Average Error: 0.9 → 0.3
Time: 35.0s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\sqrt[3]{\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right) + \cos \phi_1\right) \cdot \left(\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right) + \cos \phi_1\right) \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right) + \cos \phi_1\right)\right)}} + \lambda_1\]
\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}
↓
\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\sqrt[3]{\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right) + \cos \phi_1\right) \cdot \left(\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right) + \cos \phi_1\right) \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right) + \cos \phi_1\right)\right)}} + \lambda_1
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r2146770 = lambda1;
        double r2146771 = phi2;
        double r2146772 = cos(r2146771);
        double r2146773 = lambda2;
        double r2146774 = r2146770 - r2146773;
        double r2146775 = sin(r2146774);
        double r2146776 = r2146772 * r2146775;
        double r2146777 = phi1;
        double r2146778 = cos(r2146777);
        double r2146779 = cos(r2146774);
        double r2146780 = r2146772 * r2146779;
        double r2146781 = r2146778 + r2146780;
        double r2146782 = atan2(r2146776, r2146781);
        double r2146783 = r2146770 + r2146782;
        return r2146783;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r2146784 = phi2;
        double r2146785 = cos(r2146784);
        double r2146786 = lambda1;
        double r2146787 = sin(r2146786);
        double r2146788 = lambda2;
        double r2146789 = cos(r2146788);
        double r2146790 = r2146787 * r2146789;
        double r2146791 = cos(r2146786);
        double r2146792 = sin(r2146788);
        double r2146793 = r2146791 * r2146792;
        double r2146794 = r2146790 - r2146793;
        double r2146795 = r2146785 * r2146794;
        double r2146796 = r2146791 * r2146789;
        double r2146797 = r2146792 * r2146787;
        double r2146798 = r2146796 + r2146797;
        double r2146799 = r2146785 * r2146798;
        double r2146800 = phi1;
        double r2146801 = cos(r2146800);
        double r2146802 = r2146799 + r2146801;
        double r2146803 = r2146802 * r2146802;
        double r2146804 = r2146802 * r2146803;
        double r2146805 = cbrt(r2146804);
        double r2146806 = atan2(r2146795, r2146805);
        double r2146807 = r2146806 + r2146786;
        return r2146807;
}

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 0.9

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

  3. Applied sin-diff0.8

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

  5. Applied cos-diff0.2

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

  7. Applied add-cbrt-cube0.3

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

  8. Final simplification0.3

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

Reproduce

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