Average Error: 39.3 → 3.8
Time: 12.8s
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 r101105 = R;
        double r101106 = lambda1;
        double r101107 = lambda2;
        double r101108 = r101106 - r101107;
        double r101109 = phi1;
        double r101110 = phi2;
        double r101111 = r101109 + r101110;
        double r101112 = 2.0;
        double r101113 = r101111 / r101112;
        double r101114 = cos(r101113);
        double r101115 = r101108 * r101114;
        double r101116 = r101115 * r101115;
        double r101117 = r101109 - r101110;
        double r101118 = r101117 * r101117;
        double r101119 = r101116 + r101118;
        double r101120 = sqrt(r101119);
        double r101121 = r101105 * r101120;
        return r101121;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r101122 = R;
        double r101123 = lambda1;
        double r101124 = lambda2;
        double r101125 = r101123 - r101124;
        double r101126 = phi1;
        double r101127 = phi2;
        double r101128 = r101126 + r101127;
        double r101129 = 2.0;
        double r101130 = r101128 / r101129;
        double r101131 = cos(r101130);
        double r101132 = r101125 * r101131;
        double r101133 = r101126 - r101127;
        double r101134 = hypot(r101132, r101133);
        double r101135 = r101122 * r101134;
        return r101135;
}

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 39.3

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

  2. Simplified3.8

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

  4. Applied *-commutative3.8

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

  5. Final simplification3.8

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

Reproduce

herbie shell --seed 2020046 +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))))))