Average Error: 13.6 → 10.7
Time: 1.2m
Precision: 64
Internal Precision: 576
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;D \le -2.8651473920027 \cdot 10^{+113} \lor \neg \left(D \le 1.4560986628752805 \cdot 10^{+106}\right):\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\sqrt[3]{\frac{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}{\ell}} \cdot \sqrt[3]{\frac{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}{\ell}}\right) \cdot \left(h \cdot \left(\sqrt[3]{\sqrt[3]{\frac{\frac{D}{2} \cdot \frac{M}{d}}{\frac{\ell}{\frac{D}{2} \cdot \frac{M}{d}}}}} \cdot \left(\sqrt[3]{\sqrt[3]{\frac{\frac{D}{2} \cdot \frac{M}{d}}{\frac{\ell}{\frac{D}{2} \cdot \frac{M}{d}}}}} \cdot \sqrt[3]{\sqrt[3]{\frac{\frac{D}{2} \cdot \frac{M}{d}}{\frac{\ell}{\frac{D}{2} \cdot \frac{M}{d}}}}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \left(\frac{D}{\frac{2}{h}} \cdot \frac{M}{d}\right)}\\ \end{array}\]

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 D < -2.8651473920027e+113 or 1.4560986628752805e+106 < D

    1. Initial program 23.8

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

    2. Initial simplification23.5

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

    4. Applied div-inv23.5

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

    5. Applied associate-/r*22.8

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

    7. Applied *-un-lft-identity22.8

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

    8. Applied add-cube-cbrt22.8

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

    9. Applied times-frac22.8

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

    10. Simplified24.5

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

    11. Simplified23.4

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

    13. Applied add-cube-cbrt23.4

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

    if -2.8651473920027e+113 < D < 1.4560986628752805e+106

    1. Initial program 10.7

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

    2. Initial simplification10.3

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

    4. Applied div-inv10.3

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

    5. Applied times-frac5.2

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

    6. Simplified7.2

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

  4. Final simplification10.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;D \le -2.8651473920027 \cdot 10^{+113} \lor \neg \left(D \le 1.4560986628752805 \cdot 10^{+106}\right):\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\sqrt[3]{\frac{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}{\ell}} \cdot \sqrt[3]{\frac{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}{\ell}}\right) \cdot \left(h \cdot \left(\sqrt[3]{\sqrt[3]{\frac{\frac{D}{2} \cdot \frac{M}{d}}{\frac{\ell}{\frac{D}{2} \cdot \frac{M}{d}}}}} \cdot \left(\sqrt[3]{\sqrt[3]{\frac{\frac{D}{2} \cdot \frac{M}{d}}{\frac{\ell}{\frac{D}{2} \cdot \frac{M}{d}}}}} \cdot \sqrt[3]{\sqrt[3]{\frac{\frac{D}{2} \cdot \frac{M}{d}}{\frac{\ell}{\frac{D}{2} \cdot \frac{M}{d}}}}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \left(\frac{D}{\frac{2}{h}} \cdot \frac{M}{d}\right)}\\ \end{array}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018248 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))