Error

Derivation

1. Initial program 0.2

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]

2. Initial simplification0.2

   \[\leadsto \frac{1}{\sin B} - \frac{x}{\tan B}\]

3. Taylor expanded around -inf 0.2

   \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}}\]
4. Using strategy rm

5. Applied add-sqr-sqrt31.2

   \[\leadsto \frac{1}{\sin B} - \color{blue}{\sqrt{\frac{x \cdot \cos B}{\sin B}} \cdot \sqrt{\frac{x \cdot \cos B}{\sin B}}}\]

6. Applied add-sqr-sqrt46.7

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\sin B}} \cdot \sqrt{\frac{1}{\sin B}}} - \sqrt{\frac{x \cdot \cos B}{\sin B}} \cdot \sqrt{\frac{x \cdot \cos B}{\sin B}}\]

7. Applied prod-diff46.7

   \[\leadsto \color{blue}{(\left(\sqrt{\frac{1}{\sin B}}\right) \cdot \left(\sqrt{\frac{1}{\sin B}}\right) + \left(-\sqrt{\frac{x \cdot \cos B}{\sin B}} \cdot \sqrt{\frac{x \cdot \cos B}{\sin B}}\right))_* + (\left(-\sqrt{\frac{x \cdot \cos B}{\sin B}}\right) \cdot \left(\sqrt{\frac{x \cdot \cos B}{\sin B}}\right) + \left(\sqrt{\frac{x \cdot \cos B}{\sin B}} \cdot \sqrt{\frac{x \cdot \cos B}{\sin B}}\right))_*}\]

8. Simplified31.1

   \[\leadsto \color{blue}{(\left(\frac{-x}{\sin B}\right) \cdot \left(\cos B\right) + \left(\frac{1}{\sin B}\right))_*} + (\left(-\sqrt{\frac{x \cdot \cos B}{\sin B}}\right) \cdot \left(\sqrt{\frac{x \cdot \cos B}{\sin B}}\right) + \left(\sqrt{\frac{x \cdot \cos B}{\sin B}} \cdot \sqrt{\frac{x \cdot \cos B}{\sin B}}\right))_*\]

9. Simplified0.2

   \[\leadsto (\left(\frac{-x}{\sin B}\right) \cdot \left(\cos B\right) + \left(\frac{1}{\sin B}\right))_* + \color{blue}{0}\]

10. Final simplification0.2

    \[\leadsto (\left(\frac{-x}{\sin B}\right) \cdot \left(\cos B\right) + \left(\frac{1}{\sin B}\right))_*\]

Runtime

Time bar (total: 28.0s)Debug logProfile

herbie shell --seed 2018273 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))