Henrywood and Agarwal, Equation (13)

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 74.2%

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

      1. associate-*l*73.3%

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

      2. difference-of-squares73.3%

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

      3. associate-*l*71.9%

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

      4. associate-*l*74.2%

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

   3. Simplified74.2%

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

   4. Taylor expanded in c0 around inf 76.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)} \]
   5. Step-by-step derivation

      1. *-commutative76.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{c0 \cdot {d}^{2}}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]

      2. *-commutative76.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{\left(w \cdot h\right) \cdot {D}^{2}}}\right) \]

      3. associate-*r*76.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{w \cdot \left(h \cdot {D}^{2}\right)}}\right) \]

      4. unpow276.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{w \cdot \left(h \cdot \color{blue}{\left(D \cdot D\right)}\right)}\right) \]

      5. times-frac76.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{c0}{w} \cdot \frac{{d}^{2}}{h \cdot \left(D \cdot D\right)}\right)}\right) \]

      6. unpow276.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{\color{blue}{d \cdot d}}{h \cdot \left(D \cdot D\right)}\right)\right) \]

      7. associate-/l*76.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \color{blue}{\frac{d}{\frac{h \cdot \left(D \cdot D\right)}{d}}}\right)\right) \]

      8. associate-*r*76.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(h \cdot D\right) \cdot D}}{d}}\right)\right) \]

      9. *-commutative76.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(D \cdot h\right)} \cdot D}{d}}\right)\right) \]

   6. Simplified76.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\left(D \cdot h\right) \cdot D}{d}}\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*l/76.4%

         \[\leadsto \color{blue}{\frac{c0 \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\left(D \cdot h\right) \cdot D}{d}}\right)\right)}{2 \cdot w}} \]

      2. *-commutative76.4%

         \[\leadsto \frac{c0 \cdot \color{blue}{\left(\left(\frac{c0}{w} \cdot \frac{d}{\frac{\left(D \cdot h\right) \cdot D}{d}}\right) \cdot 2\right)}}{2 \cdot w} \]

      3. associate-*l*76.4%

         \[\leadsto \frac{c0 \cdot \left(\left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{D \cdot \left(h \cdot D\right)}}{d}}\right) \cdot 2\right)}{2 \cdot w} \]

      4. *-commutative76.4%

         \[\leadsto \frac{c0 \cdot \left(\left(\frac{c0}{w} \cdot \frac{d}{\frac{D \cdot \color{blue}{\left(D \cdot h\right)}}{d}}\right) \cdot 2\right)}{2 \cdot w} \]

      5. *-commutative76.4%

         \[\leadsto \frac{c0 \cdot \left(\left(\frac{c0}{w} \cdot \frac{d}{\frac{D \cdot \left(D \cdot h\right)}{d}}\right) \cdot 2\right)}{\color{blue}{w \cdot 2}} \]

   8. Applied egg-rr76.4%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\left(\frac{c0}{w} \cdot \frac{d}{\frac{D \cdot \left(D \cdot h\right)}{d}}\right) \cdot 2\right)}{w \cdot 2}} \]
   9. Step-by-step derivation

      1. *-commutative76.4%

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{c0}{w} \cdot \frac{d}{\frac{D \cdot \left(D \cdot h\right)}{d}}\right) \cdot 2\right) \cdot c0}}{w \cdot 2} \]

      2. associate-*r/77.6%

         \[\leadsto \frac{\left(\color{blue}{\frac{\frac{c0}{w} \cdot d}{\frac{D \cdot \left(D \cdot h\right)}{d}}} \cdot 2\right) \cdot c0}{w \cdot 2} \]

      3. associate-/l*77.6%

         \[\leadsto \frac{\left(\frac{\frac{c0}{w} \cdot d}{\color{blue}{\frac{D}{\frac{d}{D \cdot h}}}} \cdot 2\right) \cdot c0}{w \cdot 2} \]

      4. associate-*r/77.5%

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{c0}{w} \cdot \frac{d}{\frac{D}{\frac{d}{D \cdot h}}}\right)} \cdot 2\right) \cdot c0}{w \cdot 2} \]

      5. associate-/r/77.5%

         \[\leadsto \frac{\left(\left(\frac{c0}{w} \cdot \color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D \cdot h}\right)}\right) \cdot 2\right) \cdot c0}{w \cdot 2} \]

      6. *-commutative77.5%

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \left(\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D \cdot h}\right)\right)\right)} \cdot c0}{w \cdot 2} \]

      7. *-commutative77.5%

         \[\leadsto \frac{\left(2 \cdot \left(\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D \cdot h}\right)\right)\right) \cdot c0}{\color{blue}{2 \cdot w}} \]

      8. associate-*r/78.6%

         \[\leadsto \color{blue}{\left(2 \cdot \left(\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D \cdot h}\right)\right)\right) \cdot \frac{c0}{2 \cdot w}} \]

      9. add-cbrt-cube75.4%

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

   10. Applied egg-rr79.6%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 0.0%

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

      1. associate-*l*0.0%

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

      2. difference-of-squares5.3%

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

      3. associate-*l*5.3%

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

      4. associate-*l*8.4%

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

   3. Simplified8.4%

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

   4. Taylor expanded in c0 around -inf 0.6%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}} \]
   5. Step-by-step derivation

      1. associate-/l*0.6%

         \[\leadsto -0.5 \cdot \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}{\frac{w}{{c0}^{2}}}} \]

      2. distribute-rgt1-in0.6%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}}{\frac{w}{{c0}^{2}}} \]

      3. metadata-eval0.6%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}{\frac{w}{{c0}^{2}}} \]

      4. mul0-lft31.4%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{0}}{\frac{w}{{c0}^{2}}} \]

      5. unpow231.4%

         \[\leadsto -0.5 \cdot \frac{0}{\frac{w}{\color{blue}{c0 \cdot c0}}} \]

   6. Simplified31.4%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{0}{\frac{w}{c0 \cdot c0}}} \]

   7. Taylor expanded in w around 0 45.9%

      \[\leadsto -0.5 \cdot \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\sqrt[3]{\frac{c0}{w} \cdot \frac{2 \cdot \left(\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{h \cdot D}\right)}{2}} \cdot \left(\sqrt[3]{\frac{c0}{w} \cdot \frac{2 \cdot \left(\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{h \cdot D}\right)}{2}} \cdot \sqrt[3]{\frac{c0}{w} \cdot \frac{2 \cdot \left(\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{h \cdot D}\right)}{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 74.2%

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

      1. associate-*l*73.3%

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

      2. difference-of-squares73.3%

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

      3. associate-*l*71.9%

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

      4. associate-*l*74.2%

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

   3. Simplified74.2%

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

   4. Taylor expanded in c0 around inf 76.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)} \]
   5. Step-by-step derivation

      1. *-commutative76.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{c0 \cdot {d}^{2}}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]

      2. *-commutative76.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{\left(w \cdot h\right) \cdot {D}^{2}}}\right) \]

      3. associate-*r*76.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{w \cdot \left(h \cdot {D}^{2}\right)}}\right) \]

      4. unpow276.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{w \cdot \left(h \cdot \color{blue}{\left(D \cdot D\right)}\right)}\right) \]

      5. times-frac76.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{c0}{w} \cdot \frac{{d}^{2}}{h \cdot \left(D \cdot D\right)}\right)}\right) \]

      6. unpow276.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{\color{blue}{d \cdot d}}{h \cdot \left(D \cdot D\right)}\right)\right) \]

      7. associate-/l*76.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \color{blue}{\frac{d}{\frac{h \cdot \left(D \cdot D\right)}{d}}}\right)\right) \]

      8. associate-*r*76.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(h \cdot D\right) \cdot D}}{d}}\right)\right) \]

      9. *-commutative76.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(D \cdot h\right)} \cdot D}{d}}\right)\right) \]

   6. Simplified76.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\left(D \cdot h\right) \cdot D}{d}}\right)\right)} \]

   7. Taylor expanded in D around 0 76.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\color{blue}{\frac{{D}^{2} \cdot h}{d}}}\right)\right) \]
   8. Step-by-step derivation

      1. unpow276.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(D \cdot D\right)} \cdot h}{d}}\right)\right) \]

      2. associate-*r*76.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{D \cdot \left(D \cdot h\right)}}{d}}\right)\right) \]

      3. associate-/l*78.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\color{blue}{\frac{D}{\frac{d}{D \cdot h}}}}\right)\right) \]

   9. Simplified78.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\color{blue}{\frac{D}{\frac{d}{D \cdot h}}}}\right)\right) \]
   10. Step-by-step derivation

       1. associate-*l/77.5%

          \[\leadsto \color{blue}{\frac{c0 \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{D}{\frac{d}{D \cdot h}}}\right)\right)}{2 \cdot w}} \]

       2. associate-*r/77.6%

          \[\leadsto \frac{c0 \cdot \left(2 \cdot \color{blue}{\frac{\frac{c0}{w} \cdot d}{\frac{D}{\frac{d}{D \cdot h}}}}\right)}{2 \cdot w} \]

       3. associate-/l*77.6%

          \[\leadsto \frac{c0 \cdot \left(2 \cdot \frac{\frac{c0}{w} \cdot d}{\color{blue}{\frac{D \cdot \left(D \cdot h\right)}{d}}}\right)}{2 \cdot w} \]

       4. associate-*r/76.4%

          \[\leadsto \frac{c0 \cdot \left(2 \cdot \color{blue}{\left(\frac{c0}{w} \cdot \frac{d}{\frac{D \cdot \left(D \cdot h\right)}{d}}\right)}\right)}{2 \cdot w} \]

       5. *-commutative76.4%

          \[\leadsto \frac{c0 \cdot \color{blue}{\left(\left(\frac{c0}{w} \cdot \frac{d}{\frac{D \cdot \left(D \cdot h\right)}{d}}\right) \cdot 2\right)}}{2 \cdot w} \]

       6. *-commutative76.4%

          \[\leadsto \frac{c0 \cdot \left(\left(\frac{c0}{w} \cdot \frac{d}{\frac{D \cdot \left(D \cdot h\right)}{d}}\right) \cdot 2\right)}{\color{blue}{w \cdot 2}} \]

       7. associate-/l*75.5%

          \[\leadsto \color{blue}{\frac{c0}{\frac{w \cdot 2}{\left(\frac{c0}{w} \cdot \frac{d}{\frac{D \cdot \left(D \cdot h\right)}{d}}\right) \cdot 2}}} \]

       8. *-commutative75.5%

          \[\leadsto \frac{c0}{\frac{w \cdot 2}{\color{blue}{2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{D \cdot \left(D \cdot h\right)}{d}}\right)}}} \]

       9. associate-/l*77.7%

          \[\leadsto \frac{c0}{\frac{w \cdot 2}{2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\color{blue}{\frac{D}{\frac{d}{D \cdot h}}}}\right)}} \]

       10. associate-/r/77.7%

           \[\leadsto \frac{c0}{\frac{w \cdot 2}{2 \cdot \left(\frac{c0}{w} \cdot \color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D \cdot h}\right)}\right)}} \]

       11. associate-*r*78.9%

           \[\leadsto \frac{c0}{\frac{w \cdot 2}{2 \cdot \color{blue}{\left(\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D \cdot h}\right)}}} \]

   11. Applied egg-rr78.9%

       \[\leadsto \color{blue}{\frac{c0}{\frac{w \cdot 2}{2 \cdot \left(\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D \cdot h}\right)}}} \]

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 0.0%

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

      1. associate-*l*0.0%

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

      2. difference-of-squares5.3%

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

      3. associate-*l*5.3%

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

      4. associate-*l*8.4%

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

   3. Simplified8.4%

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

   4. Taylor expanded in c0 around -inf 0.6%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}} \]
   5. Step-by-step derivation

      1. associate-/l*0.6%

         \[\leadsto -0.5 \cdot \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}{\frac{w}{{c0}^{2}}}} \]

      2. distribute-rgt1-in0.6%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}}{\frac{w}{{c0}^{2}}} \]

      3. metadata-eval0.6%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}{\frac{w}{{c0}^{2}}} \]

      4. mul0-lft31.4%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{0}}{\frac{w}{{c0}^{2}}} \]

      5. unpow231.4%

         \[\leadsto -0.5 \cdot \frac{0}{\frac{w}{\color{blue}{c0 \cdot c0}}} \]

   6. Simplified31.4%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{0}{\frac{w}{c0 \cdot c0}}} \]

   7. Taylor expanded in w around 0 45.9%

      \[\leadsto -0.5 \cdot \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{\frac{2 \cdot w}{2 \cdot \left(\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{h \cdot D}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

1. Split input into 5 regimes

2. if c0 < -1.95000000000000008e204 or -4.99999999999999961e133 < c0 < -8.50000000000000012e-50

   1. Initial program 34.3%

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

      1. associate-*l*34.4%

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

      2. difference-of-squares43.2%

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

      3. associate-*l*43.2%

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

      4. associate-*l*46.1%

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

   3. Simplified46.1%

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

   4. Taylor expanded in c0 around inf 46.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)} \]
   5. Step-by-step derivation

      1. *-commutative46.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{c0 \cdot {d}^{2}}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]

      2. *-commutative46.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{\left(w \cdot h\right) \cdot {D}^{2}}}\right) \]

      3. associate-*r*49.2%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{w \cdot \left(h \cdot {D}^{2}\right)}}\right) \]

      4. unpow249.2%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{w \cdot \left(h \cdot \color{blue}{\left(D \cdot D\right)}\right)}\right) \]

      5. times-frac49.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{c0}{w} \cdot \frac{{d}^{2}}{h \cdot \left(D \cdot D\right)}\right)}\right) \]

      6. unpow249.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{\color{blue}{d \cdot d}}{h \cdot \left(D \cdot D\right)}\right)\right) \]

      7. associate-/l*54.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \color{blue}{\frac{d}{\frac{h \cdot \left(D \cdot D\right)}{d}}}\right)\right) \]

      8. associate-*r*54.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(h \cdot D\right) \cdot D}}{d}}\right)\right) \]

      9. *-commutative54.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(D \cdot h\right)} \cdot D}{d}}\right)\right) \]

   6. Simplified54.3%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\left(D \cdot h\right) \cdot D}{d}}\right)\right)} \]

   7. Taylor expanded in D around 0 54.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\color{blue}{\frac{{D}^{2} \cdot h}{d}}}\right)\right) \]
   8. Step-by-step derivation

      1. unpow254.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(D \cdot D\right)} \cdot h}{d}}\right)\right) \]

      2. associate-*r*54.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{D \cdot \left(D \cdot h\right)}}{d}}\right)\right) \]

      3. associate-/l*54.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\color{blue}{\frac{D}{\frac{d}{D \cdot h}}}}\right)\right) \]

   9. Simplified54.3%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\color{blue}{\frac{D}{\frac{d}{D \cdot h}}}}\right)\right) \]
   10. Step-by-step derivation

       1. frac-times55.4%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{c0 \cdot d}{w \cdot \frac{D}{\frac{d}{D \cdot h}}}}\right) \]

       2. associate-/l*53.9%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot d}{w \cdot \color{blue}{\frac{D \cdot \left(D \cdot h\right)}{d}}}\right) \]

       3. associate-/l*55.4%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot d}{w \cdot \color{blue}{\frac{D}{\frac{d}{D \cdot h}}}}\right) \]

       4. associate-/r/56.8%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot d}{w \cdot \color{blue}{\left(\frac{D}{d} \cdot \left(D \cdot h\right)\right)}}\right) \]

   11. Applied egg-rr56.8%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{c0 \cdot d}{w \cdot \left(\frac{D}{d} \cdot \left(D \cdot h\right)\right)}}\right) \]

   if -1.95000000000000008e204 < c0 < -4.99999999999999961e133 or -1.01999999999999992e-238 < c0 < 1.50000000000000005e-83

   1. Initial program 10.6%

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

      1. associate-*l*10.6%

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

      2. difference-of-squares10.6%

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

      3. associate-*l*10.6%

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

      4. associate-*l*10.7%

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

   3. Simplified10.7%

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

   4. Taylor expanded in c0 around -inf 4.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}} \]
   5. Step-by-step derivation

      1. associate-/l*4.3%

         \[\leadsto -0.5 \cdot \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}{\frac{w}{{c0}^{2}}}} \]

      2. distribute-rgt1-in4.3%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}}{\frac{w}{{c0}^{2}}} \]

      3. metadata-eval4.3%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}{\frac{w}{{c0}^{2}}} \]

      4. mul0-lft46.9%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{0}}{\frac{w}{{c0}^{2}}} \]

      5. unpow246.9%

         \[\leadsto -0.5 \cdot \frac{0}{\frac{w}{\color{blue}{c0 \cdot c0}}} \]

   6. Simplified46.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{0}{\frac{w}{c0 \cdot c0}}} \]

   7. Taylor expanded in w around 0 54.7%

      \[\leadsto -0.5 \cdot \color{blue}{0} \]

   if -8.50000000000000012e-50 < c0 < -4.6000000000000004e-196

   1. Initial program 34.2%

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

      1. associate-*l*31.2%

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

      2. difference-of-squares38.6%

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

      3. associate-*l*37.9%

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

      4. associate-*l*41.5%

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

   3. Simplified41.5%

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

   4. Taylor expanded in c0 around inf 42.3%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)} \]
   5. Step-by-step derivation

      1. *-commutative42.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{c0 \cdot {d}^{2}}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]

      2. *-commutative42.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{\left(w \cdot h\right) \cdot {D}^{2}}}\right) \]

      3. associate-*r*42.2%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{w \cdot \left(h \cdot {D}^{2}\right)}}\right) \]

      4. unpow242.2%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{w \cdot \left(h \cdot \color{blue}{\left(D \cdot D\right)}\right)}\right) \]

      5. times-frac39.2%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{c0}{w} \cdot \frac{{d}^{2}}{h \cdot \left(D \cdot D\right)}\right)}\right) \]

      6. unpow239.2%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{\color{blue}{d \cdot d}}{h \cdot \left(D \cdot D\right)}\right)\right) \]

      7. associate-/l*43.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \color{blue}{\frac{d}{\frac{h \cdot \left(D \cdot D\right)}{d}}}\right)\right) \]

      8. associate-*r*43.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(h \cdot D\right) \cdot D}}{d}}\right)\right) \]

      9. *-commutative43.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(D \cdot h\right)} \cdot D}{d}}\right)\right) \]

   6. Simplified43.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\left(D \cdot h\right) \cdot D}{d}}\right)\right)} \]

   7. Taylor expanded in D around 0 43.3%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\color{blue}{\frac{{D}^{2} \cdot h}{d}}}\right)\right) \]
   8. Step-by-step derivation

      1. unpow243.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(D \cdot D\right)} \cdot h}{d}}\right)\right) \]

      2. associate-*r*43.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{D \cdot \left(D \cdot h\right)}}{d}}\right)\right) \]

      3. associate-/l*46.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\color{blue}{\frac{D}{\frac{d}{D \cdot h}}}}\right)\right) \]

   9. Simplified46.9%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\color{blue}{\frac{D}{\frac{d}{D \cdot h}}}}\right)\right) \]

   10. Taylor expanded in c0 around 0 45.3%

       \[\leadsto \color{blue}{\frac{{d}^{2} \cdot {c0}^{2}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)}} \]
   11. Step-by-step derivation

       1. associate-/r*41.8%

          \[\leadsto \color{blue}{\frac{\frac{{d}^{2} \cdot {c0}^{2}}{{D}^{2}}}{{w}^{2} \cdot h}} \]

       2. unpow241.8%

          \[\leadsto \frac{\frac{{d}^{2} \cdot {c0}^{2}}{{D}^{2}}}{\color{blue}{\left(w \cdot w\right)} \cdot h} \]

       3. unpow241.8%

          \[\leadsto \frac{\frac{\color{blue}{\left(d \cdot d\right)} \cdot {c0}^{2}}{{D}^{2}}}{\left(w \cdot w\right) \cdot h} \]

       4. unpow241.8%

          \[\leadsto \frac{\frac{\left(d \cdot d\right) \cdot \color{blue}{\left(c0 \cdot c0\right)}}{{D}^{2}}}{\left(w \cdot w\right) \cdot h} \]

       5. associate-*l*42.0%

          \[\leadsto \frac{\frac{\color{blue}{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}}{{D}^{2}}}{\left(w \cdot w\right) \cdot h} \]

       6. unpow242.0%

          \[\leadsto \frac{\frac{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}{\color{blue}{D \cdot D}}}{\left(w \cdot w\right) \cdot h} \]

       7. times-frac46.6%

          \[\leadsto \frac{\color{blue}{\frac{d}{D} \cdot \frac{d \cdot \left(c0 \cdot c0\right)}{D}}}{\left(w \cdot w\right) \cdot h} \]

       8. associate-*l/48.2%

          \[\leadsto \frac{\frac{d}{D} \cdot \color{blue}{\left(\frac{d}{D} \cdot \left(c0 \cdot c0\right)\right)}}{\left(w \cdot w\right) \cdot h} \]

       9. *-commutative48.2%

          \[\leadsto \frac{\frac{d}{D} \cdot \left(\frac{d}{D} \cdot \left(c0 \cdot c0\right)\right)}{\color{blue}{h \cdot \left(w \cdot w\right)}} \]

       10. times-frac62.6%

           \[\leadsto \color{blue}{\frac{\frac{d}{D}}{h} \cdot \frac{\frac{d}{D} \cdot \left(c0 \cdot c0\right)}{w \cdot w}} \]

       11. associate-/r*62.6%

           \[\leadsto \color{blue}{\frac{d}{D \cdot h}} \cdot \frac{\frac{d}{D} \cdot \left(c0 \cdot c0\right)}{w \cdot w} \]

   12. Simplified62.6%

       \[\leadsto \color{blue}{\frac{d}{D \cdot h} \cdot \frac{\frac{d}{D} \cdot \left(c0 \cdot c0\right)}{w \cdot w}} \]

   if -4.6000000000000004e-196 < c0 < -1.01999999999999992e-238

   1. Initial program 29.1%

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

      1. associate-*l*29.1%

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

      2. difference-of-squares29.1%

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

      3. associate-*l*29.1%

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

      4. associate-*l*43.4%

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

   3. Simplified43.4%

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

   4. Taylor expanded in c0 around inf 29.8%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)} \]
   5. Step-by-step derivation

      1. *-commutative29.8%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{c0 \cdot {d}^{2}}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]

      2. *-commutative29.8%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{\left(w \cdot h\right) \cdot {D}^{2}}}\right) \]

      3. associate-*r*44.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{w \cdot \left(h \cdot {D}^{2}\right)}}\right) \]

      4. unpow244.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{w \cdot \left(h \cdot \color{blue}{\left(D \cdot D\right)}\right)}\right) \]

      5. times-frac57.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{c0}{w} \cdot \frac{{d}^{2}}{h \cdot \left(D \cdot D\right)}\right)}\right) \]

      6. unpow257.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{\color{blue}{d \cdot d}}{h \cdot \left(D \cdot D\right)}\right)\right) \]

      7. associate-/l*57.7%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \color{blue}{\frac{d}{\frac{h \cdot \left(D \cdot D\right)}{d}}}\right)\right) \]

      8. associate-*r*44.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(h \cdot D\right) \cdot D}}{d}}\right)\right) \]

      9. *-commutative44.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(D \cdot h\right)} \cdot D}{d}}\right)\right) \]

   6. Simplified44.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\left(D \cdot h\right) \cdot D}{d}}\right)\right)} \]

   7. Taylor expanded in c0 around 0 29.8%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}}\right) \]
   8. Step-by-step derivation

      1. times-frac43.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{{d}^{2}}{{D}^{2}} \cdot \frac{c0}{w \cdot h}\right)}\right) \]

      2. unpow243.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\color{blue}{d \cdot d}}{{D}^{2}} \cdot \frac{c0}{w \cdot h}\right)\right) \]

      3. unpow243.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{d \cdot d}{\color{blue}{D \cdot D}} \cdot \frac{c0}{w \cdot h}\right)\right) \]

      4. associate-/r*57.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{d \cdot d}{D \cdot D} \cdot \color{blue}{\frac{\frac{c0}{w}}{h}}\right)\right) \]

   9. Simplified57.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{d \cdot d}{D \cdot D} \cdot \frac{\frac{c0}{w}}{h}\right)}\right) \]

   if 1.50000000000000005e-83 < c0

   1. Initial program 24.9%

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

      1. associate-*l*25.0%

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

      2. difference-of-squares26.2%

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

      3. associate-*l*24.9%

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

      4. associate-*l*28.7%

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

   3. Simplified28.7%

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

   4. Taylor expanded in c0 around inf 27.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)} \]
   5. Step-by-step derivation

      1. *-commutative27.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{c0 \cdot {d}^{2}}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]

      2. *-commutative27.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{\left(w \cdot h\right) \cdot {D}^{2}}}\right) \]

      3. associate-*r*29.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{w \cdot \left(h \cdot {D}^{2}\right)}}\right) \]

      4. unpow229.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{w \cdot \left(h \cdot \color{blue}{\left(D \cdot D\right)}\right)}\right) \]

      5. times-frac32.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{c0}{w} \cdot \frac{{d}^{2}}{h \cdot \left(D \cdot D\right)}\right)}\right) \]

      6. unpow232.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{\color{blue}{d \cdot d}}{h \cdot \left(D \cdot D\right)}\right)\right) \]

      7. associate-/l*36.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \color{blue}{\frac{d}{\frac{h \cdot \left(D \cdot D\right)}{d}}}\right)\right) \]

      8. associate-*r*39.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(h \cdot D\right) \cdot D}}{d}}\right)\right) \]

      9. *-commutative39.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(D \cdot h\right)} \cdot D}{d}}\right)\right) \]

   6. Simplified39.3%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\left(D \cdot h\right) \cdot D}{d}}\right)\right)} \]

   7. Taylor expanded in d around 0 32.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \color{blue}{\frac{{d}^{2}}{{D}^{2} \cdot h}}\right)\right) \]
   8. Step-by-step derivation

      1. unpow232.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{\color{blue}{d \cdot d}}{{D}^{2} \cdot h}\right)\right) \]

      2. unpow232.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d \cdot d}{\color{blue}{\left(D \cdot D\right)} \cdot h}\right)\right) \]

      3. associate-*r*34.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d \cdot d}{\color{blue}{D \cdot \left(D \cdot h\right)}}\right)\right) \]

      4. times-frac40.7%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D \cdot h}\right)}\right)\right) \]

   9. Simplified40.7%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D \cdot h}\right)}\right)\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -1.95 \cdot 10^{+204}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot d}{w \cdot \left(\left(h \cdot D\right) \cdot \frac{D}{d}\right)}\right)\\ \mathbf{elif}\;c0 \leq -5 \cdot 10^{+133}:\\ \;\;\;\;0\\ \mathbf{elif}\;c0 \leq -8.5 \cdot 10^{-50}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot d}{w \cdot \left(\left(h \cdot D\right) \cdot \frac{D}{d}\right)}\right)\\ \mathbf{elif}\;c0 \leq -4.6 \cdot 10^{-196}:\\ \;\;\;\;\frac{d}{h \cdot D} \cdot \frac{\frac{d}{D} \cdot \left(c0 \cdot c0\right)}{w \cdot w}\\ \mathbf{elif}\;c0 \leq -1.02 \cdot 10^{-238}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{d \cdot d}{D \cdot D} \cdot \frac{\frac{c0}{w}}{h}\right)\right)\\ \mathbf{elif}\;c0 \leq 1.5 \cdot 10^{-83}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{h \cdot D}\right)\right)\right)\\ \end{array} \]

1. Split input into 2 regimes

2. if c0 < -1.6500000000000001e205 or -6.6e133 < c0 < -2.49999999999999981e-164 or 1.45e-83 < c0

   1. Initial program 30.3%

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

      1. associate-*l*29.9%

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

      2. difference-of-squares35.0%

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

      3. associate-*l*34.3%

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

      4. associate-*l*37.9%

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

   3. Simplified37.9%

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

   4. Taylor expanded in c0 around inf 37.3%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)} \]
   5. Step-by-step derivation

      1. *-commutative37.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{c0 \cdot {d}^{2}}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]

      2. *-commutative37.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{\left(w \cdot h\right) \cdot {D}^{2}}}\right) \]

      3. associate-*r*39.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{w \cdot \left(h \cdot {D}^{2}\right)}}\right) \]

      4. unpow239.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{w \cdot \left(h \cdot \color{blue}{\left(D \cdot D\right)}\right)}\right) \]

      5. times-frac41.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{c0}{w} \cdot \frac{{d}^{2}}{h \cdot \left(D \cdot D\right)}\right)}\right) \]

      6. unpow241.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{\color{blue}{d \cdot d}}{h \cdot \left(D \cdot D\right)}\right)\right) \]

      7. associate-/l*45.8%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \color{blue}{\frac{d}{\frac{h \cdot \left(D \cdot D\right)}{d}}}\right)\right) \]

      8. associate-*r*47.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(h \cdot D\right) \cdot D}}{d}}\right)\right) \]

      9. *-commutative47.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(D \cdot h\right)} \cdot D}{d}}\right)\right) \]

   6. Simplified47.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\left(D \cdot h\right) \cdot D}{d}}\right)\right)} \]

   7. Taylor expanded in D around 0 45.8%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\color{blue}{\frac{{D}^{2} \cdot h}{d}}}\right)\right) \]
   8. Step-by-step derivation

      1. unpow245.8%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(D \cdot D\right)} \cdot h}{d}}\right)\right) \]

      2. associate-*r*47.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{D \cdot \left(D \cdot h\right)}}{d}}\right)\right) \]

      3. associate-/l*48.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\color{blue}{\frac{D}{\frac{d}{D \cdot h}}}}\right)\right) \]

   9. Simplified48.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\color{blue}{\frac{D}{\frac{d}{D \cdot h}}}}\right)\right) \]

   10. Taylor expanded in d around 0 41.0%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \color{blue}{\frac{{d}^{2}}{{D}^{2} \cdot h}}\right)\right) \]
   11. Step-by-step derivation

       1. unpow241.0%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{\color{blue}{d \cdot d}}{{D}^{2} \cdot h}\right)\right) \]

       2. associate-*r/45.9%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \color{blue}{\left(d \cdot \frac{d}{{D}^{2} \cdot h}\right)}\right)\right) \]

       3. unpow245.9%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \left(d \cdot \frac{d}{\color{blue}{\left(D \cdot D\right)} \cdot h}\right)\right)\right) \]

       4. associate-*r*47.0%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \left(d \cdot \frac{d}{\color{blue}{D \cdot \left(D \cdot h\right)}}\right)\right)\right) \]

       5. associate-/r*48.1%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \left(d \cdot \color{blue}{\frac{\frac{d}{D}}{D \cdot h}}\right)\right)\right) \]

   12. Simplified48.1%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \color{blue}{\left(d \cdot \frac{\frac{d}{D}}{D \cdot h}\right)}\right)\right) \]

   if -1.6500000000000001e205 < c0 < -6.6e133 or -2.49999999999999981e-164 < c0 < 1.45e-83

   1. Initial program 12.7%

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

      1. associate-*l*12.7%

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

      2. difference-of-squares12.7%

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

      3. associate-*l*12.7%

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

      4. associate-*l*14.0%

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

   3. Simplified14.0%

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

   4. Taylor expanded in c0 around -inf 4.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}} \]
   5. Step-by-step derivation

      1. associate-/l*4.9%

         \[\leadsto -0.5 \cdot \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}{\frac{w}{{c0}^{2}}}} \]

      2. distribute-rgt1-in4.9%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}}{\frac{w}{{c0}^{2}}} \]

      3. metadata-eval4.9%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}{\frac{w}{{c0}^{2}}} \]

      4. mul0-lft45.9%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{0}}{\frac{w}{{c0}^{2}}} \]

      5. unpow245.9%

         \[\leadsto -0.5 \cdot \frac{0}{\frac{w}{\color{blue}{c0 \cdot c0}}} \]

   6. Simplified45.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{0}{\frac{w}{c0 \cdot c0}}} \]

   7. Taylor expanded in w around 0 52.5%

      \[\leadsto -0.5 \cdot \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -1.65 \cdot 10^{+205} \lor \neg \left(c0 \leq -6.6 \cdot 10^{+133} \lor \neg \left(c0 \leq -2.5 \cdot 10^{-164}\right) \land c0 \leq 1.45 \cdot 10^{-83}\right):\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \left(d \cdot \frac{\frac{d}{D}}{h \cdot D}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

1. Split input into 3 regimes

2. if c0 < -1.3999999999999999e206 or -4.7999999999999997e133 < c0 < -2.09999999999999998e-163

   1. Initial program 35.4%

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

      1. associate-*l*34.5%

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

      2. difference-of-squares43.5%

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

      3. associate-*l*43.3%

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

      4. associate-*l*46.7%

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

   3. Simplified46.7%

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

   4. Taylor expanded in c0 around inf 47.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)} \]
   5. Step-by-step derivation

      1. *-commutative47.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{c0 \cdot {d}^{2}}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]

      2. *-commutative47.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{\left(w \cdot h\right) \cdot {D}^{2}}}\right) \]

      3. associate-*r*49.2%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{w \cdot \left(h \cdot {D}^{2}\right)}}\right) \]

      4. unpow249.2%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{w \cdot \left(h \cdot \color{blue}{\left(D \cdot D\right)}\right)}\right) \]

      5. times-frac49.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{c0}{w} \cdot \frac{{d}^{2}}{h \cdot \left(D \cdot D\right)}\right)}\right) \]

      6. unpow249.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{\color{blue}{d \cdot d}}{h \cdot \left(D \cdot D\right)}\right)\right) \]

      7. associate-/l*54.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \color{blue}{\frac{d}{\frac{h \cdot \left(D \cdot D\right)}{d}}}\right)\right) \]

      8. associate-*r*54.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(h \cdot D\right) \cdot D}}{d}}\right)\right) \]

      9. *-commutative54.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(D \cdot h\right)} \cdot D}{d}}\right)\right) \]

   6. Simplified54.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\left(D \cdot h\right) \cdot D}{d}}\right)\right)} \]

   7. Taylor expanded in D around 0 54.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\color{blue}{\frac{{D}^{2} \cdot h}{d}}}\right)\right) \]
   8. Step-by-step derivation

      1. unpow254.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(D \cdot D\right)} \cdot h}{d}}\right)\right) \]

      2. associate-*r*54.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{D \cdot \left(D \cdot h\right)}}{d}}\right)\right) \]

      3. associate-/l*55.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\color{blue}{\frac{D}{\frac{d}{D \cdot h}}}}\right)\right) \]

   9. Simplified55.3%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\color{blue}{\frac{D}{\frac{d}{D \cdot h}}}}\right)\right) \]

   10. Taylor expanded in d around 0 49.4%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \color{blue}{\frac{{d}^{2}}{{D}^{2} \cdot h}}\right)\right) \]
   11. Step-by-step derivation

       1. unpow249.4%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{\color{blue}{d \cdot d}}{{D}^{2} \cdot h}\right)\right) \]

       2. associate-*r/54.4%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \color{blue}{\left(d \cdot \frac{d}{{D}^{2} \cdot h}\right)}\right)\right) \]

       3. unpow254.4%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \left(d \cdot \frac{d}{\color{blue}{\left(D \cdot D\right)} \cdot h}\right)\right)\right) \]

       4. associate-*r*54.3%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \left(d \cdot \frac{d}{\color{blue}{D \cdot \left(D \cdot h\right)}}\right)\right)\right) \]

       5. associate-/r*55.3%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \left(d \cdot \color{blue}{\frac{\frac{d}{D}}{D \cdot h}}\right)\right)\right) \]

   12. Simplified55.3%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \color{blue}{\left(d \cdot \frac{\frac{d}{D}}{D \cdot h}\right)}\right)\right) \]

   if -1.3999999999999999e206 < c0 < -4.7999999999999997e133 or -2.09999999999999998e-163 < c0 < 1e-83

   1. Initial program 12.7%

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

      1. associate-*l*12.7%

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

      2. difference-of-squares12.7%

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

      3. associate-*l*12.7%

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

      4. associate-*l*14.0%

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

   3. Simplified14.0%

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

   4. Taylor expanded in c0 around -inf 4.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}} \]
   5. Step-by-step derivation

      1. associate-/l*4.9%

         \[\leadsto -0.5 \cdot \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}{\frac{w}{{c0}^{2}}}} \]

      2. distribute-rgt1-in4.9%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}}{\frac{w}{{c0}^{2}}} \]

      3. metadata-eval4.9%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}{\frac{w}{{c0}^{2}}} \]

      4. mul0-lft45.9%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{0}}{\frac{w}{{c0}^{2}}} \]

      5. unpow245.9%

         \[\leadsto -0.5 \cdot \frac{0}{\frac{w}{\color{blue}{c0 \cdot c0}}} \]

   6. Simplified45.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{0}{\frac{w}{c0 \cdot c0}}} \]

   7. Taylor expanded in w around 0 52.5%

      \[\leadsto -0.5 \cdot \color{blue}{0} \]

   if 1e-83 < c0

   1. Initial program 24.9%

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

      1. associate-*l*25.0%

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

      2. difference-of-squares26.2%

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

      3. associate-*l*24.9%

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

      4. associate-*l*28.7%

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

   3. Simplified28.7%

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

   4. Taylor expanded in c0 around inf 27.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)} \]
   5. Step-by-step derivation

      1. *-commutative27.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{c0 \cdot {d}^{2}}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]

      2. *-commutative27.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{\left(w \cdot h\right) \cdot {D}^{2}}}\right) \]

      3. associate-*r*29.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{w \cdot \left(h \cdot {D}^{2}\right)}}\right) \]

      4. unpow229.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{w \cdot \left(h \cdot \color{blue}{\left(D \cdot D\right)}\right)}\right) \]

      5. times-frac32.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{c0}{w} \cdot \frac{{d}^{2}}{h \cdot \left(D \cdot D\right)}\right)}\right) \]

      6. unpow232.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{\color{blue}{d \cdot d}}{h \cdot \left(D \cdot D\right)}\right)\right) \]

      7. associate-/l*36.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \color{blue}{\frac{d}{\frac{h \cdot \left(D \cdot D\right)}{d}}}\right)\right) \]

      8. associate-*r*39.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(h \cdot D\right) \cdot D}}{d}}\right)\right) \]

      9. *-commutative39.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(D \cdot h\right)} \cdot D}{d}}\right)\right) \]

   6. Simplified39.3%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\left(D \cdot h\right) \cdot D}{d}}\right)\right)} \]

   7. Taylor expanded in d around 0 32.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \color{blue}{\frac{{d}^{2}}{{D}^{2} \cdot h}}\right)\right) \]
   8. Step-by-step derivation

      1. unpow232.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{\color{blue}{d \cdot d}}{{D}^{2} \cdot h}\right)\right) \]

      2. unpow232.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d \cdot d}{\color{blue}{\left(D \cdot D\right)} \cdot h}\right)\right) \]

      3. associate-*r*34.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d \cdot d}{\color{blue}{D \cdot \left(D \cdot h\right)}}\right)\right) \]

      4. times-frac40.7%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D \cdot h}\right)}\right)\right) \]

   9. Simplified40.7%

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

4. Final simplification49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -1.4 \cdot 10^{+206}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \left(d \cdot \frac{\frac{d}{D}}{h \cdot D}\right)\right)\right)\\ \mathbf{elif}\;c0 \leq -4.8 \cdot 10^{+133}:\\ \;\;\;\;0\\ \mathbf{elif}\;c0 \leq -2.1 \cdot 10^{-163}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \left(d \cdot \frac{\frac{d}{D}}{h \cdot D}\right)\right)\right)\\ \mathbf{elif}\;c0 \leq 10^{-83}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{h \cdot D}\right)\right)\right)\\ \end{array} \]

1. Split input into 3 regimes

2. if c0 < -1.05e205 or -8.0000000000000002e133 < c0 < -1.25000000000000008e-159

   1. Initial program 35.4%

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

      1. associate-*l*34.5%

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

      2. difference-of-squares43.5%

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

      3. associate-*l*43.3%

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

      4. associate-*l*46.7%

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

   3. Simplified46.7%

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

   4. Taylor expanded in c0 around inf 47.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)} \]
   5. Step-by-step derivation

      1. *-commutative47.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{c0 \cdot {d}^{2}}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]

      2. *-commutative47.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{\left(w \cdot h\right) \cdot {D}^{2}}}\right) \]

      3. associate-*r*49.2%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{w \cdot \left(h \cdot {D}^{2}\right)}}\right) \]

      4. unpow249.2%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{w \cdot \left(h \cdot \color{blue}{\left(D \cdot D\right)}\right)}\right) \]

      5. times-frac49.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{c0}{w} \cdot \frac{{d}^{2}}{h \cdot \left(D \cdot D\right)}\right)}\right) \]

      6. unpow249.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{\color{blue}{d \cdot d}}{h \cdot \left(D \cdot D\right)}\right)\right) \]

      7. associate-/l*54.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \color{blue}{\frac{d}{\frac{h \cdot \left(D \cdot D\right)}{d}}}\right)\right) \]

      8. associate-*r*54.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(h \cdot D\right) \cdot D}}{d}}\right)\right) \]

      9. *-commutative54.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(D \cdot h\right)} \cdot D}{d}}\right)\right) \]

   6. Simplified54.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\left(D \cdot h\right) \cdot D}{d}}\right)\right)} \]

   7. Taylor expanded in D around 0 54.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\color{blue}{\frac{{D}^{2} \cdot h}{d}}}\right)\right) \]
   8. Step-by-step derivation

      1. unpow254.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(D \cdot D\right)} \cdot h}{d}}\right)\right) \]

      2. associate-*r*54.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{D \cdot \left(D \cdot h\right)}}{d}}\right)\right) \]

      3. associate-/l*55.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\color{blue}{\frac{D}{\frac{d}{D \cdot h}}}}\right)\right) \]

   9. Simplified55.3%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\color{blue}{\frac{D}{\frac{d}{D \cdot h}}}}\right)\right) \]

   if -1.05e205 < c0 < -8.0000000000000002e133 or -1.25000000000000008e-159 < c0 < 1.14999999999999995e-83

   1. Initial program 12.7%

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

      1. associate-*l*12.7%

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

      2. difference-of-squares12.7%

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

      3. associate-*l*12.7%

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

      4. associate-*l*14.0%

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

   3. Simplified14.0%

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

   4. Taylor expanded in c0 around -inf 4.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}} \]
   5. Step-by-step derivation

      1. associate-/l*4.9%

         \[\leadsto -0.5 \cdot \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}{\frac{w}{{c0}^{2}}}} \]

      2. distribute-rgt1-in4.9%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}}{\frac{w}{{c0}^{2}}} \]

      3. metadata-eval4.9%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}{\frac{w}{{c0}^{2}}} \]

      4. mul0-lft45.9%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{0}}{\frac{w}{{c0}^{2}}} \]

      5. unpow245.9%

         \[\leadsto -0.5 \cdot \frac{0}{\frac{w}{\color{blue}{c0 \cdot c0}}} \]

   6. Simplified45.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{0}{\frac{w}{c0 \cdot c0}}} \]

   7. Taylor expanded in w around 0 52.5%

      \[\leadsto -0.5 \cdot \color{blue}{0} \]

   if 1.14999999999999995e-83 < c0

   1. Initial program 24.9%

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

      1. associate-*l*25.0%

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

      2. difference-of-squares26.2%

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

      3. associate-*l*24.9%

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

      4. associate-*l*28.7%

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

   3. Simplified28.7%

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

   4. Taylor expanded in c0 around inf 27.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)} \]
   5. Step-by-step derivation

      1. *-commutative27.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{c0 \cdot {d}^{2}}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]

      2. *-commutative27.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{\left(w \cdot h\right) \cdot {D}^{2}}}\right) \]

      3. associate-*r*29.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{w \cdot \left(h \cdot {D}^{2}\right)}}\right) \]

      4. unpow229.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{w \cdot \left(h \cdot \color{blue}{\left(D \cdot D\right)}\right)}\right) \]

      5. times-frac32.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{c0}{w} \cdot \frac{{d}^{2}}{h \cdot \left(D \cdot D\right)}\right)}\right) \]

      6. unpow232.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{\color{blue}{d \cdot d}}{h \cdot \left(D \cdot D\right)}\right)\right) \]

      7. associate-/l*36.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \color{blue}{\frac{d}{\frac{h \cdot \left(D \cdot D\right)}{d}}}\right)\right) \]

      8. associate-*r*39.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(h \cdot D\right) \cdot D}}{d}}\right)\right) \]

      9. *-commutative39.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(D \cdot h\right)} \cdot D}{d}}\right)\right) \]

   6. Simplified39.3%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\left(D \cdot h\right) \cdot D}{d}}\right)\right)} \]

   7. Taylor expanded in d around 0 32.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \color{blue}{\frac{{d}^{2}}{{D}^{2} \cdot h}}\right)\right) \]
   8. Step-by-step derivation

      1. unpow232.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{\color{blue}{d \cdot d}}{{D}^{2} \cdot h}\right)\right) \]

      2. unpow232.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d \cdot d}{\color{blue}{\left(D \cdot D\right)} \cdot h}\right)\right) \]

      3. associate-*r*34.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d \cdot d}{\color{blue}{D \cdot \left(D \cdot h\right)}}\right)\right) \]

      4. times-frac40.7%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D \cdot h}\right)}\right)\right) \]

   9. Simplified40.7%

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

4. Final simplification49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -1.05 \cdot 10^{+205}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{D}{\frac{d}{h \cdot D}}}\right)\right)\\ \mathbf{elif}\;c0 \leq -8 \cdot 10^{+133}:\\ \;\;\;\;0\\ \mathbf{elif}\;c0 \leq -1.25 \cdot 10^{-159}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{D}{\frac{d}{h \cdot D}}}\right)\right)\\ \mathbf{elif}\;c0 \leq 1.15 \cdot 10^{-83}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{h \cdot D}\right)\right)\right)\\ \end{array} \]

1. Split input into 4 regimes

2. if c0 < -1.95000000000000008e204

   1. Initial program 27.1%

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

      1. associate-*l*27.1%

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

      2. difference-of-squares38.6%

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

      3. associate-*l*38.6%

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

      4. associate-*l*42.5%

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

   3. Simplified42.5%

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

   4. Taylor expanded in c0 around inf 42.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)} \]
   5. Step-by-step derivation

      1. *-commutative42.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{c0 \cdot {d}^{2}}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]

      2. *-commutative42.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{\left(w \cdot h\right) \cdot {D}^{2}}}\right) \]

      3. associate-*r*42.8%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{w \cdot \left(h \cdot {D}^{2}\right)}}\right) \]

      4. unpow242.8%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{w \cdot \left(h \cdot \color{blue}{\left(D \cdot D\right)}\right)}\right) \]

      5. times-frac47.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{c0}{w} \cdot \frac{{d}^{2}}{h \cdot \left(D \cdot D\right)}\right)}\right) \]

      6. unpow247.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{\color{blue}{d \cdot d}}{h \cdot \left(D \cdot D\right)}\right)\right) \]

      7. associate-/l*55.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \color{blue}{\frac{d}{\frac{h \cdot \left(D \cdot D\right)}{d}}}\right)\right) \]

      8. associate-*r*54.8%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(h \cdot D\right) \cdot D}}{d}}\right)\right) \]

      9. *-commutative54.8%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(D \cdot h\right)} \cdot D}{d}}\right)\right) \]

   6. Simplified54.8%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\left(D \cdot h\right) \cdot D}{d}}\right)\right)} \]

   if -1.95000000000000008e204 < c0 < -2.7000000000000002e133 or -9.4999999999999997e-159 < c0 < 1.89999999999999988e-83

   1. Initial program 12.7%

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

      1. associate-*l*12.7%

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

      2. difference-of-squares12.7%

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

      3. associate-*l*12.7%

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

      4. associate-*l*14.0%

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

   3. Simplified14.0%

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

   4. Taylor expanded in c0 around -inf 4.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}} \]
   5. Step-by-step derivation

      1. associate-/l*4.9%

         \[\leadsto -0.5 \cdot \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}{\frac{w}{{c0}^{2}}}} \]

      2. distribute-rgt1-in4.9%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}}{\frac{w}{{c0}^{2}}} \]

      3. metadata-eval4.9%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}{\frac{w}{{c0}^{2}}} \]

      4. mul0-lft45.9%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{0}}{\frac{w}{{c0}^{2}}} \]

      5. unpow245.9%

         \[\leadsto -0.5 \cdot \frac{0}{\frac{w}{\color{blue}{c0 \cdot c0}}} \]

   6. Simplified45.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{0}{\frac{w}{c0 \cdot c0}}} \]

   7. Taylor expanded in w around 0 52.5%

      \[\leadsto -0.5 \cdot \color{blue}{0} \]

   if -2.7000000000000002e133 < c0 < -9.4999999999999997e-159

   1. Initial program 38.8%

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

      1. associate-*l*37.6%

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

      2. difference-of-squares45.6%

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

      3. associate-*l*45.3%

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

      4. associate-*l*48.4%

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

   3. Simplified48.4%

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

   4. Taylor expanded in c0 around inf 49.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)} \]
   5. Step-by-step derivation

      1. *-commutative49.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{c0 \cdot {d}^{2}}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]

      2. *-commutative49.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{\left(w \cdot h\right) \cdot {D}^{2}}}\right) \]

      3. associate-*r*51.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{w \cdot \left(h \cdot {D}^{2}\right)}}\right) \]

      4. unpow251.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{w \cdot \left(h \cdot \color{blue}{\left(D \cdot D\right)}\right)}\right) \]

      5. times-frac50.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{c0}{w} \cdot \frac{{d}^{2}}{h \cdot \left(D \cdot D\right)}\right)}\right) \]

      6. unpow250.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{\color{blue}{d \cdot d}}{h \cdot \left(D \cdot D\right)}\right)\right) \]

      7. associate-/l*54.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \color{blue}{\frac{d}{\frac{h \cdot \left(D \cdot D\right)}{d}}}\right)\right) \]

      8. associate-*r*54.2%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(h \cdot D\right) \cdot D}}{d}}\right)\right) \]

      9. *-commutative54.2%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(D \cdot h\right)} \cdot D}{d}}\right)\right) \]

   6. Simplified54.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\left(D \cdot h\right) \cdot D}{d}}\right)\right)} \]

   7. Taylor expanded in D around 0 54.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\color{blue}{\frac{{D}^{2} \cdot h}{d}}}\right)\right) \]
   8. Step-by-step derivation

      1. unpow254.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(D \cdot D\right)} \cdot h}{d}}\right)\right) \]

      2. associate-*r*54.2%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{D \cdot \left(D \cdot h\right)}}{d}}\right)\right) \]

      3. associate-/l*55.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\color{blue}{\frac{D}{\frac{d}{D \cdot h}}}}\right)\right) \]

   9. Simplified55.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\color{blue}{\frac{D}{\frac{d}{D \cdot h}}}}\right)\right) \]

   if 1.89999999999999988e-83 < c0

   1. Initial program 24.9%

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

      1. associate-*l*25.0%

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

      2. difference-of-squares26.2%

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

      3. associate-*l*24.9%

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

      4. associate-*l*28.7%

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

   3. Simplified28.7%

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

   4. Taylor expanded in c0 around inf 27.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)} \]
   5. Step-by-step derivation

      1. *-commutative27.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{c0 \cdot {d}^{2}}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]

      2. *-commutative27.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{\left(w \cdot h\right) \cdot {D}^{2}}}\right) \]

      3. associate-*r*29.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{w \cdot \left(h \cdot {D}^{2}\right)}}\right) \]

      4. unpow229.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{w \cdot \left(h \cdot \color{blue}{\left(D \cdot D\right)}\right)}\right) \]

      5. times-frac32.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{c0}{w} \cdot \frac{{d}^{2}}{h \cdot \left(D \cdot D\right)}\right)}\right) \]

      6. unpow232.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{\color{blue}{d \cdot d}}{h \cdot \left(D \cdot D\right)}\right)\right) \]

      7. associate-/l*36.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \color{blue}{\frac{d}{\frac{h \cdot \left(D \cdot D\right)}{d}}}\right)\right) \]

      8. associate-*r*39.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(h \cdot D\right) \cdot D}}{d}}\right)\right) \]

      9. *-commutative39.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(D \cdot h\right)} \cdot D}{d}}\right)\right) \]

   6. Simplified39.3%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\left(D \cdot h\right) \cdot D}{d}}\right)\right)} \]

   7. Taylor expanded in d around 0 32.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \color{blue}{\frac{{d}^{2}}{{D}^{2} \cdot h}}\right)\right) \]
   8. Step-by-step derivation

      1. unpow232.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{\color{blue}{d \cdot d}}{{D}^{2} \cdot h}\right)\right) \]

      2. unpow232.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d \cdot d}{\color{blue}{\left(D \cdot D\right)} \cdot h}\right)\right) \]

      3. associate-*r*34.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d \cdot d}{\color{blue}{D \cdot \left(D \cdot h\right)}}\right)\right) \]

      4. times-frac40.7%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D \cdot h}\right)}\right)\right) \]

   9. Simplified40.7%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D \cdot h}\right)}\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -1.95 \cdot 10^{+204}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{D \cdot \left(h \cdot D\right)}{d}}\right)\right)\\ \mathbf{elif}\;c0 \leq -2.7 \cdot 10^{+133}:\\ \;\;\;\;0\\ \mathbf{elif}\;c0 \leq -9.5 \cdot 10^{-159}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{D}{\frac{d}{h \cdot D}}}\right)\right)\\ \mathbf{elif}\;c0 \leq 1.9 \cdot 10^{-83}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{h \cdot D}\right)\right)\right)\\ \end{array} \]

1. Split input into 2 regimes

2. if d < -2.19999999999999986e118 or 1.05e-7 < d

   1. Initial program 21.0%

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

      1. associate-*l*20.5%

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

      2. difference-of-squares23.8%

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

      3. associate-*l*23.7%

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

      4. associate-*l*26.3%

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

   3. Simplified26.3%

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

   4. Taylor expanded in c0 around -inf 3.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}} \]
   5. Step-by-step derivation

      1. associate-/l*3.3%

         \[\leadsto -0.5 \cdot \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}{\frac{w}{{c0}^{2}}}} \]

      2. distribute-rgt1-in3.3%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}}{\frac{w}{{c0}^{2}}} \]

      3. metadata-eval3.3%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}{\frac{w}{{c0}^{2}}} \]

      4. mul0-lft28.6%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{0}}{\frac{w}{{c0}^{2}}} \]

      5. unpow228.6%

         \[\leadsto -0.5 \cdot \frac{0}{\frac{w}{\color{blue}{c0 \cdot c0}}} \]

   6. Simplified28.6%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{0}{\frac{w}{c0 \cdot c0}}} \]

   7. Taylor expanded in w around 0 42.3%

      \[\leadsto -0.5 \cdot \color{blue}{0} \]

   if -2.19999999999999986e118 < d < 1.05e-7

   1. Initial program 30.2%

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

      1. associate-*l*30.2%

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

      2. difference-of-squares34.2%

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

      3. associate-*l*33.2%

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

      4. associate-*l*36.2%

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

   3. Simplified36.2%

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

   4. Taylor expanded in c0 around inf 35.7%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)} \]
   5. Step-by-step derivation

      1. *-commutative35.7%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{c0 \cdot {d}^{2}}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]

      2. *-commutative35.7%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{\left(w \cdot h\right) \cdot {D}^{2}}}\right) \]

      3. associate-*r*37.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{w \cdot \left(h \cdot {D}^{2}\right)}}\right) \]

      4. unpow237.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{w \cdot \left(h \cdot \color{blue}{\left(D \cdot D\right)}\right)}\right) \]

      5. times-frac39.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{c0}{w} \cdot \frac{{d}^{2}}{h \cdot \left(D \cdot D\right)}\right)}\right) \]

      6. unpow239.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{\color{blue}{d \cdot d}}{h \cdot \left(D \cdot D\right)}\right)\right) \]

      7. associate-/l*42.8%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \color{blue}{\frac{d}{\frac{h \cdot \left(D \cdot D\right)}{d}}}\right)\right) \]

      8. associate-*r*42.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(h \cdot D\right) \cdot D}}{d}}\right)\right) \]

      9. *-commutative42.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(D \cdot h\right)} \cdot D}{d}}\right)\right) \]

   6. Simplified42.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\left(D \cdot h\right) \cdot D}{d}}\right)\right)} \]

   7. Taylor expanded in D around 0 42.8%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\color{blue}{\frac{{D}^{2} \cdot h}{d}}}\right)\right) \]
   8. Step-by-step derivation

      1. unpow242.8%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{\left(D \cdot D\right)} \cdot h}{d}}\right)\right) \]

      2. associate-*r*42.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\frac{\color{blue}{D \cdot \left(D \cdot h\right)}}{d}}\right)\right) \]

      3. associate-/l*45.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\color{blue}{\frac{D}{\frac{d}{D \cdot h}}}}\right)\right) \]

   9. Simplified45.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d}{\color{blue}{\frac{D}{\frac{d}{D \cdot h}}}}\right)\right) \]

   10. Taylor expanded in c0 around 0 32.4%

       \[\leadsto \color{blue}{\frac{{d}^{2} \cdot {c0}^{2}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)}} \]
   11. Step-by-step derivation

       1. associate-/r*32.5%

          \[\leadsto \color{blue}{\frac{\frac{{d}^{2} \cdot {c0}^{2}}{{D}^{2}}}{{w}^{2} \cdot h}} \]

       2. unpow232.5%

          \[\leadsto \frac{\frac{{d}^{2} \cdot {c0}^{2}}{{D}^{2}}}{\color{blue}{\left(w \cdot w\right)} \cdot h} \]

       3. unpow232.5%

          \[\leadsto \frac{\frac{\color{blue}{\left(d \cdot d\right)} \cdot {c0}^{2}}{{D}^{2}}}{\left(w \cdot w\right) \cdot h} \]

       4. unpow232.5%

          \[\leadsto \frac{\frac{\left(d \cdot d\right) \cdot \color{blue}{\left(c0 \cdot c0\right)}}{{D}^{2}}}{\left(w \cdot w\right) \cdot h} \]

       5. associate-*l*35.6%

          \[\leadsto \frac{\frac{\color{blue}{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}}{{D}^{2}}}{\left(w \cdot w\right) \cdot h} \]

       6. unpow235.6%

          \[\leadsto \frac{\frac{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}{\color{blue}{D \cdot D}}}{\left(w \cdot w\right) \cdot h} \]

       7. times-frac44.3%

          \[\leadsto \frac{\color{blue}{\frac{d}{D} \cdot \frac{d \cdot \left(c0 \cdot c0\right)}{D}}}{\left(w \cdot w\right) \cdot h} \]

       8. associate-*l/45.7%

          \[\leadsto \frac{\frac{d}{D} \cdot \color{blue}{\left(\frac{d}{D} \cdot \left(c0 \cdot c0\right)\right)}}{\left(w \cdot w\right) \cdot h} \]

       9. *-commutative45.7%

          \[\leadsto \frac{\frac{d}{D} \cdot \left(\frac{d}{D} \cdot \left(c0 \cdot c0\right)\right)}{\color{blue}{h \cdot \left(w \cdot w\right)}} \]

       10. times-frac48.1%

           \[\leadsto \color{blue}{\frac{\frac{d}{D}}{h} \cdot \frac{\frac{d}{D} \cdot \left(c0 \cdot c0\right)}{w \cdot w}} \]

       11. associate-/r*47.1%

           \[\leadsto \color{blue}{\frac{d}{D \cdot h}} \cdot \frac{\frac{d}{D} \cdot \left(c0 \cdot c0\right)}{w \cdot w} \]

   12. Simplified47.1%

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

4. Final simplification44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.2 \cdot 10^{+118}:\\ \;\;\;\;0\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-7}:\\ \;\;\;\;\frac{d}{h \cdot D} \cdot \frac{\frac{d}{D} \cdot \left(c0 \cdot c0\right)}{w \cdot w}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

1. Initial program 24.6%

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

   1. associate-*l*24.4%

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

   2. difference-of-squares27.9%

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

   3. associate-*l*27.4%

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

   4. associate-*l*30.2%

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

3. Simplified30.2%

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

4. Taylor expanded in c0 around -inf 4.5%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}} \]
5. Step-by-step derivation

   1. associate-/l*4.5%

      \[\leadsto -0.5 \cdot \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}{\frac{w}{{c0}^{2}}}} \]

   2. distribute-rgt1-in4.5%

      \[\leadsto -0.5 \cdot \frac{\color{blue}{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}}{\frac{w}{{c0}^{2}}} \]

   3. metadata-eval4.5%

      \[\leadsto -0.5 \cdot \frac{\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}{\frac{w}{{c0}^{2}}} \]

   4. mul0-lft25.8%

      \[\leadsto -0.5 \cdot \frac{\color{blue}{0}}{\frac{w}{{c0}^{2}}} \]

   5. unpow225.8%

      \[\leadsto -0.5 \cdot \frac{0}{\frac{w}{\color{blue}{c0 \cdot c0}}} \]

6. Simplified25.8%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{0}{\frac{w}{c0 \cdot c0}}} \]

7. Taylor expanded in w around 0 35.6%

   \[\leadsto -0.5 \cdot \color{blue}{0} \]

8. Final simplification35.6%

   \[\leadsto 0 \]

herbie shell --seed 2023167 
(FPCore (c0 w h D d M)
  :name "Henrywood and Agarwal, Equation (13)"
  :precision binary64
  (* (/ c0 (* 2.0 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))))))