Average Error: 18.6 → 12.0
Time: 12.1s
Precision: 64
Internal Precision: 128
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \le -1.0714601080388011 \cdot 10^{-247}:\\ \;\;\;\;\sqrt{\frac{A}{V \cdot \ell}} \cdot c0\\ \mathbf{elif}\;V \cdot \ell \le -0.0:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \cdot c0\\ \mathbf{elif}\;V \cdot \ell \le 3.639051990373028 \cdot 10^{+298}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array}\]

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

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Derivation

  1. Split input into 4 regimes

  2. if (* V l) < -1.0714601080388011e-247

    1. Initial program 13.6

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

    3. Applied div-inv13.6

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

    5. Applied associate-/r*13.4

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

    6. Taylor expanded around -inf 13.6

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

    if -1.0714601080388011e-247 < (* V l) < -0.0

    1. Initial program 52.0

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

    3. Applied div-inv53.4

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

    5. Applied associate-/r*53.4

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

    7. Applied associate-*r/33.5

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

    8. Applied sqrt-div38.1

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

    9. Simplified38.1

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

    if -0.0 < (* V l) < 3.639051990373028e+298

    1. Initial program 10.3

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

    3. Applied div-inv10.6

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

    5. Applied un-div-inv10.3

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

    6. Applied sqrt-div0.7

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

    if 3.639051990373028e+298 < (* V l)

    1. Initial program 42.0

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

    3. Applied associate-/r*23.7

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

  4. Final simplification12.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \le -1.0714601080388011 \cdot 10^{-247}:\\ \;\;\;\;\sqrt{\frac{A}{V \cdot \ell}} \cdot c0\\ \mathbf{elif}\;V \cdot \ell \le -0.0:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \cdot c0\\ \mathbf{elif}\;V \cdot \ell \le 3.639051990373028 \cdot 10^{+298}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array}\]

Runtime

Time bar (total: 12.1s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes18.612.06.911.756.8%
herbie shell --seed 2018355 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))