Average Error: 36.8 → 3.8
Time: 32.1s
Precision: 64
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[R \cdot \sqrt{\left(\sqrt[3]{\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right) \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)\right) \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right)^2 + \left(\phi_1 - \phi_2\right)^2}^*\]
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r14566976 = R;
        double r14566977 = lambda1;
        double r14566978 = lambda2;
        double r14566979 = r14566977 - r14566978;
        double r14566980 = phi1;
        double r14566981 = phi2;
        double r14566982 = r14566980 + r14566981;
        double r14566983 = 2.0;
        double r14566984 = r14566982 / r14566983;
        double r14566985 = cos(r14566984);
        double r14566986 = r14566979 * r14566985;
        double r14566987 = r14566986 * r14566986;
        double r14566988 = r14566980 - r14566981;
        double r14566989 = r14566988 * r14566988;
        double r14566990 = r14566987 + r14566989;
        double r14566991 = sqrt(r14566990);
        double r14566992 = r14566976 * r14566991;
        return r14566992;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r14566993 = R;
        double r14566994 = phi1;
        double r14566995 = phi2;
        double r14566996 = r14566994 + r14566995;
        double r14566997 = 0.5;
        double r14566998 = r14566996 * r14566997;
        double r14566999 = cos(r14566998);
        double r14567000 = r14566999 * r14566999;
        double r14567001 = r14567000 * r14566999;
        double r14567002 = cbrt(r14567001);
        double r14567003 = lambda1;
        double r14567004 = lambda2;
        double r14567005 = r14567003 - r14567004;
        double r14567006 = r14567002 * r14567005;
        double r14567007 = r14566994 - r14566995;
        double r14567008 = hypot(r14567006, r14567007);
        double r14567009 = r14566993 * r14567008;
        return r14567009;
}


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 \sqrt{\left(\sqrt[3]{\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right) \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)\right) \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right)^2 + \left(\phi_1 - \phi_2\right)^2}^*

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 36.8

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

  2. Simplified3.8

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

  3. Taylor expanded around -inf 3.8

    \[\leadsto \sqrt{\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \lambda_1 - \lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)}^2 + \left(\phi_1 - \phi_2\right)^2}^* \cdot R\]

  4. Simplified3.8

    \[\leadsto \sqrt{\color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)\right)}^2 + \left(\phi_1 - \phi_2\right)^2}^* \cdot R\]
  5. Using strategy rm

  6. Applied add-cbrt-cube3.8

    \[\leadsto \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sqrt[3]{\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right) \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)\right) \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)}}\right)^2 + \left(\phi_1 - \phi_2\right)^2}^* \cdot R\]

  7. Final simplification3.8

    \[\leadsto R \cdot \sqrt{\left(\sqrt[3]{\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right) \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)\right) \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right)^2 + \left(\phi_1 - \phi_2\right)^2}^*\]

Reproduce

herbie shell --seed 2019101 +o rules:numerics
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Equirectangular approximation to distance on a great circle"
  (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2))))))