Error

Derivation

1. Split input into 2 regimes

2. if (* D D) < 1.5007410882524968e+285

   1. Initial program 58.1

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]

   2. Initial simplification56.8

      \[\leadsto (\left(\frac{c0}{w \cdot 2}\right) \cdot \left(\sqrt{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) + \left(M \cdot \left(-M\right)\right))_*}\right) + \left(\frac{c0}{w \cdot 2} \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right)\right))_*\]

   3. Taylor expanded around -inf 30.2

      \[\leadsto \color{blue}{0}\]

   if 1.5007410882524968e+285 < (* D D)

   1. Initial program 60.8

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]

   2. Initial simplification50.0

      \[\leadsto (\left(\frac{c0}{w \cdot 2}\right) \cdot \left(\sqrt{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) + \left(M \cdot \left(-M\right)\right))_*}\right) + \left(\frac{c0}{w \cdot 2} \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right)\right))_*\]
   3. Using strategy rm

   4. Applied fma-udef45.6

      \[\leadsto \color{blue}{\frac{c0}{w \cdot 2} \cdot \sqrt{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) + \left(M \cdot \left(-M\right)\right))_*} + \frac{c0}{w \cdot 2} \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right)}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt48.8

      \[\leadsto \frac{c0}{w \cdot 2} \cdot \sqrt{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) + \left(M \cdot \left(-M\right)\right))_*} + \frac{c0}{w \cdot 2} \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{\frac{c0}{h}}{w}} \cdot \sqrt[3]{\frac{\frac{c0}{h}}{w}}\right) \cdot \sqrt[3]{\frac{\frac{c0}{h}}{w}}\right)}\right)\]

   7. Applied associate-*r*49.0

      \[\leadsto \frac{c0}{w \cdot 2} \cdot \sqrt{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) + \left(M \cdot \left(-M\right)\right))_*} + \frac{c0}{w \cdot 2} \cdot \color{blue}{\left(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \left(\sqrt[3]{\frac{\frac{c0}{h}}{w}} \cdot \sqrt[3]{\frac{\frac{c0}{h}}{w}}\right)\right) \cdot \sqrt[3]{\frac{\frac{c0}{h}}{w}}\right)}\]
   8. Using strategy rm

   9. Applied cbrt-div48.8

      \[\leadsto \frac{c0}{w \cdot 2} \cdot \sqrt{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) + \left(M \cdot \left(-M\right)\right))_*} + \frac{c0}{w \cdot 2} \cdot \left(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \left(\sqrt[3]{\frac{\frac{c0}{h}}{w}} \cdot \color{blue}{\frac{\sqrt[3]{\frac{c0}{h}}}{\sqrt[3]{w}}}\right)\right) \cdot \sqrt[3]{\frac{\frac{c0}{h}}{w}}\right)\]

   10. Applied associate-*r/48.8

       \[\leadsto \frac{c0}{w \cdot 2} \cdot \sqrt{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) + \left(M \cdot \left(-M\right)\right))_*} + \frac{c0}{w \cdot 2} \cdot \left(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \color{blue}{\frac{\sqrt[3]{\frac{\frac{c0}{h}}{w}} \cdot \sqrt[3]{\frac{c0}{h}}}{\sqrt[3]{w}}}\right) \cdot \sqrt[3]{\frac{\frac{c0}{h}}{w}}\right)\]

   11. Applied associate-*r/48.7

       \[\leadsto \frac{c0}{w \cdot 2} \cdot \sqrt{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) + \left(M \cdot \left(-M\right)\right))_*} + \frac{c0}{w \cdot 2} \cdot \left(\color{blue}{\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \left(\sqrt[3]{\frac{\frac{c0}{h}}{w}} \cdot \sqrt[3]{\frac{c0}{h}}\right)}{\sqrt[3]{w}}} \cdot \sqrt[3]{\frac{\frac{c0}{h}}{w}}\right)\]

   12. Applied associate-*l/48.8

       \[\leadsto \frac{c0}{w \cdot 2} \cdot \sqrt{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) + \left(M \cdot \left(-M\right)\right))_*} + \frac{c0}{w \cdot 2} \cdot \color{blue}{\frac{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \left(\sqrt[3]{\frac{\frac{c0}{h}}{w}} \cdot \sqrt[3]{\frac{c0}{h}}\right)\right) \cdot \sqrt[3]{\frac{\frac{c0}{h}}{w}}}{\sqrt[3]{w}}}\]

   13. Applied associate-*r/48.8

       \[\leadsto \frac{c0}{w \cdot 2} \cdot \sqrt{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) + \left(M \cdot \left(-M\right)\right))_*} + \color{blue}{\frac{\frac{c0}{w \cdot 2} \cdot \left(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \left(\sqrt[3]{\frac{\frac{c0}{h}}{w}} \cdot \sqrt[3]{\frac{c0}{h}}\right)\right) \cdot \sqrt[3]{\frac{\frac{c0}{h}}{w}}\right)}{\sqrt[3]{w}}}\]

   14. Applied associate-*l/49.4

       \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) + \left(M \cdot \left(-M\right)\right))_*}}{w \cdot 2}} + \frac{\frac{c0}{w \cdot 2} \cdot \left(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \left(\sqrt[3]{\frac{\frac{c0}{h}}{w}} \cdot \sqrt[3]{\frac{c0}{h}}\right)\right) \cdot \sqrt[3]{\frac{\frac{c0}{h}}{w}}\right)}{\sqrt[3]{w}}\]

   15. Applied frac-add50.6

       \[\leadsto \color{blue}{\frac{\left(c0 \cdot \sqrt{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) + \left(M \cdot \left(-M\right)\right))_*}\right) \cdot \sqrt[3]{w} + \left(w \cdot 2\right) \cdot \left(\frac{c0}{w \cdot 2} \cdot \left(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \left(\sqrt[3]{\frac{\frac{c0}{h}}{w}} \cdot \sqrt[3]{\frac{c0}{h}}\right)\right) \cdot \sqrt[3]{\frac{\frac{c0}{h}}{w}}\right)\right)}{\left(w \cdot 2\right) \cdot \sqrt[3]{w}}}\]

   16. Simplified51.4

       \[\leadsto \frac{\color{blue}{(c0 \cdot \left(\sqrt{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) + \left(M \cdot \left(-M\right)\right))_*} \cdot \sqrt[3]{w}\right) + \left(\left(\frac{w \cdot 2}{\frac{w \cdot 2}{c0}} \cdot \sqrt[3]{\frac{\frac{c0}{h}}{w}}\right) \cdot \left(\sqrt[3]{\frac{\frac{c0}{h}}{w}} \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \sqrt[3]{\frac{c0}{h}}\right)\right)\right))_*}}{\left(w \cdot 2\right) \cdot \sqrt[3]{w}}\]
3. Recombined 2 regimes into one program.

4. Final simplification32.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot D \le 1.5007410882524968 \cdot 10^{+285}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{(c0 \cdot \left(\sqrt{(\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) + \left(\left(-M\right) \cdot M\right))_*} \cdot \sqrt[3]{w}\right) + \left(\left(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \sqrt[3]{\frac{c0}{h}}\right) \cdot \sqrt[3]{\frac{\frac{c0}{h}}{w}}\right) \cdot \left(\frac{2 \cdot w}{\frac{2 \cdot w}{c0}} \cdot \sqrt[3]{\frac{\frac{c0}{h}}{w}}\right)\right))_*}{\sqrt[3]{w} \cdot \left(2 \cdot w\right)}\\ \end{array}\]

Runtime

Time bar (total: 2.6m)Debug logProfile

herbie shell --seed 2018252 +o rules:numerics
(FPCore (c0 w h D d M)
  :name "Henrywood and Agarwal, Equation (13)"
  (* (/ c0 (* 2 w)) (+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))))