Midpoint on a great circle

Average Error: 0.9 → 0.3

Time: 29.5s

Precision: 64

Math

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

↓

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

C

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r969972 = lambda1;
        double r969973 = phi2;
        double r969974 = cos(r969973);
        double r969975 = lambda2;
        double r969976 = r969972 - r969975;
        double r969977 = sin(r969976);
        double r969978 = r969974 * r969977;
        double r969979 = phi1;
        double r969980 = cos(r969979);
        double r969981 = cos(r969976);
        double r969982 = r969974 * r969981;
        double r969983 = r969980 + r969982;
        double r969984 = atan2(r969978, r969983);
        double r969985 = r969972 + r969984;
        return r969985;
}

↓

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r969986 = lambda1;
        double r969987 = sin(r969986);
        double r969988 = lambda2;
        double r969989 = cos(r969988);
        double r969990 = r969987 * r969989;
        double r969991 = cos(r969986);
        double r969992 = sin(r969988);
        double r969993 = r969991 * r969992;
        double r969994 = r969990 - r969993;
        double r969995 = phi2;
        double r969996 = cos(r969995);
        double r969997 = r969994 * r969996;
        double r969998 = r969991 * r969989;
        double r969999 = fma(r969992, r969987, r969998);
        double r970000 = phi1;
        double r970001 = cos(r970000);
        double r970002 = fma(r969999, r969996, r970001);
        double r970003 = r970002 * r970002;
        double r970004 = r970002 * r970003;
        double r970005 = cbrt(r970004);
        double r970006 = atan2(r969997, r970005);
        double r970007 = r970006 + r969986;
        return r970007;
}

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. Simplified 0.9

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

4. Applied cos-diff 0.9

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

6. Applied sin-diff 0.2

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

8. Applied add-cbrt-cube 0.3

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

9. Simplified 0.3

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

10. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(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)))))))