Average Error: 19.1 → 13.3
Time: 33.9s
Precision: 64
Internal Precision: 320
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt{V \cdot \ell} \le 1.039736147640242 \cdot 10^{-146}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\sqrt{\frac{\frac{A}{\ell}}{V}}} \cdot \sqrt{\sqrt{\frac{\frac{A}{\ell}}{V}}}\right)\\ \mathbf{if}\;\sqrt{V \cdot \ell} \le +\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{1}}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \end{array}\]

Error

Bits error versus c0

Bits error versus A

Bits error versus V

Bits error versus l

Derivation

  1. Split input into 3 regimes

  2. if (sqrt (* V l)) < 1.039736147640242e-146

    1. Initial program 57.7

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

    3. Applied div-inv59.3

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

    5. Applied add-sqr-sqrt59.3

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

    6. Applied simplify59.3

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

    7. Applied simplify35.7

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

    if 1.039736147640242e-146 < (sqrt (* V l)) < +inf.0

    1. Initial program 14.3

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

    3. Applied sqrt-div6.5

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

    if +inf.0 < (sqrt (* V l))

    1. Initial program 15.4

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

    3. Applied clear-num15.7

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

    5. Applied sqrt-div15.2

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

Runtime

Time bar (total: 33.9s)Debug logProfile

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))