Midpoint on a great circle

Average Error: 0.7 → 0.3

Time: 18.9s

Precision: 64

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

Math

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

↓

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

C

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r51880 = lambda1;
        double r51881 = phi2;
        double r51882 = cos(r51881);
        double r51883 = lambda2;
        double r51884 = r51880 - r51883;
        double r51885 = sin(r51884);
        double r51886 = r51882 * r51885;
        double r51887 = phi1;
        double r51888 = cos(r51887);
        double r51889 = cos(r51884);
        double r51890 = r51882 * r51889;
        double r51891 = r51888 + r51890;
        double r51892 = atan2(r51886, r51891);
        double r51893 = r51880 + r51892;
        return r51893;
}

↓

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r51894 = lambda1;
        double r51895 = phi2;
        double r51896 = cos(r51895);
        double r51897 = sin(r51894);
        double r51898 = lambda2;
        double r51899 = cos(r51898);
        double r51900 = r51897 * r51899;
        double r51901 = cos(r51894);
        double r51902 = -r51898;
        double r51903 = sin(r51902);
        double r51904 = r51901 * r51903;
        double r51905 = r51900 + r51904;
        double r51906 = r51896 * r51905;
        double r51907 = sin(r51898);
        double r51908 = r51901 * r51899;
        double r51909 = fma(r51897, r51907, r51908);
        double r51910 = r51909 * r51896;
        double r51911 = 3.0;
        double r51912 = pow(r51910, r51911);
        double r51913 = phi1;
        double r51914 = cos(r51913);
        double r51915 = pow(r51914, r51911);
        double r51916 = r51912 + r51915;
        double r51917 = r51914 - r51910;
        double r51918 = r51910 * r51917;
        double r51919 = -r51918;
        double r51920 = fma(r51914, r51914, r51919);
        double r51921 = r51916 / r51920;
        double r51922 = atan2(r51906, r51921);
        double r51923 = r51894 + r51922;
        return r51923;
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

1. Initial program 0.7

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

2. Simplified 0.7

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

4. Applied sub-neg 0.7

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

5. Applied sin-sum 0.7

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

6. Simplified 0.7

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

8. Applied cos-diff 0.2

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

10. Applied fma-udef 0.2

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

11. Simplified 0.2

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

13. Applied flip3-+ 0.3

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

14. Simplified 0.3

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

15. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020045 +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)))))))