Error

Derivation

1. Split input into 3 regimes

2. if (/ h l) < -inf.0 or -0.0 < (/ h l)

   1. Initial program 14.5

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

   2. Initial simplification14.4

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

   3. Taylor expanded around 0 5.9

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

   if -inf.0 < (/ h l) < -4.5657420158628364e-175

   1. Initial program 13.3

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

   2. Initial simplification13.2

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

   if -4.5657420158628364e-175 < (/ h l) < -0.0

   1. Initial program 12.9

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

   2. Initial simplification13.0

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

   3. Taylor expanded around inf 31.5

      \[\leadsto \sqrt{\color{blue}{1 - \frac{1}{4} \cdot \frac{{M}^{2} \cdot \left({D}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}} \cdot w0\]

   4. Simplified22.9

      \[\leadsto \sqrt{\color{blue}{(\left(\frac{\frac{M \cdot M}{\frac{\ell}{h}}}{\frac{d}{D} \cdot \frac{d}{D}}\right) \cdot \frac{-1}{4} + 1)_*}} \cdot w0\]
   5. Using strategy rm

   6. Applied div-inv22.9

      \[\leadsto \sqrt{(\left(\frac{\frac{M \cdot M}{\color{blue}{\ell \cdot \frac{1}{h}}}}{\frac{d}{D} \cdot \frac{d}{D}}\right) \cdot \frac{-1}{4} + 1)_*} \cdot w0\]

   7. Applied times-frac13.2

      \[\leadsto \sqrt{(\left(\frac{\color{blue}{\frac{M}{\ell} \cdot \frac{M}{\frac{1}{h}}}}{\frac{d}{D} \cdot \frac{d}{D}}\right) \cdot \frac{-1}{4} + 1)_*} \cdot w0\]

   8. Applied times-frac7.0

      \[\leadsto \sqrt{(\color{blue}{\left(\frac{\frac{M}{\ell}}{\frac{d}{D}} \cdot \frac{\frac{M}{\frac{1}{h}}}{\frac{d}{D}}\right)} \cdot \frac{-1}{4} + 1)_*} \cdot w0\]

   9. Simplified9.4

      \[\leadsto \sqrt{(\left(\frac{\frac{M}{\ell}}{\frac{d}{D}} \cdot \color{blue}{\left(\frac{h}{d} \cdot \left(M \cdot D\right)\right)}\right) \cdot \frac{-1}{4} + 1)_*} \cdot w0\]
3. Recombined 3 regimes into one program.

4. Final simplification9.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} = -\infty:\\ \;\;\;\;w0\\ \mathbf{elif}\;\frac{h}{\ell} \le -4.5657420158628364 \cdot 10^{-175}:\\ \;\;\;\;w0 \cdot \sqrt{(\left(\frac{\frac{M}{2}}{\frac{d}{D}} \cdot \frac{\frac{M}{2}}{\frac{d}{D}}\right) \cdot \left(-\frac{h}{\ell}\right) + 1)_*}\\ \mathbf{elif}\;\frac{h}{\ell} \le -0.0:\\ \;\;\;\;\sqrt{(\left(\left(\left(D \cdot M\right) \cdot \frac{h}{d}\right) \cdot \frac{\frac{M}{\ell}}{\frac{d}{D}}\right) \cdot \frac{-1}{4} + 1)_*} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array}\]

Runtime

Time bar (total: 34.8s)Debug logProfile

herbie shell --seed 2018353 +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))))))