NAME¶

std::numeric_limits::epsilon - std::numeric_limits::epsilon

Synopsis¶

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.

Return value¶

T std::numeric_limits<T>::epsilon()
/* non-specialized */ T()
bool false
char 0
signed char 0
unsigned char 0
wchar_t 0
char8_t (since C++20) 0
char16_t (since C++11) 0
char32_t (since C++11) 0
short 0
unsigned short 0
int 0
unsigned int 0
long 0
unsigned long 0
long long (since C++11) 0
unsigned long long(since C++11) 0
float FLT_EPSILON
double DBL_EPSILON
long double LDBL_EPSILON

Example¶

Demonstrates the use of machine epsilon to compare floating-point values for
equality:

// Run this code

#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

See also¶

nextafter
nextafterf
nextafterl
nexttoward
nexttowardf
nexttowardl next representable floating-point value towards the given value
(C++11) (function)
(C++11)
(C++11)
(C++11)
(C++11)
(C++11)