Average Error: 38.9 → 3.6
Time: 8.2s
Precision: 64
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)\]
R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}
↓
R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r98061 = R;
        double r98062 = lambda1;
        double r98063 = lambda2;
        double r98064 = r98062 - r98063;
        double r98065 = phi1;
        double r98066 = phi2;
        double r98067 = r98065 + r98066;
        double r98068 = 2.0;
        double r98069 = r98067 / r98068;
        double r98070 = cos(r98069);
        double r98071 = r98064 * r98070;
        double r98072 = r98071 * r98071;
        double r98073 = r98065 - r98066;
        double r98074 = r98073 * r98073;
        double r98075 = r98072 + r98074;
        double r98076 = sqrt(r98075);
        double r98077 = r98061 * r98076;
        return r98077;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r98078 = R;
        double r98079 = lambda1;
        double r98080 = lambda2;
        double r98081 = r98079 - r98080;
        double r98082 = phi1;
        double r98083 = phi2;
        double r98084 = r98082 + r98083;
        double r98085 = 2.0;
        double r98086 = r98084 / r98085;
        double r98087 = cos(r98086);
        double r98088 = r98081 * r98087;
        double r98089 = r98082 - r98083;
        double r98090 = hypot(r98088, r98089);
        double r98091 = r98078 * r98090;
        return r98091;
}

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 38.9

    \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

  2. Simplified3.6

    \[\leadsto \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right) \cdot R}\]
  3. Using strategy rm

  4. Applied *-commutative3.6

    \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)}\]

  5. Final simplification3.6

    \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)\]

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Equirectangular approximation to distance on a great circle"
  :precision binary64
  (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2))))))