Spherical law of cosines

Average Error: 17.3 → 4.1

Time: 22.6s

Precision: 64

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

Math

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

↓

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

C

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r25301 = phi1;
        double r25302 = sin(r25301);
        double r25303 = phi2;
        double r25304 = sin(r25303);
        double r25305 = r25302 * r25304;
        double r25306 = cos(r25301);
        double r25307 = cos(r25303);
        double r25308 = r25306 * r25307;
        double r25309 = lambda1;
        double r25310 = lambda2;
        double r25311 = r25309 - r25310;
        double r25312 = cos(r25311);
        double r25313 = r25308 * r25312;
        double r25314 = r25305 + r25313;
        double r25315 = acos(r25314);
        double r25316 = R;
        double r25317 = r25315 * r25316;
        return r25317;
}

↓

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r25318 = phi1;
        double r25319 = sin(r25318);
        double r25320 = phi2;
        double r25321 = sin(r25320);
        double r25322 = r25319 * r25321;
        double r25323 = 3.0;
        double r25324 = pow(r25322, r25323);
        double r25325 = cbrt(r25324);
        double r25326 = cos(r25318);
        double r25327 = cos(r25320);
        double r25328 = r25326 * r25327;
        double r25329 = lambda1;
        double r25330 = cos(r25329);
        double r25331 = lambda2;
        double r25332 = cos(r25331);
        double r25333 = r25330 * r25332;
        double r25334 = sin(r25329);
        double r25335 = -r25331;
        double r25336 = sin(r25335);
        double r25337 = r25334 * r25336;
        double r25338 = pow(r25337, r25323);
        double r25339 = cbrt(r25338);
        double r25340 = r25333 - r25339;
        double r25341 = r25328 * r25340;
        double r25342 = r25325 + r25341;
        double r25343 = acos(r25342);
        double r25344 = R;
        double r25345 = r25343 * r25344;
        return r25345;
}

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

1. Initial program 17.3

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

3. Applied sub-neg 17.3

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

4. Applied cos-sum 4.1

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

5. Simplified 4.1

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

7. Applied add-cbrt-cube 4.1

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

8. Applied add-cbrt-cube 4.1

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

9. Applied cbrt-unprod 4.1

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

10. Simplified 4.1

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

12. Applied add-cbrt-cube 4.1

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

13. Applied add-cbrt-cube 4.1

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

14. Applied cbrt-unprod 4.1

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

15. Simplified 4.1

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

16. Final simplification 4.1

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

Reproduce

herbie shell --seed 2020047 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Spherical law of cosines"
  :precision binary64
  (* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R))