Midpoint on a great circle

Average Error: 0.9 → 0.3

Time: 38.8s

Precision: 64

Math

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

↓

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

C

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r1642094 = lambda1;
        double r1642095 = phi2;
        double r1642096 = cos(r1642095);
        double r1642097 = lambda2;
        double r1642098 = r1642094 - r1642097;
        double r1642099 = sin(r1642098);
        double r1642100 = r1642096 * r1642099;
        double r1642101 = phi1;
        double r1642102 = cos(r1642101);
        double r1642103 = cos(r1642098);
        double r1642104 = r1642096 * r1642103;
        double r1642105 = r1642102 + r1642104;
        double r1642106 = atan2(r1642100, r1642105);
        double r1642107 = r1642094 + r1642106;
        return r1642107;
}

↓

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r1642108 = lambda1;
        double r1642109 = sin(r1642108);
        double r1642110 = lambda2;
        double r1642111 = cos(r1642110);
        double r1642112 = r1642109 * r1642111;
        double r1642113 = cos(r1642108);
        double r1642114 = sin(r1642110);
        double r1642115 = r1642113 * r1642114;
        double r1642116 = r1642112 - r1642115;
        double r1642117 = phi2;
        double r1642118 = cos(r1642117);
        double r1642119 = r1642116 * r1642118;
        double r1642120 = r1642114 * r1642109;
        double r1642121 = fma(r1642113, r1642111, r1642120);
        double r1642122 = phi1;
        double r1642123 = cos(r1642122);
        double r1642124 = fma(r1642121, r1642118, r1642123);
        double r1642125 = r1642124 * r1642124;
        double r1642126 = r1642124 * r1642125;
        double r1642127 = cbrt(r1642126);
        double r1642128 = atan2(r1642119, r1642127);
        double r1642129 = r1642128 + r1642108;
        return r1642129;
}

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 0.8

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

4. Applied sin-diff 0.8

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

6. Applied cos-diff 0.2

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

8. Applied add-log-exp 0.3

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

9. Simplified 0.3

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

11. Applied add-cbrt-cube 0.3

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

12. Simplified 0.3

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

13. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019151 +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)))))))