Error

Derivation

1. Initial program 0.2

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

2. Simplified0.2

   \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}}\]
3. Using strategy rm

4. Applied tan-quot0.2

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

5. Applied associate-/r/0.2

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

6. Applied add-cube-cbrt0.9

   \[\leadsto \frac{1}{\color{blue}{\left(\sqrt[3]{\sin B} \cdot \sqrt[3]{\sin B}\right) \cdot \sqrt[3]{\sin B}}} - \frac{x}{\sin B} \cdot \cos B\]

7. Applied add-cube-cbrt0.9

   \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\left(\sqrt[3]{\sin B} \cdot \sqrt[3]{\sin B}\right) \cdot \sqrt[3]{\sin B}} - \frac{x}{\sin B} \cdot \cos B\]

8. Applied times-frac0.9

   \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{\sin B} \cdot \sqrt[3]{\sin B}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\sin B}}} - \frac{x}{\sin B} \cdot \cos B\]

9. Applied prod-diff0.9

   \[\leadsto \color{blue}{(\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{\sin B} \cdot \sqrt[3]{\sin B}}\right) \cdot \left(\frac{\sqrt[3]{1}}{\sqrt[3]{\sin B}}\right) + \left(-\cos B \cdot \frac{x}{\sin B}\right))_* + (\left(-\cos B\right) \cdot \left(\frac{x}{\sin B}\right) + \left(\cos B \cdot \frac{x}{\sin B}\right))_*}\]

10. Simplified0.2

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

11. Simplified0.2

    \[\leadsto \left(\frac{1}{\sin B} - \frac{x}{\frac{\sin B}{\cos B}}\right) + \color{blue}{0}\]
12. Using strategy rm

13. Applied div-inv0.3

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

14. Applied add-sqr-sqrt31.2

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

15. Applied prod-diff31.2

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

16. Simplified0.3

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

17. Simplified0.2

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

18. Final simplification0.2

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

Reproduce

herbie shell --seed 2019089 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))