Error

Derivation

1. Split input into 2 regimes

2. if (/ h l) < -inf.0 or -6.372538479045205e-73 < (/ h l)

   1. Initial program 13.2

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]

   2. Initial simplification13.1

      \[\leadsto \sqrt{(\left({\left(\frac{\frac{M}{2}}{\frac{d}{D}}\right)}^{2}\right) \cdot \left(-\frac{h}{\ell}\right) + 1)_*} \cdot w0\]

   3. Taylor expanded around -inf 62.5

      \[\leadsto \sqrt{\color{blue}{1 - \frac{e^{2 \cdot \left(\left(\log \left(\frac{-1}{d}\right) + \log \frac{-1}{2}\right) - \left(\log \left(\frac{-1}{M}\right) + \log \left(\frac{-1}{D}\right)\right)\right)} \cdot h}{\ell}}} \cdot w0\]

   4. Simplified8.2

      \[\leadsto \sqrt{\color{blue}{(\left(\frac{{\left(\frac{\frac{-1}{d} \cdot \frac{-1}{2}}{\frac{-1}{D} \cdot \frac{-1}{M}}\right)}^{2}}{\ell}\right) \cdot \left(-h\right) + 1)_*}} \cdot w0\]
   5. Using strategy rm

   6. Applied unpow28.2

      \[\leadsto \sqrt{(\left(\frac{\color{blue}{\frac{\frac{-1}{d} \cdot \frac{-1}{2}}{\frac{-1}{D} \cdot \frac{-1}{M}} \cdot \frac{\frac{-1}{d} \cdot \frac{-1}{2}}{\frac{-1}{D} \cdot \frac{-1}{M}}}}{\ell}\right) \cdot \left(-h\right) + 1)_*} \cdot w0\]

   7. Applied associate-/l*6.1

      \[\leadsto \sqrt{(\color{blue}{\left(\frac{\frac{\frac{-1}{d} \cdot \frac{-1}{2}}{\frac{-1}{D} \cdot \frac{-1}{M}}}{\frac{\ell}{\frac{\frac{-1}{d} \cdot \frac{-1}{2}}{\frac{-1}{D} \cdot \frac{-1}{M}}}}\right)} \cdot \left(-h\right) + 1)_*} \cdot w0\]

   8. Simplified7.1

      \[\leadsto \sqrt{(\left(\frac{\color{blue}{\frac{\frac{-\frac{-1}{2}}{\frac{-1}{D}}}{\frac{-d}{M}}}}{\frac{\ell}{\frac{\frac{-1}{d} \cdot \frac{-1}{2}}{\frac{-1}{D} \cdot \frac{-1}{M}}}}\right) \cdot \left(-h\right) + 1)_*} \cdot w0\]
   9. Using strategy rm

   10. Applied *-un-lft-identity7.1

       \[\leadsto \sqrt{(\left(\frac{\frac{\frac{-\frac{-1}{2}}{\frac{-1}{D}}}{\color{blue}{1 \cdot \frac{-d}{M}}}}{\frac{\ell}{\frac{\frac{-1}{d} \cdot \frac{-1}{2}}{\frac{-1}{D} \cdot \frac{-1}{M}}}}\right) \cdot \left(-h\right) + 1)_*} \cdot w0\]

   11. Applied div-inv7.1

       \[\leadsto \sqrt{(\left(\frac{\frac{\frac{-\frac{-1}{2}}{\color{blue}{-1 \cdot \frac{1}{D}}}}{1 \cdot \frac{-d}{M}}}{\frac{\ell}{\frac{\frac{-1}{d} \cdot \frac{-1}{2}}{\frac{-1}{D} \cdot \frac{-1}{M}}}}\right) \cdot \left(-h\right) + 1)_*} \cdot w0\]

   12. Applied add-cube-cbrt7.1

       \[\leadsto \sqrt{(\left(\frac{\frac{\frac{\color{blue}{\left(\sqrt[3]{-\frac{-1}{2}} \cdot \sqrt[3]{-\frac{-1}{2}}\right) \cdot \sqrt[3]{-\frac{-1}{2}}}}{-1 \cdot \frac{1}{D}}}{1 \cdot \frac{-d}{M}}}{\frac{\ell}{\frac{\frac{-1}{d} \cdot \frac{-1}{2}}{\frac{-1}{D} \cdot \frac{-1}{M}}}}\right) \cdot \left(-h\right) + 1)_*} \cdot w0\]

   13. Applied times-frac7.1

       \[\leadsto \sqrt{(\left(\frac{\frac{\color{blue}{\frac{\sqrt[3]{-\frac{-1}{2}} \cdot \sqrt[3]{-\frac{-1}{2}}}{-1} \cdot \frac{\sqrt[3]{-\frac{-1}{2}}}{\frac{1}{D}}}}{1 \cdot \frac{-d}{M}}}{\frac{\ell}{\frac{\frac{-1}{d} \cdot \frac{-1}{2}}{\frac{-1}{D} \cdot \frac{-1}{M}}}}\right) \cdot \left(-h\right) + 1)_*} \cdot w0\]

   14. Applied times-frac7.1

       \[\leadsto \sqrt{(\left(\frac{\color{blue}{\frac{\frac{\sqrt[3]{-\frac{-1}{2}} \cdot \sqrt[3]{-\frac{-1}{2}}}{-1}}{1} \cdot \frac{\frac{\sqrt[3]{-\frac{-1}{2}}}{\frac{1}{D}}}{\frac{-d}{M}}}}{\frac{\ell}{\frac{\frac{-1}{d} \cdot \frac{-1}{2}}{\frac{-1}{D} \cdot \frac{-1}{M}}}}\right) \cdot \left(-h\right) + 1)_*} \cdot w0\]

   15. Simplified7.1

       \[\leadsto \sqrt{(\left(\frac{\color{blue}{\left(\frac{\sqrt[3]{-\frac{-1}{2}}}{-1} \cdot \sqrt[3]{-\frac{-1}{2}}\right)} \cdot \frac{\frac{\sqrt[3]{-\frac{-1}{2}}}{\frac{1}{D}}}{\frac{-d}{M}}}{\frac{\ell}{\frac{\frac{-1}{d} \cdot \frac{-1}{2}}{\frac{-1}{D} \cdot \frac{-1}{M}}}}\right) \cdot \left(-h\right) + 1)_*} \cdot w0\]

   16. Simplified6.2

       \[\leadsto \sqrt{(\left(\frac{\left(\frac{\sqrt[3]{-\frac{-1}{2}}}{-1} \cdot \sqrt[3]{-\frac{-1}{2}}\right) \cdot \color{blue}{\left(\frac{D \cdot M}{-d} \cdot \sqrt[3]{-\frac{-1}{2}}\right)}}{\frac{\ell}{\frac{\frac{-1}{d} \cdot \frac{-1}{2}}{\frac{-1}{D} \cdot \frac{-1}{M}}}}\right) \cdot \left(-h\right) + 1)_*} \cdot w0\]

   if -inf.0 < (/ h l) < -6.372538479045205e-73

   1. Initial program 13.5

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]

   2. Initial simplification13.5

      \[\leadsto \sqrt{(\left({\left(\frac{\frac{M}{2}}{\frac{d}{D}}\right)}^{2}\right) \cdot \left(-\frac{h}{\ell}\right) + 1)_*} \cdot w0\]
3. Recombined 2 regimes into one program.

4. Final simplification8.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} = -\infty \lor \neg \left(\frac{h}{\ell} \le -6.372538479045205 \cdot 10^{-73}\right):\\ \;\;\;\;\sqrt{(\left(\frac{\left(\frac{\sqrt[3]{-\frac{-1}{2}}}{-1} \cdot \sqrt[3]{-\frac{-1}{2}}\right) \cdot \left(\sqrt[3]{-\frac{-1}{2}} \cdot \frac{D \cdot M}{-d}\right)}{\frac{\ell}{\frac{\frac{-1}{2} \cdot \frac{-1}{d}}{\frac{-1}{D} \cdot \frac{-1}{M}}}}\right) \cdot \left(-h\right) + 1)_*} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{(\left({\left(\frac{\frac{M}{2}}{\frac{d}{D}}\right)}^{2}\right) \cdot \left(\frac{-h}{\ell}\right) + 1)_*} \cdot w0\\ \end{array}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018221 +o rules:numerics
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))