Midpoint on a great circle

Average Error: 0.8 → 0.3

Time: 16.6s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r42587 = lambda1;
        double r42588 = phi2;
        double r42589 = cos(r42588);
        double r42590 = lambda2;
        double r42591 = r42587 - r42590;
        double r42592 = sin(r42591);
        double r42593 = r42589 * r42592;
        double r42594 = phi1;
        double r42595 = cos(r42594);
        double r42596 = cos(r42591);
        double r42597 = r42589 * r42596;
        double r42598 = r42595 + r42597;
        double r42599 = atan2(r42593, r42598);
        double r42600 = r42587 + r42599;
        return r42600;
}

↓

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r42601 = lambda1;
        double r42602 = phi2;
        double r42603 = cos(r42602);
        double r42604 = sin(r42601);
        double r42605 = lambda2;
        double r42606 = cos(r42605);
        double r42607 = r42604 * r42606;
        double r42608 = cos(r42601);
        double r42609 = -r42605;
        double r42610 = sin(r42609);
        double r42611 = r42608 * r42610;
        double r42612 = r42607 + r42611;
        double r42613 = r42603 * r42612;
        double r42614 = r42603 * r42606;
        double r42615 = r42608 * r42614;
        double r42616 = phi1;
        double r42617 = cos(r42616);
        double r42618 = r42615 - r42617;
        double r42619 = r42615 * r42618;
        double r42620 = 2.0;
        double r42621 = pow(r42617, r42620);
        double r42622 = r42619 + r42621;
        double r42623 = sqrt(r42622);
        double r42624 = r42622 / r42623;
        double r42625 = r42615 + r42617;
        double r42626 = r42625 / r42623;
        double r42627 = r42624 * r42626;
        double r42628 = sin(r42605);
        double r42629 = r42604 * r42628;
        double r42630 = r42629 * r42603;
        double r42631 = r42627 + r42630;
        double r42632 = atan2(r42613, r42631);
        double r42633 = r42601 + r42632;
        return r42633;
}

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 cos-diff 0.8

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

4. Applied distribute-rgt-in 0.8

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

5. Applied associate-+r+ 0.8

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

7. Applied sub-neg 0.8

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

8. Applied sin-sum 0.2

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

9. Simplified 0.2

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

11. Applied flip3-+ 0.2

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

12. Simplified 0.2

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

14. Applied add-sqr-sqrt 0.3

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

15. Applied sum-cubes 0.3

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

16. Applied times-frac 0.3

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

17. Simplified 0.3

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

18. Simplified 0.3

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

19. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020042 
(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)))))))