Hypot

Defined in header <cmath>.

Description

Declarations 1-3 (C++23)

Computes the square root of the sum of the squares of x and y, without undue overflow or underflow at intermediate stages of the computation. The library provides overloads of std::hypot for all cv-unqualified floating-point types as the type of the parameters x and y.

Declaration 4 (C++23)

Computes the square root of the sum of the squares of x, y, and z, without undue overflow or underflow at intermediate stages of the computation. The library provides overloads of std::hypot for all cv-unqualified floating-point types as the type of the parameters x, y and z.

The value computed by the two-argument version of this function is the length of the hypotenuse of a right-angled triangle with sides of length x and y, or the distance of the point (x,y) from the origin (0,0), or the magnitude of a complex number x+iy.

The value computed by the three-argument version of this function is length of the body diagonal of a rectangular parallelepiped with sides of length x, y and z or the distance of the point (x,y,z) from the origin (0,0,0).

Declarations

C++23
C++17
C++11

// 1)
/* floating-point-type */ hypot( /* floating-point-type */ x,
                                 /* floating-point-type */ y );
// 2)
float hypotf( float x, float y );
// 3)
long double hypotl( long double x, long double y );
// 4)
/* floating-point-type */ hypot( /* floating-point-type */ x,
                                 /* floating-point-type */ y,
                                 /* floating-point-type */ z );

Additional Overloads
// 5)
template< class Arithmetic1, Arithmetic2 >
/* common-floating-point-type */ hypot( Arithmetic1 x, Arithmetic2 y );
// 6)
template< class Arithmetic1, Arithmetic2, Arithmetic3 >
/* common-floating-point-type */
    hypot( Arithmetic1 x, Arithmetic2 y, Arithmetic3 z );

// 1)
float hypot ( float x, float y );
// 2)
double hypot ( double x, double y );
// 3)
long double hypot ( long double x, long double y );
// 4)
float hypotf( float x, float y );
// 5)
long double hypotl( long double x, long double y );
// 6)
float hypot ( float x, float y, float z );
// 7)
double hypot ( double x, double y, double z );
// 8)
long double hypot ( long double x, long double y, long double z );

Additional Overloads
// 9)
template< class Arithmetic1, Arithmetic2 >
/* common-floating-point-type */ hypot( Arithmetic1 x, Arithmetic2 y );
// 10)
template< class Arithmetic1, Arithmetic2, Arithmetic3 >
/* common-floating-point-type */
    hypot( Arithmetic1 x, Arithmetic2 y, Arithmetic3 z );

// 1)
float hypot ( float x, float y );
// 2)
double hypot ( double x, double y );
// 3)
long double hypot ( long double x, long double y );
// 4)
float hypotf( float x, float y );
// 5)
long double hypotl( long double x, long double y );

Additional Overloads
// 6)
template< class Arithmetic1, Arithmetic2 >
/* common-floating-point-type */ hypot( Arithmetic1 x, Arithmetic2 y );

Parameters

x,y,z - floating-point or integer values

Return value

1-3,A) If no errors occur, the hypotenuse of a right-angled triangle(√(x²+y²)) is returned.
4,B) If no errors occur, the distance from origin in 3D space (√(x²+y²+z²)) is returned.
If a range error due to overflow occurs, +HUGE_VAL, +HUGE_VALF, or +HUGE_VALL is returned.

If a range error due to underflow occurs, the correct result (after rounding) is returned.

Error handling

Errors are reported as specified in math_errhandling

If the implementation supports IEEE floating-point arithmetic (IEC 60559),

std::hypot(x, y), std::hypot(y, x), and std::hypot(x, -y) are equivalent
if one of the arguments is ±0, std::hypot(x, y) is equivalent to std::fabs called with the non-zero argument
if one of the arguments is ±∞, std::hypot(x, y) returns +∞ even if the other argument is NaN
otherwise, if any of the arguments is NaN, NaN is returned

Notes

Implementations usually guarantee precision of less than 1 ulp (Unit in the Last Place — Unit of Least Precision): GNU, BSD.

std::hypot(x, y) is equivalent to std::abs(std::complex<double>(x, y)).

POSIX specifies that underflow may only occur when both arguments are subnormal and the correct result is also subnormal (this forbids naive implementations).

Distance between two points (x1,y1,z1) and (x2,y2,z2) on 3D space can be calculated using 3-argument overload of std::hypot:
std::hypot(x2 - x1, y2 - y1, z2 - z1). (since C++17)

The additional overloads are not required to be provided exactly as Additional Overloads. They only need to be sufficient to ensure that for their first argument num1, second argument num2 and the optional third argument num3:

If num1, num2 or num3 has type long double, then
std::hypot(num1, num2) has the same effect as
std::hypot(static_cast<long double>(num1), static_cast<long double>(num2)).

and std::hypot(num1, num2, num3) has the same effect as
std::hypot(static_cast<long double>(num1), static_cast<long double>(num2), static_cast<long double>(num3)).

Otherwise, if num1, num2 and/or num3 has type double or an integer type, then
std::hypot(num1, num2) has the same effect as
std::hypot(static_cast<double>(num1), static_cast<double>(num2)).

and std::hypot(num1, num2, num3) has the same effect as
std::hypot(static_cast<double>(num1), static_cast<double>(num2), static_cast<double>(num3)).

Otherwise, if num1, num2 or num3 has type float, then
std::hypot(num1, num2) has the same effect as
std::hypot(static_cast<float>(num1), static_cast<float>(num2)).

and std::hypot(num1, num2, num3) has the same effect as
std::hypot(static_cast<float>(num1), static_cast<float>(num2), static_cast<float>(num3)). (until C++23)

If num1, num2 and num3 have arithmetic types, then

std::hypot(num1, num2) has the same effect as
std::hypot(static_cast</* common-floating-point-type */>(num1), static_cast</* common-floating-point-type */>(num2)) .

and std::hypot(num1, num2, num3) has the same effect as
std::hypot(static_cast</* common-floating-point-type */>(num1), static_cast</* common-floating-point-type */>(num2), static_cast</* common-floating-point-type */>(num3))

where /* common-floating-point-type */ is the floating-point type with the greatest floating-point conversion rank and greatest floating-point conversion subrank among the types of num1, num2 and num3, arguments of integer type are considered to have the same floating-point conversion rank as double.

If no such floating-point type with the greatest rank and subrank exists, then overload resolution does not result in a usable candidate from the overloads provided.

Examples

#include <cerrno>
#include <cfenv>
#include <cfloat>
#include <cmath>
#include <cstring>
#include <iostream>
 
// #pragma STDC FENV_ACCESS ON
 
struct Point3D { float x, y, z; };
 
int main()
{
    // typical usage
    std::cout 
        << "(1,1) cartesian is (" 
        << std::hypot(1,1)
        << ',' << std::atan2(1,1) 
        << ") polar\n";
 
    Point3D a{3.14, 2.71, 9.87}, b{1.14, 5.71, 3.87};
    // C++17 has 3-argument hypot overload:
    std::cout 
        << "distance(a,b) = "
        << std::hypot(a.x - b.x, a.y - b.y, a.z - b.z) 
        << '\n';
 
    // special values
    std::cout 
        << "hypot(NAN,INFINITY) = " 
        << std::hypot(NAN, INFINITY) << '\n';
 
    // error handling
    errno = 0;
    std::feclearexcept(FE_ALL_EXCEPT);
    std::cout 
        << "hypot(DBL_MAX,DBL_MAX) = " 
        << std::hypot(DBL_MAX, DBL_MAX) << '\n';
 
    if (errno == ERANGE)
        std::cout 
            << "errno = ERANGE " 
            << std::strerror(errno) << '\n';
    if (std::fetestexcept(FE_OVERFLOW))
        std::cout 
            << "FE_OVERFLOW raised\n";
}

Result
(1,1) cartesian is (1.41421,0.785398) polar
distance(a,b) = 7
hypot(NAN,INFINITY) = inf
hypot(DBL_MAX,DBL_MAX) = inf
errno = ERANGE Numerical result out of range
FE_OVERFLOW raised