Average Error: 18.3 → 14.8
Time: 14.9s
Precision: 64
Internal Precision: 128
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -3.53330846117507 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\sqrt{\frac{\ell}{\frac{A}{V}}}}}}{\sqrt{\sqrt{\frac{\ell}{\frac{A}{V}}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{1}{\frac{A}{V}}}}\\ \end{array}\]

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 2 regimes

  2. if l < -3.53330846117507e-310

    1. Initial program 18.5

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

    2. Initial simplification17.8

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

    4. Applied clear-num18.2

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

    6. Applied sqrt-div17.9

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

    7. Applied associate-*r/17.8

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

    9. Applied add-sqr-sqrt18.0

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

    10. Applied associate-/r*18.0

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

    if -3.53330846117507e-310 < l

    1. Initial program 18.2

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

    2. Initial simplification18.5

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

    4. Applied clear-num18.7

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

    6. Applied sqrt-div18.3

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

    7. Applied associate-*r/18.3

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

    9. Applied div-inv18.5

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

    10. Applied sqrt-prod11.4

      \[\leadsto \frac{c0 \cdot \sqrt{1}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\frac{1}{\frac{A}{V}}}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification14.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -3.53330846117507 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\sqrt{\frac{\ell}{\frac{A}{V}}}}}}{\sqrt{\sqrt{\frac{\ell}{\frac{A}{V}}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{1}{\frac{A}{V}}}}\\ \end{array}\]

Runtime

Time bar (total: 14.9s)Debug logProfile

herbie shell --seed 2018348 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))