Average Error: 13.0 → 0.4
Time: 31.4s
Precision: 64
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\left(\left(\sqrt[3]{\cos \phi_2} \cdot \sqrt[3]{\cos \phi_2}\right) \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot \sqrt[3]{\cos \phi_2}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)}\]
\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(\left(\sqrt[3]{\cos \phi_2} \cdot \sqrt[3]{\cos \phi_2}\right) \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot \sqrt[3]{\cos \phi_2}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r78311 = lambda1;
        double r78312 = lambda2;
        double r78313 = r78311 - r78312;
        double r78314 = sin(r78313);
        double r78315 = phi2;
        double r78316 = cos(r78315);
        double r78317 = r78314 * r78316;
        double r78318 = phi1;
        double r78319 = cos(r78318);
        double r78320 = sin(r78315);
        double r78321 = r78319 * r78320;
        double r78322 = sin(r78318);
        double r78323 = r78322 * r78316;
        double r78324 = cos(r78313);
        double r78325 = r78323 * r78324;
        double r78326 = r78321 - r78325;
        double r78327 = atan2(r78317, r78326);
        return r78327;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r78328 = phi2;
        double r78329 = cos(r78328);
        double r78330 = cbrt(r78329);
        double r78331 = r78330 * r78330;
        double r78332 = lambda1;
        double r78333 = sin(r78332);
        double r78334 = lambda2;
        double r78335 = cos(r78334);
        double r78336 = r78333 * r78335;
        double r78337 = cos(r78332);
        double r78338 = sin(r78334);
        double r78339 = r78337 * r78338;
        double r78340 = r78336 - r78339;
        double r78341 = r78331 * r78340;
        double r78342 = r78341 * r78330;
        double r78343 = phi1;
        double r78344 = cos(r78343);
        double r78345 = sin(r78328);
        double r78346 = r78344 * r78345;
        double r78347 = sin(r78343);
        double r78348 = r78347 * r78329;
        double r78349 = r78335 * r78337;
        double r78350 = r78338 * r78333;
        double r78351 = r78349 + r78350;
        double r78352 = r78348 * r78351;
        double r78353 = r78346 - r78352;
        double r78354 = atan2(r78342, r78353);
        return r78354;
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 13.0

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

  3. Applied sin-diff6.6

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

  5. Applied cos-diff0.2

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

  6. Simplified0.2

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

  7. Simplified0.2

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

  9. Applied add-cube-cbrt0.4

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

  10. Applied associate-*r*0.4

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

  11. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019208 
(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))))))