Average Error: 58.7 → 28.1
Time: 6.9m
Precision: 64
Internal Precision: 7232
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;M \cdot \frac{\frac{c0 \cdot M}{w \cdot 2}}{\frac{\frac{\frac{c0}{h}}{w}}{\frac{D}{d} \cdot \frac{D}{d}} - \sqrt{(\left(\frac{\frac{\frac{c0}{h}}{w}}{\frac{D}{d} \cdot \frac{D}{d}}\right) \cdot \left(\frac{\frac{\frac{c0}{h}}{w}}{\frac{D}{d} \cdot \frac{D}{d}}\right) + \left(-M \cdot M\right))_*}} \le -2.750775893256393 \cdot 10^{+87}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \sqrt[3]{{\left(\sqrt{(\left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) + \left(-M \cdot M\right))_*} + \frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{3}}\\ \mathbf{if}\;M \cdot \frac{\frac{c0 \cdot M}{w \cdot 2}}{\frac{\frac{\frac{c0}{h}}{w}}{\frac{D}{d} \cdot \frac{D}{d}} - \sqrt{(\left(\frac{\frac{\frac{c0}{h}}{w}}{\frac{D}{d} \cdot \frac{D}{d}}\right) \cdot \left(\frac{\frac{\frac{c0}{h}}{w}}{\frac{D}{d} \cdot \frac{D}{d}}\right) + \left(-M \cdot M\right))_*}} \le 6.942428153254334 \cdot 10^{+205}:\\ \;\;\;\;M \cdot \frac{\frac{c0 \cdot M}{w \cdot 2}}{\frac{\frac{\frac{c0}{h}}{w}}{\frac{D}{d} \cdot \frac{D}{d}} - \sqrt{(\left(\frac{\frac{\frac{c0}{h}}{w}}{\frac{D}{d} \cdot \frac{D}{d}}\right) \cdot \left(\frac{\frac{\frac{c0}{h}}{w}}{\frac{D}{d} \cdot \frac{D}{d}}\right) + \left(-M \cdot M\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

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 (* M (/ (/ (* c0 M) (* w 2)) (- (/ (/ (/ c0 h) w) (* (/ D d) (/ D d))) (sqrt (fma (/ (/ (/ c0 h) w) (* (/ D d) (/ D d))) (/ (/ (/ c0 h) w) (* (/ D d) (/ D d))) (- (* M M))))))) < -2.750775893256393e+87

    1. Initial program 48.6

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

    3. Applied add-cbrt-cube52.3

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

    4. Applied simplify32.6

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

    if -2.750775893256393e+87 < (* M (/ (/ (* c0 M) (* w 2)) (- (/ (/ (/ c0 h) w) (* (/ D d) (/ D d))) (sqrt (fma (/ (/ (/ c0 h) w) (* (/ D d) (/ D d))) (/ (/ (/ c0 h) w) (* (/ D d) (/ D d))) (- (* M M))))))) < 6.942428153254334e+205

    1. Initial program 59.9

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

    3. Applied flip-+60.6

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

    4. Applied simplify29.4

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

    5. Taylor expanded around 0 29.4

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

    6. Applied simplify13.3

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

    8. Applied *-un-lft-identity13.3

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

    9. Applied times-frac12.2

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

    10. Applied simplify12.2

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

    if 6.942428153254334e+205 < (* M (/ (/ (* c0 M) (* w 2)) (- (/ (/ (/ c0 h) w) (* (/ D d) (/ D d))) (sqrt (fma (/ (/ (/ c0 h) w) (* (/ D d) (/ D d))) (/ (/ (/ c0 h) w) (* (/ D d) (/ D d))) (- (* M M)))))))

    1. Initial program 58.2

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

    2. Taylor expanded around inf 48.0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{0}\]

    3. Applied simplify44.4

      \[\leadsto \color{blue}{0}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 6.9m)Debug logProfile

herbie shell --seed 2019053 +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))))))