Average Error: 0.8 → 0.3
Time: 40.7s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\frac{\left(3 \cdot \left(\sin \lambda_1 \cdot \left({\left(\cos \phi_2\right)}^{3} \cdot \left(\sin \lambda_2 \cdot \left({\left(\cos \lambda_1\right)}^{2} \cdot {\left(\cos \lambda_2\right)}^{2}\right)\right)\right)\right) + \left({\left(\cos \phi_2\right)}^{3} \cdot {\left(\sin \lambda_1 \cdot \sin \lambda_2\right)}^{3} + {\left(\cos \phi_1\right)}^{3}\right)\right) + \left({\left(\cos \lambda_1 \cdot \cos \lambda_2\right)}^{3} \cdot {\left(\cos \phi_2\right)}^{3} + 3 \cdot \left(\cos \lambda_1 \cdot \left({\left(\cos \phi_2\right)}^{3} \cdot \left({\left(\sin \lambda_2\right)}^{2} \cdot \left({\left(\sin \lambda_1\right)}^{2} \cdot \cos \lambda_2\right)\right)\right)\right)\right)}{\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) - \cos \phi_1\right) + \cos \phi_1 \cdot \cos \phi_1}}\]
\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)}
↓
\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(3 \cdot \left(\sin \lambda_1 \cdot \left({\left(\cos \phi_2\right)}^{3} \cdot \left(\sin \lambda_2 \cdot \left({\left(\cos \lambda_1\right)}^{2} \cdot {\left(\cos \lambda_2\right)}^{2}\right)\right)\right)\right) + \left({\left(\cos \phi_2\right)}^{3} \cdot {\left(\sin \lambda_1 \cdot \sin \lambda_2\right)}^{3} + {\left(\cos \phi_1\right)}^{3}\right)\right) + \left({\left(\cos \lambda_1 \cdot \cos \lambda_2\right)}^{3} \cdot {\left(\cos \phi_2\right)}^{3} + 3 \cdot \left(\cos \lambda_1 \cdot \left({\left(\cos \phi_2\right)}^{3} \cdot \left({\left(\sin \lambda_2\right)}^{2} \cdot \left({\left(\sin \lambda_1\right)}^{2} \cdot \cos \lambda_2\right)\right)\right)\right)\right)}{\left(\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_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) - \cos \phi_1\right) + \cos \phi_1 \cdot \cos \phi_1}}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r63483 = lambda1;
        double r63484 = phi2;
        double r63485 = cos(r63484);
        double r63486 = lambda2;
        double r63487 = r63483 - r63486;
        double r63488 = sin(r63487);
        double r63489 = r63485 * r63488;
        double r63490 = phi1;
        double r63491 = cos(r63490);
        double r63492 = cos(r63487);
        double r63493 = r63485 * r63492;
        double r63494 = r63491 + r63493;
        double r63495 = atan2(r63489, r63494);
        double r63496 = r63483 + r63495;
        return r63496;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r63497 = lambda1;
        double r63498 = phi2;
        double r63499 = cos(r63498);
        double r63500 = sin(r63497);
        double r63501 = lambda2;
        double r63502 = cos(r63501);
        double r63503 = r63500 * r63502;
        double r63504 = cos(r63497);
        double r63505 = sin(r63501);
        double r63506 = r63504 * r63505;
        double r63507 = r63503 - r63506;
        double r63508 = r63499 * r63507;
        double r63509 = 3.0;
        double r63510 = pow(r63499, r63509);
        double r63511 = 2.0;
        double r63512 = pow(r63504, r63511);
        double r63513 = pow(r63502, r63511);
        double r63514 = r63512 * r63513;
        double r63515 = r63505 * r63514;
        double r63516 = r63510 * r63515;
        double r63517 = r63500 * r63516;
        double r63518 = r63509 * r63517;
        double r63519 = r63500 * r63505;
        double r63520 = pow(r63519, r63509);
        double r63521 = r63510 * r63520;
        double r63522 = phi1;
        double r63523 = cos(r63522);
        double r63524 = pow(r63523, r63509);
        double r63525 = r63521 + r63524;
        double r63526 = r63518 + r63525;
        double r63527 = r63504 * r63502;
        double r63528 = pow(r63527, r63509);
        double r63529 = r63528 * r63510;
        double r63530 = pow(r63505, r63511);
        double r63531 = pow(r63500, r63511);
        double r63532 = r63531 * r63502;
        double r63533 = r63530 * r63532;
        double r63534 = r63510 * r63533;
        double r63535 = r63504 * r63534;
        double r63536 = r63509 * r63535;
        double r63537 = r63529 + r63536;
        double r63538 = r63526 + r63537;
        double r63539 = r63527 + r63519;
        double r63540 = r63499 * r63539;
        double r63541 = r63540 - r63523;
        double r63542 = r63540 * r63541;
        double r63543 = r63523 * r63523;
        double r63544 = r63542 + r63543;
        double r63545 = r63538 / r63544;
        double r63546 = atan2(r63508, r63545);
        double r63547 = r63497 + r63546;
        return r63547;
}

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 cos-diff0.8

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

  5. Applied sin-diff0.2

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

  7. Applied flip3-+0.3

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

  8. Simplified0.3

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

  9. Taylor expanded around -inf 0.3

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

  10. Simplified0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019323 
(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)))))))