Average Error: 0 → 0
Time: 8.3m
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{(\left(\cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right)\right) + \left(\cos \phi_1\right))_*}\]
\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}
↓
\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{(\left(\cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right)\right) + \left(\cos \phi_1\right))_*}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r2568405 = lambda1;
        double r2568406 = phi2;
        double r2568407 = cos(r2568406);
        double r2568408 = lambda2;
        double r2568409 = r2568405 - r2568408;
        double r2568410 = sin(r2568409);
        double r2568411 = r2568407 * r2568410;
        double r2568412 = phi1;
        double r2568413 = cos(r2568412);
        double r2568414 = cos(r2568409);
        double r2568415 = r2568407 * r2568414;
        double r2568416 = r2568413 + r2568415;
        double r2568417 = atan2(r2568411, r2568416);
        double r2568418 = r2568405 + r2568417;
        return r2568418;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r2568419 = lambda1;
        double r2568420 = lambda2;
        double r2568421 = r2568419 - r2568420;
        double r2568422 = sin(r2568421);
        double r2568423 = phi2;
        double r2568424 = cos(r2568423);
        double r2568425 = r2568422 * r2568424;
        double r2568426 = cos(r2568421);
        double r2568427 = phi1;
        double r2568428 = cos(r2568427);
        double r2568429 = fma(r2568424, r2568426, r2568428);
        double r2568430 = atan2(r2568425, r2568429);
        double r2568431 = r2568419 + r2568430;
        return r2568431;
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 0

    \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]

  2. Simplified0

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

  3. Final simplification0

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{(\left(\cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right)\right) + \left(\cos \phi_1\right))_*}\]

Reproduce

herbie shell --seed 2019119 +o rules:numerics
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Midpoint on a great circle"
  (+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))