Error

Derivation

1. Split input into 2 regimes

2. if M < 7.661994230609251e-164 or 4.642757572049068e-129 < M

   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. Taylor expanded around inf 35.4

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

   3. Taylor expanded around inf 33.8

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

   if 7.661994230609251e-164 < M < 4.642757572049068e-129

   1. Initial program 57.2

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

   3. Applied div-inv57.5

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \color{blue}{\left(\left(c0 \cdot \left(d \cdot d\right)\right) \cdot \frac{1}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)} - M \cdot M}\right)\]
   4. Using strategy rm

   5. Applied *-un-lft-identity57.5

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

   6. Applied *-un-lft-identity57.5

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

   7. Applied distribute-lft-out57.5

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

   8. Simplified51.7

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

4. Final simplification34.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \le 7.661994230609251 \cdot 10^{-164} \lor \neg \left(M \le 4.642757572049068 \cdot 10^{-129}\right):\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{(\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) + \left(M \cdot \left(-M\right)\right))_*} + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \frac{c0}{2 \cdot w}\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 +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))))))

Details

Time bar (total: 57.9s)Debug log