Average Error: 58.4 → 47.0
Time: 1.0m
Precision: 64
Internal Precision: 128
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;d \le -1.164042777697057 \cdot 10^{-134}:\\ \;\;\;\;\frac{\frac{\left(\sqrt[3]{\frac{d}{D} \cdot c0} \cdot \sqrt[3]{\frac{1}{w}}\right) \cdot \left(\sqrt[3]{\frac{\frac{d}{D} \cdot c0}{w}} \cdot \sqrt[3]{\frac{\frac{d}{D} \cdot c0}{w}}\right)}{\frac{h}{\frac{\frac{d}{D} \cdot c0}{w}}} + \frac{\frac{\frac{d}{D} \cdot c0}{w}}{\frac{h}{\frac{\frac{d}{D} \cdot c0}{w}}}}{2}\\ \mathbf{elif}\;d \le 9.826801793869819 \cdot 10^{-272}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{\sqrt{\left(\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h} + M\right) \cdot \left(\frac{\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h} - M\right)} + \frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{d}{D} \cdot c0}{w}}{\frac{h}{\frac{\frac{d}{D} \cdot c0}{w}}} + \frac{\frac{\frac{d}{D} \cdot c0}{w}}{\frac{h}{\frac{\frac{d}{D} \cdot c0}{w}}}}{2}\\ \end{array}\]

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 3 regimes

  2. if d < -1.164042777697057e-134

    1. Initial program 58.1

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

    2. Simplified54.0

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

    3. Taylor expanded around 0 59.1

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

    4. Simplified53.8

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

    6. Applied pow153.8

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

    7. Applied pow153.8

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

    8. Applied pow-prod-down53.8

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

    9. Simplified47.2

      \[\leadsto {\color{blue}{\left(\frac{\frac{\frac{\frac{d}{D} \cdot c0}{w}}{\frac{h}{\frac{\frac{d}{D} \cdot c0}{w}}} + \frac{\frac{\frac{d}{D} \cdot c0}{w}}{\frac{h}{\frac{\frac{d}{D} \cdot c0}{w}}}}{2}\right)}}^{1}\]
    10. Using strategy rm

    11. Applied add-cube-cbrt47.3

      \[\leadsto {\left(\frac{\frac{\frac{\frac{d}{D} \cdot c0}{w}}{\frac{h}{\frac{\frac{d}{D} \cdot c0}{w}}} + \frac{\color{blue}{\left(\sqrt[3]{\frac{\frac{d}{D} \cdot c0}{w}} \cdot \sqrt[3]{\frac{\frac{d}{D} \cdot c0}{w}}\right) \cdot \sqrt[3]{\frac{\frac{d}{D} \cdot c0}{w}}}}{\frac{h}{\frac{\frac{d}{D} \cdot c0}{w}}}}{2}\right)}^{1}\]
    12. Using strategy rm

    13. Applied div-inv47.3

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

    14. Applied cbrt-prod47.3

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

    if -1.164042777697057e-134 < d < 9.826801793869819e-272

    1. Initial program 60.8

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

    2. Simplified44.2

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

    4. Applied associate-*r*45.5

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

    if 9.826801793869819e-272 < d

    1. Initial program 58.3

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

    2. Simplified52.4

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

    3. Taylor expanded around 0 59.1

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

    4. Simplified52.8

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

    6. Applied pow152.8

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

    7. Applied pow152.8

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

    8. Applied pow-prod-down52.8

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

    9. Simplified47.0

      \[\leadsto {\color{blue}{\left(\frac{\frac{\frac{\frac{d}{D} \cdot c0}{w}}{\frac{h}{\frac{\frac{d}{D} \cdot c0}{w}}} + \frac{\frac{\frac{d}{D} \cdot c0}{w}}{\frac{h}{\frac{\frac{d}{D} \cdot c0}{w}}}}{2}\right)}}^{1}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification47.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;d \le -1.164042777697057 \cdot 10^{-134}:\\ \;\;\;\;\frac{\frac{\left(\sqrt[3]{\frac{d}{D} \cdot c0} \cdot \sqrt[3]{\frac{1}{w}}\right) \cdot \left(\sqrt[3]{\frac{\frac{d}{D} \cdot c0}{w}} \cdot \sqrt[3]{\frac{\frac{d}{D} \cdot c0}{w}}\right)}{\frac{h}{\frac{\frac{d}{D} \cdot c0}{w}}} + \frac{\frac{\frac{d}{D} \cdot c0}{w}}{\frac{h}{\frac{\frac{d}{D} \cdot c0}{w}}}}{2}\\ \mathbf{elif}\;d \le 9.826801793869819 \cdot 10^{-272}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{\sqrt{\left(\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h} + M\right) \cdot \left(\frac{\left(\frac{c0}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h} - M\right)} + \frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{d}{D} \cdot c0}{w}}{\frac{h}{\frac{\frac{d}{D} \cdot c0}{w}}} + \frac{\frac{\frac{d}{D} \cdot c0}{w}}{\frac{h}{\frac{\frac{d}{D} \cdot c0}{w}}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019053 
(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))))))