Average Error: 0.2 → 0.2
Time: 29.4s
Precision: 64
Internal Precision: 576
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[(\left(\frac{-x}{\sin B}\right) \cdot \left(\cos B\right) + \left(\frac{1}{\sin B}\right))_* + 0\]

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. Applied simplify0.2

    \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}}\]
  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-cbrt0.8

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

  7. Applied prod-diff0.8

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

  8. Applied simplify0.2

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

  9. Applied simplify0.2

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

Runtime

Time bar (total: 29.4s)Debug logProfile

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))