Average Error: 0.8 → 0.3
Time: 25.3s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\frac{{\left(\cos \phi_1\right)}^{3} + {\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2\right)}^{3}}{\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2\right) \cdot \frac{{\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2\right)}^{2} - {\left(\cos \phi_1\right)}^{2}}{\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2 + \cos \phi_1} + \cos \phi_1 \cdot \cos \phi_1} + \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)}\]
\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)}
↓
\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)}{\frac{{\left(\cos \phi_1\right)}^{3} + {\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2\right)}^{3}}{\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2\right) \cdot \frac{{\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2\right)}^{2} - {\left(\cos \phi_1\right)}^{2}}{\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2 + \cos \phi_1} + \cos \phi_1 \cdot \cos \phi_1} + \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r43570 = lambda1;
        double r43571 = phi2;
        double r43572 = cos(r43571);
        double r43573 = lambda2;
        double r43574 = r43570 - r43573;
        double r43575 = sin(r43574);
        double r43576 = r43572 * r43575;
        double r43577 = phi1;
        double r43578 = cos(r43577);
        double r43579 = cos(r43574);
        double r43580 = r43572 * r43579;
        double r43581 = r43578 + r43580;
        double r43582 = atan2(r43576, r43581);
        double r43583 = r43570 + r43582;
        return r43583;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r43584 = lambda1;
        double r43585 = phi2;
        double r43586 = cos(r43585);
        double r43587 = sin(r43584);
        double r43588 = lambda2;
        double r43589 = cos(r43588);
        double r43590 = r43587 * r43589;
        double r43591 = cos(r43584);
        double r43592 = sin(r43588);
        double r43593 = r43591 * r43592;
        double r43594 = r43590 - r43593;
        double r43595 = r43586 * r43594;
        double r43596 = phi1;
        double r43597 = cos(r43596);
        double r43598 = 3.0;
        double r43599 = pow(r43597, r43598);
        double r43600 = r43591 * r43589;
        double r43601 = r43600 * r43586;
        double r43602 = pow(r43601, r43598);
        double r43603 = r43599 + r43602;
        double r43604 = 2.0;
        double r43605 = pow(r43601, r43604);
        double r43606 = pow(r43597, r43604);
        double r43607 = r43605 - r43606;
        double r43608 = r43601 + r43597;
        double r43609 = r43607 / r43608;
        double r43610 = r43601 * r43609;
        double r43611 = r43597 * r43597;
        double r43612 = r43610 + r43611;
        double r43613 = r43603 / r43612;
        double r43614 = r43587 * r43592;
        double r43615 = r43586 * r43614;
        double r43616 = r43613 + r43615;
        double r43617 = atan2(r43595, r43616);
        double r43618 = r43584 + r43617;
        return r43618;
}

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 0.8

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

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

  7. Applied associate-+r+0.2

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

  8. Simplified0.2

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

  10. Applied flip3-+0.3

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

  11. Simplified0.3

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

  13. Applied flip--0.3

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

  14. Simplified0.3

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\frac{{\left(\cos \phi_1\right)}^{3} + {\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2\right)}^{3}}{\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2\right) \cdot \frac{\color{blue}{{\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2\right)}^{2} - {\left(\cos \phi_1\right)}^{2}}}{\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2 + \cos \phi_1} + \cos \phi_1 \cdot \cos \phi_1} + \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)}\]

  15. Final simplification0.3

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\frac{{\left(\cos \phi_1\right)}^{3} + {\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2\right)}^{3}}{\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2\right) \cdot \frac{{\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2\right)}^{2} - {\left(\cos \phi_1\right)}^{2}}{\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2 + \cos \phi_1} + \cos \phi_1 \cdot \cos \phi_1} + \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Midpoint on a great circle"
  :precision binary64
  (+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))