Average Error: 37.5 → 28.7
Time: 8.1m
Precision: 64
Internal Precision: 1344
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\phi_1 - \phi_2 \le -7.793891452890422 \cdot 10^{+86}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \mathbf{if}\;\phi_1 - \phi_2 \le -1.7876931619331914 \cdot 10^{-126} \lor \neg \left(\phi_1 - \phi_2 \le -1.4167013807936541 \cdot 10^{-160}\right):\\ \;\;\;\;\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\lambda_1 \cdot R\right) \cdot \cos \left(\left(\phi_2 + \phi_1\right) \cdot \frac{1}{2}\right)\\ \end{array}\]

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Split input into 3 regimes

  2. if (- phi1 phi2) < -7.793891452890422e+86

    1. Initial program 50.5

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

    2. Taylor expanded around 0 16.5

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

    if -7.793891452890422e+86 < (- phi1 phi2) < -1.7876931619331914e-126 or -1.4167013807936541e-160 < (- phi1 phi2)

    1. Initial program 33.0

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

    if -1.7876931619331914e-126 < (- phi1 phi2) < -1.4167013807936541e-160

    1. Initial program 21.9

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

    3. Applied add-exp-log24.6

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

    4. Taylor expanded around inf 45.1

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

    5. Applied simplify42.1

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

  4. Applied simplify28.7

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\phi_1 - \phi_2 \le -7.793891452890422 \cdot 10^{+86}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \mathbf{if}\;\phi_1 - \phi_2 \le -1.7876931619331914 \cdot 10^{-126} \lor \neg \left(\phi_1 - \phi_2 \le -1.4167013807936541 \cdot 10^{-160}\right):\\ \;\;\;\;\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\lambda_1 \cdot R\right) \cdot \cos \left(\left(\phi_2 + \phi_1\right) \cdot \frac{1}{2}\right)\\ \end{array}}\]

Runtime

Time bar (total: 8.1m)Debug logProfile

herbie shell --seed '#(1071501266 3581234924 1086666455 2685055582 1243441566 1802958749)' 
(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))))))