Bearing on a great circle

Average Error: 13.6 → 0.2

Time: 40.1s

Precision: 64

Math

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

↓

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

C

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r3876549 = lambda1;
        double r3876550 = lambda2;
        double r3876551 = r3876549 - r3876550;
        double r3876552 = sin(r3876551);
        double r3876553 = phi2;
        double r3876554 = cos(r3876553);
        double r3876555 = r3876552 * r3876554;
        double r3876556 = phi1;
        double r3876557 = cos(r3876556);
        double r3876558 = sin(r3876553);
        double r3876559 = r3876557 * r3876558;
        double r3876560 = sin(r3876556);
        double r3876561 = r3876560 * r3876554;
        double r3876562 = cos(r3876551);
        double r3876563 = r3876561 * r3876562;
        double r3876564 = r3876559 - r3876563;
        double r3876565 = atan2(r3876555, r3876564);
        return r3876565;
}

↓

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r3876566 = lambda2;
        double r3876567 = cos(r3876566);
        double r3876568 = lambda1;
        double r3876569 = sin(r3876568);
        double r3876570 = r3876567 * r3876569;
        double r3876571 = cos(r3876568);
        double r3876572 = cbrt(r3876571);
        double r3876573 = sin(r3876566);
        double r3876574 = r3876572 * r3876573;
        double r3876575 = r3876572 * r3876572;
        double r3876576 = r3876574 * r3876575;
        double r3876577 = r3876570 - r3876576;
        double r3876578 = phi2;
        double r3876579 = cos(r3876578);
        double r3876580 = r3876577 * r3876579;
        double r3876581 = phi1;
        double r3876582 = cos(r3876581);
        double r3876583 = sin(r3876578);
        double r3876584 = r3876582 * r3876583;
        double r3876585 = sin(r3876581);
        double r3876586 = r3876585 * r3876579;
        double r3876587 = r3876573 * r3876569;
        double r3876588 = r3876587 * r3876587;
        double r3876589 = r3876587 * r3876588;
        double r3876590 = r3876567 * r3876571;
        double r3876591 = r3876590 * r3876590;
        double r3876592 = r3876591 * r3876590;
        double r3876593 = r3876589 + r3876592;
        double r3876594 = r3876586 * r3876593;
        double r3876595 = r3876590 * r3876587;
        double r3876596 = r3876588 - r3876595;
        double r3876597 = r3876591 + r3876596;
        double r3876598 = r3876594 / r3876597;
        double r3876599 = r3876584 - r3876598;
        double r3876600 = atan2(r3876580, r3876599);
        return r3876600;
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

1. Initial program 13.6

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

3. Applied sin-diff 7.0

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

5. Applied cos-diff 0.2

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

7. Applied add-cube-cbrt 0.2

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

8. Applied associate-*l* 0.2

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

10. Applied flip3-+ 0.2

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

11. Applied associate-*r/ 0.2

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

12. Simplified 0.2

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

13. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019171 
(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))))))