Bearing on a great circle

Average Error: 13.6 → 0.2

Time: 40.0s

Precision: 64

Math

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

↓

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

C

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r4518216 = lambda1;
        double r4518217 = lambda2;
        double r4518218 = r4518216 - r4518217;
        double r4518219 = sin(r4518218);
        double r4518220 = phi2;
        double r4518221 = cos(r4518220);
        double r4518222 = r4518219 * r4518221;
        double r4518223 = phi1;
        double r4518224 = cos(r4518223);
        double r4518225 = sin(r4518220);
        double r4518226 = r4518224 * r4518225;
        double r4518227 = sin(r4518223);
        double r4518228 = r4518227 * r4518221;
        double r4518229 = cos(r4518218);
        double r4518230 = r4518228 * r4518229;
        double r4518231 = r4518226 - r4518230;
        double r4518232 = atan2(r4518222, r4518231);
        return r4518232;
}

↓

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r4518233 = phi2;
        double r4518234 = cos(r4518233);
        double r4518235 = lambda2;
        double r4518236 = cos(r4518235);
        double r4518237 = lambda1;
        double r4518238 = sin(r4518237);
        double r4518239 = r4518236 * r4518238;
        double r4518240 = cos(r4518237);
        double r4518241 = sin(r4518235);
        double r4518242 = r4518240 * r4518241;
        double r4518243 = log1p(r4518242);
        double r4518244 = expm1(r4518243);
        double r4518245 = r4518239 - r4518244;
        double r4518246 = r4518234 * r4518245;
        double r4518247 = phi1;
        double r4518248 = cos(r4518247);
        double r4518249 = sin(r4518233);
        double r4518250 = r4518248 * r4518249;
        double r4518251 = r4518236 * r4518240;
        double r4518252 = fma(r4518241, r4518238, r4518251);
        double r4518253 = sin(r4518247);
        double r4518254 = r4518253 * r4518234;
        double r4518255 = r4518252 * r4518254;
        double r4518256 = expm1(r4518255);
        double r4518257 = log1p(r4518256);
        double r4518258 = r4518250 - r4518257;
        double r4518259 = atan2(r4518246, r4518258);
        return r4518259;
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

1. Initial program 13.6

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

3. Applied sin-diff 7.0

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

5. Applied cos-diff 0.2

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

7. Applied expm1-log1p-u 0.2

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

9. Applied log1p-expm1-u 0.2

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

10. Simplified 0.2

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

11. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(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))))))