Average Error: 58.3 → 28.3
Time: 4.3m
Precision: 64
Internal Precision: 7232
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{\frac{c0}{w \cdot 2}}}{\frac{c0}{h \cdot w}} \cdot \sqrt{\frac{c0}{w \cdot 2}} \le 1.1218416008262068 \cdot 10^{-266} \lor \neg \left(\frac{\sqrt{\frac{c0}{w \cdot 2}}}{\frac{c0}{h \cdot w}} \cdot \sqrt{\frac{c0}{w \cdot 2}} \le +\infty\right):\\ \;\;\;\;\left(\frac{M}{\frac{d}{D}} \cdot \frac{M}{\frac{d}{D}}\right) \cdot \left(\sqrt[3]{\frac{h}{2}} \cdot \left(\sqrt[3]{\frac{h}{2}} \cdot \sqrt[3]{\frac{h}{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\sqrt{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) + M\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right)} + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{3}} \cdot \frac{c0}{w \cdot 2}\\ \end{array}\]

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 2 regimes

  2. if (* (sqrt (/ c0 (* 2 w))) (/ (sqrt (/ c0 (* w 2))) (/ c0 (* h w)))) < 1.1218416008262068e-266 or +inf.0 < (* (sqrt (/ c0 (* 2 w))) (/ (sqrt (/ c0 (* w 2))) (/ c0 (* h w))))

    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. 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 simplify42.0

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

    5. Taylor expanded around 0 42.1

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

    6. Applied simplify36.3

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

    8. Applied add-cube-cbrt36.3

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

    9. Applied simplify36.3

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

    10. Applied simplify26.6

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

    if 1.1218416008262068e-266 < (* (sqrt (/ c0 (* 2 w))) (/ (sqrt (/ c0 (* w 2))) (/ c0 (* h w)))) < +inf.0

    1. Initial program 50.8

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

    3. Applied add-cbrt-cube52.9

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

    4. Applied simplify43.8

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

  4. Applied simplify28.3

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{\sqrt{\frac{c0}{w \cdot 2}}}{\frac{c0}{h \cdot w}} \cdot \sqrt{\frac{c0}{w \cdot 2}} \le 1.1218416008262068 \cdot 10^{-266} \lor \neg \left(\frac{\sqrt{\frac{c0}{w \cdot 2}}}{\frac{c0}{h \cdot w}} \cdot \sqrt{\frac{c0}{w \cdot 2}} \le +\infty\right):\\ \;\;\;\;\left(\frac{M}{\frac{d}{D}} \cdot \frac{M}{\frac{d}{D}}\right) \cdot \left(\sqrt[3]{\frac{h}{2}} \cdot \left(\sqrt[3]{\frac{h}{2}} \cdot \sqrt[3]{\frac{h}{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\sqrt{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) + M\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right)} + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{3}} \cdot \frac{c0}{w \cdot 2}\\ \end{array}}\]

Runtime

Time bar (total: 4.3m)Debug logProfile

herbie shell --seed '#(1072107073 2127697367 3936270018 2300570620 2134894798 4023771849)' 
(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))))))