Average Error: 13.4 → 7.8
Time: 53.7s
Precision: 64
Internal Precision: 576
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;M \cdot D \le -0.0:\\ \;\;\;\;\sqrt{1 - \frac{\frac{M}{\frac{d}{D}} \cdot h}{\frac{\ell}{\frac{\frac{\frac{M}{\sqrt[3]{\frac{d}{D}} \cdot \sqrt[3]{\frac{d}{D}}}}{\sqrt[3]{\frac{d}{D}}}}{4}}}} \cdot w0\\ \mathbf{elif}\;M \cdot D \le 1.8217782890704009 \cdot 10^{+25}:\\ \;\;\;\;\sqrt{1 - h \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{\ell}{\frac{M \cdot D}{2 \cdot d}}}} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{M}{\frac{d}{D}} \cdot h}{\frac{\ell}{\frac{\frac{1}{\frac{\frac{d}{D}}{M}}}{4}}}}\\ \end{array}\]

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) < -0.0

    1. Initial program 12.9

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

    2. Initial simplification12.5

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

    4. Applied associate-/r/9.5

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

    6. Applied associate-*l/9.5

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

    7. Simplified7.8

      \[\leadsto \sqrt{1 - \frac{\color{blue}{\left(h \cdot \frac{M}{\frac{d}{D}}\right) \cdot \frac{\frac{M}{\frac{d}{D}}}{4}}}{\ell}} \cdot w0\]
    8. Using strategy rm

    9. Applied associate-/l*7.0

      \[\leadsto \sqrt{1 - \color{blue}{\frac{h \cdot \frac{M}{\frac{d}{D}}}{\frac{\ell}{\frac{\frac{M}{\frac{d}{D}}}{4}}}}} \cdot w0\]
    10. Using strategy rm

    11. Applied add-cube-cbrt7.1

      \[\leadsto \sqrt{1 - \frac{h \cdot \frac{M}{\frac{d}{D}}}{\frac{\ell}{\frac{\frac{M}{\color{blue}{\left(\sqrt[3]{\frac{d}{D}} \cdot \sqrt[3]{\frac{d}{D}}\right) \cdot \sqrt[3]{\frac{d}{D}}}}}{4}}}} \cdot w0\]

    12. Applied associate-/r*7.1

      \[\leadsto \sqrt{1 - \frac{h \cdot \frac{M}{\frac{d}{D}}}{\frac{\ell}{\frac{\color{blue}{\frac{\frac{M}{\sqrt[3]{\frac{d}{D}} \cdot \sqrt[3]{\frac{d}{D}}}}{\sqrt[3]{\frac{d}{D}}}}}{4}}}} \cdot w0\]

    if -0.0 < (* M D) < 1.8217782890704009e+25

    1. Initial program 8.3

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

    2. Initial simplification7.9

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

    4. Applied associate-/r/5.0

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

    6. Applied associate-/l*3.9

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

    if 1.8217782890704009e+25 < (* M D)

    1. Initial program 25.1

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

    2. Initial simplification24.9

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

    4. Applied associate-/r/24.6

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

    6. Applied associate-*l/25.1

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

    7. Simplified20.2

      \[\leadsto \sqrt{1 - \frac{\color{blue}{\left(h \cdot \frac{M}{\frac{d}{D}}\right) \cdot \frac{\frac{M}{\frac{d}{D}}}{4}}}{\ell}} \cdot w0\]
    8. Using strategy rm

    9. Applied associate-/l*18.4

      \[\leadsto \sqrt{1 - \color{blue}{\frac{h \cdot \frac{M}{\frac{d}{D}}}{\frac{\ell}{\frac{\frac{M}{\frac{d}{D}}}{4}}}}} \cdot w0\]
    10. Using strategy rm

    11. Applied clear-num18.4

      \[\leadsto \sqrt{1 - \frac{h \cdot \frac{M}{\frac{d}{D}}}{\frac{\ell}{\frac{\color{blue}{\frac{1}{\frac{\frac{d}{D}}{M}}}}{4}}}} \cdot w0\]
  3. Recombined 3 regimes into one program.

  4. Final simplification7.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \le -0.0:\\ \;\;\;\;\sqrt{1 - \frac{\frac{M}{\frac{d}{D}} \cdot h}{\frac{\ell}{\frac{\frac{\frac{M}{\sqrt[3]{\frac{d}{D}} \cdot \sqrt[3]{\frac{d}{D}}}}{\sqrt[3]{\frac{d}{D}}}}{4}}}} \cdot w0\\ \mathbf{elif}\;M \cdot D \le 1.8217782890704009 \cdot 10^{+25}:\\ \;\;\;\;\sqrt{1 - h \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{\ell}{\frac{M \cdot D}{2 \cdot d}}}} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{M}{\frac{d}{D}} \cdot h}{\frac{\ell}{\frac{\frac{1}{\frac{\frac{d}{D}}{M}}}{4}}}}\\ \end{array}\]

Runtime

Time bar (total: 53.7s)Debug logProfile

herbie shell --seed 2018254 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))