Average Error: 0.8 → 0.9
Time: 35.8s
Precision: 64
Internal Precision: 1408
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\log \left(e^{(\left(\cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right)\right) + \left(\cos \phi_1\right))_*}\right)}\]

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 0.8

    \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
  2. Using strategy rm

  3. Applied add-log-exp0.9

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \color{blue}{\log \left(e^{\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)}}\]

  4. Applied add-log-exp0.9

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\log \left(e^{\cos \phi_1}\right)} + \log \left(e^{\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)}\]

  5. Applied sum-log0.9

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\log \left(e^{\cos \phi_1} \cdot e^{\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)}}\]

  6. Applied simplify0.9

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\log \color{blue}{\left(e^{(\left(\cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right)\right) + \left(\cos \phi_1\right))_*}\right)}}\]

Runtime

Time bar (total: 35.8s)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' +o rules:numerics
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Midpoint on a great circle"
  (+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))