Error

Try it out

Derivation

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 simplification13.1

   \[\leadsto \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\frac{\ell}{h}}} \cdot w0\]
3. Using strategy rm

4. Applied div-inv13.1

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

5. Applied associate-/r*10.4

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

7. Applied associate-/l*9.0

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

9. Applied associate-/r/9.0

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

10. Applied associate-/l*8.2

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

11. Final simplification8.2

    \[\leadsto \sqrt{1 - \frac{\frac{\frac{D \cdot M}{2 \cdot d}}{\ell}}{\frac{\frac{1}{h}}{\frac{D \cdot M}{2 \cdot d}}}} \cdot w0\]

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed 2018234 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))