Error

Derivation

1. Split input into 3 regimes

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

   1. Initial program 61.5

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

   3. Applied div-inv61.5

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

   4. Applied associate-*r*24.8

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

   6. Applied unpow224.8

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

   7. Applied associate-*l*21.3

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

   9. Applied add-cube-cbrt21.4

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

   if -inf.0 < (/ h l) < -4.485396369295968e-252

   1. Initial program 12.9

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

   3. Applied div-inv12.9

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

   4. Applied associate-*r*14.3

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

   6. Applied unpow214.3

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

   7. Applied associate-*l*13.7

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

   9. Applied add-sqr-sqrt13.7

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

   10. Applied rem-sqrt-square13.7

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

   11. Simplified11.6

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

   if -4.485396369295968e-252 < (/ h l)

   1. Initial program 8.8

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

   2. Taylor expanded around 0 3.9

      \[\leadsto \color{blue}{w0}\]
3. Recombined 3 regimes into one program.

4. Final simplification8.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} = -\infty:\\ \;\;\;\;\sqrt{1 - \sqrt[3]{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot h\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{1}{\ell}} \cdot \left(\sqrt[3]{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot h\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{1}{\ell}} \cdot \sqrt[3]{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot h\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{1}{\ell}}\right)} \cdot w0\\ \mathbf{elif}\;\frac{h}{\ell} \le -4.485396369295968 \cdot 10^{-252}:\\ \;\;\;\;\left|\sqrt{(\left(\frac{M}{d} \cdot \frac{D}{2}\right) \cdot \left(\left(-\frac{h}{\ell}\right) \cdot \left(\frac{M}{d} \cdot \frac{D}{2}\right)\right) + 1)_*}\right| \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 +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))))))

Details

Time bar (total: 38.1s)Debug log