Equirectangular approximation to distance on a great circle

Average Error: 38.6 → 20.1

Time: 19.7s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)\\ t_1 := \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ \mathbf{if}\;\begin{array}{l} t_2 := R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot t_1\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot t_1\right) + t_0}\\ t_3 := \left(\lambda_1 - \lambda_2\right) \cdot t_1\\ t_2 \leq -\infty \lor \neg \left(t_2 \leq 9.584291413480771 \cdot 10^{+240}\right) \end{array}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \sqrt{t_0 + {t_1}^{2} \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}\\ \end{array} \]

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
 (let* ((t_0 (* (- phi1 phi2) (- phi1 phi2)))
        (t_1 (cos (/ (+ phi1 phi2) 2.0))))
   (if (let* ((t_2
               (*
                R
                (sqrt
                 (+
                  (* (* (- lambda1 lambda2) t_1) (* (- lambda1 lambda2) t_1))
                  t_0))))
              (t_3 (* (- lambda1 lambda2) t_1)))
         (or (<= t_2 (- INFINITY)) (not (<= t_2 9.584291413480771e+240))))
     (* R (- phi2 phi1))
     (*
      R
      (sqrt
       (+
        t_0
        (* (pow t_1 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 t_0 = (phi1 - phi2) * (phi1 - phi2);
	double t_1 = cos((phi1 + phi2) / 2.0);
	double t_2 = R * sqrt((((lambda1 - lambda2) * t_1) * ((lambda1 - lambda2) * t_1)) + t_0);
	double t_3 = (lambda1 - lambda2) * t_1;
	double tmp;
	if ((t_2 <= -((double) INFINITY)) || !(t_2 <= 9.584291413480771e+240)) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = R * sqrt(t_0 + (pow(t_1, 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 2 regimes

2. if (*.f64 R (sqrt.f64 (+.f64 (*.f64 (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) 2))) (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) 2)))) (*.f64 (-.f64 phi1 phi2) (-.f64 phi1 phi2))))) < -inf.0 or 9.58429141348077142e240 < (*.f64 R (sqrt.f64 (+.f64 (*.f64 (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) 2))) (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) 2)))) (*.f64 (-.f64 phi1 phi2) (-.f64 phi1 phi2)))))

   1. Initial program 62.0

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

   2. Taylor expanded around -inf 31.6

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

   if -inf.0 < (*.f64 R (sqrt.f64 (+.f64 (*.f64 (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) 2))) (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) 2)))) (*.f64 (-.f64 phi1 phi2) (-.f64 phi1 phi2))))) < 9.58429141348077142e240

   1. Initial program 1.9

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

   3. Applied add-cbrt-cube_binary64 2.0

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

   4. Simplified 2.0

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

   6. Applied *-un-lft-identity_binary64 2.0

      \[\leadsto R \cdot \sqrt{\color{blue}{1 \cdot \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt[3]{{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}^{3}}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)\right)}} \]

   7. Applied sqrt-prod_binary64 2.0

      \[\leadsto R \cdot \color{blue}{\left(\sqrt{1} \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt[3]{{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}^{3}}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)} \]

   8. Simplified 2.0

      \[\leadsto R \cdot \left(\color{blue}{1} \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt[3]{{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}^{3}}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right) \]

   9. Simplified 2.0

      \[\leadsto R \cdot \left(1 \cdot \color{blue}{\sqrt{{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}^{2} \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 20.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \leq -\infty \lor \neg \left(R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \leq 9.584291413480771 \cdot 10^{+240}\right):\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + {\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}^{2} \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021205 
(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))))))