Mathematical constants

 

C++

Compiler support

Freestanding and hosted

Language

Standard library

Standard library headers

Named requirements

Feature test macros (C++20)

Language support library

Concepts library (C++20)

Metaprogramming library (C++11)

Diagnostics library

General utilities library

Strings library

Containers library

Iterators library

Ranges library (C++20)

Algorithms library

Numerics library

Localizations library

Input/output library

Filesystem library (C++17)

Regular expressions library (C++11)

Concurrency support library (C++11)

Technical specifications

Symbols index

External libraries

 

Numerics library

Common mathematical functions

Mathematical special functions (C++17)

Mathematical constants (C++20)

Floating-point environment (C++11)

Complex numbers

Numeric arrays

Pseudo-random number generation

Factor operations

gcd (C++17)

lcm (C++17)

Interpolations

midpoint (C++20)

lerp (C++20)

Generic numeric operations

iota (C++11)

ranges::iota (C++23)

accumulate

inner_product

adjacent_difference

partial_sum

reduce (C++17)

transform_reduce (C++17)

inclusive_scan (C++17)

exclusive_scan (C++17)

transform_inclusive_scan (C++17)

transform_exclusive_scan (C++17)

Bit operations

bit_cast (C++20)

byteswap (C++23)

has_single_bit (C++20)

bit_ceil (C++20)

bit_floor (C++20)

bit_width (C++20)

rotl (C++20)

rotr (C++20)

countl_zero (C++20)

countl_one (C++20)

countr_zero (C++20)

countr_one (C++20)

popcount (C++20)

endian (C++20)

 

Mathematical constants

 

Constants (since C++20)

Defined in header <numbers>
Defined in namespace std::numbers
e_v	the mathematical constant \(\small e\)e (variable template)
log2e_v	\(\log_{2}e\)log 2e (variable template)
log10e_v	\(\log_{10}e\)log 10e (variable template)
pi_v	the mathematical constant \(\pi\)π (variable template)
inv_pi_v	\(\frac1\pi\) 1 π (variable template)
inv_sqrtpi_v	\(\frac1{\sqrt\pi}\) 1 √π (variable template)
ln2_v	\(\ln{2}\)ln 2 (variable template)
ln10_v	\(\ln{10}\)ln 10 (variable template)
sqrt2_v	\(\sqrt2\)√2 (variable template)
sqrt3_v	\(\sqrt3\)√3 (variable template)
inv_sqrt3_v	\(\frac1{\sqrt3}\) 1 √3 (variable template)
egamma_v	the Euler–Mascheroni constant γ (variable template)
phi_v	the golden ratio Φ (\(\frac{1+\sqrt5}2\) 1 + √5 2 ) (variable template)
inline constexpr double e	e_v<double> (constant)
inline constexpr double log2e	log2e_v<double> (constant)
inline constexpr double log10e	log10e_v<double> (constant)
inline constexpr double pi	pi_v<double> (constant)
inline constexpr double inv_pi	inv_pi_v<double> (constant)
inline constexpr double inv_sqrtpi	inv_sqrtpi_v<double> (constant)
inline constexpr double ln2	ln2_v<double> (constant)
inline constexpr double ln10	ln10_v<double> (constant)
inline constexpr double sqrt2	sqrt2_v<double> (constant)
inline constexpr double sqrt3	sqrt3_v<double> (constant)
inline constexpr double inv_sqrt3	inv_sqrt3_v<double> (constant)
inline constexpr double egamma	egamma_v<double> (constant)
inline constexpr double phi	phi_v<double> (constant)

Notes

A program that instantiates a primary template of a mathematical constant variable template is ill-formed.

The standard library specializes mathematical constant variable templates for all floating-point types (i.e. float, double and long double).

A program may partially or explicitly specialize a mathematical constant variable template provided that the specialization depends on a program-defined type.

Feature-test macro	Value	Std	Comment
__cpp_lib_math_constants	201907L	(c++20)	Mathematical constants

Example

Run this code

#include <cmath>
#include <iomanip>
#include <iostream>
#include <limits>
#include <numbers>
#include <string_view>
 
auto egamma_aprox(const unsigned iterations)
{
    long double s = 0;
    for (unsigned m = 2; m < iterations; ++m)
    {
        if (const long double t = std::riemann_zetal(m) / m; m % 2)
            s -= t;
        else
            s += t;
    }
    return s;
};
 
int main()
{
    using namespace std;
    using namespace std::numbers;
 
    const auto 
        x = sqrt(inv_pi)/inv_sqrtpi + ceil(exp2(log2e)) + sqrt3*inv_sqrt3 + exp(0),
        v = (phi*phi - phi) + 1/log2(sqrt2) + log10e*ln10 + pow(e, ln2) - cos(pi);
 
    std::cout << "The answer is " << x*v << '\n';
 
    using namespace std::string_view_literals;
    constexpr auto γ = "0.577215664901532860606512090082402"sv;
 
    std::cout
        << "γ as 10⁶ sums of ±ζ(m)/m   = "
        << egamma_aprox(1'000'000) << '\n'
        << "γ as egamma_v<float>       = "
        << std::setprecision(std::numeric_limits<float>::digits10 + 1)
        << egamma_v<float> << '\n'
        << "γ as egamma_v<double>      = "
        << std::setprecision(std::numeric_limits<double>::digits10 + 1)
        << egamma_v<double> << '\n'
        << "γ as egamma_v<long double> = "
        << std::setprecision(std::numeric_limits<long double>::digits10 + 1)
        << egamma_v<long double> << '\n'
        << "γ with " << γ.length() - 1 << " digits precision = " << γ << '\n'
    ;
}

Possible output:

The answer is 42
γ as 10⁶ sums of ±ζ(m)/m   = 0.577215
γ as egamma_v<float>       = 0.5772157
γ as egamma_v<double>      = 0.5772156649015329
γ as egamma_v<long double> = 0.5772156649015328606
γ with 34 digits precision = 0.577215664901532860606512090082402

See also

ratio (C++11)

represents exact rational fraction (class template)