Average Error: 13.9 → 9.7
Time: 38.9s
Precision: 64
Internal Precision: 576
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \le -1.0167463028438718 \cdot 10^{+104}:\\ \;\;\;\;\sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{\frac{\frac{\ell}{h}}{\sqrt[3]{\frac{M \cdot D}{2 \cdot d}} \cdot \sqrt[3]{\frac{M \cdot D}{2 \cdot d}}}}{\sqrt[3]{\frac{M \cdot D}{2 \cdot d}}}}} \cdot w0\\ \mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \le 3.1243219844813356 \cdot 10^{-44}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\ell}}{\frac{1}{h}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{2 \cdot d}{\left(h \cdot M\right) \cdot D} \cdot \ell}} \cdot w0\\ \end{array}\]

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 3 regimes

  2. if (/ (* M D) (* 2 d)) < -1.0167463028438718e+104

    1. Initial program 51.0

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

    2. Initial simplification51.1

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

    4. Applied associate-/l*41.9

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

    6. Applied add-cube-cbrt42.0

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

    7. Applied associate-/r*42.0

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

    if -1.0167463028438718e+104 < (/ (* M D) (* 2 d)) < 3.1243219844813356e-44

    1. Initial program 6.4

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

    2. Initial simplification5.9

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

    4. Applied div-inv5.9

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

    5. Applied associate-/r*1.6

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

    if 3.1243219844813356e-44 < (/ (* M D) (* 2 d))

    1. Initial program 28.6

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

    2. Initial simplification28.6

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

    4. Applied associate-/l*25.3

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

    6. Applied *-un-lft-identity25.3

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

    7. Applied div-inv25.3

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

    8. Applied times-frac24.5

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

    9. Simplified24.5

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

    10. Simplified29.5

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

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \le -1.0167463028438718 \cdot 10^{+104}:\\ \;\;\;\;\sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{\frac{\frac{\ell}{h}}{\sqrt[3]{\frac{M \cdot D}{2 \cdot d}} \cdot \sqrt[3]{\frac{M \cdot D}{2 \cdot d}}}}{\sqrt[3]{\frac{M \cdot D}{2 \cdot d}}}}} \cdot w0\\ \mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \le 3.1243219844813356 \cdot 10^{-44}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\ell}}{\frac{1}{h}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{2 \cdot d}{\left(h \cdot M\right) \cdot D} \cdot \ell}} \cdot w0\\ \end{array}\]

Runtime

Time bar (total: 38.9s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes12.29.78.04.258.6%
herbie shell --seed 2018295 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))