Average Error: 0.8 → 0.4
Time: 29.4s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\mathsf{fma}\left(\sqrt[3]{\cos \phi_1} \cdot \sqrt[3]{\cos \phi_1}, \sqrt[3]{\sqrt[3]{\cos \phi_1}} \cdot \left(\sqrt[3]{\sqrt[3]{\cos \phi_1}} \cdot \sqrt[3]{\sqrt[3]{\cos \phi_1}}\right), \cos \phi_2 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)} + \lambda_1\]
\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)}
↓
\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(\sqrt[3]{\cos \phi_1} \cdot \sqrt[3]{\cos \phi_1}, \sqrt[3]{\sqrt[3]{\cos \phi_1}} \cdot \left(\sqrt[3]{\sqrt[3]{\cos \phi_1}} \cdot \sqrt[3]{\sqrt[3]{\cos \phi_1}}\right), \cos \phi_2 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)} + \lambda_1
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r1909709 = lambda1;
        double r1909710 = phi2;
        double r1909711 = cos(r1909710);
        double r1909712 = lambda2;
        double r1909713 = r1909709 - r1909712;
        double r1909714 = sin(r1909713);
        double r1909715 = r1909711 * r1909714;
        double r1909716 = phi1;
        double r1909717 = cos(r1909716);
        double r1909718 = cos(r1909713);
        double r1909719 = r1909711 * r1909718;
        double r1909720 = r1909717 + r1909719;
        double r1909721 = atan2(r1909715, r1909720);
        double r1909722 = r1909709 + r1909721;
        return r1909722;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r1909723 = phi2;
        double r1909724 = cos(r1909723);
        double r1909725 = lambda1;
        double r1909726 = sin(r1909725);
        double r1909727 = lambda2;
        double r1909728 = cos(r1909727);
        double r1909729 = r1909726 * r1909728;
        double r1909730 = cos(r1909725);
        double r1909731 = sin(r1909727);
        double r1909732 = r1909730 * r1909731;
        double r1909733 = r1909729 - r1909732;
        double r1909734 = r1909724 * r1909733;
        double r1909735 = phi1;
        double r1909736 = cos(r1909735);
        double r1909737 = cbrt(r1909736);
        double r1909738 = r1909737 * r1909737;
        double r1909739 = cbrt(r1909737);
        double r1909740 = r1909739 * r1909739;
        double r1909741 = r1909739 * r1909740;
        double r1909742 = r1909731 * r1909726;
        double r1909743 = r1909730 * r1909728;
        double r1909744 = r1909742 + r1909743;
        double r1909745 = r1909724 * r1909744;
        double r1909746 = fma(r1909738, r1909741, r1909745);
        double r1909747 = atan2(r1909734, r1909746);
        double r1909748 = r1909747 + r1909725;
        return r1909748;
}

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. Using strategy rm

  3. Applied sin-diff0.8

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

  5. Applied cos-diff0.2

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

  7. Applied add-cube-cbrt0.3

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\color{blue}{\left(\sqrt[3]{\cos \phi_1} \cdot \sqrt[3]{\cos \phi_1}\right) \cdot \sqrt[3]{\cos \phi_1}} + \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\]

  8. Applied fma-def0.3

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

  10. Applied add-cube-cbrt0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019164 +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)))))))