Average Error: 39.3 → 24.7
Time: 33.4s
Precision: binary64
\[[lambda1, lambda2]=\mathsf{sort}([lambda1, lambda2])\]
\[[phi1, phi2]=\mathsf{sort}([phi1, phi2])\]
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -6.7854680350671286 \cdot 10^{+190}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) - \lambda_1 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -8.528961825021155 \cdot 10^{+163}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -1.868496750933575 \cdot 10^{+149}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) - \lambda_1 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -3.565954069687537 \cdot 10^{+139}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -2.3353426615490326 \cdot 10^{+106}:\\ \;\;\;\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -6.376671243128646 \cdot 10^{+80}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -2.2960510655886167 \cdot 10^{+23}:\\ \;\;\;\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \end{array}\]
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}\;\lambda_1 - \lambda_2 \leq -6.7854680350671286 \cdot 10^{+190}:\\
\;\;\;\;R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) - \lambda_1 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\\

\mathbf{elif}\;\lambda_1 - \lambda_2 \leq -8.528961825021155 \cdot 10^{+163}:\\
\;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\

\mathbf{elif}\;\lambda_1 - \lambda_2 \leq -1.868496750933575 \cdot 10^{+149}:\\
\;\;\;\;R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) - \lambda_1 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\\

\mathbf{elif}\;\lambda_1 - \lambda_2 \leq -3.565954069687537 \cdot 10^{+139}:\\
\;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\

\mathbf{elif}\;\lambda_1 - \lambda_2 \leq -2.3353426615490326 \cdot 10^{+106}:\\
\;\;\;\;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 \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\

\mathbf{elif}\;\lambda_1 - \lambda_2 \leq -6.376671243128646 \cdot 10^{+80}:\\
\;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\

\mathbf{elif}\;\lambda_1 - \lambda_2 \leq -2.2960510655886167 \cdot 10^{+23}:\\
\;\;\;\;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 \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\

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

\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) -6.7854680350671286e+190)
   (*
    R
    (-
     (* lambda2 (cos (* 0.5 (+ phi2 phi1))))
     (* lambda1 (cos (* 0.5 (+ phi2 phi1))))))
   (if (<= (- lambda1 lambda2) -8.528961825021155e+163)
     (* R (- phi2 phi1))
     (if (<= (- lambda1 lambda2) -1.868496750933575e+149)
       (*
        R
        (-
         (* lambda2 (cos (* 0.5 (+ phi2 phi1))))
         (* lambda1 (cos (* 0.5 (+ phi2 phi1))))))
       (if (<= (- lambda1 lambda2) -3.565954069687537e+139)
         (* R (- phi2 phi1))
         (if (<= (- lambda1 lambda2) -2.3353426615490326e+106)
           (*
            R
            (sqrt
             (+
              (*
               (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2.0)))
               (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2.0))))
              (* (- phi1 phi2) (- phi1 phi2)))))
           (if (<= (- lambda1 lambda2) -6.376671243128646e+80)
             (* R (- phi2 phi1))
             (if (<= (- lambda1 lambda2) -2.2960510655886167e+23)
               (*
                R
                (sqrt
                 (+
                  (*
                   (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2.0)))
                   (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2.0))))
                  (* (- phi1 phi2) (- phi1 phi2)))))
               (- (* R phi2) (* R 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 ((lambda1 - lambda2) <= -6.7854680350671286e+190) {
		tmp = R * ((lambda2 * cos(0.5 * (phi2 + phi1))) - (lambda1 * cos(0.5 * (phi2 + phi1))));
	} else if ((lambda1 - lambda2) <= -8.528961825021155e+163) {
		tmp = R * (phi2 - phi1);
	} else if ((lambda1 - lambda2) <= -1.868496750933575e+149) {
		tmp = R * ((lambda2 * cos(0.5 * (phi2 + phi1))) - (lambda1 * cos(0.5 * (phi2 + phi1))));
	} else if ((lambda1 - lambda2) <= -3.565954069687537e+139) {
		tmp = R * (phi2 - phi1);
	} else if ((lambda1 - lambda2) <= -2.3353426615490326e+106) {
		tmp = R * sqrt((((lambda1 - lambda2) * cos((phi2 + phi1) / 2.0)) * ((lambda1 - lambda2) * cos((phi2 + phi1) / 2.0))) + ((phi1 - phi2) * (phi1 - phi2)));
	} else if ((lambda1 - lambda2) <= -6.376671243128646e+80) {
		tmp = R * (phi2 - phi1);
	} else if ((lambda1 - lambda2) <= -2.2960510655886167e+23) {
		tmp = R * sqrt((((lambda1 - lambda2) * cos((phi2 + phi1) / 2.0)) * ((lambda1 - lambda2) * cos((phi2 + phi1) / 2.0))) + ((phi1 - phi2) * (phi1 - phi2)));
	} else {
		tmp = (R * phi2) - (R * 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 4 regimes

  2. if (-.f64 lambda1 lambda2) < -6.78546803506712857e190 or -8.5289618250211547e163 < (-.f64 lambda1 lambda2) < -1.868496750933575e149

    1. Initial program 62.7

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

    2. Taylor expanded around -inf 32.8

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

    if -6.78546803506712857e190 < (-.f64 lambda1 lambda2) < -8.5289618250211547e163 or -1.868496750933575e149 < (-.f64 lambda1 lambda2) < -3.565954069687537e139 or -2.3353426615490326e106 < (-.f64 lambda1 lambda2) < -6.37667124312864593e80

    1. Initial program 42.5

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

    2. Taylor expanded around -inf 36.1

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

    if -3.565954069687537e139 < (-.f64 lambda1 lambda2) < -2.3353426615490326e106 or -6.37667124312864593e80 < (-.f64 lambda1 lambda2) < -2.29605106558861671e23

    1. Initial program 24.2

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

    if -2.29605106558861671e23 < (-.f64 lambda1 lambda2)

    1. Initial program 22.3

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

    2. Taylor expanded around -inf 10.9

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

    4. Applied sub-neg_binary64_211710.9

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

    5. Applied distribute-rgt-in_binary64_207410.9

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

    6. Simplified10.9

      \[\leadsto \phi_2 \cdot R + \color{blue}{\left(-\phi_1 \cdot R\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification24.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -6.7854680350671286 \cdot 10^{+190}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) - \lambda_1 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -8.528961825021155 \cdot 10^{+163}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -1.868496750933575 \cdot 10^{+149}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) - \lambda_1 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -3.565954069687537 \cdot 10^{+139}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -2.3353426615490326 \cdot 10^{+106}:\\ \;\;\;\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -6.376671243128646 \cdot 10^{+80}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -2.2960510655886167 \cdot 10^{+23}:\\ \;\;\;\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \end{array}\]

Reproduce

herbie shell --seed 2021093 
(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))))))