Average Error: 37.1 → 27.6
Time: 7.1m
Precision: 64
Internal Precision: 1344
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\phi_2 - \phi_1 \le -4.022296368567685 \cdot 10^{+227}:\\ \;\;\;\;-\left(\frac{\lambda_1 \cdot \left(R \cdot \left(\phi_2 \cdot \lambda_2\right)\right)}{{\phi_1}^{2}} + \left(\frac{\lambda_1 \cdot \left(R \cdot \lambda_2\right)}{\phi_1} + R \cdot \phi_2\right)\right)\\ \mathbf{if}\;\phi_2 - \phi_1 \le 5.557209968373049 \cdot 10^{-258}:\\ \;\;\;\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{if}\;\phi_2 - \phi_1 \le 1.613092116503615 \cdot 10^{-200}:\\ \;\;\;\;\left(\lambda_1 - \lambda_2\right) \cdot R\\ \mathbf{if}\;\phi_2 - \phi_1 \le 1.934980837305936 \cdot 10^{+115}:\\ \;\;\;\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array}\]

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Split input into 4 regimes

  2. if (- phi2 phi1) < -4.022296368567685e+227

    1. Initial program 60.8

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

    2. Taylor expanded around 0 60.9

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

    3. Applied simplify60.8

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

    4. Taylor expanded around inf 52.9

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

    if -4.022296368567685e+227 < (- phi2 phi1) < 5.557209968373049e-258 or 1.613092116503615e-200 < (- phi2 phi1) < 1.934980837305936e+115

    1. Initial program 27.8

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

    if 5.557209968373049e-258 < (- phi2 phi1) < 1.613092116503615e-200

    1. Initial program 22.5

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

    2. Taylor expanded around 0 22.5

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

    3. Applied simplify22.5

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

    4. Taylor expanded around 0 29.5

      \[\leadsto \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R\]

    if 1.934980837305936e+115 < (- phi2 phi1)

    1. Initial program 53.5

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

    2. Taylor expanded around 0 16.3

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

Runtime

Time bar (total: 7.1m)Debug logProfile

herbie shell --seed '#(1071215679 2002590028 935158157 1944352234 2656991306 2955288481)' 
(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))))))