Average Error: 58.6 → 29.6
Time: 4.9m
Precision: 64
Internal Precision: 7488
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{h \cdot w}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{h \cdot w}{c0}} - M \cdot M} + \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{h \cdot w}{c0}}\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{\frac{c0}{w \cdot 2}}{M \cdot M}\right) \le 4.2213177622225 \cdot 10^{-315}:\\ \;\;\;\;\frac{\frac{M}{2} \cdot \left(c0 \cdot \frac{M}{w}\right)}{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{h \cdot w}{c0}} - \sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{h \cdot w}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{h \cdot w}{c0}} - M \cdot M}}\\ \mathbf{if}\;\left(\sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{h \cdot w}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{h \cdot w}{c0}} - M \cdot M} + \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{h \cdot w}{c0}}\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{\frac{c0}{w \cdot 2}}{M \cdot M}\right) \le +\infty:\\ \;\;\;\;\left(\sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{h \cdot w}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{h \cdot w}{c0}} - M \cdot M} + \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{h \cdot w}{c0}}\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{\frac{c0}{w \cdot 2}}{M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

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 (* 2 w)) (+ 0 (* M M))) (* M M)) (+ (/ (* (/ d D) (/ d D)) (/ (* w h) c0)) (sqrt (- (* (/ (* (/ d D) (/ d D)) (/ (* w h) c0)) (/ (* (/ d D) (/ d D)) (/ (* w h) c0))) (* M M))))) < 4.2213177622225e-315

    1. Initial program 55.1

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

    3. Applied flip-+60.2

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

    4. Applied simplify50.5

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

    5. Taylor expanded around 0 50.5

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

    6. Applied simplify38.3

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

    8. Applied *-un-lft-identity38.3

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

    9. Applied associate-/r*38.3

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

    10. Applied simplify36.9

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

    if 4.2213177622225e-315 < (* (* (/ (/ c0 (* 2 w)) (+ 0 (* M M))) (* M M)) (+ (/ (* (/ 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 52.7

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

    3. Applied flip-+61.2

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

    4. Applied simplify58.4

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

    5. Taylor expanded around 0 58.4

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

    6. Applied simplify56.2

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

    8. Applied flip--60.7

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

    9. Applied associate-/r/60.7

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

    10. Applied simplify43.3

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

    if +inf.0 < (* (* (/ (/ c0 (* 2 w)) (+ 0 (* M M))) (* M M)) (+ (/ (* (/ 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 59.4

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

    2. Taylor expanded around inf 33.0

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

    3. Applied simplify27.9

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

  4. Applied simplify29.6

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\left(\sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{h \cdot w}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{h \cdot w}{c0}} - M \cdot M} + \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{h \cdot w}{c0}}\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{\frac{c0}{w \cdot 2}}{M \cdot M}\right) \le 4.2213177622225 \cdot 10^{-315}:\\ \;\;\;\;\frac{\frac{M}{2} \cdot \left(c0 \cdot \frac{M}{w}\right)}{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{h \cdot w}{c0}} - \sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{h \cdot w}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{h \cdot w}{c0}} - M \cdot M}}\\ \mathbf{if}\;\left(\sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{h \cdot w}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{h \cdot w}{c0}} - M \cdot M} + \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{h \cdot w}{c0}}\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{\frac{c0}{w \cdot 2}}{M \cdot M}\right) \le +\infty:\\ \;\;\;\;\left(\sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{h \cdot w}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{h \cdot w}{c0}} - M \cdot M} + \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{h \cdot w}{c0}}\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{\frac{c0}{w \cdot 2}}{M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}}\]

Runtime

Time bar (total: 4.9m)Debug logProfile

herbie shell --seed '#(1071246582 2318319007 2683472949 3810440501 3233274817 2724848749)' 
(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))))))