Average Error: 58.2 → 35.9
Time: 41.9s
Precision: 64
Internal Precision: 128
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;D \cdot D \le 6.230201052839926 \cdot 10^{+49}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \cdot D \le 8.488222743869225 \cdot 10^{+200}:\\ \;\;\;\;\frac{\frac{c0}{w} \cdot (\left(\sqrt{M + \frac{\frac{d}{D} \cdot \frac{d}{D}}{w} \cdot \frac{c0}{h}}\right) \cdot \left(\sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{w} \cdot \frac{c0}{h} - M}\right) + \left(\frac{\frac{d}{D} \cdot \frac{d}{D}}{w} \cdot \frac{c0}{h}\right))_*}{2}\\ \mathbf{elif}\;D \cdot D \le 1.379185921615959 \cdot 10^{+245}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c0}{w} \cdot \left(\left(\sqrt[3]{\sqrt{\left(\frac{\frac{d}{D} \cdot \frac{d}{D}}{w} \cdot \frac{c0}{h}\right) \cdot \left(\frac{\frac{d}{D} \cdot \frac{d}{D}}{w} \cdot \frac{c0}{h}\right) - M \cdot M} + \frac{\frac{d}{D} \cdot \frac{d}{D}}{w} \cdot \frac{c0}{h}} \cdot \sqrt[3]{\sqrt{\left(\frac{\frac{d}{D} \cdot \frac{d}{D}}{w} \cdot \frac{c0}{h}\right) \cdot \left(\frac{\frac{d}{D} \cdot \frac{d}{D}}{w} \cdot \frac{c0}{h}\right) - M \cdot M} + \frac{\frac{d}{D} \cdot \frac{d}{D}}{w} \cdot \frac{c0}{h}}\right) \cdot \sqrt[3]{\sqrt{\left(\frac{\frac{d}{D} \cdot \frac{d}{D}}{w} \cdot \frac{c0}{h}\right) \cdot \left(\frac{\frac{d}{D} \cdot \frac{d}{D}}{w} \cdot \frac{c0}{h}\right) - M \cdot M} + \frac{\frac{d}{D} \cdot \frac{d}{D}}{w} \cdot \frac{c0}{h}}\right)}{2}\\ \end{array}\]

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 (* D D) < 6.230201052839926e+49 or 8.488222743869225e+200 < (* D D) < 1.379185921615959e+245

    1. Initial program 58.5

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

    2. Simplified55.1

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

    3. Taylor expanded around inf 33.1

      \[\leadsto \frac{\frac{c0}{w} \cdot \color{blue}{0}}{2}\]
    4. Using strategy rm

    5. Applied mul031.4

      \[\leadsto \frac{\color{blue}{0}}{2}\]

    if 6.230201052839926e+49 < (* D D) < 8.488222743869225e+200

    1. Initial program 53.4

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

    2. Simplified50.2

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

    4. Applied difference-of-squares50.2

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

    5. Applied sqrt-prod52.6

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

    6. Applied fma-def52.6

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

    if 1.379185921615959e+245 < (* D D)

    1. Initial program 60.0

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

    2. Simplified46.0

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

    4. Applied add-cube-cbrt46.1

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

  4. Final simplification35.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot D \le 6.230201052839926 \cdot 10^{+49}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \cdot D \le 8.488222743869225 \cdot 10^{+200}:\\ \;\;\;\;\frac{\frac{c0}{w} \cdot (\left(\sqrt{M + \frac{\frac{d}{D} \cdot \frac{d}{D}}{w} \cdot \frac{c0}{h}}\right) \cdot \left(\sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{w} \cdot \frac{c0}{h} - M}\right) + \left(\frac{\frac{d}{D} \cdot \frac{d}{D}}{w} \cdot \frac{c0}{h}\right))_*}{2}\\ \mathbf{elif}\;D \cdot D \le 1.379185921615959 \cdot 10^{+245}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c0}{w} \cdot \left(\left(\sqrt[3]{\sqrt{\left(\frac{\frac{d}{D} \cdot \frac{d}{D}}{w} \cdot \frac{c0}{h}\right) \cdot \left(\frac{\frac{d}{D} \cdot \frac{d}{D}}{w} \cdot \frac{c0}{h}\right) - M \cdot M} + \frac{\frac{d}{D} \cdot \frac{d}{D}}{w} \cdot \frac{c0}{h}} \cdot \sqrt[3]{\sqrt{\left(\frac{\frac{d}{D} \cdot \frac{d}{D}}{w} \cdot \frac{c0}{h}\right) \cdot \left(\frac{\frac{d}{D} \cdot \frac{d}{D}}{w} \cdot \frac{c0}{h}\right) - M \cdot M} + \frac{\frac{d}{D} \cdot \frac{d}{D}}{w} \cdot \frac{c0}{h}}\right) \cdot \sqrt[3]{\sqrt{\left(\frac{\frac{d}{D} \cdot \frac{d}{D}}{w} \cdot \frac{c0}{h}\right) \cdot \left(\frac{\frac{d}{D} \cdot \frac{d}{D}}{w} \cdot \frac{c0}{h}\right) - M \cdot M} + \frac{\frac{d}{D} \cdot \frac{d}{D}}{w} \cdot \frac{c0}{h}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019068 +o rules:numerics
(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))))))