Average Error: 14.2 → 8.7
Time: 58.0s
Precision: 64
Internal Precision: 576
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \le 2.0051494305701082 \cdot 10^{-238} \lor \neg \left(\frac{M \cdot D}{2 \cdot d} \le 9.708777033081603 \cdot 10^{+198}\right):\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{M}{\frac{2 \cdot d}{D}}}{\ell} \cdot \frac{\left(D \cdot \frac{1}{d}\right) \cdot M}{\frac{2}{h}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\sqrt[3]{\frac{\ell}{h}} \cdot \sqrt[3]{\frac{\ell}{h}}} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\sqrt[3]{\frac{\ell}{h}}}} \cdot w0\\ \end{array}\]

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 2 regimes

  2. if (/ (* M D) (* 2 d)) < 2.0051494305701082e-238 or 9.708777033081603e+198 < (/ (* M D) (* 2 d))

    1. Initial program 16.1

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

    2. Initial simplification15.7

      \[\leadsto \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\frac{\ell}{h}}} \cdot w0\]
    3. Using strategy rm

    4. Applied div-inv15.7

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

    5. Applied times-frac9.7

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

    6. Simplified12.5

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

    8. Applied associate-/l*11.8

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

    10. Applied associate-*r/9.7

      \[\leadsto \sqrt{1 - \frac{\frac{M}{\frac{2 \cdot d}{D}}}{\ell} \cdot \color{blue}{\frac{\frac{M}{d} \cdot D}{\frac{2}{h}}}} \cdot w0\]
    11. Using strategy rm

    12. Applied div-inv9.7

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

    13. Applied associate-*l*9.2

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

    if 2.0051494305701082e-238 < (/ (* M D) (* 2 d)) < 9.708777033081603e+198

    1. Initial program 8.4

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

    2. Initial simplification8.2

      \[\leadsto \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\frac{\ell}{h}}} \cdot w0\]
    3. Using strategy rm

    4. Applied add-cube-cbrt8.3

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

    5. Applied times-frac7.1

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

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \le 2.0051494305701082 \cdot 10^{-238} \lor \neg \left(\frac{M \cdot D}{2 \cdot d} \le 9.708777033081603 \cdot 10^{+198}\right):\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{M}{\frac{2 \cdot d}{D}}}{\ell} \cdot \frac{\left(D \cdot \frac{1}{d}\right) \cdot M}{\frac{2}{h}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\sqrt[3]{\frac{\ell}{h}} \cdot \sqrt[3]{\frac{\ell}{h}}} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\sqrt[3]{\frac{\ell}{h}}}} \cdot w0\\ \end{array}\]

Runtime

Time bar (total: 58.0s)Debug logProfile

herbie shell --seed 2018256 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))