Equirectangular approximation to distance on a great circle

Average Error: 39.1 → 24.3

Time: 33.2s

Precision: binary64

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

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.9598478054591583 \cdot 10^{+65}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -1.0463587714142531 \cdot 10^{-75}:\\ \;\;\;\;R \cdot \sqrt{{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)}^{2} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\phi_1 \leq 5.231931449245959 \cdot 10^{-233}:\\ \;\;\;\;\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right) \cdot \left(R \cdot \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 9.654991117266558 \cdot 10^{-126}:\\ \;\;\;\;R \cdot \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + {\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)}^{2} \cdot \left(\lambda_1 \cdot \lambda_1\right)}\\ \mathbf{elif}\;\phi_1 \leq 2.1327685869242195 \cdot 10^{-69}:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot \left(R \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array}\]

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 -1.9598478054591583e+65)
   (* R (- phi2 phi1))
   (if (<= phi1 -1.0463587714142531e-75)
     (*
      R
      (sqrt
       (+
        (pow (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) 2.0)
        (* (- phi1 phi2) (- phi1 phi2)))))
     (if (<= phi1 5.231931449245959e-233)
       (* (cos (* (+ phi1 phi2) 0.5)) (* R (- lambda2 lambda1)))
       (if (<= phi1 9.654991117266558e-126)
         (*
          R
          (sqrt
           (+
            (* (- phi1 phi2) (- phi1 phi2))
            (* (pow (cos (* (+ phi1 phi2) 0.5)) 2.0) (* lambda1 lambda1)))))
         (if (<= phi1 2.1327685869242195e-69)
           (* (- lambda2 lambda1) (* R (cos (* (+ phi1 phi2) 0.5))))
           (* R phi2)))))))

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 <= -1.9598478054591583e+65) {
		tmp = R * (phi2 - phi1);
	} else if (phi1 <= -1.0463587714142531e-75) {
		tmp = R * sqrt(pow(((lambda1 - lambda2) * cos((phi1 + phi2) / 2.0)), 2.0) + ((phi1 - phi2) * (phi1 - phi2)));
	} else if (phi1 <= 5.231931449245959e-233) {
		tmp = cos((phi1 + phi2) * 0.5) * (R * (lambda2 - lambda1));
	} else if (phi1 <= 9.654991117266558e-126) {
		tmp = R * sqrt(((phi1 - phi2) * (phi1 - phi2)) + (pow(cos((phi1 + phi2) * 0.5), 2.0) * (lambda1 * lambda1)));
	} else if (phi1 <= 2.1327685869242195e-69) {
		tmp = (lambda2 - lambda1) * (R * cos((phi1 + phi2) * 0.5));
	} else {
		tmp = R * phi2;
	}
	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 6 regimes

2. if phi1 < -1.95984780545915831e65

   1. Initial program 51.1

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

   3. Applied add-cube-cbrt_binary64_795 51.1

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

   4. Simplified 51.1

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

   5. Simplified 51.1

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

   6. Taylor expanded around -inf 18.0

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

   if -1.95984780545915831e65 < phi1 < -1.0463587714142531e-75

   1. Initial program 28.0

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

   3. Applied pow1_binary64_821 28.0

      \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}^{1}}\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

   4. Applied pow1_binary64_821 28.0

      \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\color{blue}{{\left(\lambda_1 - \lambda_2\right)}^{1}} \cdot {\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}^{1}\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

   5. Applied pow-prod-down_binary64_831 28.0

      \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \color{blue}{{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)}^{1}} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

   6. Applied pow1_binary64_821 28.0

      \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}^{1}}\right) \cdot {\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)}^{1} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

   7. Applied pow1_binary64_821 28.0

      \[\leadsto R \cdot \sqrt{\left(\color{blue}{{\left(\lambda_1 - \lambda_2\right)}^{1}} \cdot {\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}^{1}\right) \cdot {\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)}^{1} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

   8. Applied pow-prod-down_binary64_831 28.0

      \[\leadsto R \cdot \sqrt{\color{blue}{{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)}^{1}} \cdot {\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)}^{1} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

   9. Applied pow-prod-up_binary64_830 28.0

      \[\leadsto R \cdot \sqrt{\color{blue}{{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)}^{\left(1 + 1\right)}} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

   10. Simplified 28.0

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

   if -1.0463587714142531e-75 < phi1 < 5.2319314492459587e-233

   1. Initial program 31.4

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

   2. Taylor expanded around -inf 24.7

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

   3. Simplified 24.7

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

   4. Taylor expanded around 0 24.7

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

   5. Simplified 24.7

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

   if 5.2319314492459587e-233 < phi1 < 9.6549911172665581e-126

   1. Initial program 33.3

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

   2. Taylor expanded around inf 40.4

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

   3. Simplified 40.4

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

   if 9.6549911172665581e-126 < phi1 < 2.1327685869242195e-69

   1. Initial program 38.2

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

   2. Taylor expanded around -inf 40.0

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

   3. Simplified 40.0

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

   5. Applied associate-*r*_binary64_700 40.0

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

   if 2.1327685869242195e-69 < phi1

   1. Initial program 49.1

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

   3. Applied add-cube-cbrt_binary64_795 49.1

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

   4. Simplified 49.1

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

   5. Simplified 49.1

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

   6. Taylor expanded around inf 20.6

      \[\leadsto R \cdot \color{blue}{\phi_2}\]
3. Recombined 6 regimes into one program.

4. Final simplification 24.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.9598478054591583 \cdot 10^{+65}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -1.0463587714142531 \cdot 10^{-75}:\\ \;\;\;\;R \cdot \sqrt{{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)}^{2} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\phi_1 \leq 5.231931449245959 \cdot 10^{-233}:\\ \;\;\;\;\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right) \cdot \left(R \cdot \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 9.654991117266558 \cdot 10^{-126}:\\ \;\;\;\;R \cdot \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + {\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)}^{2} \cdot \left(\lambda_1 \cdot \lambda_1\right)}\\ \mathbf{elif}\;\phi_1 \leq 2.1327685869242195 \cdot 10^{-69}:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot \left(R \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array}\]

Reproduce

herbie shell --seed 2021098 
(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))))))