Average Error: 0.9 → 0.2
Time: 26.8s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\left(\left(-\cos \lambda_1 \cdot \sin \lambda_2\right) + \cos \lambda_2 \cdot \sin \lambda_1\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\cos \phi_2, \cos \lambda_2 \cdot \cos \lambda_1 - \log \left({\left(e^{-\sin \lambda_1}\right)}^{\left(\sin \lambda_2\right)}\right), \cos \phi_1\right)}\]
\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{\left(\left(-\cos \lambda_1 \cdot \sin \lambda_2\right) + \cos \lambda_2 \cdot \sin \lambda_1\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\cos \phi_2, \cos \lambda_2 \cdot \cos \lambda_1 - \log \left({\left(e^{-\sin \lambda_1}\right)}^{\left(\sin \lambda_2\right)}\right), \cos \phi_1\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r39610 = lambda1;
        double r39611 = phi2;
        double r39612 = cos(r39611);
        double r39613 = lambda2;
        double r39614 = r39610 - r39613;
        double r39615 = sin(r39614);
        double r39616 = r39612 * r39615;
        double r39617 = phi1;
        double r39618 = cos(r39617);
        double r39619 = cos(r39614);
        double r39620 = r39612 * r39619;
        double r39621 = r39618 + r39620;
        double r39622 = atan2(r39616, r39621);
        double r39623 = r39610 + r39622;
        return r39623;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r39624 = lambda1;
        double r39625 = cos(r39624);
        double r39626 = lambda2;
        double r39627 = sin(r39626);
        double r39628 = r39625 * r39627;
        double r39629 = -r39628;
        double r39630 = cos(r39626);
        double r39631 = sin(r39624);
        double r39632 = r39630 * r39631;
        double r39633 = r39629 + r39632;
        double r39634 = phi2;
        double r39635 = cos(r39634);
        double r39636 = r39633 * r39635;
        double r39637 = r39630 * r39625;
        double r39638 = -r39631;
        double r39639 = exp(r39638);
        double r39640 = pow(r39639, r39627);
        double r39641 = log(r39640);
        double r39642 = r39637 - r39641;
        double r39643 = phi1;
        double r39644 = cos(r39643);
        double r39645 = fma(r39635, r39642, r39644);
        double r39646 = atan2(r39636, r39645);
        double r39647 = r39624 + r39646;
        return r39647;
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 0.9

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

  2. Simplified0.9

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

  4. Applied sub-neg0.9

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

  5. Applied cos-sum0.9

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

  6. Simplified0.9

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

  7. Simplified0.9

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

  9. Applied sub-neg0.9

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

  10. Applied sin-sum0.2

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

  11. Simplified0.2

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

  12. Simplified0.2

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

  14. Applied add-log-exp0.2

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

  15. Simplified0.2

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

  16. Final simplification0.2

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

Reproduce

herbie shell --seed 2019194 +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)))))))