Average Error: 0.9 → 0.3
Time: 24.3s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\log \left(e^{\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_2, \cos \phi_1\right)}\right)}\]
\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{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\log \left(e^{\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_2, \cos \phi_1\right)}\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r32447 = lambda1;
        double r32448 = phi2;
        double r32449 = cos(r32448);
        double r32450 = lambda2;
        double r32451 = r32447 - r32450;
        double r32452 = sin(r32451);
        double r32453 = r32449 * r32452;
        double r32454 = phi1;
        double r32455 = cos(r32454);
        double r32456 = cos(r32451);
        double r32457 = r32449 * r32456;
        double r32458 = r32455 + r32457;
        double r32459 = atan2(r32453, r32458);
        double r32460 = r32447 + r32459;
        return r32460;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r32461 = lambda1;
        double r32462 = phi2;
        double r32463 = cos(r32462);
        double r32464 = sin(r32461);
        double r32465 = lambda2;
        double r32466 = cos(r32465);
        double r32467 = r32464 * r32466;
        double r32468 = cos(r32461);
        double r32469 = sin(r32465);
        double r32470 = r32468 * r32469;
        double r32471 = r32467 - r32470;
        double r32472 = r32463 * r32471;
        double r32473 = r32468 * r32466;
        double r32474 = fma(r32469, r32464, r32473);
        double r32475 = phi1;
        double r32476 = cos(r32475);
        double r32477 = fma(r32474, r32463, r32476);
        double r32478 = exp(r32477);
        double r32479 = log(r32478);
        double r32480 = atan2(r32472, r32479);
        double r32481 = r32461 + r32480;
        return r32481;
}

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

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

  4. Applied sin-diff0.8

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

  6. Applied sub-neg0.8

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

  7. Applied cos-sum0.2

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

  8. Simplified0.2

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

  9. Simplified0.2

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

  11. Applied fma-udef0.2

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

  12. Simplified0.2

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

  14. Applied add-log-exp0.2

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

  15. Applied add-log-exp0.3

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

  16. Applied sum-log0.3

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

  17. Simplified0.3

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

  18. Final simplification0.3

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

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Midpoint on a great circle"
  :precision binary64
  (+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))