Average Error: 13.5 → 0.2
Time: 45.7s
Precision: 64
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)\right)\right) + \left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)\right)}\]
\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(\mathsf{log1p}\left(\mathsf{expm1}\left(\left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)\right)\right) + \left(\sin \lambda_2 \cdot \sin \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 r1576042 = lambda1;
        double r1576043 = lambda2;
        double r1576044 = r1576042 - r1576043;
        double r1576045 = sin(r1576044);
        double r1576046 = phi2;
        double r1576047 = cos(r1576046);
        double r1576048 = r1576045 * r1576047;
        double r1576049 = phi1;
        double r1576050 = cos(r1576049);
        double r1576051 = sin(r1576046);
        double r1576052 = r1576050 * r1576051;
        double r1576053 = sin(r1576049);
        double r1576054 = r1576053 * r1576047;
        double r1576055 = cos(r1576044);
        double r1576056 = r1576054 * r1576055;
        double r1576057 = r1576052 - r1576056;
        double r1576058 = atan2(r1576048, r1576057);
        return r1576058;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r1576059 = lambda2;
        double r1576060 = cos(r1576059);
        double r1576061 = lambda1;
        double r1576062 = sin(r1576061);
        double r1576063 = r1576060 * r1576062;
        double r1576064 = cos(r1576061);
        double r1576065 = sin(r1576059);
        double r1576066 = r1576064 * r1576065;
        double r1576067 = r1576063 - r1576066;
        double r1576068 = phi2;
        double r1576069 = cos(r1576068);
        double r1576070 = r1576067 * r1576069;
        double r1576071 = sin(r1576068);
        double r1576072 = phi1;
        double r1576073 = cos(r1576072);
        double r1576074 = r1576071 * r1576073;
        double r1576075 = r1576060 * r1576064;
        double r1576076 = sin(r1576072);
        double r1576077 = r1576069 * r1576076;
        double r1576078 = r1576075 * r1576077;
        double r1576079 = expm1(r1576078);
        double r1576080 = log1p(r1576079);
        double r1576081 = r1576065 * r1576062;
        double r1576082 = r1576081 * r1576077;
        double r1576083 = r1576080 + r1576082;
        double r1576084 = r1576074 - r1576083;
        double r1576085 = atan2(r1576070, r1576084);
        return r1576085;
}

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 13.5

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

  3. Applied sin-diff6.7

    \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
  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 log1p-expm1-u0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2019155 +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))))))