Average Error: 39.3 → 34.0
Time: 18.4s
Precision: binary64
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.5973862171465132 \cdot 10^{-05}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1 + \left(\cos \phi_1 \cdot \cos \phi_2 - \sin \phi_1 \cdot \sin \phi_2\right)}{2}\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array}\]
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_2 \leq 2.5973862171465132 \cdot 10^{-05}:\\
\;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1 + \left(\cos \phi_1 \cdot \cos \phi_2 - \sin \phi_1 \cdot \sin \phi_2\right)}{2}\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\

\mathbf{else}:\\
\;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\

\end{array}
(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 (<= phi2 2.5973862171465132e-05)
   (*
    R
    (sqrt
     (+
      (*
       (- lambda1 lambda2)
       (*
        (- lambda1 lambda2)
        (/
         (+ 1.0 (- (* (cos phi1) (cos phi2)) (* (sin phi1) (sin phi2))))
         2.0)))
      (* (- phi1 phi2) (- phi1 phi2)))))
   (* R (- phi2 phi1))))
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 (phi2 <= 2.5973862171465132e-05) {
		tmp = R * sqrt(((lambda1 - lambda2) * ((lambda1 - lambda2) * ((1.0 + ((cos(phi1) * cos(phi2)) - (sin(phi1) * sin(phi2)))) / 2.0))) + ((phi1 - phi2) * (phi1 - phi2)));
	} 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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if phi2 < 2.5973862171465132e-5

    1. Initial program 36.6

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

    3. Applied associate-*l*_binary64_105236.7

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

    4. Simplified36.7

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

    6. Applied cos-mult_binary64_126536.7

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

    7. Simplified36.7

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

    9. Applied cos-sum_binary64_124536.3

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

    if 2.5973862171465132e-5 < phi2

    1. Initial program 48.0

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

    2. Taylor expanded around 0 26.6

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 - \phi_1\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification34.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.5973862171465132 \cdot 10^{-05}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1 + \left(\cos \phi_1 \cdot \cos \phi_2 - \sin \phi_1 \cdot \sin \phi_2\right)}{2}\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020277 
(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))))))