Average Error: 0 → 0
Time: 8.2m
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\left(\cos \phi_2\right), \left(\cos \left(\lambda_1 - \lambda_2\right)\right), \left(\cos \phi_1\right)\right)}\]
\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}{\mathsf{fma}\left(\left(\cos \phi_2\right), \left(\cos \left(\lambda_1 - \lambda_2\right)\right), \left(\cos \phi_1\right)\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r3554405 = lambda1;
        double r3554406 = phi2;
        double r3554407 = cos(r3554406);
        double r3554408 = lambda2;
        double r3554409 = r3554405 - r3554408;
        double r3554410 = sin(r3554409);
        double r3554411 = r3554407 * r3554410;
        double r3554412 = phi1;
        double r3554413 = cos(r3554412);
        double r3554414 = cos(r3554409);
        double r3554415 = r3554407 * r3554414;
        double r3554416 = r3554413 + r3554415;
        double r3554417 = atan2(r3554411, r3554416);
        double r3554418 = r3554405 + r3554417;
        return r3554418;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r3554419 = lambda1;
        double r3554420 = lambda2;
        double r3554421 = r3554419 - r3554420;
        double r3554422 = sin(r3554421);
        double r3554423 = phi2;
        double r3554424 = cos(r3554423);
        double r3554425 = r3554422 * r3554424;
        double r3554426 = cos(r3554421);
        double r3554427 = phi1;
        double r3554428 = cos(r3554427);
        double r3554429 = fma(r3554424, r3554426, r3554428);
        double r3554430 = atan2(r3554425, r3554429);
        double r3554431 = r3554419 + r3554430;
        return r3554431;
}

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

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

  3. Final simplification0

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

Reproduce

herbie shell --seed 2019125 +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)))))))