Average Error: 39.5 → 19.4
Time: 34.1s
Precision: binary64
\[[phi1, phi2]=\mathsf{sort}([phi1, phi2])\]
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) \leq 2.3672316897985693 \cdot 10^{+307}:\\ \;\;\;\;R \cdot \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \frac{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2\right)}{\lambda_1 + \lambda_2}}\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R - \phi_1 \cdot R\\ \end{array}\]
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}\;\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 2.3672316897985693 \cdot 10^{+307}:\\
\;\;\;\;R \cdot \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \frac{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2\right)}{\lambda_1 + \lambda_2}}\\

\mathbf{else}:\\
\;\;\;\;\phi_2 \cdot R - \phi_1 \cdot R\\

\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 (<=
      (+
       (*
        (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))
        (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))))
       (* (- phi1 phi2) (- phi1 phi2)))
      2.3672316897985693e+307)
   (*
    R
    (sqrt
     (+
      (* (- phi1 phi2) (- phi1 phi2))
      (*
       (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))
       (/
        (*
         (cos (/ (+ phi1 phi2) 2.0))
         (- (* lambda1 lambda1) (* lambda2 lambda2)))
        (+ lambda1 lambda2))))))
   (- (* phi2 R) (* phi1 R))))
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 - lambda2) * cos((phi1 + phi2) / 2.0)) * ((lambda1 - lambda2) * cos((phi1 + phi2) / 2.0))) + ((phi1 - phi2) * (phi1 - phi2))) <= 2.3672316897985693e+307) {
		tmp = R * sqrt(((phi1 - phi2) * (phi1 - phi2)) + (((lambda1 - lambda2) * cos((phi1 + phi2) / 2.0)) * ((cos((phi1 + phi2) / 2.0) * ((lambda1 * lambda1) - (lambda2 * lambda2))) / (lambda1 + lambda2))));
	} else {
		tmp = (phi2 * R) - (phi1 * R);
	}
	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 (+.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))) < 2.3672316897985693e307

    1. Initial program 1.7

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

    3. Applied flip--_binary64_34631.7

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

    4. Applied associate-*l/_binary64_34311.7

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

    if 2.3672316897985693e307 < (+.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 63.8

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

    2. Taylor expanded around -inf 30.7

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 - \phi_1\right)}\]
    3. Using strategy rm

    4. Applied sub-neg_binary64_348130.7

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

    5. Applied distribute-rgt-in_binary64_343830.7

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

    6. Simplified30.7

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

  4. Final simplification19.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) \leq 2.3672316897985693 \cdot 10^{+307}:\\ \;\;\;\;R \cdot \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \frac{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2\right)}{\lambda_1 + \lambda_2}}\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R - \phi_1 \cdot R\\ \end{array}\]

Reproduce

herbie shell --seed 2021050 
(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))))))