Midpoint on a great circle

Average Error: 0.9 → 0.4

Time: 27.8s

Precision: 64

Math

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

↓

\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\sqrt[3]{{\left(\sqrt[3]{{\left(\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_1\right)\right)}^{2}} \cdot \sqrt[3]{\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_1\right)}\right)}^{3}}}\]

C

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r46497 = lambda1;
        double r46498 = phi2;
        double r46499 = cos(r46498);
        double r46500 = lambda2;
        double r46501 = r46497 - r46500;
        double r46502 = sin(r46501);
        double r46503 = r46499 * r46502;
        double r46504 = phi1;
        double r46505 = cos(r46504);
        double r46506 = cos(r46501);
        double r46507 = r46499 * r46506;
        double r46508 = r46505 + r46507;
        double r46509 = atan2(r46503, r46508);
        double r46510 = r46497 + r46509;
        return r46510;
}

↓

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r46511 = lambda1;
        double r46512 = phi2;
        double r46513 = cos(r46512);
        double r46514 = lambda2;
        double r46515 = cos(r46514);
        double r46516 = sin(r46511);
        double r46517 = r46515 * r46516;
        double r46518 = cos(r46511);
        double r46519 = -r46514;
        double r46520 = sin(r46519);
        double r46521 = r46518 * r46520;
        double r46522 = r46517 + r46521;
        double r46523 = r46513 * r46522;
        double r46524 = sin(r46514);
        double r46525 = r46518 * r46515;
        double r46526 = fma(r46516, r46524, r46525);
        double r46527 = phi1;
        double r46528 = cos(r46527);
        double r46529 = fma(r46513, r46526, r46528);
        double r46530 = 2.0;
        double r46531 = pow(r46529, r46530);
        double r46532 = cbrt(r46531);
        double r46533 = cbrt(r46529);
        double r46534 = r46532 * r46533;
        double r46535 = 3.0;
        double r46536 = pow(r46534, r46535);
        double r46537 = cbrt(r46536);
        double r46538 = atan2(r46523, r46537);
        double r46539 = r46511 + r46538;
        return r46539;
}

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. Simplified 0.9

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

4. Applied sub-neg 0.9

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

5. Applied cos-sum 0.8

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

6. Simplified 0.8

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

8. Applied sub-neg 0.8

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

9. Applied sin-sum 0.2

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

10. Simplified 0.2

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

12. Applied add-cbrt-cube 0.3

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

13. Simplified 0.3

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

15. Applied add-cube-cbrt 0.6

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

16. Simplified 0.4

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\sqrt[3]{{\left(\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_1\right)\right)}^{2}}} \cdot \sqrt[3]{\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_1\right)}\right)}^{3}}}\]

17. Final simplification 0.4

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\sqrt[3]{{\left(\sqrt[3]{{\left(\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_1\right)\right)}^{2}} \cdot \sqrt[3]{\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_1\right)}\right)}^{3}}}\]

Reproduce

herbie shell --seed 2019235 +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)))))))