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atan2, atan2f, atan2l

Defined in header <math.h> 
float       atan2f( float y, float x );
 (1) (since C99) 
double      atan2( double y, double x );
 (2) 
long double atan2l( long double y, long double x );
 (3) (since C99) 
_Decimal32  atan2d32( _Decimal32 y, _Decimal32 x );
 (4) (since C23) 
_Decimal64  atan2d64( _Decimal64 y, _Decimal64 x );
 (5) (since C23) 
_Decimal128 atan2d128( _Decimal128 y, _Decimal128 x );
 (6) (since C23)
Defined in header <tgmath.h> 
#define atan2( y, x )
 (7) (since C99)
1-6) Computes the arc tangent of y / x using the signs of arguments to determine the correct quadrant.
7) Type-generic macro: If any argument has type long double, (3) (atan2l) is called. Otherwise, if any argument has integer type or has type double, (2) (atan2) is called. Otherwise, (1) (atan2f) is called.

The functions (4-6) are declared if and only if the implementation predefines __STDC_IEC_60559_DFP__ (i.e. the implementation supports decimal floating-point numbers).

 (since C23)

Parameters

x, y - floating-point value

Return value

If no errors occur, the arc tangent of y / x (arctan(y/x)) in the range [-π ; +π] radians, is returned.
Y argument
Return value
math-atan2.png
X argument

If a domain error occurs, an implementation-defined value is returned.

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

Error handling

Errors are reported as specified in math_errhandling.

Domain error may occur if x and y are both zero.

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

  • If x and y are both zero, domain error does not occur;
  • If x and y are both zero, range error does not occur either;
  • If y is zero, pole error does not occur;
  • If y is ±0 and x is negative or -0, ±π is returned;
  • If y is ±0 and x is positive or +0, ±0 is returned;
  • If y is ±∞ and x is finite, ±π/2 is returned;
  • If y is ±∞ and x is -∞, ±3π/4 is returned;
  • If y is ±∞ and x is +∞, ±π/4 is returned;
  • If x is ±0 and y is negative, -π/2 is returned;
  • If x is ±0 and y is positive, +π/2 is returned;
  • If x is -∞ and y is finite and positive, +π is returned;
  • If x is -∞ and y is finite and negative, -π is returned;
  • If x is +∞ and y is finite and positive, +0 is returned;
  • If x is +∞ and y is finite and negative, -0 is returned;
  • If either x is NaN or y is NaN, NaN is returned.

Notes

atan2(y, x) is equivalent to carg(x + I*y).

POSIX specifies that in case of underflow, y / x is the value returned, and if that is not supported, an implementation-defined value no greater than DBL_MIN, FLT_MIN, and LDBL_MIN is returned.

Example

#include <math.h>
#include <stdio.h>
 
int main(void)
{
    // normal usage: the signs of the two arguments determine the quadrant
    // atan2(1,1) = +pi/4, Quad I
    printf("(+1,+1) cartesian is (%f,%f) polar\n", hypot( 1, 1), atan2( 1, 1));
    // atan2(1, -1) = +3pi/4, Quad II
    printf("(+1,-1) cartesian is (%f,%f) polar\n", hypot( 1,-1), atan2( 1,-1));
    // atan2(-1,-1) = -3pi/4, Quad III
    printf("(-1,-1) cartesian is (%f,%f) polar\n", hypot(-1,-1), atan2(-1,-1));
    // atan2(-1,-1) = -pi/4, Quad IV
    printf("(-1,+1) cartesian is (%f,%f) polar\n", hypot(-1, 1), atan2(-1, 1));
 
    // special values
    printf("atan2(0, 0) = %f atan2(0, -0)=%f\n", atan2(0,0), atan2(0,-0.0));
    printf("atan2(7, 0) = %f atan2(7, -0)=%f\n", atan2(7,0), atan2(7,-0.0));
}

Output:

(+1,+1) cartesian is (1.414214,0.785398) polar
(+1,-1) cartesian is (1.414214,2.356194) polar
(-1,-1) cartesian is (1.414214,-2.356194) polar
(-1,+1) cartesian is (1.414214,-0.785398) polar
atan2(0, 0) = 0.000000 atan2(0, -0)=3.141593
atan2(7, 0) = 1.570796 atan2(7, -0)=1.570796

References

  • C23 standard (ISO/IEC 9899:2023):
    • 7.12.4.4 The atan2 functions (p: TBD)
    • 7.25 Type-generic math <tgmath.h> (p: TBD)
    • F.10.1.4 The atan2 functions (p: TBD)
  • C17 standard (ISO/IEC 9899:2018):
    • 7.12.4.4 The atan2 functions (p: 174)
    • 7.25 Type-generic math <tgmath.h> (p: 272-273)
    • F.10.1.4 The atan2 functions (p: 378)
  • C11 standard (ISO/IEC 9899:2011):
    • 7.12.4.4 The atan2 functions (p: 239)
    • 7.25 Type-generic math <tgmath.h> (p: 373-375)
    • F.10.1.4 The atan2 functions (p: 519)
  • C99 standard (ISO/IEC 9899:1999):
    • 7.12.4.4 The atan2 functions (p: 219)
    • 7.22 Type-generic math <tgmath.h> (p: 335-337)
    • F.9.1.4 The atan2 functions (p: 456)
  • C89/C90 standard (ISO/IEC 9899:1990):
    • 4.5.2.4 The atan2 function

See also

(C99)(C99)
computes arc sine (\({\small\arcsin{x} }\)arcsin(x))
(function)
(C99)(C99)
computes arc cosine (\({\small\arccos{x} }\)arccos(x))
(function)
(C99)(C99)
computes arc tangent (\({\small\arctan{x} }\)arctan(x))
(function)
(C99)(C99)(C99)
computes the phase angle of a complex number
(function)
C++ documentation for atan2

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