Midpoint on a great circle

Average Error: 0.8 → 0.3

Time: 36.6s

Precision: 64

Math

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

↓

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

C

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r48909 = lambda1;
        double r48910 = phi2;
        double r48911 = cos(r48910);
        double r48912 = lambda2;
        double r48913 = r48909 - r48912;
        double r48914 = sin(r48913);
        double r48915 = r48911 * r48914;
        double r48916 = phi1;
        double r48917 = cos(r48916);
        double r48918 = cos(r48913);
        double r48919 = r48911 * r48918;
        double r48920 = r48917 + r48919;
        double r48921 = atan2(r48915, r48920);
        double r48922 = r48909 + r48921;
        return r48922;
}

↓

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r48923 = lambda1;
        double r48924 = phi2;
        double r48925 = cos(r48924);
        double r48926 = sin(r48923);
        double r48927 = lambda2;
        double r48928 = cos(r48927);
        double r48929 = r48926 * r48928;
        double r48930 = cos(r48923);
        double r48931 = sin(r48927);
        double r48932 = r48930 * r48931;
        double r48933 = r48929 - r48932;
        double r48934 = r48925 * r48933;
        double r48935 = r48926 * r48931;
        double r48936 = fma(r48930, r48928, r48935);
        double r48937 = 3.0;
        double r48938 = pow(r48936, r48937);
        double r48939 = pow(r48925, r48937);
        double r48940 = r48938 * r48939;
        double r48941 = phi1;
        double r48942 = cos(r48941);
        double r48943 = pow(r48942, r48937);
        double r48944 = r48940 + r48943;
        double r48945 = r48936 * r48925;
        double r48946 = r48945 - r48942;
        double r48947 = r48946 * r48945;
        double r48948 = fma(r48942, r48942, r48947);
        double r48949 = r48944 / r48948;
        double r48950 = atan2(r48934, r48949);
        double r48951 = r48923 + r48950;
        return r48951;
}

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. Using strategy rm

5. Applied sin-diff 0.2

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

7. Applied flip3-+ 0.3

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

8. Simplified 0.3

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

9. Simplified 0.3

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

11. Applied unpow-prod-down 0.3

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

12. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019323 +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)))))))