Average Error: 14.0 → 8.6
Time: 1.1m
Precision: 64
Internal Precision: 320
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \le -3.4183666695052727 \cdot 10^{+171}:\\ \;\;\;\;\sqrt{1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{\frac{\ell}{h}}{\frac{M}{2} \cdot \frac{D}{d}}}} \cdot w0\\ \mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \le 3.9618113535378547 \cdot 10^{+89}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\ell}}{\frac{1}{h}}}\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{1 - \frac{\frac{M}{d} \cdot \frac{D}{2}}{\frac{\ell}{\left(h \cdot \frac{D}{2}\right) \cdot \frac{M}{d}}}}\right| \cdot w0\\ \end{array}\]

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 3 regimes

  2. if (/ (* M D) (* 2 d)) < -3.4183666695052727e+171

    1. Initial program 61.1

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

    2. Initial simplification61.1

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

    4. Applied div-inv61.1

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

    5. Applied associate-/r*60.9

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

    7. Applied div-inv60.9

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

    8. Applied *-un-lft-identity60.9

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

    9. Applied times-frac60.9

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

    10. Simplified60.9

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

    11. Simplified44.9

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

    if -3.4183666695052727e+171 < (/ (* M D) (* 2 d)) < 3.9618113535378547e+89

    1. Initial program 7.0

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

    2. Initial simplification6.5

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

    4. Applied div-inv6.5

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

    5. Applied associate-/r*3.0

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

    if 3.9618113535378547e+89 < (/ (* M D) (* 2 d))

    1. Initial program 49.4

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

    2. Initial simplification49.5

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

    4. Applied div-inv49.5

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

    5. Applied associate-/r*50.9

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

    7. Applied add-sqr-sqrt50.9

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

    8. Applied rem-sqrt-square50.9

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

    9. Simplified40.0

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

    11. Applied associate-/l/38.2

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

    13. Applied associate-*l*39.2

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

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \le -3.4183666695052727 \cdot 10^{+171}:\\ \;\;\;\;\sqrt{1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{\frac{\ell}{h}}{\frac{M}{2} \cdot \frac{D}{d}}}} \cdot w0\\ \mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \le 3.9618113535378547 \cdot 10^{+89}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\ell}}{\frac{1}{h}}}\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{1 - \frac{\frac{M}{d} \cdot \frac{D}{2}}{\frac{\ell}{\left(h \cdot \frac{D}{2}\right) \cdot \frac{M}{d}}}}\right| \cdot w0\\ \end{array}\]

Runtime

Time bar (total: 1.1m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes10.98.67.03.957.4%
herbie shell --seed 2018263 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))