Average Error: 37.2 → 16.3
Time: 3.6m
Precision: 64
Internal Precision: 1408
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\phi_2 - \phi_1 \le -2.8392269565562946 \cdot 10^{+145}:\\ \;\;\;\;R \cdot \left|\phi_2 - \phi_1\right|\\ \mathbf{if}\;\phi_2 - \phi_1 \le -1.8619547323619714 \cdot 10^{-19}:\\ \;\;\;\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{if}\;\phi_2 - \phi_1 \le 2.3013880695478263 \cdot 10^{-90}:\\ \;\;\;\;R \cdot \left|\lambda_2 - \lambda_1\right|\\ \mathbf{if}\;\phi_2 - \phi_1 \le 5.398217016267653 \cdot 10^{+92}:\\ \;\;\;\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left|\phi_2 - \phi_1\right|\\ \end{array}\]

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 3 regimes

  2. if (- phi2 phi1) < -2.8392269565562946e+145 or 5.398217016267653e+92 < (- phi2 phi1)

    1. Initial program 54.6

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

    2. Taylor expanded around 0 55.2

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

    3. Applied simplify55.2

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

    4. Taylor expanded around 0 55.4

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

    5. Applied simplify15.8

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

    if -2.8392269565562946e+145 < (- phi2 phi1) < -1.8619547323619714e-19 or 2.3013880695478263e-90 < (- phi2 phi1) < 5.398217016267653e+92

    1. Initial program 22.2

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

    if -1.8619547323619714e-19 < (- phi2 phi1) < 2.3013880695478263e-90

    1. Initial program 22.4

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

    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 simplify22.4

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

    4. Taylor expanded around inf 28.7

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

    5. Applied simplify6.9

      \[\leadsto \color{blue}{R \cdot \left|\lambda_2 - \lambda_1\right|}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 3.6m)Debug logProfile

herbie shell --seed '#(1070258749 1877548225 2229079127 1588002776 3179087814 1886870650)' 
(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))))))