Equirectangular approximation to distance on a great circle

Average Error: 39.0 → 22.7

Time: 15.9s

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 \leq -4.6714712786393294 \cdot 10^{+92}:\\ \;\;\;\;R \cdot \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 - \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 \leq -318.0908199786162:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_1 \leq -0.00028256165896540234:\\ \;\;\;\;R \cdot \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 - \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 \leq -9.748496887418038 \cdot 10^{-67}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_1 \leq -2.2368391435846272 \cdot 10^{-98}:\\ \;\;\;\;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 \left(\sqrt[3]{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)} \cdot \left(\sqrt[3]{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)} \cdot \sqrt[3]{\cos \left(0.5 \cdot \phi_1\right)}\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\lambda_1 \leq 2.2078159103029802 \cdot 10^{-30}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \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 - \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)\\ \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 -4.6714712786393294e+92)
   (*
    R
    (-
     (*
      (sqrt
       (* (cos (* 0.5 (+ phi2 phi1))) (log (exp (cos (* 0.5 (+ phi2 phi1)))))))
      lambda2)
     (*
      lambda1
      (sqrt
       (*
        (cos (* 0.5 (+ phi2 phi1)))
        (log (exp (cos (* 0.5 (+ phi2 phi1))))))))))
   (if (<= lambda1 -318.0908199786162)
     (* R (- phi2 phi1))
     (if (<= lambda1 -0.00028256165896540234)
       (*
        R
        (-
         (*
          (sqrt
           (*
            (cos (* 0.5 (+ phi2 phi1)))
            (log (exp (cos (* 0.5 (+ phi2 phi1)))))))
          lambda2)
         (*
          lambda1
          (sqrt
           (*
            (cos (* 0.5 (+ phi2 phi1)))
            (log (exp (cos (* 0.5 (+ phi2 phi1))))))))))
       (if (<= lambda1 -9.748496887418038e-67)
         (* R (- phi2 phi1))
         (if (<= lambda1 -2.2368391435846272e-98)
           (*
            R
            (sqrt
             (+
              (*
               (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2.0)))
               (*
                (- lambda1 lambda2)
                (*
                 (cbrt (cos (/ (+ phi2 phi1) 2.0)))
                 (*
                  (cbrt (cos (/ (+ phi2 phi1) 2.0)))
                  (cbrt (cos (* 0.5 phi1)))))))
              (* (- phi1 phi2) (- phi1 phi2)))))
           (if (<= lambda1 2.2078159103029802e-30)
             (* R (- phi2 phi1))
             (*
              R
              (-
               (*
                (sqrt
                 (*
                  (cos (* 0.5 (+ phi2 phi1)))
                  (log (exp (cos (* 0.5 (+ phi2 phi1)))))))
                lambda2)
               (*
                lambda1
                (sqrt
                 (*
                  (cos (* 0.5 (+ phi2 phi1)))
                  (log (exp (cos (* 0.5 (+ 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 (lambda1 <= -4.6714712786393294e+92) {
		tmp = R * ((sqrt(cos(0.5 * (phi2 + phi1)) * log(exp(cos(0.5 * (phi2 + phi1))))) * lambda2) - (lambda1 * sqrt(cos(0.5 * (phi2 + phi1)) * log(exp(cos(0.5 * (phi2 + phi1)))))));
	} else if (lambda1 <= -318.0908199786162) {
		tmp = R * (phi2 - phi1);
	} else if (lambda1 <= -0.00028256165896540234) {
		tmp = R * ((sqrt(cos(0.5 * (phi2 + phi1)) * log(exp(cos(0.5 * (phi2 + phi1))))) * lambda2) - (lambda1 * sqrt(cos(0.5 * (phi2 + phi1)) * log(exp(cos(0.5 * (phi2 + phi1)))))));
	} else if (lambda1 <= -9.748496887418038e-67) {
		tmp = R * (phi2 - phi1);
	} else if (lambda1 <= -2.2368391435846272e-98) {
		tmp = R * sqrt((((lambda1 - lambda2) * cos((phi2 + phi1) / 2.0)) * ((lambda1 - lambda2) * (cbrt(cos((phi2 + phi1) / 2.0)) * (cbrt(cos((phi2 + phi1) / 2.0)) * cbrt(cos(0.5 * phi1)))))) + ((phi1 - phi2) * (phi1 - phi2)));
	} else if (lambda1 <= 2.2078159103029802e-30) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = R * ((sqrt(cos(0.5 * (phi2 + phi1)) * log(exp(cos(0.5 * (phi2 + phi1))))) * lambda2) - (lambda1 * sqrt(cos(0.5 * (phi2 + phi1)) * log(exp(cos(0.5 * (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 3 regimes

2. if lambda1 < -4.67147127863932939e92 or -318.0908199786162 < lambda1 < -2.8256165896540234e-4 or 2.2078159103029802e-30 < lambda1

   1. Initial program 52.9

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

   3. Applied add-log-exp_binary64 52.9

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

   4. Simplified 52.9

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

   5. Taylor expanded around -inf 23.5

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

   if -4.67147127863932939e92 < lambda1 < -318.0908199786162 or -2.8256165896540234e-4 < lambda1 < -9.7484968874180381e-67 or -2.2368391435846272e-98 < lambda1 < 2.2078159103029802e-30

   1. Initial program 31.3

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

   2. Taylor expanded around -inf 22.0

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

   if -9.7484968874180381e-67 < lambda1 < -2.2368391435846272e-98

   1. Initial program 25.2

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

   3. Applied add-cube-cbrt_binary64 25.2

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

   4. Simplified 25.2

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

   5. Simplified 25.2

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

   6. Taylor expanded around 0 25.7

      \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\sqrt[3]{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)} \cdot \sqrt[3]{\color{blue}{\cos \left(0.5 \cdot \phi_1\right)}}\right) \cdot \sqrt[3]{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 22.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -4.6714712786393294 \cdot 10^{+92}:\\ \;\;\;\;R \cdot \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 - \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 \leq -318.0908199786162:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_1 \leq -0.00028256165896540234:\\ \;\;\;\;R \cdot \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 - \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 \leq -9.748496887418038 \cdot 10^{-67}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_1 \leq -2.2368391435846272 \cdot 10^{-98}:\\ \;\;\;\;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 \left(\sqrt[3]{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)} \cdot \left(\sqrt[3]{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)} \cdot \sqrt[3]{\cos \left(0.5 \cdot \phi_1\right)}\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\lambda_1 \leq 2.2078159103029802 \cdot 10^{-30}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \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 - \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)\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021110 
(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))))))