Average Error: 0.8 → 0.9
Time: 31.9s
Precision: 64
Internal precision: 128
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
⬇
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\frac{{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)}^3 + {\left(\cos \phi_1\right)}^3}{{\left(\cos \phi_1\right)}^2 + \left({\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}^2 - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}\]

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 flip3-+ 0.9

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\frac{{\left(\cos \phi_1\right)}^{3} + {\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}^{3}}{{\left(\cos \phi_1\right)}^2 + \left({\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}^2 - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}}\]

  4. Applied simplify 0.9

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\frac{\color{blue}{{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)}^3 + {\left(\cos \phi_1\right)}^3}}{{\left(\cos \phi_1\right)}^2 + \left({\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}^2 - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}\]
  5. Removed slow pow expressions

Runtime

Time bar (total: 31.9s) Debug log

Please include this information when filing a bug report:

herbie --seed '#(4107763823 131681691 1519327889 4277081522 2385980247 3846704547)'
(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)))))))