Average Error: 58.2 → 46.4
Time: 4.1m
Precision: 64
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;w \le -1.1860835760269866 \cdot 10^{-84}:\\ \;\;\;\;\frac{\left(\left(\frac{d \cdot c0}{\left(w \cdot \sqrt[3]{-1}\right) \cdot D} \cdot \sqrt[3]{\frac{-1}{h}}\right) \cdot \frac{\frac{d \cdot c0}{w \cdot D}}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right) \cdot 2}{2}\\ \mathbf{elif}\;w \le 5.20312015598324 \cdot 10^{+83}:\\ \;\;\;\;\frac{\frac{\left(\frac{d}{D} \cdot \frac{c0 \cdot 2}{h}\right) \cdot \frac{d}{D}}{w} \cdot \frac{c0}{w}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\frac{d \cdot c0}{\left(w \cdot \sqrt[3]{-1}\right) \cdot D} \cdot \sqrt[3]{\frac{-1}{h}}\right) \cdot \frac{\frac{d \cdot c0}{w \cdot D}}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right) \cdot 2}{2}\\ \end{array}\]

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 w < -1.1860835760269866e-84 or 5.20312015598324e+83 < w

    1. Initial program 57.4

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

    2. Simplified51.2

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

    3. Taylor expanded around 0 58.1

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

    4. Simplified54.1

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

    5. Taylor expanded around 0 59.2

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

    6. Simplified42.6

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

    8. Applied add-cube-cbrt42.7

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

    9. Applied times-frac40.8

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

    10. Taylor expanded around -inf 52.0

      \[\leadsto \frac{\left(\frac{\frac{d \cdot c0}{w \cdot D}}{\sqrt[3]{h} \cdot \sqrt[3]{h}} \cdot \color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(\sqrt[3]{-1} \cdot w\right)} \cdot {\left(\frac{-1}{h}\right)}^{\frac{1}{3}}\right)}\right) \cdot 2}{2}\]

    11. Simplified40.7

      \[\leadsto \frac{\left(\frac{\frac{d \cdot c0}{w \cdot D}}{\sqrt[3]{h} \cdot \sqrt[3]{h}} \cdot \color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(\sqrt[3]{-1} \cdot w\right)} \cdot \sqrt[3]{\frac{-1}{h}}\right)}\right) \cdot 2}{2}\]

    if -1.1860835760269866e-84 < w < 5.20312015598324e+83

    1. Initial program 59.0

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

    2. Simplified53.6

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

    3. Taylor expanded around 0 58.7

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

    4. Simplified54.2

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

    6. Applied associate-*r*52.2

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

  4. Final simplification46.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;w \le -1.1860835760269866 \cdot 10^{-84}:\\ \;\;\;\;\frac{\left(\left(\frac{d \cdot c0}{\left(w \cdot \sqrt[3]{-1}\right) \cdot D} \cdot \sqrt[3]{\frac{-1}{h}}\right) \cdot \frac{\frac{d \cdot c0}{w \cdot D}}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right) \cdot 2}{2}\\ \mathbf{elif}\;w \le 5.20312015598324 \cdot 10^{+83}:\\ \;\;\;\;\frac{\frac{\left(\frac{d}{D} \cdot \frac{c0 \cdot 2}{h}\right) \cdot \frac{d}{D}}{w} \cdot \frac{c0}{w}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\frac{d \cdot c0}{\left(w \cdot \sqrt[3]{-1}\right) \cdot D} \cdot \sqrt[3]{\frac{-1}{h}}\right) \cdot \frac{\frac{d \cdot c0}{w \cdot D}}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right) \cdot 2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019100 
(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))))))