Average Error: 13.3 → 8.3
Time: 2.6m
Precision: 64
Internal Precision: 576
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \le 1.6276735551048187 \cdot 10^{+246}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{\ell}{\frac{M \cdot D}{2 \cdot d}}}}{\frac{1}{h}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(h \cdot \left(\frac{D}{d} \cdot \frac{M}{2}\right)\right) \cdot \frac{\frac{D}{d}}{\frac{\ell}{\frac{M}{2}}}} \cdot w0\\ \end{array}\]

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) (* 2 d)) < 1.6276735551048187e+246

    1. Initial program 11.4

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

    2. Initial simplification10.9

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

    4. Applied div-inv10.9

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

    5. Applied associate-/r*8.2

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

    7. Applied associate-/l*6.8

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

    if 1.6276735551048187e+246 < (/ (* M D) (* 2 d))

    1. Initial program 61.0

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

    2. Initial simplification61.0

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

    4. Applied div-inv61.0

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

    5. Applied associate-/r*60.8

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

    7. Applied associate-/l*59.4

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

    9. Applied *-un-lft-identity59.4

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

    10. Applied *-un-lft-identity59.4

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

    11. Applied *-un-lft-identity59.4

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

    12. Applied times-frac59.4

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

    13. Applied times-frac59.4

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

    14. Simplified59.4

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

    15. Simplified46.2

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

  4. Final simplification8.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \le 1.6276735551048187 \cdot 10^{+246}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{\ell}{\frac{M \cdot D}{2 \cdot d}}}}{\frac{1}{h}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(h \cdot \left(\frac{D}{d} \cdot \frac{M}{2}\right)\right) \cdot \frac{\frac{D}{d}}{\frac{\ell}{\frac{M}{2}}}} \cdot w0\\ \end{array}\]

Runtime

Time bar (total: 2.6m)Debug logProfile

herbie shell --seed 2018221 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))