Average Error: 58.4 → 28.6
Time: 5.0m
Precision: 64
Internal Precision: 7296
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} - M \cdot M} + \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}}\right) \cdot \frac{\frac{c0}{w}}{\frac{M \cdot M}{\frac{M \cdot M}{2}}} \le -2.061055827424937 \cdot 10^{-70}:\\ \;\;\;\;\left(\sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} - M \cdot M} + \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}}\right) \cdot \frac{\frac{c0}{w}}{\frac{M \cdot M}{\frac{M \cdot M}{2}}}\\ \mathbf{if}\;\left(\sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} - M \cdot M} + \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}}\right) \cdot \frac{\frac{c0}{w}}{\frac{M \cdot M}{\frac{M \cdot M}{2}}} \le -0.0:\\ \;\;\;\;\frac{\frac{c0 \cdot M}{2 \cdot w} \cdot M}{\frac{\frac{d}{D}}{\sqrt[3]{\frac{w \cdot h}{c0}}} \cdot \frac{\frac{d}{D}}{\sqrt[3]{\frac{w \cdot h}{c0}} \cdot \sqrt[3]{\frac{w \cdot h}{c0}}} - \sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} - M \cdot M}}\\ \mathbf{if}\;\left(\sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} - M \cdot M} + \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}}\right) \cdot \frac{\frac{c0}{w}}{\frac{M \cdot M}{\frac{M \cdot M}{2}}} \le +\infty:\\ \;\;\;\;\left(\sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} - M \cdot M} + \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}}\right) \cdot \frac{\frac{c0}{w}}{\frac{M \cdot M}{\frac{M \cdot M}{2}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

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

Derivation

  1. Split input into 3 regimes

  2. if (* (/ (/ c0 w) (/ (+ 0 (* M M)) (/ (* M M) 2))) (+ (/ (* (/ d D) (/ d D)) (/ (* w h) c0)) (sqrt (- (* (/ (* (/ d D) (/ d D)) (/ (* w h) c0)) (/ (* (/ d D) (/ d D)) (/ (* w h) c0))) (* M M))))) < -2.061055827424937e-70 or -0.0 < (* (/ (/ c0 w) (/ (+ 0 (* M M)) (/ (* M M) 2))) (+ (/ (* (/ d D) (/ d D)) (/ (* w h) c0)) (sqrt (- (* (/ (* (/ d D) (/ d D)) (/ (* w h) c0)) (/ (* (/ d D) (/ d D)) (/ (* w h) c0))) (* M M))))) < +inf.0

    1. Initial program 54.6

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

    3. Applied flip-+62.2

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

    4. Applied simplify57.8

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

    5. Taylor expanded around 0 57.8

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

    6. Applied simplify54.7

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

    8. Applied flip--61.4

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

    9. Applied associate-/r/61.4

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

    10. Applied simplify44.6

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

    if -2.061055827424937e-70 < (* (/ (/ c0 w) (/ (+ 0 (* M M)) (/ (* M M) 2))) (+ (/ (* (/ d D) (/ d D)) (/ (* w h) c0)) (sqrt (- (* (/ (* (/ d D) (/ d D)) (/ (* w h) c0)) (/ (* (/ d D) (/ d D)) (/ (* w h) c0))) (* M M))))) < -0.0

    1. Initial program 54.1

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

    3. Applied flip-+57.1

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

    4. Applied simplify40.2

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

    5. Taylor expanded around 0 40.2

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

    6. Applied simplify14.8

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

    8. Applied add-cube-cbrt12.3

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

    9. Applied times-frac12.4

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

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

    1. Initial program 59.1

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

    2. Taylor expanded around inf 31.7

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

    3. Applied simplify27.5

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

  4. Applied simplify28.6

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\left(\sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} - M \cdot M} + \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}}\right) \cdot \frac{\frac{c0}{w}}{\frac{M \cdot M}{\frac{M \cdot M}{2}}} \le -2.061055827424937 \cdot 10^{-70}:\\ \;\;\;\;\left(\sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} - M \cdot M} + \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}}\right) \cdot \frac{\frac{c0}{w}}{\frac{M \cdot M}{\frac{M \cdot M}{2}}}\\ \mathbf{if}\;\left(\sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} - M \cdot M} + \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}}\right) \cdot \frac{\frac{c0}{w}}{\frac{M \cdot M}{\frac{M \cdot M}{2}}} \le -0.0:\\ \;\;\;\;\frac{\frac{c0 \cdot M}{2 \cdot w} \cdot M}{\frac{\frac{d}{D}}{\sqrt[3]{\frac{w \cdot h}{c0}}} \cdot \frac{\frac{d}{D}}{\sqrt[3]{\frac{w \cdot h}{c0}} \cdot \sqrt[3]{\frac{w \cdot h}{c0}}} - \sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} - M \cdot M}}\\ \mathbf{if}\;\left(\sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} - M \cdot M} + \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}}\right) \cdot \frac{\frac{c0}{w}}{\frac{M \cdot M}{\frac{M \cdot M}{2}}} \le +\infty:\\ \;\;\;\;\left(\sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} - M \cdot M} + \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}}\right) \cdot \frac{\frac{c0}{w}}{\frac{M \cdot M}{\frac{M \cdot M}{2}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}}\]

Runtime

Time bar (total: 5.0m)Debug logProfile

herbie shell --seed '#(1070833653 108281690 3330367898 3632331308 3494323072 43156186)' 
(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))))))