C: How to wrap a float to the interval [-pi, pi)

Question

I'm looking for some nice C code that will accomplish effectively:

while (deltaPhase >= M_PI) deltaPhase -= M_TWOPI;
while (deltaPhase < -M_PI) deltaPhase += M_TWOPI;

What are my options?

Answer 1

Edit Apr 19, 2013:

Modulo function updated to handle boundary cases as noted by aka.nice and arr_sea:

static const double     _PI= 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348;
static const double _TWO_PI= 6.2831853071795864769252867665590057683943387987502116419498891846156328125724179972560696;

// Floating-point modulo
// The result (the remainder) has same sign as the divisor.
// Similar to matlab's mod(); Not similar to fmod() -   Mod(-3,4)= 1   fmod(-3,4)= -3
template<typename T>
T Mod(T x, T y)
{
    static_assert(!std::numeric_limits<T>::is_exact , "Mod: floating-point type expected");

    if (0. == y)
        return x;

    double m= x - y * floor(x/y);

    // handle boundary cases resulted from floating-point cut off:

    if (y > 0)              // modulo range: [0..y)
    {
        if (m>=y)           // Mod(-1e-16             , 360.    ): m= 360.
            return 0;

        if (m<0 )
        {
            if (y+m == y)
                return 0  ; // just in case...
            else
                return y+m; // Mod(106.81415022205296 , _TWO_PI ): m= -1.421e-14 
        }
    }
    else                    // modulo range: (y..0]
    {
        if (m<=y)           // Mod(1e-16              , -360.   ): m= -360.
            return 0;

        if (m>0 )
        {
            if (y+m == y)
                return 0  ; // just in case...
            else
                return y+m; // Mod(-106.81415022205296, -_TWO_PI): m= 1.421e-14 
        }
    }

    return m;
}

// wrap [rad] angle to [-PI..PI)
inline double WrapPosNegPI(double fAng)
{
    return Mod(fAng + _PI, _TWO_PI) - _PI;
}

// wrap [rad] angle to [0..TWO_PI)
inline double WrapTwoPI(double fAng)
{
    return Mod(fAng, _TWO_PI);
}

// wrap [deg] angle to [-180..180)
inline double WrapPosNeg180(double fAng)
{
    return Mod(fAng + 180., 360.) - 180.;
}

// wrap [deg] angle to [0..360)
inline double Wrap360(double fAng)
{
    return Mod(fAng ,360.);
}

Answer 2

If ever your input angle can reach arbitrarily high values, and if continuity matters, you can also try

atan2(sin(x),cos(x))

This will preserve continuity of sin(x) and cos(x) better than modulo for high values of x, especially in single precision (float).

Indeed, exact_value_of_pi - double_precision_approximation ~= 1.22e-16

On the other hand, most library/hardware use a high precision approximation of PI for applying the modulo when evaluating trigonometric functions (though x86 family is known to use a rather poor one).

Result might be in [-pi,pi], you'll have to check the exact bounds.

Personaly, I would prevent any angle to reach several revolutions by wrapping systematically and stick to a fmod solution like the one of boost.

Answer 3

One-liner constant-time solution:

Okay, it's a two-liner if you count the second function for [min,max) form, but close enough — you could merge them together anyways.

/* change to `float/fmodf` or `long double/fmodl` or `int/%` as appropriate */

/* wrap x -> [0,max) */
double wrapMax(double x, double max)
{
    /* integer math: `(max + x % max) % max` */
    return fmod(max + fmod(x, max), max);
}
/* wrap x -> [min,max) */
double wrapMinMax(double x, double min, double max)
{
    return min + wrapMax(x - min, max - min);
}

Then you can simply use deltaPhase = wrapMinMax(deltaPhase, -M_PI, +M_PI).

The solutions is constant-time, meaning that the time it takes does not depend on how far your value is from [-PI,+PI) — for better or for worse.

Verification:

Now, I don't expect you to take my word for it, so here are some examples, including boundary conditions. I'm using integers for clarity, but it works much the same with fmod() and floats:

• Positive x:
  • wrapMax(3, 5) == 3: (5 + 3 % 5) % 5 == (5 + 3) % 5 == 8 % 5 == 3
  • wrapMax(6, 5) == 1: (5 + 6 % 5) % 5 == (5 + 1) % 5 == 6 % 5 == 1
• Negative x:
  • Note: These assume that integer modulo copies left-hand sign; if not, you get the above ("Positive") case.
  • wrapMax(-3, 5) == 2: (5 + (-3) % 5) % 5 == (5 - 3) % 5 == 2 % 5 == 2
  • wrapMax(-6, 5) == 4: (5 + (-6) % 5) % 5 == (5 - 1) % 5 == 4 % 5 == 4
• Boundaries:
  • wrapMax(0, 5) == 0: (5 + 0 % 5) % 5 == (5 + 0) % 5 == 5 % 5 == 0
  • wrapMax(5, 5) == 0: (5 + 5 % 5) % 5 == (5 + 0) % 5== 5 % 5 == 0
  • wrapMax(-5, 5) == 0: (5 + (-5) % 5) % 5 == (5 + 0) % 5 == 5 % 5 == 0
    • Note: Possibly -0 instead of +0 for floating-point.

The wrapMinMax function works much the same: wrapping x to [min,max) is the same as wrapping x - min to [0,max-min), and then (re-)adding min to the result.

I don't know what would happen with a negative max, but feel free to check that yourself!

Answer 4

There is also fmod function in math.h but the sign causes trouble so that a subsequent operation is needed to make the result fir in the proper range (like you already do with the while's). For big values of deltaPhase this is probably faster than substracting/adding `M_TWOPI' hundreds of times.

deltaPhase = fmod(deltaPhase, M_TWOPI);

EDIT: I didn't try it intensively but I think you can use fmod this way by handling positive and negative values differently:

    if (deltaPhase>0)
        deltaPhase = fmod(deltaPhase+M_PI, 2.0*M_PI)-M_PI;
    else
        deltaPhase = fmod(deltaPhase-M_PI, 2.0*M_PI)+M_PI;

The computational time is constant (unlike the while solution which gets slower as the absolute value of deltaPhase increases)

Answer 5

I would do this:

double wrap(double x) {
    return x-2*M_PI*floor(x/(2*M_PI)+0.5);  
}

There will be significant numerical errors. The best solution to the numerical errors is to store your phase scaled by 1/PI or by 1/(2*PI) and depending on what you are doing store them as fixed point.

Answer 6

Here is a version for other people finding this question that can use C++ with Boost:

#include <boost/math/constants/constants.hpp>
#include <boost/math/special_functions/sign.hpp>

template<typename T>
inline T normalizeRadiansPiToMinusPi(T rad)
{
  // copy the sign of the value in radians to the value of pi
  T signedPI = boost::math::copysign(boost::math::constants::pi<T>(),rad);
  // set the value of rad to the appropriate signed value between pi and -pi
  rad = fmod(rad+signedPI,(2*boost::math::constants::pi<T>())) - signedPI;

  return rad;
} 

C++11 version, no Boost dependency:

#include <cmath>

// Bring the 'difference' between two angles into [-pi; pi].
template <typename T>
T normalizeRadiansPiToMinusPi(T rad) {
  // Copy the sign of the value in radians to the value of pi.
  T signed_pi = std::copysign(M_PI,rad);
  // Set the value of difference to the appropriate signed value between pi and -pi.
  rad = std::fmod(rad + signed_pi,(2 * M_PI)) - signed_pi;
  return rad;
}

Answer 7

Instead of working in radians, use angles scaled by 1/(2π) and use modf, floor etc. Convert back to radians to use library functions.

This also has the effect that rotating ten thousand and a half revolutions is the same as rotating half then ten thousand revolutions, which is not guaranteed if your angles are in radians, as you have an exact representation in the floating point value rather than summing approximate representations:

#include <iostream>
#include <cmath>

float wrap_rads ( float r )
{
    while ( r > M_PI ) {
        r -= 2 * M_PI;
    }

    while ( r <= -M_PI ) {
        r += 2 * M_PI;
    }

    return r;
}
float wrap_grads ( float r )
{
    float i;
    r = modff ( r, &i );

    if ( r > 0.5 ) r -= 1;
    if ( r <= -0.5 ) r += 1;

    return r;
}

int main ()
{
    for (int rotations = 1; rotations < 100000; rotations *= 10 ) {
    {
        float pi = ( float ) M_PI;
        float two_pi = 2 * pi;

        float a = pi;
        a += rotations * two_pi;

        std::cout << rotations << " and a half rotations in radians " << a << " => " << wrap_rads ( a ) / two_pi << '\n' ;
    }
    {
        float pi = ( float ) 0.5;
        float two_pi = 2 * pi;

        float a = pi;
        a += rotations * two_pi;

        std::cout << rotations << " and a half rotations in grads " << a << " => " << wrap_grads ( a ) / two_pi << '\n' ;
    }
    std::cout << '\n';
}}

Answer 8

I encountered this question when searching for how to wrap a floating point value (or a double) between two arbitrary numbers. It didn't answer specifically for my case, so I worked out my own solution which can be seen here. This will take a given value and wrap it between lowerBound and upperBound where upperBound perfectly meets lowerBound such that they are equivalent (ie: 360 degrees == 0 degrees so 360 would wrap to 0)

Hopefully this answer is helpful to others stumbling across this question looking for a more generic bounding solution.

double boundBetween(double val, double lowerBound, double upperBound){
   if(lowerBound > upperBound){std::swap(lowerBound, upperBound);}
   val-=lowerBound; //adjust to 0
   double rangeSize = upperBound - lowerBound;
   if(rangeSize == 0){return upperBound;} //avoid dividing by 0
   return val - (rangeSize * std::floor(val/rangeSize)) + lowerBound;
}

A related question for integers is available here: Clean, efficient algorithm for wrapping integers in C++

Answer 9

A two-liner, non-iterative, tested solution for normalizing arbitrary angles to [-π, π):

double normalizeAngle(double angle)
{
    double a = fmod(angle + M_PI, 2 * M_PI);
    return a >= 0 ? (a - M_PI) : (a + M_PI);
}

Similarly, for [0, 2π):

double normalizeAngle(double angle)
{
    double a = fmod(angle, 2 * M_PI);
    return a >= 0 ? a : (a + 2 * M_PI);
}

Answer 10

In the case where fmod() is implemented through truncated division and has the same sign as the dividend, it can be taken advantage of to solve the general problem thusly:

For the case of (-PI, PI]:

if (x > 0) x = x - 2PI * ceil(x/2PI)  #Shift to the negative regime
return fmod(x - PI, 2PI) + PI

And for the case of [-PI, PI):

if (x < 0) x = x - 2PI * floor(x/2PI)  #Shift to the positive regime
return fmod(x + PI, 2PI) - PI

[Note that this is pseudocode; my original was written in Tcl, and I didn't want to torture everyone with that. I needed the first case, so had to figure this out.]

Answer 11

I have used (in python):

def WrapAngle(Wrapped, UnWrapped ):
    TWOPI = math.pi * 2
    TWOPIINV = 1.0 / TWOPI
    return  UnWrapped + round((Wrapped - UnWrapped) * TWOPIINV) * TWOPI

c-code equivalent:

#define TWOPI 6.28318531

double WrapAngle(const double dWrapped, const double dUnWrapped )
{   
    const double TWOPIINV = 1.0/ TWOPI;
    return  dUnWrapped + round((dWrapped - dUnWrapped) * TWOPIINV) * TWOPI;
}

notice that this brings it in the wrapped domain +/- 2pi so for +/- pi domain you need to handle that afterward like:

if( angle > pi):
    angle -= 2*math.pi

Answer 12

The way suggested you suggested is best. It is fastest for small deflections. If angles in your program are constantly being deflected into the proper range, then you should only run into big out of range values rarely. Therefore paying the cost of a complicated modular arithmetic code every round seems wasteful. Comparisons are cheap compared to modular arithmetic (http://embeddedgurus.com/stack-overflow/2011/02/efficient-c-tip-13-use-the-modulus-operator-with-caution/).

Answer 13

In C99:

float unwindRadians( float radians )
{
   const bool radiansNeedUnwinding = radians < -M_PI || M_PI <= radians;

   if ( radiansNeedUnwinding )
   {
      if ( signbit( radians ) )
      {
         radians = -fmodf( -radians + M_PI, 2.f * M_PI ) + M_PI;
      }
      else
      {
         radians = fmodf( radians + M_PI, 2.f * M_PI ) - M_PI;
      }
   }

   return radians;
}

Answer 14

If linking against glibc's libm (including newlib's implementation) you can access __ieee754_rem_pio2f() and __ieee754_rem_pio2() private functions:

extern __int32_t __ieee754_rem_pio2f (float,float*);

float wrapToPI(float xf){
const float p[4]={0,M_PI_2,M_PI,-M_PI_2};

    float yf[2];
    int q;
    int qmod4;

    q=__ieee754_rem_pio2f(xf,yf);

/* xf = q * M_PI_2 + yf[0] + yf[1]                 /
 * yf[1] << y[0], not sure if it could be ignored */

    qmod4= q % 4;

    if (qmod4==2) 
      /* (yf[0] > 0) defines interval (-pi,pi]*/
      return ( (yf[0] > 0) ?  -p[2] : p[2] ) + yf[0] + yf[1];
    else
      return p[qmod4] + yf[0] + yf[1];
}

Edit: Just realised that you need to link to libm.a, I couldn't find the symbols declared in libm.so

Answer 15

deltaPhase -= floor(deltaPhase/M_TWOPI)*M_TWOPI;