Average Error: 37.2 → 28.9
Time: 5.1m
Precision: 64
Internal Precision: 1344
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \le 4.986782136186872 \cdot 10^{+305}:\\ \;\;\;\;\sqrt{\left(\lambda_1 \cdot \lambda_1 + \lambda_2 \cdot \left(\lambda_2 - \lambda_1 \cdot 2\right)\right) \cdot \left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right) \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array}\]

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 (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2))) < 4.986782136186872e+305

    1. Initial program 2.0

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

    2. Taylor expanded around inf 2.0

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

    3. Simplified2.0

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

    if 4.986782136186872e+305 < (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2)))

    1. Initial program 60.5

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

    2. Taylor expanded around 0 46.6

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

  4. Final simplification28.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \le 4.986782136186872 \cdot 10^{+305}:\\ \;\;\;\;\sqrt{\left(\lambda_1 \cdot \lambda_1 + \lambda_2 \cdot \left(\lambda_2 - \lambda_1 \cdot 2\right)\right) \cdot \left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right) \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array}\]

Runtime

Time bar (total: 5.1m)Debug logProfile

herbie shell --seed 2018221 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Equirectangular approximation to distance on a great circle"
  (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2))))))