Average Error: 13.4 → 8.5
Time: 1.2m
Precision: 64
Internal Precision: 320
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -3.88413374182046 \cdot 10^{+285}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot {\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{2}\right) \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \le -5.264412529417236 \cdot 10^{-255}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D \cdot M}{d \cdot 2} \cdot \left(\frac{h}{\ell} \cdot \frac{D \cdot M}{d \cdot 2}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(\left({\left(\sqrt[3]{\frac{D \cdot M}{d \cdot 2}}\right)}^{2} \cdot h\right) \cdot \left(\sqrt[3]{\frac{D}{2}} \cdot \frac{\frac{D}{2}}{\frac{d}{M}}\right)\right) \cdot \frac{1}{\ell}}{\sqrt[3]{\frac{d}{M}}}}\\ \end{array}\]

Error

Bits error versus w0

Bits error versus M

Bits error versus D

Bits error versus h

Bits error versus l

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (/ h l) < -3.88413374182046e+285

    1. Initial program 53.7

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
    2. Using strategy rm

    3. Applied div-inv53.7

      \[\leadsto w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\left(h \cdot \frac{1}{\ell}\right)}}\]

    4. Applied associate-*r*24.5

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}}}\]
    5. Using strategy rm

    6. Applied associate-/l*24.1

      \[\leadsto w0 \cdot \sqrt{1 - \left({\color{blue}{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}}^{2} \cdot h\right) \cdot \frac{1}{\ell}}\]

    if -3.88413374182046e+285 < (/ h l) < -5.264412529417236e-255

    1. Initial program 12.6

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
    2. Using strategy rm

    3. Applied unpow212.6

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}}\]

    4. Applied associate-*l*11.1

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}}\]

    if -5.264412529417236e-255 < (/ h l)

    1. Initial program 8.4

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
    2. Using strategy rm

    3. Applied div-inv8.4

      \[\leadsto w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\left(h \cdot \frac{1}{\ell}\right)}}\]

    4. Applied associate-*r*5.1

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}}}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt5.1

      \[\leadsto w0 \cdot \sqrt{1 - \left({\color{blue}{\left(\left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}} \cdot \sqrt[3]{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \sqrt[3]{\frac{M \cdot D}{2 \cdot d}}\right)}}^{2} \cdot h\right) \cdot \frac{1}{\ell}}\]

    7. Applied unpow-prod-down5.1

      \[\leadsto w0 \cdot \sqrt{1 - \left(\color{blue}{\left({\left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}} \cdot \sqrt[3]{\frac{M \cdot D}{2 \cdot d}}\right)}^{2} \cdot {\left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}}\right)}^{2}\right)} \cdot h\right) \cdot \frac{1}{\ell}}\]

    8. Applied associate-*l*3.7

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left({\left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}} \cdot \sqrt[3]{\frac{M \cdot D}{2 \cdot d}}\right)}^{2} \cdot \left({\left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}}\right)}^{2} \cdot h\right)\right)} \cdot \frac{1}{\ell}}\]

    9. Simplified4.2

      \[\leadsto w0 \cdot \sqrt{1 - \left(\color{blue}{\left(\sqrt[3]{\frac{\frac{D}{2}}{\frac{d}{M}}} \cdot \frac{\frac{D}{2}}{\frac{d}{M}}\right)} \cdot \left({\left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}}\right)}^{2} \cdot h\right)\right) \cdot \frac{1}{\ell}}\]
    10. Using strategy rm

    11. Applied cbrt-div4.2

      \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\color{blue}{\frac{\sqrt[3]{\frac{D}{2}}}{\sqrt[3]{\frac{d}{M}}}} \cdot \frac{\frac{D}{2}}{\frac{d}{M}}\right) \cdot \left({\left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}}\right)}^{2} \cdot h\right)\right) \cdot \frac{1}{\ell}}\]

    12. Applied associate-*l/4.3

      \[\leadsto w0 \cdot \sqrt{1 - \left(\color{blue}{\frac{\sqrt[3]{\frac{D}{2}} \cdot \frac{\frac{D}{2}}{\frac{d}{M}}}{\sqrt[3]{\frac{d}{M}}}} \cdot \left({\left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}}\right)}^{2} \cdot h\right)\right) \cdot \frac{1}{\ell}}\]

    13. Applied associate-*l/4.2

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(\sqrt[3]{\frac{D}{2}} \cdot \frac{\frac{D}{2}}{\frac{d}{M}}\right) \cdot \left({\left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}}\right)}^{2} \cdot h\right)}{\sqrt[3]{\frac{d}{M}}}} \cdot \frac{1}{\ell}}\]

    14. Applied associate-*l/4.1

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(\left(\sqrt[3]{\frac{D}{2}} \cdot \frac{\frac{D}{2}}{\frac{d}{M}}\right) \cdot \left({\left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}}\right)}^{2} \cdot h\right)\right) \cdot \frac{1}{\ell}}{\sqrt[3]{\frac{d}{M}}}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -3.88413374182046 \cdot 10^{+285}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot {\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{2}\right) \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \le -5.264412529417236 \cdot 10^{-255}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D \cdot M}{d \cdot 2} \cdot \left(\frac{h}{\ell} \cdot \frac{D \cdot M}{d \cdot 2}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(\left({\left(\sqrt[3]{\frac{D \cdot M}{d \cdot 2}}\right)}^{2} \cdot h\right) \cdot \left(\sqrt[3]{\frac{D}{2}} \cdot \frac{\frac{D}{2}}{\frac{d}{M}}\right)\right) \cdot \frac{1}{\ell}}{\sqrt[3]{\frac{d}{M}}}}\\ \end{array}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018251 +o rules:numerics
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))