Average Error: 13.8 → 9.4
Time: 1.2m
Precision: 64
Internal Precision: 320
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;M \cdot D \le -5.4580893614757005 \cdot 10^{-201} \lor \neg \left(M \cdot D \le 2.84563663228828 \cdot 10^{-211}\right):\\ \;\;\;\;\sqrt{1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\frac{1}{M \cdot D} \cdot \left(\frac{d}{h} \cdot \left(\ell \cdot 2\right)\right)}} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(\sqrt[3]{\frac{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}{\ell}} \cdot \sqrt[3]{\frac{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}{\ell}}\right) \cdot \left(h \cdot \sqrt[3]{\frac{\frac{D}{2} \cdot \frac{M}{d}}{\frac{\ell}{\frac{D}{2} \cdot \frac{M}{d}}}}\right)} \cdot w0\\ \end{array}\]

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) < -5.4580893614757005e-201 or 2.84563663228828e-211 < (* M D)

    1. Initial program 17.3

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

    2. Initial simplification17.0

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

    4. Applied associate-/l*14.9

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

    6. Applied div-inv14.9

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

    7. Applied *-un-lft-identity14.9

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

    8. Applied times-frac15.3

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

    9. Simplified13.7

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

    if -5.4580893614757005e-201 < (* M D) < 2.84563663228828e-211

    1. Initial program 6.8

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

    2. Initial simplification6.1

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

    4. Applied div-inv6.1

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

    5. Applied associate-/r*1.0

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

    7. Applied *-un-lft-identity1.0

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

    8. Applied add-cube-cbrt1.0

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

    9. Applied times-frac1.0

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

    10. Simplified1.0

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

    11. Simplified0.9

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

  4. Final simplification9.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \le -5.4580893614757005 \cdot 10^{-201} \lor \neg \left(M \cdot D \le 2.84563663228828 \cdot 10^{-211}\right):\\ \;\;\;\;\sqrt{1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\frac{1}{M \cdot D} \cdot \left(\frac{d}{h} \cdot \left(\ell \cdot 2\right)\right)}} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(\sqrt[3]{\frac{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}{\ell}} \cdot \sqrt[3]{\frac{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}{\ell}}\right) \cdot \left(h \cdot \sqrt[3]{\frac{\frac{D}{2} \cdot \frac{M}{d}}{\frac{\ell}{\frac{D}{2} \cdot \frac{M}{d}}}}\right)} \cdot w0\\ \end{array}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018225 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))