Average Error: 58.1 → 46.8
Time: 2.1m
Precision: 64
Internal Precision: 128
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;w \le 1.736281195403628 \cdot 10^{+124}:\\ \;\;\;\;\frac{\frac{\frac{1}{\frac{w}{c0 \cdot \frac{d}{D}}}}{\frac{h}{\frac{1}{\frac{w}{c0 \cdot \frac{d}{D}}}}} + \frac{\frac{c0 \cdot \frac{d}{D}}{w}}{\frac{h}{\frac{c0 \cdot \frac{d}{D}}{w}}}}{2}\\ \mathbf{elif}\;w \le 7.81791615961806 \cdot 10^{+280}:\\ \;\;\;\;\frac{\left(\frac{d}{w} \cdot \frac{d}{w}\right) \cdot \left(\frac{c0}{D} \cdot \frac{c0}{D}\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{\frac{c0 \cdot 2}{h \cdot w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}{2}\\ \end{array}\]

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 w < 1.736281195403628e+124

    1. Initial program 58.4

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

    2. Simplified52.1

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

    3. Taylor expanded around 0 59.3

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

    4. Simplified53.0

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

    6. Applied pow153.0

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

    7. Applied pow153.0

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

    8. Applied pow-prod-down53.0

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

    9. Simplified47.0

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

    11. Applied clear-num47.0

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

    13. Applied clear-num47.0

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

    if 1.736281195403628e+124 < w < 7.81791615961806e+280

    1. Initial program 56.7

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

    2. Simplified50.0

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

    3. Taylor expanded around 0 58.5

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

    4. Simplified49.4

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

    5. Taylor expanded around -inf 58.3

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

    6. Simplified45.0

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

    if 7.81791615961806e+280 < w

    1. Initial program 56.3

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

    2. Simplified52.0

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

    3. Taylor expanded around 0 58.4

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

    4. Simplified50.3

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

  4. Final simplification46.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;w \le 1.736281195403628 \cdot 10^{+124}:\\ \;\;\;\;\frac{\frac{\frac{1}{\frac{w}{c0 \cdot \frac{d}{D}}}}{\frac{h}{\frac{1}{\frac{w}{c0 \cdot \frac{d}{D}}}}} + \frac{\frac{c0 \cdot \frac{d}{D}}{w}}{\frac{h}{\frac{c0 \cdot \frac{d}{D}}{w}}}}{2}\\ \mathbf{elif}\;w \le 7.81791615961806 \cdot 10^{+280}:\\ \;\;\;\;\frac{\left(\frac{d}{w} \cdot \frac{d}{w}\right) \cdot \left(\frac{c0}{D} \cdot \frac{c0}{D}\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{\frac{c0 \cdot 2}{h \cdot w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019051 
(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))))))