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

↓

\[\begin{array}{l} \mathbf{if}\;D \cdot D \le 2.4703282292062 \cdot 10^{-323}:\\ \;\;\;\;\frac{\frac{c0}{2}}{w} \cdot \left(\sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} - M\right) \cdot \left(M + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right)} + \frac{d}{D} \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \frac{d}{D}\right)\right)\\ \mathbf{elif}\;D \cdot D \le 5.068742875779787 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{c0}{2}}{w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot c0}{\left(w \cdot h\right) \cdot {D}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{\frac{c0}{h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}{w} + \sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} - M\right) \cdot \left(M + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right)}} \cdot \left(\sqrt[3]{\sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} - M\right) \cdot \left(M + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right)} + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}} \cdot \sqrt[3]{\sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} - M\right) \cdot \left(M + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right)} + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}}\right)\right) \cdot \frac{\frac{c0}{2}}{w}\\ \end{array}\]

Derivation

Split input into 3 regimes
if (* D D) < 2.4703282292062e-323
1. Initial program 62.6
  \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
2. Initial simplification55.4
  \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \left(\sqrt{\left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right)} + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\]
3. Using strategy rm
4. Applied associate-*r*56.1
  \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \left(\sqrt{\left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right)} + \color{blue}{\left(\frac{\frac{c0}{h}}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}\right)\]
if 2.4703282292062e-323 < (* D D) < 5.068742875779787e+94
1. Initial program 53.8
  \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
2. Initial simplification53.2
  \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \left(\sqrt{\left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right)} + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\]
3. Taylor expanded around 0 54.1
  \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}\]
if 5.068742875779787e+94 < (* D D)
1. Initial program 57.5
  \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
2. Initial simplification46.8
  \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \left(\sqrt{\left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right)} + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\]
3. Using strategy rm
4. Applied add-cube-cbrt46.9
  \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt{\left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right)} + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)} \cdot \sqrt[3]{\sqrt{\left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right)} + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}\right) \cdot \sqrt[3]{\sqrt{\left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right)} + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}\right)}\]
5. Using strategy rm
6. Applied associate-*l/48.0
  \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \left(\left(\sqrt[3]{\sqrt{\left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right)} + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)} \cdot \sqrt[3]{\sqrt{\left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right)} + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}\right) \cdot \sqrt[3]{\sqrt{\left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right)} + \color{blue}{\frac{\frac{c0}{h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}{w}}}\right)\]
Recombined 3 regimes into one program.
Final simplification53.3
\[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot D \le 2.4703282292062 \cdot 10^{-323}:\\ \;\;\;\;\frac{\frac{c0}{2}}{w} \cdot \left(\sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} - M\right) \cdot \left(M + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right)} + \frac{d}{D} \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \frac{d}{D}\right)\right)\\ \mathbf{elif}\;D \cdot D \le 5.068742875779787 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{c0}{2}}{w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot c0}{\left(w \cdot h\right) \cdot {D}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{\frac{c0}{h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}{w} + \sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} - M\right) \cdot \left(M + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right)}} \cdot \left(\sqrt[3]{\sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} - M\right) \cdot \left(M + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right)} + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}} \cdot \sqrt[3]{\sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} - M\right) \cdot \left(M + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right)} + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}}\right)\right) \cdot \frac{\frac{c0}{2}}{w}\\ \end{array}\]

Error

Try it out

Derivation

`if (* D D) < 2.4703282292062e-323`

`if 2.4703282292062e-323 < (* D D) < 5.068742875779787e+94`

`if 5.068742875779787e+94 < (* D D)`

Runtime

In
Out