Midpoint on a great circle

Average Error: 0.9 → 0.3

Time: 21.9s

Precision: 64

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

Math

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

↓

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

C

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r36217 = lambda1;
        double r36218 = phi2;
        double r36219 = cos(r36218);
        double r36220 = lambda2;
        double r36221 = r36217 - r36220;
        double r36222 = sin(r36221);
        double r36223 = r36219 * r36222;
        double r36224 = phi1;
        double r36225 = cos(r36224);
        double r36226 = cos(r36221);
        double r36227 = r36219 * r36226;
        double r36228 = r36225 + r36227;
        double r36229 = atan2(r36223, r36228);
        double r36230 = r36217 + r36229;
        return r36230;
}

↓

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r36231 = lambda1;
        double r36232 = phi2;
        double r36233 = cos(r36232);
        double r36234 = sin(r36231);
        double r36235 = lambda2;
        double r36236 = cos(r36235);
        double r36237 = r36234 * r36236;
        double r36238 = cos(r36231);
        double r36239 = -r36235;
        double r36240 = sin(r36239);
        double r36241 = r36238 * r36240;
        double r36242 = r36237 + r36241;
        double r36243 = r36233 * r36242;
        double r36244 = r36233 * r36236;
        double r36245 = phi1;
        double r36246 = cos(r36245);
        double r36247 = fma(r36238, r36244, r36246);
        double r36248 = 3.0;
        double r36249 = pow(r36247, r36248);
        double r36250 = pow(r36249, r36248);
        double r36251 = cbrt(r36250);
        double r36252 = cbrt(r36251);
        double r36253 = sin(r36235);
        double r36254 = r36234 * r36253;
        double r36255 = r36233 * r36254;
        double r36256 = r36252 + r36255;
        double r36257 = atan2(r36243, r36256);
        double r36258 = r36231 + r36257;
        return r36258;
}

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. Using strategy rm

3. Applied sub-neg 0.9

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

4. Applied sin-sum 0.9

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

5. Simplified 0.9

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

7. Applied cos-diff 0.2

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

8. Applied distribute-lft-in 0.2

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

9. Applied associate-+r+ 0.2

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

10. Simplified 0.2

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

12. Applied add-cbrt-cube 0.3

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

13. Simplified 0.3

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

15. Applied add-cbrt-cube 0.3

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

16. Simplified 0.3

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

17. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019351 +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)))))))