Midpoint on a great circle

Average Error: 0.9 → 0.3

Time: 11.0s

Precision: 64

Math

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

↓

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

C

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r71723 = lambda1;
        double r71724 = phi2;
        double r71725 = cos(r71724);
        double r71726 = lambda2;
        double r71727 = r71723 - r71726;
        double r71728 = sin(r71727);
        double r71729 = r71725 * r71728;
        double r71730 = phi1;
        double r71731 = cos(r71730);
        double r71732 = cos(r71727);
        double r71733 = r71725 * r71732;
        double r71734 = r71731 + r71733;
        double r71735 = atan2(r71729, r71734);
        double r71736 = r71723 + r71735;
        return r71736;
}

↓

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r71737 = lambda1;
        double r71738 = phi2;
        double r71739 = cos(r71738);
        double r71740 = sin(r71737);
        double r71741 = r71739 * r71740;
        double r71742 = lambda2;
        double r71743 = cos(r71742);
        double r71744 = r71741 * r71743;
        double r71745 = cos(r71737);
        double r71746 = -r71742;
        double r71747 = sin(r71746);
        double r71748 = r71745 * r71747;
        double r71749 = r71739 * r71748;
        double r71750 = r71744 + r71749;
        double r71751 = sin(r71742);
        double r71752 = r71740 * r71751;
        double r71753 = fma(r71745, r71743, r71752);
        double r71754 = phi1;
        double r71755 = cos(r71754);
        double r71756 = fma(r71739, r71753, r71755);
        double r71757 = exp(r71756);
        double r71758 = log(r71757);
        double r71759 = atan2(r71750, r71758);
        double r71760 = r71737 + r71759;
        return r71760;
}

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 cos-diff 0.9

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

5. Applied sub-neg 0.9

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

6. Applied sin-sum 0.2

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

7. Applied distribute-lft-in 0.2

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

8. Simplified 0.2

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

10. Applied add-log-exp 0.3

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

11. Applied add-log-exp 0.3

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

12. Applied sum-log 0.3

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

13. Simplified 0.3

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

14. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019346 +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)))))))