Average Error: 13.3 → 8.5
Time: 56.4s
Precision: 64
Internal Precision: 320
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} = -\infty:\\ \;\;\;\;\sqrt{1 - \left(\frac{h}{\frac{d}{D}} \cdot \frac{M}{2}\right) \cdot \frac{\frac{D \cdot M}{2 \cdot d}}{\ell}} \cdot w0\\ \mathbf{elif}\;\frac{h}{\ell} \le -6.02549628357006 \cdot 10^{-263}:\\ \;\;\;\;\sqrt{1 - \frac{\frac{D}{d} \cdot \frac{M}{2}}{\frac{\frac{\ell}{h}}{\frac{D}{d} \cdot \frac{M}{2}}}} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array}\]

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 (/ h l) < -inf.0

    1. Initial program 61.7

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

    2. Initial simplification58.2

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

    4. Applied div-inv58.2

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

    5. Applied times-frac21.4

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

    6. Simplified27.5

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

    8. Applied associate-/l*25.4

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

    if -inf.0 < (/ h l) < -6.02549628357006e-263

    1. Initial program 13.8

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

    2. Initial simplification13.8

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

    4. Applied div-inv13.8

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

    5. Applied associate-/r*15.1

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

    7. Applied *-un-lft-identity15.1

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

    8. Applied *-un-lft-identity15.1

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

    9. Applied times-frac15.1

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

    10. Simplified15.1

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

    11. Simplified12.4

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

    if -6.02549628357006e-263 < (/ h l)

    1. Initial program 7.8

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

    2. Initial simplification7.6

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

    3. Taylor expanded around 0 3.3

      \[\leadsto \color{blue}{1} \cdot w0\]
  3. Recombined 3 regimes into one program.

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} = -\infty:\\ \;\;\;\;\sqrt{1 - \left(\frac{h}{\frac{d}{D}} \cdot \frac{M}{2}\right) \cdot \frac{\frac{D \cdot M}{2 \cdot d}}{\ell}} \cdot w0\\ \mathbf{elif}\;\frac{h}{\ell} \le -6.02549628357006 \cdot 10^{-263}:\\ \;\;\;\;\sqrt{1 - \frac{\frac{D}{d} \cdot \frac{M}{2}}{\frac{\frac{\ell}{h}}{\frac{D}{d} \cdot \frac{M}{2}}}} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array}\]

Runtime

Time bar (total: 56.4s)Debug logProfile

herbie shell --seed 2018220 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))