Error

Derivation

1. Split input into 2 regimes

2. if M < 1.8305862302335185e-62 or 4.44257150862946e+145 < M

   1. Initial program 58.0

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

   2. Taylor expanded around inf 34.8

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

   3. Applied simplify30.9

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

   if 1.8305862302335185e-62 < M < 4.44257150862946e+145

   1. Initial program 60.6

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

   3. Applied flip-+62.4

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

   4. Applied simplify40.9

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

   6. Applied add-cbrt-cube41.4

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

   7. Applied add-cbrt-cube48.8

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

   8. Applied cbrt-undiv49.0

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

   9. Applied simplify38.8

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

   11. Applied *-un-lft-identity38.8

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

   12. Applied times-frac38.8

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

   13. Applied unpow-prod-down41.4

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

   14. Applied cbrt-prod41.2

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

   15. Applied simplify38.6

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

   16. Applied simplify38.7

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

Runtime

Time bar (total: 5.3m)Debug logProfile

herbie shell --seed '#(1070258749 1877548225 2229079127 1588002776 3179087814 1886870650)' +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))))))