Distance on a great circle

Average Error: 24.5 → 14.3

Time: 53.6s

Precision: 64

Math

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

↓

\[\left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right), {\left(\cos \left(\frac{\phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1}{2}\right) - \sin \left(\frac{\phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\left(\sqrt[3]{\log \left(e^{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)} \cdot \sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right), {\left(\cos \left(\frac{\phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1}{2}\right) - \sin \left(\frac{\phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right)\right)}^{2}\right)}}\]

C

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r93145 = R;
        double r93146 = 2.0;
        double r93147 = phi1;
        double r93148 = phi2;
        double r93149 = r93147 - r93148;
        double r93150 = r93149 / r93146;
        double r93151 = sin(r93150);
        double r93152 = pow(r93151, r93146);
        double r93153 = cos(r93147);
        double r93154 = cos(r93148);
        double r93155 = r93153 * r93154;
        double r93156 = lambda1;
        double r93157 = lambda2;
        double r93158 = r93156 - r93157;
        double r93159 = r93158 / r93146;
        double r93160 = sin(r93159);
        double r93161 = r93155 * r93160;
        double r93162 = r93161 * r93160;
        double r93163 = r93152 + r93162;
        double r93164 = sqrt(r93163);
        double r93165 = 1.0;
        double r93166 = r93165 - r93163;
        double r93167 = sqrt(r93166);
        double r93168 = atan2(r93164, r93167);
        double r93169 = r93146 * r93168;
        double r93170 = r93145 * r93169;
        return r93170;
}

↓

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r93171 = 2.0;
        double r93172 = R;
        double r93173 = r93171 * r93172;
        double r93174 = lambda1;
        double r93175 = lambda2;
        double r93176 = r93174 - r93175;
        double r93177 = r93176 / r93171;
        double r93178 = sin(r93177);
        double r93179 = phi2;
        double r93180 = cos(r93179);
        double r93181 = phi1;
        double r93182 = cos(r93181);
        double r93183 = r93182 * r93178;
        double r93184 = r93180 * r93183;
        double r93185 = r93179 / r93171;
        double r93186 = cos(r93185);
        double r93187 = r93181 / r93171;
        double r93188 = sin(r93187);
        double r93189 = r93186 * r93188;
        double r93190 = sin(r93185);
        double r93191 = cos(r93187);
        double r93192 = r93190 * r93191;
        double r93193 = r93189 - r93192;
        double r93194 = pow(r93193, r93171);
        double r93195 = fma(r93178, r93184, r93194);
        double r93196 = sqrt(r93195);
        double r93197 = 1.0;
        double r93198 = exp(r93178);
        double r93199 = log(r93198);
        double r93200 = cbrt(r93199);
        double r93201 = cbrt(r93178);
        double r93202 = r93200 * r93201;
        double r93203 = r93202 * r93201;
        double r93204 = fma(r93203, r93184, r93194);
        double r93205 = r93197 - r93204;
        double r93206 = sqrt(r93205);
        double r93207 = atan2(r93196, r93206);
        double r93208 = r93173 * r93207;
        return r93208;
}

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

1. Initial program 24.5

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

2. Simplified 24.5

   \[\leadsto \color{blue}{\left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2, {\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2, {\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2}\right)}}}\]
3. Using strategy rm

4. Applied div-sub 24.5

   \[\leadsto \left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2, {\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2, {\left(\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}\right)}^{2}\right)}}\]

5. Applied sin-diff 23.9

   \[\leadsto \left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2, {\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2, {\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2}\right)}}\]

6. Simplified 23.9

   \[\leadsto \left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2, {\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2, {\left(\color{blue}{\cos \left(\frac{\phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}\right)}}\]

7. Simplified 23.9

   \[\leadsto \left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2, {\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{\phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1}{2}\right) - \color{blue}{\sin \left(\frac{\phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right)}\right)}^{2}\right)}}\]
8. Using strategy rm

9. Applied div-sub 23.9

   \[\leadsto \left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2, {\left(\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{\phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1}{2}\right) - \sin \left(\frac{\phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right)\right)}^{2}\right)}}\]

10. Applied sin-diff 14.2

    \[\leadsto \left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2, {\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{\phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1}{2}\right) - \sin \left(\frac{\phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right)\right)}^{2}\right)}}\]

11. Simplified 14.2

    \[\leadsto \left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2, {\left(\color{blue}{\cos \left(\frac{\phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{\phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1}{2}\right) - \sin \left(\frac{\phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right)\right)}^{2}\right)}}\]

12. Simplified 14.2

    \[\leadsto \left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{\phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1}{2}\right) - \color{blue}{\sin \left(\frac{\phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right)}\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{\phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1}{2}\right) - \sin \left(\frac{\phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right)\right)}^{2}\right)}}\]
13. Using strategy rm

14. Applied add-cube-cbrt 14.3

    \[\leadsto \left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{\phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1}{2}\right) - \sin \left(\frac{\phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\color{blue}{\left(\sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}, \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{\phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1}{2}\right) - \sin \left(\frac{\phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right)\right)}^{2}\right)}}\]
15. Using strategy rm

16. Applied add-log-exp 14.3

    \[\leadsto \left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{\phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1}{2}\right) - \sin \left(\frac{\phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\left(\sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)} \cdot \sqrt[3]{\color{blue}{\log \left(e^{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \cdot \sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}, \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{\phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1}{2}\right) - \sin \left(\frac{\phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right)\right)}^{2}\right)}}\]

17. Final simplification 14.3

    \[\leadsto \left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right), {\left(\cos \left(\frac{\phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1}{2}\right) - \sin \left(\frac{\phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\left(\sqrt[3]{\log \left(e^{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)} \cdot \sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right), {\left(\cos \left(\frac{\phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1}{2}\right) - \sin \left(\frac{\phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right)\right)}^{2}\right)}}\]

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Distance on a great circle"
  (* R (* 2.0 (atan2 (sqrt (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0))) (sin (/ (- lambda1 lambda2) 2.0))))) (sqrt (- 1.0 (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0))) (sin (/ (- lambda1 lambda2) 2.0))))))))))