Destination given bearing on a great circle

Average Error: 0.2 → 0.2

Time: 13.7s

Precision: 64

• unused variable phi2

Math

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

↓

\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \log \left({\left(e^{\sin \left(\frac{\pi}{2}\right) \cdot \sin \left(\sqrt[3]{{\left(\sin^{-1} \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \sin \phi_1 \cdot \cos delta\right)\right)}^{3}}\right)}\right)}^{\left(\sin \phi_1\right)}\right)}\]

C

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r85101 = lambda1;
        double r85102 = theta;
        double r85103 = sin(r85102);
        double r85104 = delta;
        double r85105 = sin(r85104);
        double r85106 = r85103 * r85105;
        double r85107 = phi1;
        double r85108 = cos(r85107);
        double r85109 = r85106 * r85108;
        double r85110 = cos(r85104);
        double r85111 = sin(r85107);
        double r85112 = r85111 * r85110;
        double r85113 = r85108 * r85105;
        double r85114 = cos(r85102);
        double r85115 = r85113 * r85114;
        double r85116 = r85112 + r85115;
        double r85117 = asin(r85116);
        double r85118 = sin(r85117);
        double r85119 = r85111 * r85118;
        double r85120 = r85110 - r85119;
        double r85121 = atan2(r85109, r85120);
        double r85122 = r85101 + r85121;
        return r85122;
}

↓

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r85123 = lambda1;
        double r85124 = theta;
        double r85125 = sin(r85124);
        double r85126 = delta;
        double r85127 = sin(r85126);
        double r85128 = r85125 * r85127;
        double r85129 = phi1;
        double r85130 = cos(r85129);
        double r85131 = r85128 * r85130;
        double r85132 = cos(r85126);
        double r85133 = atan2(1.0, 0.0);
        double r85134 = 2.0;
        double r85135 = r85133 / r85134;
        double r85136 = sin(r85135);
        double r85137 = cos(r85124);
        double r85138 = r85130 * r85137;
        double r85139 = r85127 * r85138;
        double r85140 = sin(r85129);
        double r85141 = r85140 * r85132;
        double r85142 = r85139 + r85141;
        double r85143 = asin(r85142);
        double r85144 = 3.0;
        double r85145 = pow(r85143, r85144);
        double r85146 = cbrt(r85145);
        double r85147 = sin(r85146);
        double r85148 = r85136 * r85147;
        double r85149 = exp(r85148);
        double r85150 = pow(r85149, r85140);
        double r85151 = log(r85150);
        double r85152 = r85132 - r85151;
        double r85153 = atan2(r85131, r85152);
        double r85154 = r85123 + r85153;
        return r85154;
}

Error

Bits error versus lambda1

Bits error versus phi1

Bits error versus phi2

Bits error versus delta

Bits error versus theta

Derivation

1. Initial program 0.2

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

3. Applied asin-acos 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \color{blue}{\left(\frac{\pi}{2} - \cos^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}}\]
4. Using strategy rm

5. Applied add-log-exp 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \color{blue}{\log \left(e^{\sin \phi_1 \cdot \sin \left(\frac{\pi}{2} - \cos^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\right)}}\]

6. Simplified 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \log \color{blue}{\left({\left(e^{\cos \left(\cos^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\right)}^{\left(\sin \phi_1\right)}\right)}}\]
7. Using strategy rm

8. Applied acos-asin 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \log \left({\left(e^{\cos \color{blue}{\left(\frac{\pi}{2} - \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}}\right)}^{\left(\sin \phi_1\right)}\right)}\]

9. Applied cos-diff 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \log \left({\left(e^{\color{blue}{\cos \left(\frac{\pi}{2}\right) \cdot \cos \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) + \sin \left(\frac{\pi}{2}\right) \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}}\right)}^{\left(\sin \phi_1\right)}\right)}\]

10. Applied exp-sum 0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \log \left({\color{blue}{\left(e^{\cos \left(\frac{\pi}{2}\right) \cdot \cos \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)} \cdot e^{\sin \left(\frac{\pi}{2}\right) \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\right)}}^{\left(\sin \phi_1\right)}\right)}\]

11. Applied unpow-prod-down 0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \log \color{blue}{\left({\left(e^{\cos \left(\frac{\pi}{2}\right) \cdot \cos \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\right)}^{\left(\sin \phi_1\right)} \cdot {\left(e^{\sin \left(\frac{\pi}{2}\right) \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\right)}^{\left(\sin \phi_1\right)}\right)}}\]

12. Applied log-prod 0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \color{blue}{\left(\log \left({\left(e^{\cos \left(\frac{\pi}{2}\right) \cdot \cos \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\right)}^{\left(\sin \phi_1\right)}\right) + \log \left({\left(e^{\sin \left(\frac{\pi}{2}\right) \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\right)}^{\left(\sin \phi_1\right)}\right)\right)}}\]

13. Applied associate--r+ 0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\left(\cos delta - \log \left({\left(e^{\cos \left(\frac{\pi}{2}\right) \cdot \cos \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\right)}^{\left(\sin \phi_1\right)}\right)\right) - \log \left({\left(e^{\sin \left(\frac{\pi}{2}\right) \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\right)}^{\left(\sin \phi_1\right)}\right)}}\]

14. Simplified 0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\cos delta} - \log \left({\left(e^{\sin \left(\frac{\pi}{2}\right) \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\right)}^{\left(\sin \phi_1\right)}\right)}\]
15. Using strategy rm

16. Applied add-cbrt-cube 0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \log \left({\left(e^{\sin \left(\frac{\pi}{2}\right) \cdot \sin \color{blue}{\left(\sqrt[3]{\left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right) \cdot \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) \cdot \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)}\right)}}\right)}^{\left(\sin \phi_1\right)}\right)}\]

17. Simplified 0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \log \left({\left(e^{\sin \left(\frac{\pi}{2}\right) \cdot \sin \left(\sqrt[3]{\color{blue}{{\left(\sin^{-1} \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \sin \phi_1 \cdot \cos delta\right)\right)}^{3}}}\right)}\right)}^{\left(\sin \phi_1\right)}\right)}\]

18. Final simplification 0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \log \left({\left(e^{\sin \left(\frac{\pi}{2}\right) \cdot \sin \left(\sqrt[3]{{\left(\sin^{-1} \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \sin \phi_1 \cdot \cos delta\right)\right)}^{3}}\right)}\right)}^{\left(\sin \phi_1\right)}\right)}\]

Reproduce

herbie shell --seed 2020001 
(FPCore (lambda1 phi1 phi2 delta theta)
  :name "Destination given bearing on a great circle"
  :precision binary64
  (+ lambda1 (atan2 (* (* (sin theta) (sin delta)) (cos phi1)) (- (cos delta) (* (sin phi1) (sin (asin (+ (* (sin phi1) (cos delta)) (* (* (cos phi1) (sin delta)) (cos theta))))))))))