Average Error: 0.2 → 0.8
Time: 33.7s
Precision: 64
Internal Precision: 128
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1}{\sin B} - \frac{x}{\sqrt[3]{\sqrt[3]{\tan B}} \cdot \left(\sqrt[3]{\sqrt[3]{\tan B}} \cdot \sqrt[3]{\sqrt[3]{\tan B}}\right)} \cdot \frac{1}{\sqrt[3]{\tan B} \cdot \sqrt[3]{\tan B}}\]

Error

Bits error versus B

Bits error versus x

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 clear-num0.2

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

  6. Applied *-un-lft-identity0.2

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

  7. Applied add-cube-cbrt0.7

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

  8. Applied times-frac0.7

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

  9. Applied add-cube-cbrt0.7

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

  10. Applied times-frac0.7

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

  11. Simplified0.7

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

  12. Simplified0.7

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

  14. Applied add-cube-cbrt0.8

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

  15. Final simplification0.8

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

Reproduce

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