Average Error: 37.2 → 29.4
Time: 46.9s
Precision: 64
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)} \cdot R = -\infty:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)} \cdot R \le 1.870059412379694 \cdot 10^{+298}:\\ \;\;\;\;\sqrt{\left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array}\]
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}\;\sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)} \cdot R = -\infty:\\
\;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\

\mathbf{elif}\;\sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)} \cdot R \le 1.870059412379694 \cdot 10^{+298}:\\
\;\;\;\;\sqrt{\left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \cdot R\\

\mathbf{else}:\\
\;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\

\end{array}
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r3002419 = R;
        double r3002420 = lambda1;
        double r3002421 = lambda2;
        double r3002422 = r3002420 - r3002421;
        double r3002423 = phi1;
        double r3002424 = phi2;
        double r3002425 = r3002423 + r3002424;
        double r3002426 = 2.0;
        double r3002427 = r3002425 / r3002426;
        double r3002428 = cos(r3002427);
        double r3002429 = r3002422 * r3002428;
        double r3002430 = r3002429 * r3002429;
        double r3002431 = r3002423 - r3002424;
        double r3002432 = r3002431 * r3002431;
        double r3002433 = r3002430 + r3002432;
        double r3002434 = sqrt(r3002433);
        double r3002435 = r3002419 * r3002434;
        return r3002435;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r3002436 = phi1;
        double r3002437 = phi2;
        double r3002438 = r3002436 - r3002437;
        double r3002439 = r3002438 * r3002438;
        double r3002440 = lambda1;
        double r3002441 = lambda2;
        double r3002442 = r3002440 - r3002441;
        double r3002443 = r3002437 + r3002436;
        double r3002444 = 2.0;
        double r3002445 = r3002443 / r3002444;
        double r3002446 = cos(r3002445);
        double r3002447 = r3002442 * r3002446;
        double r3002448 = r3002447 * r3002447;
        double r3002449 = r3002439 + r3002448;
        double r3002450 = sqrt(r3002449);
        double r3002451 = R;
        double r3002452 = r3002450 * r3002451;
        double r3002453 = -inf.0;
        bool r3002454 = r3002452 <= r3002453;
        double r3002455 = r3002437 - r3002436;
        double r3002456 = r3002451 * r3002455;
        double r3002457 = 1.870059412379694e+298;
        bool r3002458 = r3002452 <= r3002457;
        double r3002459 = r3002446 * r3002446;
        double r3002460 = r3002442 * r3002442;
        double r3002461 = r3002459 * r3002460;
        double r3002462 = r3002461 + r3002439;
        double r3002463 = sqrt(r3002462);
        double r3002464 = r3002463 * r3002451;
        double r3002465 = r3002458 ? r3002464 : r3002456;
        double r3002466 = r3002454 ? r3002456 : r3002465;
        return r3002466;
}

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))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2))))) < -inf.0 or 1.870059412379694e+298 < (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2)))))

    1. Initial program 60.5

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

    2. Taylor expanded around 0 47.6

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

    if -inf.0 < (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2))))) < 1.870059412379694e+298

    1. Initial program 1.7

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

    3. Applied swap-sqr1.8

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

  4. Final simplification29.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)} \cdot R = -\infty:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)} \cdot R \le 1.870059412379694 \cdot 10^{+298}:\\ \;\;\;\;\sqrt{\left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019133 
(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))))))