Error

Derivation

1. Split input into 3 regimes

2. if (* (/ c0 (* 2 w)) (* (/ M 1) (/ M (- (* (/ (/ d D) h) (/ (/ d D) (/ w c0))) (sqrt (fma (* (/ (/ d D) h) (/ (/ d D) (/ w c0))) (* (/ (/ d D) h) (/ (/ d D) (/ w c0))) (* M (- M)))))))) < -inf.0

   1. Initial program 44.4

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

   3. Applied flip-+63.3

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

   4. Applied simplify62.7

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

   5. Taylor expanded around 0 44.4

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

   6. Applied simplify23.0

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

   if -inf.0 < (* (/ c0 (* 2 w)) (* (/ M 1) (/ M (- (* (/ (/ d D) h) (/ (/ d D) (/ w c0))) (sqrt (fma (* (/ (/ d D) h) (/ (/ d D) (/ w c0))) (* (/ (/ d D) h) (/ (/ d D) (/ w c0))) (* M (- M)))))))) < +inf.0

   1. Initial program 58.2

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
   2. 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 simplify33.2

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

   6. Applied add-cbrt-cube33.9

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

   7. Applied add-cbrt-cube37.9

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

   8. Applied cbrt-undiv38.0

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

   9. Applied simplify26.8

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

   11. Applied *-un-lft-identity26.8

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

   12. Applied times-frac26.8

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

   13. Applied unpow-prod-down28.4

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

   14. Applied cbrt-prod27.9

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

   15. Applied simplify25.5

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

   16. Applied simplify20.6

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

   if +inf.0 < (* (/ c0 (* 2 w)) (* (/ M 1) (/ M (- (* (/ (/ d D) h) (/ (/ d D) (/ w c0))) (sqrt (fma (* (/ (/ d D) h) (/ (/ d D) (/ w c0))) (* (/ (/ d D) h) (/ (/ d D) (/ w c0))) (* M (- M))))))))

   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 48.4

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

   3. Applied simplify37.8

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

Runtime

Time bar (total: 3.4m)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' +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))))))