Bearing on a great circle

Average Error: 13.1 → 0.3

Time: 25.6s

Precision: 64

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

Math

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

↓

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

C

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r105203 = lambda1;
        double r105204 = lambda2;
        double r105205 = r105203 - r105204;
        double r105206 = sin(r105205);
        double r105207 = phi2;
        double r105208 = cos(r105207);
        double r105209 = r105206 * r105208;
        double r105210 = phi1;
        double r105211 = cos(r105210);
        double r105212 = sin(r105207);
        double r105213 = r105211 * r105212;
        double r105214 = sin(r105210);
        double r105215 = r105214 * r105208;
        double r105216 = cos(r105205);
        double r105217 = r105215 * r105216;
        double r105218 = r105213 - r105217;
        double r105219 = atan2(r105209, r105218);
        return r105219;
}

↓

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r105220 = lambda1;
        double r105221 = sin(r105220);
        double r105222 = lambda2;
        double r105223 = cos(r105222);
        double r105224 = r105221 * r105223;
        double r105225 = cos(r105220);
        double r105226 = sin(r105222);
        double r105227 = r105225 * r105226;
        double r105228 = r105224 - r105227;
        double r105229 = phi2;
        double r105230 = cos(r105229);
        double r105231 = r105228 * r105230;
        double r105232 = phi1;
        double r105233 = cos(r105232);
        double r105234 = sin(r105229);
        double r105235 = r105233 * r105234;
        double r105236 = r105221 * r105226;
        double r105237 = 3.0;
        double r105238 = pow(r105236, r105237);
        double r105239 = r105225 * r105223;
        double r105240 = pow(r105239, r105237);
        double r105241 = r105238 + r105240;
        double r105242 = cbrt(r105241);
        double r105243 = r105242 * r105242;
        double r105244 = r105236 * r105236;
        double r105245 = r105239 - r105236;
        double r105246 = r105239 * r105245;
        double r105247 = r105244 + r105246;
        double r105248 = cbrt(r105247);
        double r105249 = r105248 * r105248;
        double r105250 = r105243 / r105249;
        double r105251 = sin(r105232);
        double r105252 = r105250 * r105251;
        double r105253 = r105252 * r105230;
        double r105254 = -r105222;
        double r105255 = sin(r105254);
        double r105256 = r105221 * r105255;
        double r105257 = r105239 - r105256;
        double r105258 = cbrt(r105257);
        double r105259 = r105253 * r105258;
        double r105260 = r105235 - r105259;
        double r105261 = atan2(r105231, r105260);
        return r105261;
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

1. Initial program 13.1

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

3. Applied sin-diff 6.9

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

5. Applied sub-neg 6.9

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

6. Applied cos-sum 0.2

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

7. Simplified 0.2

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

9. Applied add-cube-cbrt 0.3

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

10. Applied associate-*r* 0.3

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

11. Simplified 0.3

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

13. Applied flip3-+ 0.3

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

14. Applied cbrt-div 0.3

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

15. Applied flip3-+ 0.3

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

16. Applied cbrt-div 0.3

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

17. Applied frac-times 0.3

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

18. Simplified 0.3

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

19. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019350 
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Bearing on a great circle"
  :precision binary64
  (atan2 (* (sin (- lambda1 lambda2)) (cos phi2)) (- (* (cos phi1) (sin phi2)) (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2))))))