Average Error: 37.6 → 34.4
Time: 3.9m
Precision: 64
Internal Precision: 1344
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\phi_2 \le -2.4351568138910192 \cdot 10^{+169}:\\ \;\;\;\;\left(\left(-\phi_2\right) - \frac{\left(\lambda_1 \cdot \lambda_2 + \frac{\lambda_1 \cdot \phi_2}{\frac{\phi_1}{\lambda_2}}\right) \cdot \cos \left(\left(\phi_2 + \phi_1\right) \cdot \frac{1}{2}\right)}{\frac{\phi_1}{\cos \left(\left(\phi_2 + \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \cdot R\\ \mathbf{elif}\;\phi_2 \le 2.5274423367856496 \cdot 10^{+174}:\\ \;\;\;\;\sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)} \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\left(\cos \left(\left(\phi_2 + \phi_1\right) \cdot \frac{1}{2}\right) \cdot \cos \left(\left(\phi_2 + \phi_1\right) \cdot \frac{1}{2}\right)\right) \cdot \left(\frac{\frac{\lambda_1}{\phi_1} \cdot \left(\phi_2 \cdot \lambda_2\right)}{\phi_1} + \frac{\lambda_2}{\frac{\phi_1}{\lambda_1}}\right) + \phi_2\right)\\ \end{array}\]

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 3 regimes

  2. if phi2 < -2.4351568138910192e+169

    1. Initial program 60.9

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

    2. Initial simplification60.9

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

    3. Taylor expanded around inf 45.9

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

    4. Simplified44.2

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

    if -2.4351568138910192e+169 < phi2 < 2.5274423367856496e+174

    1. Initial program 32.1

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

    2. Initial simplification32.1

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

    if 2.5274423367856496e+174 < phi2

    1. Initial program 60.9

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

    2. Initial simplification60.9

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

    4. Applied add-exp-log60.9

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

    6. Applied add-exp-log61.2

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

    7. Taylor expanded around -inf 46.1

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

    8. Simplified44.3

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

  4. Final simplification34.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \le -2.4351568138910192 \cdot 10^{+169}:\\ \;\;\;\;\left(\left(-\phi_2\right) - \frac{\left(\lambda_1 \cdot \lambda_2 + \frac{\lambda_1 \cdot \phi_2}{\frac{\phi_1}{\lambda_2}}\right) \cdot \cos \left(\left(\phi_2 + \phi_1\right) \cdot \frac{1}{2}\right)}{\frac{\phi_1}{\cos \left(\left(\phi_2 + \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \cdot R\\ \mathbf{elif}\;\phi_2 \le 2.5274423367856496 \cdot 10^{+174}:\\ \;\;\;\;\sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)} \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\left(\cos \left(\left(\phi_2 + \phi_1\right) \cdot \frac{1}{2}\right) \cdot \cos \left(\left(\phi_2 + \phi_1\right) \cdot \frac{1}{2}\right)\right) \cdot \left(\frac{\frac{\lambda_1}{\phi_1} \cdot \left(\phi_2 \cdot \lambda_2\right)}{\phi_1} + \frac{\lambda_2}{\frac{\phi_1}{\lambda_1}}\right) + \phi_2\right)\\ \end{array}\]

Runtime

Time bar (total: 3.9m)Debug logProfile

herbie shell --seed 2018273 
(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))))))