Equirectangular approximation to distance on a great circle

Average Error: 38.8 → 24.6

Time: 16.8s

Precision: binary64

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

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -106049510029.62999:\\ \;\;\;\;\phi_2 \cdot R - \phi_1 \cdot R\\ \mathbf{elif}\;\phi_1 \leq -7.172364672971488 \cdot 10^{-118}:\\ \;\;\;\;R \cdot \left(\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -2.5266019225073433 \cdot 10^{-148}:\\ \;\;\;\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)}\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\phi_1 \leq 1.7477001439068225 \cdot 10^{-203}:\\ \;\;\;\;R \cdot \left(\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 2.287702190753648 \cdot 10^{-124}:\\ \;\;\;\;\phi_2 \cdot R\\ \mathbf{elif}\;\phi_1 \leq 1.3774337930017817 \cdot 10^{-42}:\\ \;\;\;\;R \cdot \left(\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array}\]

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 (<= phi1 -106049510029.62999)
   (- (* phi2 R) (* phi1 R))
   (if (<= phi1 -7.172364672971488e-118)
     (* R (* (cos (* 0.5 (+ phi1 phi2))) (- lambda2 lambda1)))
     (if (<= phi1 -2.5266019225073433e-148)
       (*
        R
        (sqrt
         (+
          (*
           (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))
           (*
            (- lambda1 lambda2)
            (cbrt
             (*
              (cos (/ (+ phi1 phi2) 2.0))
              (* (cos (/ (+ phi1 phi2) 2.0)) (cos (/ (+ phi1 phi2) 2.0)))))))
          (* (- phi1 phi2) (- phi1 phi2)))))
       (if (<= phi1 1.7477001439068225e-203)
         (* R (* (cos (* 0.5 (+ phi1 phi2))) (- lambda2 lambda1)))
         (if (<= phi1 2.287702190753648e-124)
           (* phi2 R)
           (if (<= phi1 1.3774337930017817e-42)
             (* R (* (cos (* 0.5 (+ phi1 phi2))) (- lambda2 lambda1)))
             (* R (- phi2 phi1)))))))))

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 (phi1 <= -106049510029.62999) {
		tmp = (phi2 * R) - (phi1 * R);
	} else if (phi1 <= -7.172364672971488e-118) {
		tmp = R * (cos(0.5 * (phi1 + phi2)) * (lambda2 - lambda1));
	} else if (phi1 <= -2.5266019225073433e-148) {
		tmp = R * sqrt((((lambda1 - lambda2) * cos((phi1 + phi2) / 2.0)) * ((lambda1 - lambda2) * cbrt(cos((phi1 + phi2) / 2.0) * (cos((phi1 + phi2) / 2.0) * cos((phi1 + phi2) / 2.0))))) + ((phi1 - phi2) * (phi1 - phi2)));
	} else if (phi1 <= 1.7477001439068225e-203) {
		tmp = R * (cos(0.5 * (phi1 + phi2)) * (lambda2 - lambda1));
	} else if (phi1 <= 2.287702190753648e-124) {
		tmp = phi2 * R;
	} else if (phi1 <= 1.3774337930017817e-42) {
		tmp = R * (cos(0.5 * (phi1 + phi2)) * (lambda2 - lambda1));
	} else {
		tmp = R * (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 5 regimes

2. if phi1 < -106049510029.62999

   1. Initial program 47.5

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

   2. Taylor expanded around inf 39.2

      \[\leadsto R \cdot \color{blue}{\left(\left(\phi_2 + \left(0.5 \cdot \frac{{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)}^{2} \cdot {\lambda_2}^{2}}{\phi_2} + 0.5 \cdot \frac{{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)}^{2} \cdot {\lambda_1}^{2}}{\phi_2}\right)\right) - \left(\frac{{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)}^{2} \cdot \left(\lambda_2 \cdot \lambda_1\right)}{\phi_2} + \phi_1\right)\right)}\]

   3. Simplified 39.2

      \[\leadsto R \cdot \color{blue}{\left(\left(\phi_2 + 0.5 \cdot \left(\frac{{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)}^{2} \cdot \left(\lambda_2 \cdot \lambda_2\right)}{\phi_2} + \frac{{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)}^{2} \cdot \left(\lambda_1 \cdot \lambda_1\right)}{\phi_2}\right)\right) - \left(\phi_1 + \frac{{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)}^{2} \cdot \left(\lambda_2 \cdot \lambda_1\right)}{\phi_2}\right)\right)}\]

   4. Taylor expanded around inf 20.8

      \[\leadsto \color{blue}{\phi_2 \cdot R - \phi_1 \cdot R}\]

   if -106049510029.62999 < phi1 < -7.17236467297148783e-118 or -2.5266019225073433e-148 < phi1 < 1.7477001439068225e-203 or 2.28770219075364804e-124 < phi1 < 1.3774337930017817e-42

   1. Initial program 30.2

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

   2. Taylor expanded around -inf 27.6

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

   3. Simplified 27.6

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

   if -7.17236467297148783e-118 < phi1 < -2.5266019225073433e-148

   1. Initial program 25.7

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

   3. Applied add-cbrt-cube_binary64 25.7

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

   if 1.7477001439068225e-203 < phi1 < 2.28770219075364804e-124

   1. Initial program 36.0

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

   2. Taylor expanded around inf 38.4

      \[\leadsto R \cdot \color{blue}{\left(\left(\phi_2 + \left(0.5 \cdot \frac{{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)}^{2} \cdot {\lambda_2}^{2}}{\phi_2} + 0.5 \cdot \frac{{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)}^{2} \cdot {\lambda_1}^{2}}{\phi_2}\right)\right) - \left(\frac{{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)}^{2} \cdot \left(\lambda_2 \cdot \lambda_1\right)}{\phi_2} + \phi_1\right)\right)}\]

   3. Simplified 38.4

      \[\leadsto R \cdot \color{blue}{\left(\left(\phi_2 + 0.5 \cdot \left(\frac{{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)}^{2} \cdot \left(\lambda_2 \cdot \lambda_2\right)}{\phi_2} + \frac{{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)}^{2} \cdot \left(\lambda_1 \cdot \lambda_1\right)}{\phi_2}\right)\right) - \left(\phi_1 + \frac{{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)}^{2} \cdot \left(\lambda_2 \cdot \lambda_1\right)}{\phi_2}\right)\right)}\]

   4. Taylor expanded around inf 33.5

      \[\leadsto \color{blue}{\phi_2 \cdot R}\]

   if 1.3774337930017817e-42 < phi1

   1. Initial program 50.7

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

   2. Taylor expanded around inf 30.5

      \[\leadsto R \cdot \color{blue}{\left(\left(\phi_2 + \left(0.5 \cdot \frac{{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)}^{2} \cdot {\lambda_2}^{2}}{\phi_2} + 0.5 \cdot \frac{{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)}^{2} \cdot {\lambda_1}^{2}}{\phi_2}\right)\right) - \left(\frac{{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)}^{2} \cdot \left(\lambda_2 \cdot \lambda_1\right)}{\phi_2} + \phi_1\right)\right)}\]

   3. Simplified 30.5

      \[\leadsto R \cdot \color{blue}{\left(\left(\phi_2 + 0.5 \cdot \left(\frac{{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)}^{2} \cdot \left(\lambda_2 \cdot \lambda_2\right)}{\phi_2} + \frac{{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)}^{2} \cdot \left(\lambda_1 \cdot \lambda_1\right)}{\phi_2}\right)\right) - \left(\phi_1 + \frac{{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)}^{2} \cdot \left(\lambda_2 \cdot \lambda_1\right)}{\phi_2}\right)\right)}\]

   4. Taylor expanded around 0 26.1

      \[\leadsto \color{blue}{R \cdot \left(\left(\phi_2 + 0.5 \cdot \frac{{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)}^{2} \cdot {\lambda_1}^{2}}{\phi_2}\right) - \phi_1\right)}\]

   5. Simplified 26.1

      \[\leadsto \color{blue}{R \cdot \left(\left(\phi_2 + 0.5 \cdot \frac{{\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} \cdot \left(\lambda_1 \cdot \lambda_1\right)}{\phi_2}\right) - \phi_1\right)}\]

   6. Taylor expanded around 0 21.2

      \[\leadsto \color{blue}{\left(\phi_2 - \phi_1\right) \cdot R}\]
3. Recombined 5 regimes into one program.

4. Final simplification 24.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -106049510029.62999:\\ \;\;\;\;\phi_2 \cdot R - \phi_1 \cdot R\\ \mathbf{elif}\;\phi_1 \leq -7.172364672971488 \cdot 10^{-118}:\\ \;\;\;\;R \cdot \left(\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -2.5266019225073433 \cdot 10^{-148}:\\ \;\;\;\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)}\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\phi_1 \leq 1.7477001439068225 \cdot 10^{-203}:\\ \;\;\;\;R \cdot \left(\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 2.287702190753648 \cdot 10^{-124}:\\ \;\;\;\;\phi_2 \cdot R\\ \mathbf{elif}\;\phi_1 \leq 1.3774337930017817 \cdot 10^{-42}:\\ \;\;\;\;R \cdot \left(\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021173 
(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))))))