Average Error: 58.4 → 30.0
Time: 5.7m
Precision: 64
Internal Precision: 7488
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \sqrt[3]{{\left(\sqrt{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right) \cdot \left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)} + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{3}} \le 2.4433046726112392 \cdot 10^{-188}:\\ \;\;\;\;\frac{\left(\frac{M}{w} \cdot c0\right) \cdot \frac{M}{2}}{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} - \sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} - M \cdot M}}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \sqrt[3]{{\left(\sqrt{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right) \cdot \left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)} + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{3}} \le 1.2354806727237655 \cdot 10^{+207}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \sqrt[3]{{\left(\sqrt{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right) \cdot \left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)} + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (* (/ c0 (* 2 w)) (cbrt (pow (+ (sqrt (* (- (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) M) (+ M (* (/ (/ c0 h) w) (* (/ d D) (/ d D)))))) (* (/ (/ c0 h) w) (* (/ d D) (/ d D)))) 3))) < 2.4433046726112392e-188

    1. Initial program 51.8

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

    3. Applied flip-+57.9

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

    4. Applied simplify51.6

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

    5. Taylor expanded around 0 51.6

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

    6. Applied simplify40.4

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

    8. Applied *-un-lft-identity40.4

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

    9. Applied associate-/r*40.4

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

    10. Applied simplify38.6

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

    if 2.4433046726112392e-188 < (* (/ c0 (* 2 w)) (cbrt (pow (+ (sqrt (* (- (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) M) (+ M (* (/ (/ c0 h) w) (* (/ d D) (/ d D)))))) (* (/ (/ c0 h) w) (* (/ d D) (/ d D)))) 3))) < 1.2354806727237655e+207

    1. Initial program 43.5

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

    3. Applied add-cbrt-cube43.5

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

    4. Applied simplify2.1

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

    if 1.2354806727237655e+207 < (* (/ c0 (* 2 w)) (cbrt (pow (+ (sqrt (* (- (* (/ (/ c0 h) w) (* (/ d D) (/ d D))) M) (+ M (* (/ (/ c0 h) w) (* (/ d D) (/ d D)))))) (* (/ (/ c0 h) w) (* (/ d D) (/ d D)))) 3)))

    1. Initial program 61.1

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

    2. Taylor expanded around inf 33.3

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

    3. Applied simplify28.0

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

Runtime

Time bar (total: 5.7m)Debug logProfile

herbie shell --seed 2018198 
(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))))))