Midpoint on a great circle

Average Error: 0.8 → 0.3

Time: 10.5s

Precision: 64

Math

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

↓

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

C

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r63601 = lambda1;
        double r63602 = phi2;
        double r63603 = cos(r63602);
        double r63604 = lambda2;
        double r63605 = r63601 - r63604;
        double r63606 = sin(r63605);
        double r63607 = r63603 * r63606;
        double r63608 = phi1;
        double r63609 = cos(r63608);
        double r63610 = cos(r63605);
        double r63611 = r63603 * r63610;
        double r63612 = r63609 + r63611;
        double r63613 = atan2(r63607, r63612);
        double r63614 = r63601 + r63613;
        return r63614;
}

↓

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r63615 = lambda1;
        double r63616 = phi2;
        double r63617 = cos(r63616);
        double r63618 = cbrt(r63617);
        double r63619 = r63618 * r63618;
        double r63620 = sin(r63615);
        double r63621 = lambda2;
        double r63622 = cos(r63621);
        double r63623 = r63620 * r63622;
        double r63624 = cos(r63615);
        double r63625 = sin(r63621);
        double r63626 = r63624 * r63625;
        double r63627 = r63623 - r63626;
        double r63628 = r63618 * r63627;
        double r63629 = r63619 * r63628;
        double r63630 = r63622 * r63617;
        double r63631 = phi1;
        double r63632 = cos(r63631);
        double r63633 = fma(r63624, r63630, r63632);
        double r63634 = r63620 * r63625;
        double r63635 = r63617 * r63634;
        double r63636 = r63633 + r63635;
        double r63637 = atan2(r63629, r63636);
        double r63638 = r63615 + r63637;
        return r63638;
}

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 0.7

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

4. Applied distribute-lft-in 0.7

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

5. Applied associate-+r+ 0.7

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

6. Simplified 0.7

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

8. Applied sin-diff 0.2

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

10. Applied add-cube-cbrt 0.3

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

11. Applied associate-*l* 0.3

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

12. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020033 +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)))))))