Average Error: 58.0 → 29.5
Time: 6.0m
Precision: 64
Internal Precision: 7488
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{c0}{2 \cdot w}}{w \cdot h} \cdot \frac{c0 \cdot 2}{\frac{D}{d} \cdot \frac{D}{d}} \le -1.6040223835252562 \cdot 10^{+282}:\\ \;\;\;\;0\\ \mathbf{if}\;\frac{\frac{c0}{2 \cdot w}}{w \cdot h} \cdot \frac{c0 \cdot 2}{\frac{D}{d} \cdot \frac{D}{d}} \le 1.0731791218443217 \cdot 10^{-40}:\\ \;\;\;\;\frac{\frac{c0}{2 \cdot w}}{w \cdot h} \cdot \frac{c0 \cdot 2}{\frac{D}{d} \cdot \frac{D}{d}}\\ \mathbf{if}\;\frac{\frac{c0}{2 \cdot w}}{w \cdot h} \cdot \frac{c0 \cdot 2}{\frac{D}{d} \cdot \frac{D}{d}} \le 1.726008758591811 \cdot 10^{+182}:\\ \;\;\;\;\frac{M \cdot M}{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} - \sqrt{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) + \left(\left(-M\right) \cdot M\right))_*}} \cdot \frac{c0}{2 \cdot w}\\ \mathbf{if}\;\frac{\frac{c0}{2 \cdot w}}{w \cdot h} \cdot \frac{c0 \cdot 2}{\frac{D}{d} \cdot \frac{D}{d}} \le 2.045078630730487 \cdot 10^{+261}:\\ \;\;\;\;\frac{\frac{c0}{2 \cdot w}}{w \cdot h} \cdot \frac{c0 \cdot 2}{\frac{D}{d} \cdot \frac{D}{d}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

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 (* (/ (* 2 c0) (* (/ D d) (/ D d))) (/ (/ c0 (* w 2)) (* h w))) < -1.6040223835252562e+282 or 2.045078630730487e+261 < (* (/ (* 2 c0) (* (/ D d) (/ D d))) (/ (/ c0 (* w 2)) (* h w)))

    1. Initial program 61.4

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

    2. Taylor expanded around inf 34.5

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

    3. Applied simplify29.4

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

    if -1.6040223835252562e+282 < (* (/ (* 2 c0) (* (/ D d) (/ D d))) (/ (/ c0 (* w 2)) (* h w))) < 1.0731791218443217e-40 or 1.726008758591811e+182 < (* (/ (* 2 c0) (* (/ D d) (/ D d))) (/ (/ c0 (* w 2)) (* h w))) < 2.045078630730487e+261

    1. Initial program 46.3

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

    3. Applied flip-+57.0

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

    4. Applied simplify51.0

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

    6. Applied *-un-lft-identity51.0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{0 + M \cdot M}{\color{blue}{1 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}}\]

    7. Applied *-un-lft-identity51.0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\color{blue}{1 \cdot \left(0 + M \cdot M\right)}}{1 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}\]

    8. Applied times-frac51.0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\frac{1}{1} \cdot \frac{0 + M \cdot M}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}}\right)}\]

    9. Applied simplify51.0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{1} \cdot \frac{0 + M \cdot M}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}}\right)\]

    10. Applied simplify39.2

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

    11. Taylor expanded around 0 47.8

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

    12. Applied simplify27.9

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

    if 1.0731791218443217e-40 < (* (/ (* 2 c0) (* (/ D d) (/ D d))) (/ (/ c0 (* w 2)) (* h w))) < 1.726008758591811e+182

    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 flip-+56.3

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

    4. Applied simplify47.1

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

    6. Applied *-un-lft-identity47.1

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{0 + M \cdot M}{\color{blue}{1 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}}\]

    7. Applied *-un-lft-identity47.1

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\color{blue}{1 \cdot \left(0 + M \cdot M\right)}}{1 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}\]

    8. Applied times-frac47.1

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\frac{1}{1} \cdot \frac{0 + M \cdot M}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}}\right)}\]

    9. Applied simplify47.1

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{1} \cdot \frac{0 + M \cdot M}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}}\right)\]

    10. Applied simplify38.3

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

  4. Applied simplify29.5

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{\frac{c0}{2 \cdot w}}{w \cdot h} \cdot \frac{c0 \cdot 2}{\frac{D}{d} \cdot \frac{D}{d}} \le -1.6040223835252562 \cdot 10^{+282}:\\ \;\;\;\;0\\ \mathbf{if}\;\frac{\frac{c0}{2 \cdot w}}{w \cdot h} \cdot \frac{c0 \cdot 2}{\frac{D}{d} \cdot \frac{D}{d}} \le 1.0731791218443217 \cdot 10^{-40}:\\ \;\;\;\;\frac{\frac{c0}{2 \cdot w}}{w \cdot h} \cdot \frac{c0 \cdot 2}{\frac{D}{d} \cdot \frac{D}{d}}\\ \mathbf{if}\;\frac{\frac{c0}{2 \cdot w}}{w \cdot h} \cdot \frac{c0 \cdot 2}{\frac{D}{d} \cdot \frac{D}{d}} \le 1.726008758591811 \cdot 10^{+182}:\\ \;\;\;\;\frac{M \cdot M}{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} - \sqrt{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) + \left(\left(-M\right) \cdot M\right))_*}} \cdot \frac{c0}{2 \cdot w}\\ \mathbf{if}\;\frac{\frac{c0}{2 \cdot w}}{w \cdot h} \cdot \frac{c0 \cdot 2}{\frac{D}{d} \cdot \frac{D}{d}} \le 2.045078630730487 \cdot 10^{+261}:\\ \;\;\;\;\frac{\frac{c0}{2 \cdot w}}{w \cdot h} \cdot \frac{c0 \cdot 2}{\frac{D}{d} \cdot \frac{D}{d}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}}\]

Runtime

Time bar (total: 6.0m)Debug logProfile

herbie shell --seed '#(1072361757 3390613284 2339397988 1175251238 145061547 3101881848)' +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))))))