Average Error: 58.6 → 30.8
Time: 4.5m
Precision: 64
Internal Precision: 7296
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{d}{D} \le -8.446721792383403 \cdot 10^{+101}:\\ \;\;\;\;0\\ \mathbf{if}\;\frac{d}{D} \le -5.311478142534634 \cdot 10^{+61}:\\ \;\;\;\;\frac{\frac{c0}{1}}{w} \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\\ \mathbf{if}\;\frac{d}{D} \le -723.8580649124664:\\ \;\;\;\;c0 \cdot \frac{\frac{M \cdot M}{\frac{\frac{d}{D}}{w \cdot h} \cdot \left(c0 \cdot \frac{d}{D}\right) - \sqrt{(\left(\frac{\frac{d}{D}}{w \cdot h} \cdot \left(c0 \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{d}{D}}{w \cdot h} \cdot \left(c0 \cdot \frac{d}{D}\right)\right) + \left(M \cdot \left(-M\right)\right))_*}}}{w + w}\\ \mathbf{if}\;\frac{d}{D} \le -9.50190409094772 \cdot 10^{-51}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{M}{1} \cdot \frac{M}{\frac{\frac{d}{D}}{h} \cdot \frac{\frac{d}{D}}{\frac{w}{c0}} - \sqrt{(\left(\frac{\frac{d}{D}}{h} \cdot \frac{\frac{d}{D}}{\frac{w}{c0}}\right) \cdot \left(\frac{\frac{d}{D}}{h} \cdot \frac{\frac{d}{D}}{\frac{w}{c0}}\right) + \left(M \cdot \left(-M\right)\right))_*}}\right)\\ \mathbf{if}\;\frac{d}{D} \le 7.152545808073298 \cdot 10^{-16}:\\ \;\;\;\;c0 \cdot \frac{\frac{M \cdot M}{\frac{\frac{d}{D}}{w \cdot h} \cdot \left(c0 \cdot \frac{d}{D}\right) - \sqrt{(\left(\frac{\frac{d}{D}}{w \cdot h} \cdot \left(c0 \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{d}{D}}{w \cdot h} \cdot \left(c0 \cdot \frac{d}{D}\right)\right) + \left(M \cdot \left(-M\right)\right))_*}}}{w + w}\\ \mathbf{if}\;\frac{d}{D} \le 1.92232024099854 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{c0}{1}}{w} \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\\ \mathbf{if}\;\frac{d}{D} \le 3.0974883790727277 \cdot 10^{+118}:\\ \;\;\;\;c0 \cdot \frac{\frac{M \cdot M}{\frac{\frac{d}{D}}{w \cdot h} \cdot \left(c0 \cdot \frac{d}{D}\right) - \sqrt{(\left(\frac{\frac{d}{D}}{w \cdot h} \cdot \left(c0 \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{d}{D}}{w \cdot h} \cdot \left(c0 \cdot \frac{d}{D}\right)\right) + \left(M \cdot \left(-M\right)\right))_*}}}{w + w}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Error

Bits error versus c0

Bits error versus w

Bits error versus h

Bits error versus D

Bits error versus d

Bits error versus M

Derivation

  1. Split input into 4 regimes

  2. if (/ d D) < -8.446721792383403e+101 or 3.0974883790727277e+118 < (/ d D)

    1. Initial program 61.2

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

    2. Taylor expanded around inf 28.4

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

    3. Applied simplify24.0

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

    if -8.446721792383403e+101 < (/ d D) < -5.311478142534634e+61 or 7.152545808073298e-16 < (/ d D) < 1.92232024099854e+29

    1. Initial program 54.5

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

    3. Applied flip-+59.0

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

    4. Applied simplify50.6

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

    5. Taylor expanded around 0 55.9

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

    6. Applied simplify43.1

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

    if -5.311478142534634e+61 < (/ d D) < -723.8580649124664 or -9.50190409094772e-51 < (/ d D) < 7.152545808073298e-16 or 1.92232024099854e+29 < (/ d D) < 3.0974883790727277e+118

    1. Initial program 54.5

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

    3. Applied flip-+59.9

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

    4. Applied simplify53.9

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

    6. Applied add-cbrt-cube55.0

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

    7. Applied add-cbrt-cube56.0

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

    8. Applied cbrt-undiv56.1

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

    9. Applied simplify47.2

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

    11. Applied div-inv47.2

      \[\leadsto \color{blue}{\left(c0 \cdot \frac{1}{2 \cdot w}\right)} \cdot \sqrt[3]{{\left(\frac{M \cdot M}{\frac{\frac{d}{D}}{w \cdot h} \cdot \left(\frac{d}{D} \cdot c0\right) - \sqrt{(\left(\frac{\frac{d}{D}}{w \cdot h} \cdot \left(\frac{d}{D} \cdot c0\right)\right) \cdot \left(\frac{\frac{d}{D}}{w \cdot h} \cdot \left(\frac{d}{D} \cdot c0\right)\right) + \left(\left(-M\right) \cdot M\right))_*}}\right)}^{3}}\]

    12. Applied associate-*l*44.5

      \[\leadsto \color{blue}{c0 \cdot \left(\frac{1}{2 \cdot w} \cdot \sqrt[3]{{\left(\frac{M \cdot M}{\frac{\frac{d}{D}}{w \cdot h} \cdot \left(\frac{d}{D} \cdot c0\right) - \sqrt{(\left(\frac{\frac{d}{D}}{w \cdot h} \cdot \left(\frac{d}{D} \cdot c0\right)\right) \cdot \left(\frac{\frac{d}{D}}{w \cdot h} \cdot \left(\frac{d}{D} \cdot c0\right)\right) + \left(\left(-M\right) \cdot M\right))_*}}\right)}^{3}}\right)}\]

    13. Applied simplify40.9

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

    if -723.8580649124664 < (/ d D) < -9.50190409094772e-51

    1. Initial program 55.5

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

    3. Applied flip-+60.3

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

    4. Applied simplify55.1

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

    6. Applied add-cbrt-cube56.0

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

    7. Applied add-cbrt-cube57.2

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

    8. Applied cbrt-undiv57.2

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

    9. Applied simplify44.0

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

    11. Applied *-un-lft-identity44.0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \sqrt[3]{{\left(\frac{M \cdot M}{\color{blue}{1 \cdot \left(\frac{\frac{d}{D}}{w \cdot h} \cdot \left(\frac{d}{D} \cdot c0\right) - \sqrt{(\left(\frac{\frac{d}{D}}{w \cdot h} \cdot \left(\frac{d}{D} \cdot c0\right)\right) \cdot \left(\frac{\frac{d}{D}}{w \cdot h} \cdot \left(\frac{d}{D} \cdot c0\right)\right) + \left(\left(-M\right) \cdot M\right))_*}\right)}}\right)}^{3}}\]

    12. Applied times-frac43.9

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \sqrt[3]{{\color{blue}{\left(\frac{M}{1} \cdot \frac{M}{\frac{\frac{d}{D}}{w \cdot h} \cdot \left(\frac{d}{D} \cdot c0\right) - \sqrt{(\left(\frac{\frac{d}{D}}{w \cdot h} \cdot \left(\frac{d}{D} \cdot c0\right)\right) \cdot \left(\frac{\frac{d}{D}}{w \cdot h} \cdot \left(\frac{d}{D} \cdot c0\right)\right) + \left(\left(-M\right) \cdot M\right))_*}}\right)}}^{3}}\]

    13. Applied unpow-prod-down44.0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \sqrt[3]{\color{blue}{{\left(\frac{M}{1}\right)}^{3} \cdot {\left(\frac{M}{\frac{\frac{d}{D}}{w \cdot h} \cdot \left(\frac{d}{D} \cdot c0\right) - \sqrt{(\left(\frac{\frac{d}{D}}{w \cdot h} \cdot \left(\frac{d}{D} \cdot c0\right)\right) \cdot \left(\frac{\frac{d}{D}}{w \cdot h} \cdot \left(\frac{d}{D} \cdot c0\right)\right) + \left(\left(-M\right) \cdot M\right))_*}}\right)}^{3}}}\]

    14. Applied cbrt-prod43.3

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\sqrt[3]{{\left(\frac{M}{1}\right)}^{3}} \cdot \sqrt[3]{{\left(\frac{M}{\frac{\frac{d}{D}}{w \cdot h} \cdot \left(\frac{d}{D} \cdot c0\right) - \sqrt{(\left(\frac{\frac{d}{D}}{w \cdot h} \cdot \left(\frac{d}{D} \cdot c0\right)\right) \cdot \left(\frac{\frac{d}{D}}{w \cdot h} \cdot \left(\frac{d}{D} \cdot c0\right)\right) + \left(\left(-M\right) \cdot M\right))_*}}\right)}^{3}}\right)}\]

    15. Applied simplify41.5

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{M}{1}} \cdot \sqrt[3]{{\left(\frac{M}{\frac{\frac{d}{D}}{w \cdot h} \cdot \left(\frac{d}{D} \cdot c0\right) - \sqrt{(\left(\frac{\frac{d}{D}}{w \cdot h} \cdot \left(\frac{d}{D} \cdot c0\right)\right) \cdot \left(\frac{\frac{d}{D}}{w \cdot h} \cdot \left(\frac{d}{D} \cdot c0\right)\right) + \left(\left(-M\right) \cdot M\right))_*}}\right)}^{3}}\right)\]

    16. Applied simplify39.1

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{M}{1} \cdot \color{blue}{\frac{M}{\frac{\frac{d}{D}}{h} \cdot \frac{\frac{d}{D}}{\frac{w}{c0}} - \sqrt{(\left(\frac{\frac{d}{D}}{h} \cdot \frac{\frac{d}{D}}{\frac{w}{c0}}\right) \cdot \left(\frac{\frac{d}{D}}{h} \cdot \frac{\frac{d}{D}}{\frac{w}{c0}}\right) + \left(M \cdot \left(-M\right)\right))_*}}}\right)\]
  3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 4.5m)Debug logProfile

herbie shell --seed '#(1064269945 2896236262 301053905 1701069080 1701464310 1614783279)' +o rules:numerics
(FPCore (c0 w h D d M)
  :name "Henrywood and Agarwal, Equation (13)"
  (* (/ c0 (* 2 w)) (+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))))