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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)
Defined in header <tgmath.h> 
#define atan2( y, x )
 (4) (since C99)
1-3) Computes the arc tangent of y/x using the signs of arguments to determine the correct quadrant.
4) Type-generic macro: If any argument has type long double, atan2l is called. Otherwise, if any argument has integer type or has type double, atan2 is called. Otherwise, atan2f is called.

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 <stdio.h>
#include <math.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

  • 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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