Average Error: 37.5 → 3.7
Time: 41.0s
Precision: 64
Internal Precision: 1344
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\sqrt{\left(\log_* (1 + (e^{\sqrt[3]{{\left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)}^{3}}} - 1)^*) \cdot \left(\lambda_1 - \lambda_2\right)\right)^2 + \left(\phi_1 - \phi_2\right)^2}^* \cdot R\]

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 37.5

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

  2. Applied simplify3.6

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

  4. Applied log1p-expm1-u3.6

    \[\leadsto \sqrt{\left(\color{blue}{\log_* (1 + (e^{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)} - 1)^*)} \cdot \left(\lambda_1 - \lambda_2\right)\right)^2 + \left(\phi_1 - \phi_2\right)^2}^* \cdot R\]
  5. Using strategy rm

  6. Applied add-cbrt-cube3.7

    \[\leadsto \sqrt{\left(\log_* (1 + (e^{\color{blue}{\sqrt[3]{\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)}}} - 1)^*) \cdot \left(\lambda_1 - \lambda_2\right)\right)^2 + \left(\phi_1 - \phi_2\right)^2}^* \cdot R\]

  7. Applied simplify3.7

    \[\leadsto \sqrt{\left(\log_* (1 + (e^{\sqrt[3]{\color{blue}{{\left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)}^{3}}}} - 1)^*) \cdot \left(\lambda_1 - \lambda_2\right)\right)^2 + \left(\phi_1 - \phi_2\right)^2}^* \cdot R\]

Runtime

Time bar (total: 41.0s)Debug logProfile

herbie shell --seed '#(1071501266 3581234924 1086666455 2685055582 1243441566 1802958749)' +o rules:numerics
(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))))))