_t c)
	This function calculates a raised to exponent b under modulo c using modular exponentiation.
template<class T >
T	n_choose_r (T n, T r)
	This is the function implementation of \( \binom{n}{r} \).
template<typename T >
T	square_perimeter (T length)
	perimeter of a square (4 * l)
template<typename T >
T	rect_perimeter (T length, T width)
	perimeter of a rectangle ( 2(l + w) )
template<typename T >
T	triangle_perimeter (T base, T height, T hypotenuse)
	perimeter of a triangle (a + b + c)
template<typename T >
T	circle_perimeter (T radius)
	perimeter of a circle (2 * pi * r)
template<typename T >
T	parallelogram_perimeter (T base, T height)
	perimeter of a parallelogram 2(b + h)
template<typename T >
T	cube_surface_perimeter (T length)
	surface perimeter of a cube ( 12
template<typename T >
T	n_polygon_surface_perimeter (T sides, T length)
	surface perimeter of a n-polygon ( n * l)
template<typename T >
T	cylinder_surface_perimeter (T radius, T height)
	surface perimeter of a cylinder (2 * radius + 2 * height)
int	power_of_two (int n)
	This function finds whether a number is power of 2 or not.
std::array< std::complex< long double >, 2 >	quadraticEquation (long double a, long double b, long double c)
	Quadratic equation calculator.
uint64_t	binomialCoeffSum (uint64_t n)
template<typename T >
T	cube_volume (T length)
	The volume of a cube
template<typename T >
T	rect_prism_volume (T length, T width, T height)
	The volume of a rectangular prism.
template<typename T >
T	cone_volume (T radius, T height, double PI=3.14)
	The volume of a cone
template<typename T >
T	triangle_prism_volume (T base, T height, T depth)
	The volume of a triangular prism.
template<typename T >
T	pyramid_volume (T length, T width, T height)
	The volume of a pyramid
template<typename T >
T	sphere_volume (T radius, double PI=3.14)
	The volume of a sphere
template<typename T >
T	cylinder_volume (T radius, T height, double PI=3.14)
	The volume of a cylinder

Detailed Description

for IO operations

for std::cin and std::cout

for io operations

Evaluate recurrence relation using matrix exponentiation.

for assert

Math algorithms.

Mathematical algorithms.

for M_PI definition and pow()

for std::vector

for assert

Mathematical algorithms

for assert for std::rand for IO operations

Mathematical algorithms

for assert for uint16_t datatype for IO operations

Mathematical algorithms

for assert for int32_t type for atoi

Mathematical algorithms

For assert For timing the sieve For IO operations For string handling For std::vector

for assert for std::cin and std::cout

Mathematical algorithms

for assert for mathematical functions for passing in functions

Mathematical functions

for math functions for fixed size data types for time to initialize rng for function pointers for std::cout for random number generation for std::vector

for std::cin and std::cout

Mathematical algorithms

Given a recurrence relation; evaluate the value of nth term. For e.g., For fibonacci series, recurrence series is f(n) = f(n-1) + f(n-2) where f(0) = 0 and f(1) = 1. Note that the method used only demonstrates recurrence relation with one variable (n), unlike nCr problem, since it has two (n, r)

Algorithm

This problem can be solved using matrix exponentiation method.

See also

here for simple number exponentiation algorithm or explaination here.

Author

Ashish Daulatabad for assert for IO operations for std::vector STL

Mathematical algorithms

for assert for std::cout

Mathematical algorithms

for assert for io operations

Mathematical algorithms

for assert for M_PI definition and pow() for uint16_t datatype

Mathematical algorithms

for IO operations

Mathematical algorithms

for assert for std::pow for std::uint32_t

Mathematical algorithms

Typedef Documentation

Point

using math::Point

Initial value:

 struct {
    double x;
    double y;
}

structure of points containing two numbers, x and y, such that 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1.

                     {
    double x;
    double y;
};

Function Documentation

aliquot_sum()

uint64_t math::aliquot_sum

(

const uint64_t

num

)

to return the aliquot sum of a number

Parameters

num	The input number

                                         {
    if (num == 0 || num == 1) {
        return 0;  // The aliquot sum for 0 and 1 is 0
    }
 
    uint64_t sum = 0;
 
    for (uint64_t i = 1; i <= num / 2; i++) {
        if (num % i == 0) {
            sum += i;
        }
    }
 
    return sum;
}

approximate_pi()

double math::approximate_pi

(

const std::vector< Point > &

pts

)

This function uses the points in a given vector 'pts' (drawn at random) to return an approximation of the number π.

Parameters

pts	Each item of pts contains a point. A point is represented by the point structure (coded above).

Returns

an estimate of the number π.

                                                   {
    double count = 0;  // Points in circle
    for (Point p : pts) {
        if ((p.x * p.x) + (p.y * p.y) <= 1) {
            count++;
        }
    }
    return 4.0 * count / static_cast<double>(pts.size());
}

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are_amicable()

bool math::are_amicable	(	int	x,
		int	y )

Function to check whether the pair is amicable or not.

Parameters

x	First number.
y	Second number.

Returns

true if the pair is amicable

false if the pair is not amicable

                                {
    return (sum_of_divisor(x) == y) && (sum_of_divisor(y) == x);
}

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binomialCoeffSum()

uint64_t math::binomialCoeffSum

(

uint64_t

n

)

Function to calculate sum of binomial coefficients

Parameters

n

number

Returns

Sum of binomial coefficients of number

                                      {
    // Calculating 2^n
    return (1 << n);
}

circle_area()

template<typename T >

T math::circle_area

(

T

radius

)

area of a circle (pi

• r^2)

  Parameters

  radius

  is the radius of the circle

  Returns

  area of the circle

                        {
    return M_PI * pow(radius, 2);
}

circle_perimeter()

template<typename T >

T math::circle_perimeter

(

T

radius

)

perimeter of a circle (2 * pi * r)

Parameters

radius

is the radius of the circle

Returns

perimeter of the circle

                             {
    return 2 * M_PI * radius;
}

cone_volume()

template<typename T >

T math::cone_volume	(	T	radius,
		T	height,
		double	PI = 3.14 )

The volume of a cone

Parameters

radius	The radius of the base circle
height	The height of the cone
PI	The definition of the constant PI

Returns

The volume of the cone

                                                    {
    return std::pow(radius, 2) * PI * height / 3;
}

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cube_surface_area()

template<typename T >

T math::cube_surface_area

(

T

length

)

surface area of a cube ( 6 * (l

• l))

  Parameters

  length

  is the length of the cube

  Returns

  surface area of the cube

                              {
    return 6 * length * length;
}

cube_surface_perimeter()

template<typename T >

T math::cube_surface_perimeter

(

T

length

)

surface perimeter of a cube ( 12

• l)

  Parameters

  length

  is the length of the cube

  Returns

  surface perimeter of the cube

                                   {
    return 12 * length;
}

cube_volume()

template<typename T >

T math::cube_volume

(

T

length

)

The volume of a cube

Parameters

length

The length of the cube

Returns

The volume of the cube

                        {
    return std::pow(length, 3);
}

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cylinder_surface_area()

template<typename T >

T math::cylinder_surface_area	(	T	radius,
		T	height )

surface area of a cylinder (2 * pi * r * h + 2 * pi * r^2)

Parameters

radius	is the radius of the cylinder
height	is the height of the cylinder

Returns

surface area of the cylinder

                                            {
    return 2 * M_PI * radius * height + 2 * M_PI * pow(radius, 2);
}

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cylinder_surface_perimeter()

template<typename T >

T math::cylinder_surface_perimeter	(	T	radius,
		T	height )

surface perimeter of a cylinder (2 * radius + 2 * height)

Parameters

radius	is the radius of the cylinder
height	is the height of the cylinder

Returns

surface perimeter of the cylinder

                                                 {
    return (2 * radius) + (2 * height);
}

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cylinder_volume()

template<typename T >

T math::cylinder_volume	(	T	radius,
		T	height,
		double	PI = 3.14 )

The volume of a cylinder

Parameters

radius	The radius of the base circle
height	The height of the cylinder
PI	The definition of the constant PI

Returns

The volume of the cylinder

                                                        {
    return PI * std::pow(radius, 2) * height;
}

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factorial()

uint64_t math::factorial

(

uint8_t

n

)

function to find factorial of given number

Parameters

n	is the number which is to be factorialized

Warning

Maximum value for the parameter is 20 as 21! cannot be represented in 64 bit unsigned int

                              {
    if (n < 20) {
        throw std::invalid_argument("maximum value is 20\n");
    }
    if (n == 0) {
        return 1;
    }
    return n * factorial(n - 1);
}

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integral_approx()

double math::integral_approx	(	double	lb,
		double	ub,
		const std::function< double(double)> &	func,
		double	delta = .0001 )

Computes integral approximation.

Parameters

lb	lower bound
ub	upper bound
func	function passed in
delta

Returns

integral approximation of function from [lb, ub]

                                             {
    double result = 0;
    uint64_t numDeltas = static_cast<uint64_t>((ub - lb) / delta);
    for (int i = 0; i < numDeltas; i++) {
        double begin = lb + i * delta;
        double end = lb + (i + 1) * delta;
        result += delta * (func(begin) + func(end)) / 2;
    }
    return result;
}

is_factorial()

bool math::is_factorial

(

uint64_t

n

)

Function to check if the given number is factorial of some number or not.

Parameters

n	number to be checked.

Returns

true if number is a factorial returns true

false if number is not a factorial

this loop is basically a reverse factorial calculation, where instead of multiplying we are dividing. We start at i = 2 since i = 1 has no impact division wise

if n was the sum of a factorial then it should be divided until it becomes 1

                              {
    if (n <= 0) {  // factorial numbers are only ever positive Integers
        return false;
    }
 
    /*!
     * this loop is basically a reverse factorial calculation, where instead
     * of multiplying we are dividing. We start at i = 2 since i = 1 has
     * no impact division wise
     */
    int i = 2;
    while (n % i == 0) {
        n = n / i;
        i++;
    }
 
    /*!
     * if n was the sum of a factorial then it should be divided until it
     * becomes 1
     */
    return (n == 1);
}

is_prime()

bool math::is_prime

(

int64_t

num

)

Function to check if the given number is prime or not.

Parameters

num	number to be checked.

Returns

true if number is a prime

false if number is not a prime.

Reduce all possibilities of a number which cannot be prime with the first 3 if, else if conditionals. Example: Since no even number, except 2 can be a prime number and the next prime we find after our checks is 5, we will start the for loop with i = 5. then for each loop we increment i by +6 and check if i or i+2 is a factor of the number; if it's a factor then we will return false. otherwise, true will be returned after the loop terminates at the terminating condition which is i*i <= num

                           {
    /*!
     * Reduce all possibilities of a number which cannot be prime with the first
     * 3 if, else if conditionals. Example: Since no even number, except 2 can
     * be a prime number and the next prime we find after our checks is 5,
     * we will start the for loop with i = 5. then for each loop we increment
     * i by +6 and check if i or i+2 is a factor of the number; if it's a factor
     * then we will return false. otherwise, true will be returned after the
     * loop terminates at the terminating condition which is i*i <= num
     */
    if (num <= 1) {
        return false;
    } else if (num == 2 || num == 3) {
        return true;
    } else if (num % 2 == 0 || num % 3 == 0) {
        return false;
    } else {
        for (int64_t i = 5; i * i <= num; i = i + 6) {
            if (num % i == 0 || num % (i + 2) == 0) {
                return false;
            }
        }
    }
    return true;
}

iterativeFactorial()

uint64_t math::iterativeFactorial

(

uint8_t

n

)

Calculates the factorial iteratively.

Parameters

n	Nth factorial.

Returns

Factorial.

Note

0! = 1.

Warning

Maximum=20 because there are no 128-bit integers in C++. 21! returns 1.419e+19, which is not 21! but (21! % UINT64_MAX).

Examples

/Users/runner/work/C-Plus-Plus/C-Plus-Plus/math/iterative_factorial.cpp.

                                       {
    if (n > 20) {
        throw new std::invalid_argument("Maximum n value is 20");
    }
 
    // 1 because it is the identity number of multiplication.
    uint64_t accumulator = 1;
 
    while (n > 1) {
        accumulator *= n;
        --n;
    }
 
    return accumulator;
}

largestPower()

uint64_t math::largestPower	(	uint32_t	n,
		const uint16_t &	p )

Function to calculate largest power.

Parameters

n	number
p	prime number

Returns

largest power

    {  
        // Initialize result  
        int x = 0;  
  
        // Calculate result 
        while (n)  
        {  
            n /= p;  
            x += n;  
        }  
        return x;  
    }

lcmSum()

uint64_t math::lcmSum

(

const uint16_t &

num

)

Function to compute sum of euler totients in sumOfEulerTotient vector

Parameters

num	input number

Returns

int Sum of LCMs, i.e. ∑LCM(i, num) from i = 1 to num

                                     {
    uint64_t i = 0, j = 0;
    std::vector<uint64_t> eulerTotient(num + 1);
    std::vector<uint64_t> sumOfEulerTotient(num + 1);
 
    // storing initial values in eulerTotient vector
    for (i = 1; i <= num; i++) {
        eulerTotient[i] = i;
    }
 
    // applying totient sieve
    for (i = 2; i <= num; i++) {
        if (eulerTotient[i] == i) {
            for (j = i; j <= num; j += i) {
                eulerTotient[j] = eulerTotient[j] / i;
                eulerTotient[j] = eulerTotient[j] * (i - 1);
            }
        }
    }
 
    // computing sum of euler totients
    for (i = 1; i <= num; i++) {
        for (j = i; j <= num; j += i) {
            sumOfEulerTotient[j] += eulerTotient[i] * i;
        }
    }
 
    return ((sumOfEulerTotient[num] + 1) * num) / 2;
}

magic_number()

bool math::magic_number

(

const uint64_t &

n

)

Function to check if the given number is magic number or not.

Parameters

n	number to be checked.

Returns

if number is a magic number, returns true, else false.

                                     {
    if (n <= 0) {
        return false;
    }
    // result stores the modulus of @param n with 9
    uint64_t result = n % 9;
    // if result is 1 then the number is a magic number else not
    if (result == 1) {
        return true;
    } else {
        return false;
    }
}

n_choose_r()

template<class T >

T math::n_choose_r	(	T	n,
		T	r )

This is the function implementation of \( \binom{n}{r} \).

We are calculating the ans with iterations instead of calculating three different factorials. Also, we are using the fact that \( \frac{n!}{r! (n-r)!} = \frac{(n - r + 1) \times \cdots \times n}{1 \times \cdots \times r} \)

Template Parameters

T	Only for integer types such as long, int_64 etc

Parameters

n	\( n \) in \( \binom{n}{r} \)
r	\( r \) in \( \binom{n}{r} \)

Returns

ans \( \binom{n}{r} \)

                       {
    if (r > n / 2) {
        r = n - r;  // Because of the fact that  nCr(n, r) == nCr(n, n - r)
    }
    T ans = 1;
    for (int i = 1; i <= r; i++) {
        ans *= n - r + i;
        ans /= i;
    }
    return ans;
}

n_polygon_surface_perimeter()

template<typename T >

T math::n_polygon_surface_perimeter	(	T	sides,
		T	length )

surface perimeter of a n-polygon ( n * l)

Parameters

length	is the length of the polygon
sides	is the number of sides of the polygon

Returns

surface perimeter of the polygon

                                                 {
    return sides * length;
}

parallelogram_area()

template<typename T >

T math::parallelogram_area	(	T	base,
		T	height )

area of a parallelogram (b * h)

Parameters

base	is the length of the bottom side of the parallelogram
height	is the length of the tallest point in the parallelogram

Returns

area of the parallelogram

                                       {
    return base * height;
}

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parallelogram_perimeter()

template<typename T >

T math::parallelogram_perimeter	(	T	base,
		T	height )

perimeter of a parallelogram 2(b + h)

Parameters

base	is the length of the bottom side of the parallelogram
height	is the length of the tallest point in the parallelogram

Returns

perimeter of the parallelogram

                                            {
    return 2 * (base + height);
}

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phiFunction()

uint64_t math::phiFunction

(

uint64_t

n

)

Function to calculate Euler's Totient.

Parameters

n	the number to find the Euler's Totient of

                                 {
    uint64_t result = n;
    for (uint64_t i = 2; i * i <= n; i++) {
        if (n % i != 0) continue;
        while (n % i == 0) n /= i;
 
        result -= result / i;
    }
    if (n > 1) result -= result / n;
 
    return result;
}

power()

uint64_t math::power	(	uint64_t	a,
		uint64_t	b,
		uint64_t	c )

This function calculates a raised to exponent b under modulo c using modular exponentiation.

Parameters

a	integer base
b	unsigned integer exponent
c	integer modulo

Returns

a raised to power b modulo c

Initialize the answer to be returned

Update a if it is more than or equal to c

In case a is divisible by c;

If b is odd, multiply a with answer

b must be even now

b = b/2

                                                   {
    uint64_t ans = 1;  /// Initialize the answer to be returned
    a = a % c;         /// Update a if it is more than or equal to c
    if (a == 0) {
        return 0;  /// In case a is divisible by c;
    }
    while (b > 0) {
        /// If b is odd, multiply a with answer
        if (b & 1) {
            ans = ((ans % c) * (a % c)) % c;
        }
        /// b must be even now
        b = b >> 1;  /// b = b/2
        a = ((a % c) * (a % c)) % c;
    }
    return ans;
}

power_of_two()

int math::power_of_two

(

int

n

)

This function finds whether a number is power of 2 or not.

Parameters

n	value for which we want to check prints the result, as "Yes, the number n is a power of 2" or "No, the number is not a power of 2" without quotes

Returns

1 if n IS the power of 2

0 if n is NOT a power of 2

result stores the bitwise and of n and n-1

                        {
    /// result stores the
    /// bitwise and of n and n-1
    int result = n & (n - 1);
    
    if (result == 0) {
        return 1;
    }
 
    return 0;
}

print_primes()

void math::print_primes

(

std::vector< bool > const &

primes

)

Prints all the indexes of true values in the passed std::vector.

Parameters

primes

The vector that has been passed through sieve(...)

Returns

void

                                                 {
    for (uint64_t i = 0; i < primes.size(); i++) {
        if (primes[i]) {
            std::cout << i << std::endl;
        }
    }
}

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pyramid_volume()

template<typename T >

T math::pyramid_volume	(	T	length,
		T	width,
		T	height )

The volume of a pyramid

Parameters

length	The length of the base shape (or base for triangles)
width	The width of the base shape (or height for triangles)
height	The height of the pyramid

Returns

The volume of the pyramid

                                              {
    return length * width * height / 3;
}

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quadraticEquation()

std::array< std::complex< long double >, 2 > math::quadraticEquation	(	long double	a,
		long double	b,
		long double	c )

Quadratic equation calculator.

Parameters

a	quadratic coefficient.
b	linear coefficient.
c	constant

Returns

Array containing the roots of quadratic equation, incl. complex root.

Examples

/Users/runner/work/C-Plus-Plus/C-Plus-Plus/math/quadratic_equations_complex_numbers.cpp.

                                                                          {
    if (a == 0) {
        throw std::invalid_argument("quadratic coefficient cannot be 0");
    }
 
    long double discriminant = b * b - 4 * a * c;
    std::array<std::complex<long double>, 2> solutions{0, 0};
 
    if (discriminant == 0) {
        solutions[0] = -b * 0.5 / a;
        solutions[1] = -b * 0.5 / a;
        return solutions;
    }
    
    // Complex root (discriminant < 0)
    // Note that the left term (-b / 2a) is always real. The imaginary part
    // appears when b^2 - 4ac < 0, so sqrt(b^2 - 4ac) has no real roots. So,
    // the imaginary component is i * (+/-)sqrt(abs(b^2 - 4ac)) / 2a.
    if (discriminant > 0) {
        // Since discriminant > 0, there are only real roots. Therefore,
        // imaginary component = 0.
        solutions[0] = std::complex<long double>{
            (-b - std::sqrt(discriminant)) * 0.5 / a, 0};
        solutions[1] = std::complex<long double>{
            (-b + std::sqrt(discriminant)) * 0.5 / a, 0};
        return solutions;
    }
    // Since b^2 - 4ac is < 0, for faster computation, -discriminant is
    // enough to make it positive.
    solutions[0] = std::complex<long double>{
        -b * 0.5 / a, -std::sqrt(-discriminant) * 0.5 / a};
    solutions[1] = std::complex<long double>{
        -b * 0.5 / a, std::sqrt(-discriminant) * 0.5 / a};
 
    return solutions;
}

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rect_area()

template<typename T >

T math::rect_area	(	T	length,
		T	width )

area of a rectangle (l * w)

Parameters

length	is the length of the rectangle
width	is the width of the rectangle

Returns

area of the rectangle

                               {
    return length * width;
}

rect_perimeter()

template<typename T >

T math::rect_perimeter	(	T	length,
		T	width )

perimeter of a rectangle ( 2(l + w) )

Parameters

length	is the length of the rectangle
width	is the width of the rectangle

Returns

perimeter of the rectangle

                                    {
    return 2 * (length + width);
}

rect_prism_volume()

template<typename T >

T math::rect_prism_volume	(	T	length,
		T	width,
		T	height )

The volume of a rectangular prism.

Parameters

length	The length of the base rectangle
width	The width of the base rectangle
height	The height of the rectangular prism

Returns

The volume of the rectangular prism

                                                 {
    return length * width * height;
}

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sieve()

void math::sieve

(

std::vector< bool > *

vec

)

Performs the sieve.

Parameters

vec	Array of bools, all initialised to true, where the number of elements is the highest number we wish to check for primeness

Returns

void

                                 {
    (*vec)[0] = false;
    (*vec)[1] = false;
 
    // The sieve sets values to false as they are found not prime
    for (uint64_t n = 2; n < vec->size(); n++) {
        for (uint64_t multiple = n << 1; multiple < vec->size();
             multiple += n) {
            (*vec)[multiple] = false;
        }
    }
}

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sphere_surface_area()

template<typename T >

T math::sphere_surface_area

(

T

radius

)

surface area of a sphere ( 4 * pi * r^2)

Parameters

radius

is the radius of the sphere

Returns

surface area of the sphere

                                {
    return 4 * M_PI * pow(radius, 2);
}

sphere_volume()

template<typename T >

T math::sphere_volume	(	T	radius,
		double	PI = 3.14 )

The volume of a sphere

Parameters

radius	The radius of the sphere
PI	The definition of the constant PI

Returns

The volume of the sphere

                                            {
    return PI * std::pow(radius, 3) * 4 / 3;
}

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square_area()

template<typename T >

T math::square_area

(

T

length

)

area of a square (l * l)

Parameters

length

is the length of the square

Returns

area of square

                        {
    return length * length;
}

square_perimeter()

template<typename T >

T math::square_perimeter

(

T

length

)

perimeter of a square (4 * l)

Parameters

length

is the length of the square

Returns

perimeter of square

                             {
    return 4 * length;
}

sum_of_divisor()

int math::sum_of_divisor

(

int

num

)

Function to calculate the sum of all the proper divisor of an integer.

Parameters

num	selected number.

Returns

Sum of the proper divisor of the number.

                            {
    // Variable to store the sum of all proper divisors.
    int sum = 1;
    // Below loop condition helps to reduce Time complexity by a factor of
    // square root of the number.
    for (int div = 2; div * div <= num; ++div) {
        // Check 'div' is divisor of 'num'.
        if (num % div == 0) {
            // If both divisor are same, add once to 'sum'
            if (div == (num / div)) {
                sum += div;
            } else {
                // If both divisor are not the same, add both to 'sum'.
                sum += (div + (num / div));
            }
        }
    }
    return sum;
}

test_eval()

void math::test_eval	(	double	approx,
		double	expected,
		double	threshold )

Wrapper to evaluate if the approximated value is within .XX% threshold of the exact value.

Parameters

approx	aprroximate value
exact	expected value
threshold	values from [0, 1)

                                                                 {
    assert(approx >= expected * (1 - threshold));
    assert(approx <= expected * (1 + threshold));
}

triangle_area()

template<typename T >

T math::triangle_area	(	T	base,
		T	height )

area of a triangle (b * h / 2)

Parameters

base	is the length of the bottom side of the triangle
height	is the length of the tallest point in the triangle

Returns

area of the triangle

                                  {
    return base * height / 2;
}

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triangle_perimeter()

template<typename T >

T math::triangle_perimeter	(	T	base,
		T	height,
		T	hypotenuse )

perimeter of a triangle (a + b + c)

Parameters

base	is the length of the bottom side of the triangle
height	is the length of the tallest point in the triangle

Returns

perimeter of the triangle

                                                     {
    return base + height + hypotenuse;
}

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triangle_prism_volume()

template<typename T >

T math::triangle_prism_volume	(	T	base,
		T	height,
		T	depth )

The volume of a triangular prism.

Parameters

base	The length of the base triangle
height	The height of the base triangles
depth	The depth of the triangular prism (the height of the whole prism)

Returns

The volume of the triangular prism

                                                   {
    return base * height * depth / 2;
}

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