Equirectangular approximation to distance on a great circle

Average Error: 39.2 → 33.8

Time: 13.6s

Precision: binary64

Math

\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.902488693416451 \cdot 10^{+84}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left({\left(\sqrt[3]{\sqrt[3]{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}}\right)}^{6} \cdot \left({\left(\sqrt[3]{\left|\sqrt[3]{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}\right|}\right)}^{6} \cdot {\left(\sqrt[3]{\left|\sqrt[3]{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}\right|}\right)}^{6}\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array}\]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (sqrt
   (+
    (*
     (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))
     (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))))
    (* (- phi1 phi2) (- phi1 phi2))))))

↓

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.902488693416451e+84)
   (*
    R
    (sqrt
     (+
      (*
       (- lambda1 lambda2)
       (*
        (- lambda1 lambda2)
        (*
         (pow (cbrt (cbrt (cos (/ (+ phi2 phi1) 2.0)))) 6.0)
         (*
          (pow (cbrt (fabs (cbrt (cos (/ (+ phi2 phi1) 2.0))))) 6.0)
          (pow (cbrt (fabs (cbrt (cos (/ (+ phi2 phi1) 2.0))))) 6.0)))))
      (* (- phi1 phi2) (- phi1 phi2)))))
   (* R (- phi2 phi1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return ((double) (R * ((double) sqrt(((double) (((double) (((double) (((double) (lambda1 - lambda2)) * ((double) cos((((double) (phi1 + phi2)) / 2.0))))) * ((double) (((double) (lambda1 - lambda2)) * ((double) cos((((double) (phi1 + phi2)) / 2.0))))))) + ((double) (((double) (phi1 - phi2)) * ((double) (phi1 - phi2))))))))));
}

↓

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi2 <= 1.902488693416451e+84)) {
		tmp = ((double) (R * ((double) sqrt(((double) (((double) (((double) (lambda1 - lambda2)) * ((double) (((double) (lambda1 - lambda2)) * ((double) (((double) pow(((double) cbrt(((double) cbrt(((double) cos((((double) (phi2 + phi1)) / 2.0))))))), 6.0)) * ((double) (((double) pow(((double) cbrt(((double) fabs(((double) cbrt(((double) cos((((double) (phi2 + phi1)) / 2.0))))))))), 6.0)) * ((double) pow(((double) cbrt(((double) fabs(((double) cbrt(((double) cos((((double) (phi2 + phi1)) / 2.0))))))))), 6.0)))))))))) + ((double) (((double) (phi1 - phi2)) * ((double) (phi1 - phi2))))))))));
	} else {
		tmp = ((double) (R * ((double) (phi2 - phi1))));
	}
	return tmp;
}

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.902488693416451e84

   1. Initial program Error: 36.4 bits

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

   2. Simplified Error: 36.4 bits

      \[\leadsto \color{blue}{R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}}\]
   3. Using strategy rm

   4. Applied add-log-exp Error: 36.4 bits

      \[\leadsto R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\log \left(e^{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)}\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

   5. Simplified Error: 36.4 bits

      \[\leadsto R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \log \color{blue}{\left(e^{{\left(\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)}^{6}}\right)}\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
   6. Using strategy rm

   7. Applied add-cube-cbrt Error: 36.4 bits

      \[\leadsto R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{{\left(\sqrt[3]{\color{blue}{\left(\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}}}\right)}^{6}}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

   8. Applied cbrt-prod Error: 36.5 bits

      \[\leadsto R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{{\color{blue}{\left(\sqrt[3]{\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}} \cdot \sqrt[3]{\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}}\right)}}^{6}}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

   9. Applied unpow-prod-down Error: 36.5 bits

      \[\leadsto R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\color{blue}{{\left(\sqrt[3]{\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}}\right)}^{6} \cdot {\left(\sqrt[3]{\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}}\right)}^{6}}}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

   10. Applied exp-prod Error: 36.5 bits

       \[\leadsto R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \log \color{blue}{\left({\left(e^{{\left(\sqrt[3]{\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}}\right)}^{6}}\right)}^{\left({\left(\sqrt[3]{\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}}\right)}^{6}\right)}\right)}\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

   11. Applied log-pow Error: 36.4 bits

       \[\leadsto R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}}\right)}^{6} \cdot \log \left(e^{{\left(\sqrt[3]{\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}}\right)}^{6}}\right)\right)}\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

   12. Simplified Error: 36.4 bits

       \[\leadsto R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left({\left(\sqrt[3]{\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}}\right)}^{6} \cdot \color{blue}{{\left(\sqrt[3]{{\left(\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)}^{2}}\right)}^{6}}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
   13. Using strategy rm

   14. Applied add-sqr-sqrt Error: 36.4 bits

       \[\leadsto R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left({\left(\sqrt[3]{\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}}\right)}^{6} \cdot {\left(\sqrt[3]{\color{blue}{\sqrt{{\left(\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)}^{2}} \cdot \sqrt{{\left(\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)}^{2}}}}\right)}^{6}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

   15. Applied cbrt-prod Error: 36.5 bits

       \[\leadsto R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left({\left(\sqrt[3]{\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}}\right)}^{6} \cdot {\color{blue}{\left(\sqrt[3]{\sqrt{{\left(\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)}^{2}}} \cdot \sqrt[3]{\sqrt{{\left(\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)}^{2}}}\right)}}^{6}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

   16. Applied unpow-prod-down Error: 36.5 bits

       \[\leadsto R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left({\left(\sqrt[3]{\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}}\right)}^{6} \cdot \color{blue}{\left({\left(\sqrt[3]{\sqrt{{\left(\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)}^{2}}}\right)}^{6} \cdot {\left(\sqrt[3]{\sqrt{{\left(\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)}^{2}}}\right)}^{6}\right)}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

   17. Simplified Error: 36.5 bits

       \[\leadsto R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left({\left(\sqrt[3]{\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}}\right)}^{6} \cdot \left(\color{blue}{{\left(\sqrt[3]{\left|\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right|}\right)}^{6}} \cdot {\left(\sqrt[3]{\sqrt{{\left(\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)}^{2}}}\right)}^{6}\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

   18. Simplified Error: 36.5 bits

       \[\leadsto R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left({\left(\sqrt[3]{\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}}\right)}^{6} \cdot \left({\left(\sqrt[3]{\left|\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right|}\right)}^{6} \cdot \color{blue}{{\left(\sqrt[3]{\left|\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right|}\right)}^{6}}\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

   if 1.902488693416451e84 < phi2

   1. Initial program Error: 53.6 bits

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

   2. Simplified Error: 53.6 bits

      \[\leadsto \color{blue}{R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}}\]

   3. Taylor expanded around 0 Error: 20.3 bits

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 - \phi_1\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification Error: 33.8 bits

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.902488693416451 \cdot 10^{+84}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left({\left(\sqrt[3]{\sqrt[3]{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}}\right)}^{6} \cdot \left({\left(\sqrt[3]{\left|\sqrt[3]{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}\right|}\right)}^{6} \cdot {\left(\sqrt[3]{\left|\sqrt[3]{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}\right|}\right)}^{6}\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Equirectangular approximation to distance on a great circle"
  :precision binary64
  (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))) (* (- phi1 phi2) (- phi1 phi2))))))