Bearing on a great circle

Average Error: 13.0 → 0.4

Time: 44.5s

Precision: 64

Math

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

↓

\[\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left({\left({\left(\sqrt[3]{\left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \left(\cos \phi_2 \cdot \cos \lambda_1\right)}\right)}^{6}\right)}^{\frac{1}{3}} \cdot \sqrt[3]{\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right)} + \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right)\right)}\]

C

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r101103 = lambda1;
        double r101104 = lambda2;
        double r101105 = r101103 - r101104;
        double r101106 = sin(r101105);
        double r101107 = phi2;
        double r101108 = cos(r101107);
        double r101109 = r101106 * r101108;
        double r101110 = phi1;
        double r101111 = cos(r101110);
        double r101112 = sin(r101107);
        double r101113 = r101111 * r101112;
        double r101114 = sin(r101110);
        double r101115 = r101114 * r101108;
        double r101116 = cos(r101105);
        double r101117 = r101115 * r101116;
        double r101118 = r101113 - r101117;
        double r101119 = atan2(r101109, r101118);
        return r101119;
}

↓

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r101120 = lambda1;
        double r101121 = sin(r101120);
        double r101122 = lambda2;
        double r101123 = cos(r101122);
        double r101124 = r101121 * r101123;
        double r101125 = cos(r101120);
        double r101126 = sin(r101122);
        double r101127 = r101125 * r101126;
        double r101128 = r101124 - r101127;
        double r101129 = phi2;
        double r101130 = cos(r101129);
        double r101131 = r101128 * r101130;
        double r101132 = phi1;
        double r101133 = cos(r101132);
        double r101134 = sin(r101129);
        double r101135 = r101133 * r101134;
        double r101136 = sin(r101132);
        double r101137 = r101123 * r101136;
        double r101138 = r101130 * r101125;
        double r101139 = r101137 * r101138;
        double r101140 = cbrt(r101139);
        double r101141 = 6.0;
        double r101142 = pow(r101140, r101141);
        double r101143 = 0.3333333333333333;
        double r101144 = pow(r101142, r101143);
        double r101145 = r101125 * r101123;
        double r101146 = r101136 * r101130;
        double r101147 = r101145 * r101146;
        double r101148 = cbrt(r101147);
        double r101149 = r101144 * r101148;
        double r101150 = r101121 * r101126;
        double r101151 = r101150 * r101146;
        double r101152 = r101149 + r101151;
        double r101153 = r101135 - r101152;
        double r101154 = atan2(r101131, r101153);
        return r101154;
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

1. Initial program 13.0

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

3. Applied sin-diff 6.6

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
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. Applied distribute-lft-in 0.2

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

7. Simplified 0.2

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

8. Simplified 0.2

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

10. Applied add-cube-cbrt 0.3

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

12. Applied pow1/3 32.6

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

13. Applied pow1/3 32.6

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

14. Applied pow-prod-down 0.3

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

15. Simplified 0.4

    \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left({\color{blue}{\left({\left(\sqrt[3]{\left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \left(\cos \phi_2 \cdot \cos \lambda_1\right)}\right)}^{6}\right)}}^{\frac{1}{3}} \cdot \sqrt[3]{\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right)} + \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right)\right)}\]

16. Final simplification 0.4

    \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left({\left({\left(\sqrt[3]{\left(\cos \lambda_2 \cdot \sin \phi_1\right) \cdot \left(\cos \phi_2 \cdot \cos \lambda_1\right)}\right)}^{6}\right)}^{\frac{1}{3}} \cdot \sqrt[3]{\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right)} + \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right)\right)}\]

Reproduce

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