Average Error: 0.8 → 0.4
Time: 31.6s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\log \left(\sqrt{e^{\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_1\right)}}\right) + \log \left(\sqrt{e^{\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_1\right)}}\right)}\]
\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{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\log \left(\sqrt{e^{\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_1\right)}}\right) + \log \left(\sqrt{e^{\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_1\right)}}\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r1413710 = lambda1;
        double r1413711 = phi2;
        double r1413712 = cos(r1413711);
        double r1413713 = lambda2;
        double r1413714 = r1413710 - r1413713;
        double r1413715 = sin(r1413714);
        double r1413716 = r1413712 * r1413715;
        double r1413717 = phi1;
        double r1413718 = cos(r1413717);
        double r1413719 = cos(r1413714);
        double r1413720 = r1413712 * r1413719;
        double r1413721 = r1413718 + r1413720;
        double r1413722 = atan2(r1413716, r1413721);
        double r1413723 = r1413710 + r1413722;
        return r1413723;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r1413724 = lambda1;
        double r1413725 = sin(r1413724);
        double r1413726 = lambda2;
        double r1413727 = cos(r1413726);
        double r1413728 = r1413725 * r1413727;
        double r1413729 = cos(r1413724);
        double r1413730 = sin(r1413726);
        double r1413731 = r1413729 * r1413730;
        double r1413732 = r1413728 - r1413731;
        double r1413733 = phi2;
        double r1413734 = cos(r1413733);
        double r1413735 = r1413732 * r1413734;
        double r1413736 = r1413729 * r1413727;
        double r1413737 = fma(r1413725, r1413730, r1413736);
        double r1413738 = phi1;
        double r1413739 = cos(r1413738);
        double r1413740 = fma(r1413734, r1413737, r1413739);
        double r1413741 = exp(r1413740);
        double r1413742 = sqrt(r1413741);
        double r1413743 = log(r1413742);
        double r1413744 = r1413743 + r1413743;
        double r1413745 = atan2(r1413735, r1413744);
        double r1413746 = r1413724 + r1413745;
        return r1413746;
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 0.8

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

  2. Simplified0.8

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

  4. Applied sin-diff0.8

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

  6. Applied cos-diff0.2

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

  8. Applied add-log-exp0.3

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

  9. Simplified0.3

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

  11. Applied add-sqr-sqrt0.4

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

  12. Applied log-prod0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2019165 +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)))))))