Average Error: 38.1 → 28.2
Time: 10.2s
Precision: 64
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} = -\infty \lor \neg \left(R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \le 2.6174821950400801 \cdot 10^{286}\right):\\ \;\;\;\;R \cdot \phi_2 + R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \end{array}\]
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)}
↓
\begin{array}{l}
\mathbf{if}\;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)} = -\infty \lor \neg \left(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)} \le 2.6174821950400801 \cdot 10^{286}\right):\\
\;\;\;\;R \cdot \phi_2 + R \cdot \left(-\phi_1\right)\\

\mathbf{else}:\\
\;\;\;\;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)}\\

\end{array}
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r86976 = R;
        double r86977 = lambda1;
        double r86978 = lambda2;
        double r86979 = r86977 - r86978;
        double r86980 = phi1;
        double r86981 = phi2;
        double r86982 = r86980 + r86981;
        double r86983 = 2.0;
        double r86984 = r86982 / r86983;
        double r86985 = cos(r86984);
        double r86986 = r86979 * r86985;
        double r86987 = r86986 * r86986;
        double r86988 = r86980 - r86981;
        double r86989 = r86988 * r86988;
        double r86990 = r86987 + r86989;
        double r86991 = sqrt(r86990);
        double r86992 = r86976 * r86991;
        return r86992;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r86993 = R;
        double r86994 = lambda1;
        double r86995 = lambda2;
        double r86996 = r86994 - r86995;
        double r86997 = phi1;
        double r86998 = phi2;
        double r86999 = r86997 + r86998;
        double r87000 = 2.0;
        double r87001 = r86999 / r87000;
        double r87002 = cos(r87001);
        double r87003 = r86996 * r87002;
        double r87004 = r87003 * r87003;
        double r87005 = r86997 - r86998;
        double r87006 = r87005 * r87005;
        double r87007 = r87004 + r87006;
        double r87008 = sqrt(r87007);
        double r87009 = r86993 * r87008;
        double r87010 = -inf.0;
        bool r87011 = r87009 <= r87010;
        double r87012 = 2.61748219504008e+286;
        bool r87013 = r87009 <= r87012;
        double r87014 = !r87013;
        bool r87015 = r87011 || r87014;
        double r87016 = r86993 * r86998;
        double r87017 = -r86997;
        double r87018 = r86993 * r87017;
        double r87019 = r87016 + r87018;
        double r87020 = r87015 ? r87019 : r87009;
        return r87020;
}

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. Split input into 2 regimes

  2. if (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))) (* (- phi1 phi2) (- phi1 phi2))))) < -inf.0 or 2.61748219504008e+286 < (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))) (* (- phi1 phi2) (- phi1 phi2)))))

    1. Initial program 63.4

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

    2. Taylor expanded around 0 46.7

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 - \phi_1\right)}\]
    3. Using strategy rm

    4. Applied sub-neg46.7

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 + \left(-\phi_1\right)\right)}\]

    5. Applied distribute-lft-in46.7

      \[\leadsto \color{blue}{R \cdot \phi_2 + R \cdot \left(-\phi_1\right)}\]

    if -inf.0 < (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))) (* (- phi1 phi2) (- phi1 phi2))))) < 2.61748219504008e+286

    1. Initial program 1.8

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification28.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} = -\infty \lor \neg \left(R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \le 2.6174821950400801 \cdot 10^{286}\right):\\ \;\;\;\;R \cdot \phi_2 + R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(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))))))