Average Error: 13.1 → 0.2
Time: 14.0s
Precision: 64
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \phi_1 \cdot \sin \phi_2\right)\right) - \left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \sin \lambda_1\right) \cdot \sin \lambda_2\right)}\]
\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(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \phi_1 \cdot \sin \phi_2\right)\right) - \left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \sin \lambda_1\right) \cdot \sin \lambda_2\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r127304 = lambda1;
        double r127305 = lambda2;
        double r127306 = r127304 - r127305;
        double r127307 = sin(r127306);
        double r127308 = phi2;
        double r127309 = cos(r127308);
        double r127310 = r127307 * r127309;
        double r127311 = phi1;
        double r127312 = cos(r127311);
        double r127313 = sin(r127308);
        double r127314 = r127312 * r127313;
        double r127315 = sin(r127311);
        double r127316 = r127315 * r127309;
        double r127317 = cos(r127306);
        double r127318 = r127316 * r127317;
        double r127319 = r127314 - r127318;
        double r127320 = atan2(r127310, r127319);
        return r127320;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r127321 = lambda1;
        double r127322 = sin(r127321);
        double r127323 = lambda2;
        double r127324 = cos(r127323);
        double r127325 = r127322 * r127324;
        double r127326 = cos(r127321);
        double r127327 = sin(r127323);
        double r127328 = r127326 * r127327;
        double r127329 = r127325 - r127328;
        double r127330 = phi2;
        double r127331 = cos(r127330);
        double r127332 = r127329 * r127331;
        double r127333 = phi1;
        double r127334 = cos(r127333);
        double r127335 = sin(r127330);
        double r127336 = r127334 * r127335;
        double r127337 = log1p(r127336);
        double r127338 = expm1(r127337);
        double r127339 = sin(r127333);
        double r127340 = r127339 * r127331;
        double r127341 = r127326 * r127324;
        double r127342 = r127340 * r127341;
        double r127343 = r127340 * r127322;
        double r127344 = r127343 * r127327;
        double r127345 = r127342 + r127344;
        double r127346 = r127338 - r127345;
        double r127347 = atan2(r127332, r127346);
        return r127347;
}

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 13.1

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

  3. Applied sin-diff6.5

    \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
  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. Applied distribute-lft-in0.2

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

  8. Applied associate-*r*0.2

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

  10. Applied expm1-log1p-u0.2

    \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \phi_1 \cdot \sin \phi_2\right)\right)} - \left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \sin \lambda_1\right) \cdot \sin \lambda_2\right)}\]

  11. Final simplification0.2

    \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \phi_1 \cdot \sin \phi_2\right)\right) - \left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \sin \lambda_1\right) \cdot \sin \lambda_2\right)}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(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))))))