Midpoint on a great circle

Average Error: 0.8 → 0.3

Time: 31.4s

Precision: 64

Math

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

↓

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

C

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r2004216 = lambda1;
        double r2004217 = phi2;
        double r2004218 = cos(r2004217);
        double r2004219 = lambda2;
        double r2004220 = r2004216 - r2004219;
        double r2004221 = sin(r2004220);
        double r2004222 = r2004218 * r2004221;
        double r2004223 = phi1;
        double r2004224 = cos(r2004223);
        double r2004225 = cos(r2004220);
        double r2004226 = r2004218 * r2004225;
        double r2004227 = r2004224 + r2004226;
        double r2004228 = atan2(r2004222, r2004227);
        double r2004229 = r2004216 + r2004228;
        return r2004229;
}

↓

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r2004230 = phi2;
        double r2004231 = cos(r2004230);
        double r2004232 = lambda1;
        double r2004233 = sin(r2004232);
        double r2004234 = lambda2;
        double r2004235 = cos(r2004234);
        double r2004236 = r2004233 * r2004235;
        double r2004237 = cos(r2004232);
        double r2004238 = sin(r2004234);
        double r2004239 = r2004237 * r2004238;
        double r2004240 = r2004236 - r2004239;
        double r2004241 = r2004231 * r2004240;
        double r2004242 = r2004231 * r2004235;
        double r2004243 = phi1;
        double r2004244 = cos(r2004243);
        double r2004245 = fma(r2004242, r2004237, r2004244);
        double r2004246 = r2004245 * r2004245;
        double r2004247 = r2004238 * r2004233;
        double r2004248 = r2004247 * r2004231;
        double r2004249 = r2004248 * r2004248;
        double r2004250 = r2004246 - r2004249;
        double r2004251 = r2004237 * r2004235;
        double r2004252 = fma(r2004251, r2004231, r2004244);
        double r2004253 = r2004252 - r2004248;
        double r2004254 = r2004250 / r2004253;
        double r2004255 = atan2(r2004241, r2004254);
        double r2004256 = r2004255 + r2004232;
        return r2004256;
}

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. Applied distribute-rgt-in 0.8

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

5. Applied associate-+r+ 0.8

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

6. Simplified 0.8

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

8. Applied sin-diff 0.2

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

10. Applied flip-+ 0.2

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

11. Taylor expanded around inf 0.3

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\frac{\color{blue}{{\left(\cos \phi_1 + \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)}^{2}} - \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2\right) \cdot \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2\right)}{\mathsf{fma}\left(\cos \lambda_2 \cdot \cos \lambda_1, \cos \phi_2, \cos \phi_1\right) - \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}}\]

12. Simplified 0.3

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

13. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019162 +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)))))))