Equirectangular approximation to distance on a great circle

Average Error: 39.0 → 20.6

Time: 59.1s

Precision: binary64

\[[lambda1, lambda2]=\mathsf{sort}([lambda1, lambda2])\]

\[[phi1, phi2]=\mathsf{sort}([phi1, phi2])\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -1.9759046280288937 \cdot 10^{+177}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \sqrt{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \log \left(e^{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)}\right)} - \lambda_1 \cdot \sqrt{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \log \left(e^{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)}\right)}\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -3.34465013438458 \cdot 10^{+88}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -3.0263213508390754 \cdot 10^{+71}:\\ \;\;\;\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt[3]{{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}^{3}}\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -1.1864276945228447 \cdot 10^{+25} \lor \neg \left(\lambda_1 - \lambda_2 \leq -1.320279479473137 \cdot 10^{-59}\right):\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \frac{\left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \left(\left(\left(\lambda_1 + \lambda_2\right) \cdot \left(\lambda_1 + \lambda_2\right)\right) \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}\right)}{\left(\lambda_1 + \lambda_2\right) \cdot \left(\lambda_1 + \lambda_2\right)}}\\ \end{array}\]

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 (<= (- lambda1 lambda2) -1.9759046280288937e+177)
   (*
    R
    (-
     (*
      lambda2
      (sqrt
       (*
        (cos (* 0.5 (+ phi2 phi1)))
        (log (exp (cos (* 0.5 (+ phi2 phi1))))))))
     (*
      lambda1
      (sqrt
       (*
        (cos (* 0.5 (+ phi2 phi1)))
        (log (exp (cos (* 0.5 (+ phi2 phi1))))))))))
   (if (<= (- lambda1 lambda2) -3.34465013438458e+88)
     (* R (- phi2 phi1))
     (if (<= (- lambda1 lambda2) -3.0263213508390754e+71)
       (*
        R
        (sqrt
         (+
          (*
           (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2.0)))
           (*
            (- lambda1 lambda2)
            (cbrt (pow (cos (/ (+ phi2 phi1) 2.0)) 3.0))))
          (* (- phi1 phi2) (- phi1 phi2)))))
       (if (or (<= (- lambda1 lambda2) -1.1864276945228447e+25)
               (not (<= (- lambda1 lambda2) -1.320279479473137e-59)))
         (* R (- phi2 phi1))
         (*
          R
          (sqrt
           (+
            (* (- phi1 phi2) (- phi1 phi2))
            (/
             (*
              (* (cos (/ (+ phi2 phi1) 2.0)) (cos (/ (+ phi2 phi1) 2.0)))
              (*
               (* (+ lambda1 lambda2) (+ lambda1 lambda2))
               (pow (- lambda1 lambda2) 2.0)))
             (* (+ lambda1 lambda2) (+ lambda1 lambda2)))))))))))

C

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

↓

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda1 - lambda2) <= -1.9759046280288937e+177) {
		tmp = R * ((lambda2 * sqrt(cos(0.5 * (phi2 + phi1)) * log(exp(cos(0.5 * (phi2 + phi1)))))) - (lambda1 * sqrt(cos(0.5 * (phi2 + phi1)) * log(exp(cos(0.5 * (phi2 + phi1)))))));
	} else if ((lambda1 - lambda2) <= -3.34465013438458e+88) {
		tmp = R * (phi2 - phi1);
	} else if ((lambda1 - lambda2) <= -3.0263213508390754e+71) {
		tmp = R * sqrt((((lambda1 - lambda2) * cos((phi2 + phi1) / 2.0)) * ((lambda1 - lambda2) * cbrt(pow(cos((phi2 + phi1) / 2.0), 3.0)))) + ((phi1 - phi2) * (phi1 - phi2)));
	} else if (((lambda1 - lambda2) <= -1.1864276945228447e+25) || !((lambda1 - lambda2) <= -1.320279479473137e-59)) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = R * sqrt(((phi1 - phi2) * (phi1 - phi2)) + (((cos((phi2 + phi1) / 2.0) * cos((phi2 + phi1) / 2.0)) * (((lambda1 + lambda2) * (lambda1 + lambda2)) * pow((lambda1 - lambda2), 2.0))) / ((lambda1 + lambda2) * (lambda1 + lambda2))));
	}
	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 4 regimes

2. if (-.f64 lambda1 lambda2) < -1.9759046280288937e177

   1. Initial program 64.0

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

   3. Applied add-log-exp_binary64_1481 64.0

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

   4. Simplified 64.0

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

   5. Taylor expanded around -inf 20.3

      \[\leadsto R \cdot \color{blue}{\left(\sqrt{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \log \left(e^{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)}\right)} \cdot \lambda_2 - \sqrt{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \log \left(e^{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)}\right)} \cdot \lambda_1\right)}\]

   if -1.9759046280288937e177 < (-.f64 lambda1 lambda2) < -3.34465013438458023e88 or -3.0263213508390754e71 < (-.f64 lambda1 lambda2) < -1.18642769452284472e25 or -1.320279479473137e-59 < (-.f64 lambda1 lambda2)

   1. Initial program 29.5

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

   2. Taylor expanded around -inf 20.2

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 - \phi_1\right)}\]

   if -3.34465013438458023e88 < (-.f64 lambda1 lambda2) < -3.0263213508390754e71

   1. Initial program 23.6

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

   3. Applied add-cbrt-cube_binary64_1478 23.6

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

   4. Simplified 23.6

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

   if -1.18642769452284472e25 < (-.f64 lambda1 lambda2) < -1.320279479473137e-59

   1. Initial program 21.7

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

   3. Applied flip--_binary64_1417 21.7

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

   4. Applied associate-*l/_binary64_1385 21.7

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

   5. Applied flip--_binary64_1417 21.7

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

   6. Applied associate-*l/_binary64_1385 21.7

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

   7. Applied frac-times_binary64_1452 21.7

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

   8. Simplified 21.7

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

   9. Simplified 21.7

      \[\leadsto R \cdot \sqrt{\frac{\left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \left(\left(\left(\lambda_2 + \lambda_1\right) \cdot \left(\lambda_2 + \lambda_1\right)\right) \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}\right)}{\color{blue}{\left(\lambda_2 + \lambda_1\right) \cdot \left(\lambda_2 + \lambda_1\right)}} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification 20.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -1.9759046280288937 \cdot 10^{+177}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \sqrt{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \log \left(e^{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)}\right)} - \lambda_1 \cdot \sqrt{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \log \left(e^{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)}\right)}\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -3.34465013438458 \cdot 10^{+88}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -3.0263213508390754 \cdot 10^{+71}:\\ \;\;\;\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt[3]{{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}^{3}}\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -1.1864276945228447 \cdot 10^{+25} \lor \neg \left(\lambda_1 - \lambda_2 \leq -1.320279479473137 \cdot 10^{-59}\right):\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \frac{\left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \left(\left(\left(\lambda_1 + \lambda_2\right) \cdot \left(\lambda_1 + \lambda_2\right)\right) \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}\right)}{\left(\lambda_1 + \lambda_2\right) \cdot \left(\lambda_1 + \lambda_2\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021060 
(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))))))