# Calculation accuracy

## Question:

I solve Olympiad problems of various kinds, when I worked in Pascal I never thought about the operation of the "=" or ">=" operator in working with real numbers, I thought that there could be no problems with the accuracy of comparisons. But then I switched to C ++ and one teacher told me that "==" or ">=" cannot be used when working with real numbers in C, you need to write fabs(a - b) > eps . And here is the question:

1. Is it true?
2. How to choose the right eps?
3. And why could some tasks not pass all tests with such a feature, but when I simply wrote "==" they passed.

Here is a specific example. Task that passed after I changed LessEqual to "<=". Given a real number a and a natural number n. Calculate the nth root of a. The program should display a single number: the answer to the problem with an accuracy of at least 6 decimal places

#include <cstdio>
#include <iostream>
#include <cmath>

using namespace std;
const double eps = 1e-8;
const double eps2 = 1e-12;

bool Equal(double a, double b)
{
return fabs(a - b) < eps2;
}

bool lessEqual(double a, double b)
{
return ((a < b) || Equal(a, b));
}

double a;
int n;

void solve()
{
double l = 0,
r = a + 1,
mid = 0;
while (r - l > eps)
{
mid = (l + r) / 2;
if (lessEqual(pow(mid, n), a))
l = mid;
else
r = mid;
}
printf("%.7lf", l);
}

int main()
{
freopen("input.txt", "r", stdin);
freopen("output.txt", "w", stdout);

scanf("%lf\n", &a);
scanf("%d", &n);
solve();

return 0;
}

## Answer:

The problem here is not an "inaccurate" comparison by the processor hardware, but that some decimals cannot be accurately converted to binary floating point numbers.

Another problem is the loss of the least significant digits of the mantissa when calculating in cycles.

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