Average Error: 13.2 → 11.2
Time: 1.1m
Precision: 64
Internal Precision: 576
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\left(\sqrt[3]{\frac{\frac{F}{{\left(\sqrt{(x \cdot 2 + \left((F \cdot F + 2)_*\right))_*}\right)}^{\left(\frac{1}{2}\right)}}}{\sin B \cdot {\left(\sqrt{(x \cdot 2 + \left((F \cdot F + 2)_*\right))_*}\right)}^{\left(\frac{1}{2}\right)}} - \frac{x}{\tan B}} \cdot \sqrt[3]{\frac{\frac{F}{{\left(\sqrt{(x \cdot 2 + \left((F \cdot F + 2)_*\right))_*}\right)}^{\left(\frac{1}{2}\right)}}}{\sin B \cdot {\left(\sqrt{(x \cdot 2 + \left((F \cdot F + 2)_*\right))_*}\right)}^{\left(\frac{1}{2}\right)}} - \frac{x}{\tan B}}\right) \cdot \sqrt[3]{\frac{\frac{F}{{\left(\sqrt{(x \cdot 2 + \left((F \cdot F + 2)_*\right))_*}\right)}^{\left(\frac{1}{2}\right)}}}{{\left(\sqrt{(x \cdot 2 + \left((F \cdot F + 2)_*\right))_*}\right)}^{\left(\frac{1}{2}\right)} \cdot \sin B} - \frac{x}{\tan B}}\]

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

  1. Initial program 13.2

    \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]

  2. Applied simplify13.2

    \[\leadsto \color{blue}{(\left({\left((x \cdot 2 + \left((F \cdot F + 2)_*\right))_*\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \left(\frac{F}{\sin B}\right) + \left(\frac{-x}{\tan B}\right))_*}\]
  3. Using strategy rm

  4. Applied pow-neg13.2

    \[\leadsto (\color{blue}{\left(\frac{1}{{\left((x \cdot 2 + \left((F \cdot F + 2)_*\right))_*\right)}^{\left(\frac{1}{2}\right)}}\right)} \cdot \left(\frac{F}{\sin B}\right) + \left(\frac{-x}{\tan B}\right))_*\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt13.2

    \[\leadsto (\left(\frac{1}{{\color{blue}{\left(\sqrt{(x \cdot 2 + \left((F \cdot F + 2)_*\right))_*} \cdot \sqrt{(x \cdot 2 + \left((F \cdot F + 2)_*\right))_*}\right)}}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(\frac{F}{\sin B}\right) + \left(\frac{-x}{\tan B}\right))_*\]

  7. Applied unpow-prod-down13.2

    \[\leadsto (\left(\frac{1}{\color{blue}{{\left(\sqrt{(x \cdot 2 + \left((F \cdot F + 2)_*\right))_*}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\sqrt{(x \cdot 2 + \left((F \cdot F + 2)_*\right))_*}\right)}^{\left(\frac{1}{2}\right)}}}\right) \cdot \left(\frac{F}{\sin B}\right) + \left(\frac{-x}{\tan B}\right))_*\]

  8. Applied associate-/r*13.2

    \[\leadsto (\color{blue}{\left(\frac{\frac{1}{{\left(\sqrt{(x \cdot 2 + \left((F \cdot F + 2)_*\right))_*}\right)}^{\left(\frac{1}{2}\right)}}}{{\left(\sqrt{(x \cdot 2 + \left((F \cdot F + 2)_*\right))_*}\right)}^{\left(\frac{1}{2}\right)}}\right)} \cdot \left(\frac{F}{\sin B}\right) + \left(\frac{-x}{\tan B}\right))_*\]
  9. Using strategy rm

  10. Applied add-cube-cbrt14.0

    \[\leadsto \color{blue}{\left(\sqrt[3]{(\left(\frac{\frac{1}{{\left(\sqrt{(x \cdot 2 + \left((F \cdot F + 2)_*\right))_*}\right)}^{\left(\frac{1}{2}\right)}}}{{\left(\sqrt{(x \cdot 2 + \left((F \cdot F + 2)_*\right))_*}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(\frac{F}{\sin B}\right) + \left(\frac{-x}{\tan B}\right))_*} \cdot \sqrt[3]{(\left(\frac{\frac{1}{{\left(\sqrt{(x \cdot 2 + \left((F \cdot F + 2)_*\right))_*}\right)}^{\left(\frac{1}{2}\right)}}}{{\left(\sqrt{(x \cdot 2 + \left((F \cdot F + 2)_*\right))_*}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(\frac{F}{\sin B}\right) + \left(\frac{-x}{\tan B}\right))_*}\right) \cdot \sqrt[3]{(\left(\frac{\frac{1}{{\left(\sqrt{(x \cdot 2 + \left((F \cdot F + 2)_*\right))_*}\right)}^{\left(\frac{1}{2}\right)}}}{{\left(\sqrt{(x \cdot 2 + \left((F \cdot F + 2)_*\right))_*}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(\frac{F}{\sin B}\right) + \left(\frac{-x}{\tan B}\right))_*}}\]

  11. Applied simplify14.0

    \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\frac{F}{{\left(\sqrt{(x \cdot 2 + \left((F \cdot F + 2)_*\right))_*}\right)}^{\left(\frac{1}{2}\right)}}}{\sin B \cdot {\left(\sqrt{(x \cdot 2 + \left((F \cdot F + 2)_*\right))_*}\right)}^{\left(\frac{1}{2}\right)}} - \frac{x}{\tan B}} \cdot \sqrt[3]{\frac{\frac{F}{{\left(\sqrt{(x \cdot 2 + \left((F \cdot F + 2)_*\right))_*}\right)}^{\left(\frac{1}{2}\right)}}}{\sin B \cdot {\left(\sqrt{(x \cdot 2 + \left((F \cdot F + 2)_*\right))_*}\right)}^{\left(\frac{1}{2}\right)}} - \frac{x}{\tan B}}\right)} \cdot \sqrt[3]{(\left(\frac{\frac{1}{{\left(\sqrt{(x \cdot 2 + \left((F \cdot F + 2)_*\right))_*}\right)}^{\left(\frac{1}{2}\right)}}}{{\left(\sqrt{(x \cdot 2 + \left((F \cdot F + 2)_*\right))_*}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(\frac{F}{\sin B}\right) + \left(\frac{-x}{\tan B}\right))_*}\]

  12. Applied simplify11.2

    \[\leadsto \left(\sqrt[3]{\frac{\frac{F}{{\left(\sqrt{(x \cdot 2 + \left((F \cdot F + 2)_*\right))_*}\right)}^{\left(\frac{1}{2}\right)}}}{\sin B \cdot {\left(\sqrt{(x \cdot 2 + \left((F \cdot F + 2)_*\right))_*}\right)}^{\left(\frac{1}{2}\right)}} - \frac{x}{\tan B}} \cdot \sqrt[3]{\frac{\frac{F}{{\left(\sqrt{(x \cdot 2 + \left((F \cdot F + 2)_*\right))_*}\right)}^{\left(\frac{1}{2}\right)}}}{\sin B \cdot {\left(\sqrt{(x \cdot 2 + \left((F \cdot F + 2)_*\right))_*}\right)}^{\left(\frac{1}{2}\right)}} - \frac{x}{\tan B}}\right) \cdot \color{blue}{\sqrt[3]{\frac{\frac{F}{{\left(\sqrt{(x \cdot 2 + \left((F \cdot F + 2)_*\right))_*}\right)}^{\left(\frac{1}{2}\right)}}}{{\left(\sqrt{(x \cdot 2 + \left((F \cdot F + 2)_*\right))_*}\right)}^{\left(\frac{1}{2}\right)} \cdot \sin B} - \frac{x}{\tan B}}}\]

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed '#(1071979731 1496239409 439705970 2863295848 982327776 189749553)' +o rules:numerics
(FPCore (F B x)
  :name "VandenBroeck and Keller, Equation (23)"
  (+ (- (* x (/ 1 (tan B)))) (* (/ F (sin B)) (pow (+ (+ (* F F) 2) (* 2 x)) (- (/ 1 2))))))