Average Error: 12.8 → 0.2
Time: 43.1s
Precision: 64
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sqrt[3]{\sin \lambda_2 \cdot \sin \lambda_1} \cdot \left(\sqrt[3]{\sin \lambda_2 \cdot \sin \lambda_1} \cdot \sqrt[3]{\sin \lambda_2 \cdot \sin \lambda_1}\right)\right)}\]
\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(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sqrt[3]{\sin \lambda_2 \cdot \sin \lambda_1} \cdot \left(\sqrt[3]{\sin \lambda_2 \cdot \sin \lambda_1} \cdot \sqrt[3]{\sin \lambda_2 \cdot \sin \lambda_1}\right)\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r4521273 = lambda1;
        double r4521274 = lambda2;
        double r4521275 = r4521273 - r4521274;
        double r4521276 = sin(r4521275);
        double r4521277 = phi2;
        double r4521278 = cos(r4521277);
        double r4521279 = r4521276 * r4521278;
        double r4521280 = phi1;
        double r4521281 = cos(r4521280);
        double r4521282 = sin(r4521277);
        double r4521283 = r4521281 * r4521282;
        double r4521284 = sin(r4521280);
        double r4521285 = r4521284 * r4521278;
        double r4521286 = cos(r4521275);
        double r4521287 = r4521285 * r4521286;
        double r4521288 = r4521283 - r4521287;
        double r4521289 = atan2(r4521279, r4521288);
        return r4521289;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r4521290 = lambda2;
        double r4521291 = cos(r4521290);
        double r4521292 = lambda1;
        double r4521293 = sin(r4521292);
        double r4521294 = r4521291 * r4521293;
        double r4521295 = cos(r4521292);
        double r4521296 = sin(r4521290);
        double r4521297 = r4521295 * r4521296;
        double r4521298 = r4521294 - r4521297;
        double r4521299 = phi2;
        double r4521300 = cos(r4521299);
        double r4521301 = r4521298 * r4521300;
        double r4521302 = sin(r4521299);
        double r4521303 = phi1;
        double r4521304 = cos(r4521303);
        double r4521305 = r4521302 * r4521304;
        double r4521306 = sin(r4521303);
        double r4521307 = r4521306 * r4521300;
        double r4521308 = r4521291 * r4521295;
        double r4521309 = r4521296 * r4521293;
        double r4521310 = cbrt(r4521309);
        double r4521311 = r4521310 * r4521310;
        double r4521312 = r4521310 * r4521311;
        double r4521313 = r4521308 + r4521312;
        double r4521314 = r4521307 * r4521313;
        double r4521315 = r4521305 - r4521314;
        double r4521316 = atan2(r4521301, r4521315);
        return r4521316;
}

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 12.8

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

  7. Applied add-cube-cbrt0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019158 
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Bearing on a great circle"
  (atan2 (* (sin (- lambda1 lambda2)) (cos phi2)) (- (* (cos phi1) (sin phi2)) (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2))))))