Error

Derivation

1. Split input into 2 regimes

2. if (/ h l) < -inf.0 or -6.833578653749818e-259 < (/ h l)

   1. Initial program 12.8

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

   2. Initial simplification12.7

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

   3. Taylor expanded around 0 6.1

      \[\leadsto \color{blue}{1} \cdot w0\]

   if -inf.0 < (/ h l) < -6.833578653749818e-259

   1. Initial program 13.7

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

   2. Initial simplification13.7

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

   4. Applied add-cube-cbrt13.8

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

   5. Applied sqrt-prod13.8

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

   6. Simplified13.8

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

   8. Applied clear-num13.7

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

   10. Applied *-un-lft-identity13.7

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

   11. Applied div-inv13.7

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

   12. Applied times-frac13.9

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

   13. Simplified13.9

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

   14. Simplified13.8

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

4. Final simplification9.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} = -\infty \lor \neg \left(\frac{h}{\ell} \le -6.833578653749818 \cdot 10^{-259}\right):\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(\left|\sqrt[3]{(\left(\frac{\frac{M}{2}}{\frac{d}{D}} \cdot \left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \cdot \left(-\frac{h}{\ell}\right) + 1)_*}\right| \cdot \sqrt{\sqrt[3]{(\left(\frac{1}{\frac{\frac{d}{D}}{\frac{M}{2}}} \cdot \frac{\frac{M}{2}}{\frac{d}{D}}\right) \cdot \left(-\frac{h}{\ell}\right) + 1)_*}}\right)\\ \end{array}\]

Runtime

Time bar (total: 35.0s)Debug logProfile

herbie shell --seed 2018355 +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))))))