static T epsilon() throw();
(until C++11)static constexpr T epsilon() noexcept;
(since C++11)Returns the machine epsilon, that is, the difference between 1.0 and the next value representable by the floating-point type T. It is only meaningful if std::numeric_limits<T>::is_integer == false.
T std::numeric_limits<T>::epsilon() /* non-specialized */ T() bool false char â0â signed char â0â unsigned char â0â wchar_t â0â char8_t (since C++20) â0â char16_t (since C++11) â0â char32_t (since C++11) â0â short â0â unsigned short â0â int â0â unsigned int â0â long â0â unsigned long â0â long long (since C++11) â0â unsigned long long(since C++11) â0â float FLT_EPSILON double DBL_EPSILON long double LDBL_EPSILON [edit] Example
Demonstrates the use of machine epsilon to compare floating-point values for equality:
#include <algorithm>
#include <cmath>
#include <cstddef>
#include <iomanip>
#include <iostream>
#include <limits>
#include <type_traits>
template <class T>
std::enable_if_t<not std::numeric_limits<T>::is_integer, bool>
equal_within_ulps(T x, T y, std::size_t n)
{
// Since `epsilon()` is the gap size (ULP, unit in the last place)
// of floating-point numbers in interval [1, 2), we can scale it to
// the gap size in interval [2^e, 2^{e+1}), where `e` is the exponent
// of `x` and `y`.
// If `x` and `y` have different gap sizes (which means they have
// different exponents), we take the smaller one. Taking the bigger
// one is also reasonable, I guess.
const T m = std::min(std::fabs(x), std::fabs(y));
// Subnormal numbers have fixed exponent, which is `min_exponent - 1`.
const int exp = m < std::numeric_limits<T>::min()
? std::numeric_limits<T>::min_exponent - 1
: std::ilogb(m);
// We consider `x` and `y` equal if the difference between them is
// within `n` ULPs.
return std::fabs(x - y) <= n * std::ldexp(std::numeric_limits<T>::epsilon(), exp);
}
int main()
{
double x = 0.3;
double y = 0.1 + 0.2;
std::cout << std::hexfloat;
std::cout << "x = " << x << '\n';
std::cout << "y = " << y << '\n';
std::cout << (x == y ? "x == y" : "x != y") << '\n';
for (std::size_t n = 0; n <= 10; ++n)
if (equal_within_ulps(x, y, n))
{
std::cout << "x equals y within " << n << " ulps" << '\n';
break;
}
}
Output:
x = 0x1.3333333333333p-2 y = 0x1.3333333333334p-2 x != y x equals y within 1 ulps[edit] See also
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