Midpoint on a great circle

Average Error: 0.9 → 0.3

Time: 29.9s

Precision: 64

Math

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

↓

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

C

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r2096671 = lambda1;
        double r2096672 = phi2;
        double r2096673 = cos(r2096672);
        double r2096674 = lambda2;
        double r2096675 = r2096671 - r2096674;
        double r2096676 = sin(r2096675);
        double r2096677 = r2096673 * r2096676;
        double r2096678 = phi1;
        double r2096679 = cos(r2096678);
        double r2096680 = cos(r2096675);
        double r2096681 = r2096673 * r2096680;
        double r2096682 = r2096679 + r2096681;
        double r2096683 = atan2(r2096677, r2096682);
        double r2096684 = r2096671 + r2096683;
        return r2096684;
}

↓

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r2096685 = lambda1;
        double r2096686 = phi2;
        double r2096687 = cos(r2096686);
        double r2096688 = sin(r2096685);
        double r2096689 = lambda2;
        double r2096690 = cos(r2096689);
        double r2096691 = r2096688 * r2096690;
        double r2096692 = cos(r2096685);
        double r2096693 = sin(r2096689);
        double r2096694 = r2096692 * r2096693;
        double r2096695 = r2096691 - r2096694;
        double r2096696 = r2096687 * r2096695;
        double r2096697 = r2096693 * r2096688;
        double r2096698 = r2096687 * r2096697;
        double r2096699 = phi1;
        double r2096700 = cos(r2096699);
        double r2096701 = r2096692 * r2096687;
        double r2096702 = r2096701 * r2096690;
        double r2096703 = r2096700 + r2096702;
        double r2096704 = r2096703 * r2096703;
        double r2096705 = r2096700 * r2096700;
        double r2096706 = r2096705 * r2096700;
        double r2096707 = r2096702 * r2096702;
        double r2096708 = r2096707 * r2096702;
        double r2096709 = r2096706 + r2096708;
        double r2096710 = r2096704 * r2096709;
        double r2096711 = cbrt(r2096710);
        double r2096712 = r2096702 - r2096700;
        double r2096713 = r2096702 * r2096712;
        double r2096714 = r2096705 + r2096713;
        double r2096715 = cbrt(r2096714);
        double r2096716 = r2096711 / r2096715;
        double r2096717 = r2096698 + r2096716;
        double r2096718 = atan2(r2096696, r2096717);
        double r2096719 = r2096685 + r2096718;
        return r2096719;
}

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. Using strategy rm

3. Applied sin-diff 0.8

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

5. Applied cos-diff 0.2

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

6. Applied distribute-rgt-in 0.2

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

7. Applied associate-+r+ 0.2

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

9. Applied add-cbrt-cube 0.3

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

11. Applied flip3-+ 0.3

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

12. Applied associate-*r/ 0.3

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

13. Applied cbrt-div 0.3

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

14. Simplified 0.3

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

15. Simplified 0.3

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

16. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019168 
(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)))))))