Average Error: 13.1 → 0.2
Time: 41.2s
Precision: 64
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \left(\log \left(e^{\left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\right) + \left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)\right)}\]
\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(\log \left(e^{\left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\right) + \left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r4624123 = lambda1;
        double r4624124 = lambda2;
        double r4624125 = r4624123 - r4624124;
        double r4624126 = sin(r4624125);
        double r4624127 = phi2;
        double r4624128 = cos(r4624127);
        double r4624129 = r4624126 * r4624128;
        double r4624130 = phi1;
        double r4624131 = cos(r4624130);
        double r4624132 = sin(r4624127);
        double r4624133 = r4624131 * r4624132;
        double r4624134 = sin(r4624130);
        double r4624135 = r4624134 * r4624128;
        double r4624136 = cos(r4624125);
        double r4624137 = r4624135 * r4624136;
        double r4624138 = r4624133 - r4624137;
        double r4624139 = atan2(r4624129, r4624138);
        return r4624139;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r4624140 = lambda2;
        double r4624141 = cos(r4624140);
        double r4624142 = lambda1;
        double r4624143 = sin(r4624142);
        double r4624144 = r4624141 * r4624143;
        double r4624145 = cos(r4624142);
        double r4624146 = sin(r4624140);
        double r4624147 = r4624145 * r4624146;
        double r4624148 = r4624144 - r4624147;
        double r4624149 = phi2;
        double r4624150 = cos(r4624149);
        double r4624151 = r4624148 * r4624150;
        double r4624152 = sin(r4624149);
        double r4624153 = phi1;
        double r4624154 = cos(r4624153);
        double r4624155 = r4624152 * r4624154;
        double r4624156 = r4624146 * r4624143;
        double r4624157 = sin(r4624153);
        double r4624158 = r4624150 * r4624157;
        double r4624159 = r4624156 * r4624158;
        double r4624160 = exp(r4624159);
        double r4624161 = log(r4624160);
        double r4624162 = r4624141 * r4624145;
        double r4624163 = r4624162 * r4624158;
        double r4624164 = r4624161 + r4624163;
        double r4624165 = r4624155 - r4624164;
        double r4624166 = atan2(r4624151, r4624165);
        return r4624166;
}

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-diff6.9

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

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

  8. Applied add-log-exp0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2019162 +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))))))