Average Error: 18.6 → 13.3
Time: 12.5s
Precision: 64
Internal Precision: 576
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \le -1.834726847548682 \cdot 10^{-303}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \le 0.0:\\ \;\;\;\;\frac{\sqrt{\frac{A}{\ell}}}{\sqrt{V}} \cdot c0\\ \mathbf{elif}\;V \cdot \ell \le 5.374417065528295 \cdot 10^{+235}:\\ \;\;\;\;\frac{\sqrt{A} \cdot c0}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\sqrt{\frac{\frac{A}{\ell}}{V}}} \cdot c0\right) \cdot \sqrt{\sqrt{\frac{\frac{A}{\ell}}{V}}}\\ \end{array}\]

Error

Bits error versus c0

Bits error versus A

Bits error versus V

Bits error versus l

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if (* V l) < -1.834726847548682e-303

    1. Initial program 14.7

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
    2. Using strategy rm

    3. Applied div-inv14.7

      \[\leadsto c0 \cdot \sqrt{\color{blue}{A \cdot \frac{1}{V \cdot \ell}}}\]
    4. Using strategy rm

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

      \[\leadsto c0 \cdot \color{blue}{\left(1 \cdot \sqrt{A \cdot \frac{1}{V \cdot \ell}}\right)}\]

    6. Applied associate-*r*14.7

      \[\leadsto \color{blue}{\left(c0 \cdot 1\right) \cdot \sqrt{A \cdot \frac{1}{V \cdot \ell}}}\]

    7. Simplified16.9

      \[\leadsto \left(c0 \cdot 1\right) \cdot \color{blue}{\sqrt{\frac{\frac{A}{\ell}}{V}}}\]

    8. Taylor expanded around -inf 14.7

      \[\leadsto \left(c0 \cdot 1\right) \cdot \sqrt{\color{blue}{\frac{A}{\ell \cdot V}}}\]

    if -1.834726847548682e-303 < (* V l) < 0.0

    1. Initial program 59.5

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
    2. Using strategy rm

    3. Applied div-inv60.7

      \[\leadsto c0 \cdot \sqrt{\color{blue}{A \cdot \frac{1}{V \cdot \ell}}}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity60.7

      \[\leadsto c0 \cdot \color{blue}{\left(1 \cdot \sqrt{A \cdot \frac{1}{V \cdot \ell}}\right)}\]

    6. Applied associate-*r*60.7

      \[\leadsto \color{blue}{\left(c0 \cdot 1\right) \cdot \sqrt{A \cdot \frac{1}{V \cdot \ell}}}\]

    7. Simplified37.5

      \[\leadsto \left(c0 \cdot 1\right) \cdot \color{blue}{\sqrt{\frac{\frac{A}{\ell}}{V}}}\]
    8. Using strategy rm

    9. Applied sqrt-div38.6

      \[\leadsto \left(c0 \cdot 1\right) \cdot \color{blue}{\frac{\sqrt{\frac{A}{\ell}}}{\sqrt{V}}}\]

    if 0.0 < (* V l) < 5.374417065528295e+235

    1. Initial program 9.2

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
    2. Using strategy rm

    3. Applied div-inv9.5

      \[\leadsto c0 \cdot \sqrt{\color{blue}{A \cdot \frac{1}{V \cdot \ell}}}\]
    4. Using strategy rm

    5. Applied un-div-inv9.2

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}}\]

    6. Applied sqrt-div0.7

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}}\]

    7. Applied associate-*r/2.9

      \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}}\]

    if 5.374417065528295e+235 < (* V l)

    1. Initial program 33.2

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
    2. Using strategy rm

    3. Applied div-inv33.2

      \[\leadsto c0 \cdot \sqrt{\color{blue}{A \cdot \frac{1}{V \cdot \ell}}}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity33.2

      \[\leadsto c0 \cdot \color{blue}{\left(1 \cdot \sqrt{A \cdot \frac{1}{V \cdot \ell}}\right)}\]

    6. Applied associate-*r*33.2

      \[\leadsto \color{blue}{\left(c0 \cdot 1\right) \cdot \sqrt{A \cdot \frac{1}{V \cdot \ell}}}\]

    7. Simplified22.2

      \[\leadsto \left(c0 \cdot 1\right) \cdot \color{blue}{\sqrt{\frac{\frac{A}{\ell}}{V}}}\]
    8. Using strategy rm

    9. Applied add-sqr-sqrt22.2

      \[\leadsto \left(c0 \cdot 1\right) \cdot \sqrt{\color{blue}{\sqrt{\frac{\frac{A}{\ell}}{V}} \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}}}\]

    10. Applied sqrt-prod22.3

      \[\leadsto \left(c0 \cdot 1\right) \cdot \color{blue}{\left(\sqrt{\sqrt{\frac{\frac{A}{\ell}}{V}}} \cdot \sqrt{\sqrt{\frac{\frac{A}{\ell}}{V}}}\right)}\]

    11. Applied associate-*r*22.3

      \[\leadsto \color{blue}{\left(\left(c0 \cdot 1\right) \cdot \sqrt{\sqrt{\frac{\frac{A}{\ell}}{V}}}\right) \cdot \sqrt{\sqrt{\frac{\frac{A}{\ell}}{V}}}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification13.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \le -1.834726847548682 \cdot 10^{-303}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \le 0.0:\\ \;\;\;\;\frac{\sqrt{\frac{A}{\ell}}}{\sqrt{V}} \cdot c0\\ \mathbf{elif}\;V \cdot \ell \le 5.374417065528295 \cdot 10^{+235}:\\ \;\;\;\;\frac{\sqrt{A} \cdot c0}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\sqrt{\frac{\frac{A}{\ell}}{V}}} \cdot c0\right) \cdot \sqrt{\sqrt{\frac{\frac{A}{\ell}}{V}}}\\ \end{array}\]

Runtime

Time bar (total: 12.5s)Debug logProfile

herbie shell --seed 2018234 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))