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Derivation

1. Split input into 3 regimes

2. if (/ c0 (* (/ (/ w M) (/ M 2)) (- (/ (* c0 (/ d D)) (/ (* h w) (/ d D))) (sqrt (* (+ M (/ (* c0 (/ d D)) (/ (* h w) (/ d D)))) (- (/ (* c0 (/ d D)) (/ (* h w) (/ d D))) M)))))) < -1.8998674666154518e+145

   1. Initial program 45.1

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
   2. 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 simplify61.8

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

   5. Taylor expanded around 0 61.8

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

   6. Applied simplify57.8

      \[\leadsto \color{blue}{\frac{\frac{c0}{\frac{w \cdot 2}{M \cdot M}}}{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} - \sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} - M \cdot M}}}\]
   7. Using strategy rm

   8. Applied flip--61.6

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

   9. Applied associate-/r/61.7

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

   10. Applied simplify37.7

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

   if -1.8998674666154518e+145 < (/ c0 (* (/ (/ w M) (/ M 2)) (- (/ (* c0 (/ d D)) (/ (* h w) (/ d D))) (sqrt (* (+ M (/ (* c0 (/ d D)) (/ (* h w) (/ d D)))) (- (/ (* c0 (/ d D)) (/ (* h w) (/ d D))) M)))))) < 1.8311289264330715e-66

   1. Initial program 59.7

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
   2. 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 simplify32.8

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

   5. Taylor expanded around 0 32.8

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

   6. Applied simplify24.5

      \[\leadsto \color{blue}{\frac{\frac{c0}{\frac{w \cdot 2}{M \cdot M}}}{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} - \sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} - M \cdot M}}}\]
   7. Using strategy rm

   8. Applied div-inv24.6

      \[\leadsto \frac{\color{blue}{c0 \cdot \frac{1}{\frac{w \cdot 2}{M \cdot M}}}}{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} - \sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} - M \cdot M}}\]

   9. Applied associate-/l*23.0

      \[\leadsto \color{blue}{\frac{c0}{\frac{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} - \sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} - M \cdot M}}{\frac{1}{\frac{w \cdot 2}{M \cdot M}}}}}\]

   10. Applied simplify17.5

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

   if 1.8311289264330715e-66 < (/ c0 (* (/ (/ w M) (/ M 2)) (- (/ (* c0 (/ d D)) (/ (* h w) (/ d D))) (sqrt (* (+ M (/ (* c0 (/ d D)) (/ (* h w) (/ d D)))) (- (/ (* c0 (/ d D)) (/ (* h w) (/ d D))) M))))))

   1. Initial program 57.9

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

   2. Taylor expanded around inf 48.9

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

   3. Applied simplify47.4

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

4. Applied simplify29.6

   \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{c0}{\frac{\frac{w}{M}}{\frac{M}{2}} \cdot \left(\frac{\frac{d}{D} \cdot c0}{\frac{h \cdot w}{\frac{d}{D}}} - \sqrt{\left(\frac{\frac{d}{D} \cdot c0}{\frac{h \cdot w}{\frac{d}{D}}} - M\right) \cdot \left(\frac{\frac{d}{D} \cdot c0}{\frac{h \cdot w}{\frac{d}{D}}} + M\right)}\right)} \le -1.8998674666154518 \cdot 10^{+145}:\\ \;\;\;\;\left(\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{h \cdot w}{c0}} + \sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{h \cdot w}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{h \cdot w}{c0}} - M \cdot M}\right) \cdot \left(\frac{\frac{c0}{w \cdot 2}}{M \cdot M} \cdot \left(M \cdot M\right)\right)\\ \mathbf{if}\;\frac{c0}{\frac{\frac{w}{M}}{\frac{M}{2}} \cdot \left(\frac{\frac{d}{D} \cdot c0}{\frac{h \cdot w}{\frac{d}{D}}} - \sqrt{\left(\frac{\frac{d}{D} \cdot c0}{\frac{h \cdot w}{\frac{d}{D}}} - M\right) \cdot \left(\frac{\frac{d}{D} \cdot c0}{\frac{h \cdot w}{\frac{d}{D}}} + M\right)}\right)} \le 1.8311289264330715 \cdot 10^{-66}:\\ \;\;\;\;\frac{c0}{\frac{\frac{w}{M}}{\frac{M}{2}} \cdot \left(\frac{\frac{d}{D} \cdot c0}{\frac{h \cdot w}{\frac{d}{D}}} - \sqrt{\left(\frac{\frac{d}{D} \cdot c0}{\frac{h \cdot w}{\frac{d}{D}}} - M\right) \cdot \left(\frac{\frac{d}{D} \cdot c0}{\frac{h \cdot w}{\frac{d}{D}}} + M\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}}\]

Runtime

Time bar (total: 6.4m)Debug logProfile

herbie shell --seed 2018178 
(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))))))