Average Error: 18.8 → 12.3
Time: 38.1s
Precision: 64
Internal Precision: 576
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{V \cdot \ell} \le -6.086047157823372 \cdot 10^{+243}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{\frac{A}{\ell}}}{\sqrt{V}}\\ \mathbf{if}\;\frac{1}{V \cdot \ell} \le -8.983466433386799 \cdot 10^{-150}:\\ \;\;\;\;\left(\sqrt[3]{c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}} \cdot \sqrt[3]{c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}}\right) \cdot \sqrt[3]{c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}}\\ \mathbf{if}\;\frac{1}{V \cdot \ell} \le 2.670367123990877 \cdot 10^{-305}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{if}\;\frac{1}{V \cdot \ell} \le 2.07818662014676 \cdot 10^{+291}:\\ \;\;\;\;\left(c0 \cdot \sqrt{A}\right) \cdot \sqrt{\frac{1}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array}\]

Error

Bits error versus c0

Bits error versus A

Bits error versus V

Bits error versus l

Derivation

  1. Split input into 5 regimes

  2. if (/ 1 (* V l)) < -6.086047157823372e+243

    1. Initial program 45.0

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

    3. Applied associate-/r*31.5

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

    5. Applied div-inv31.5

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

    7. Applied associate-*l/31.4

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

    8. Applied sqrt-div37.4

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

    9. Applied associate-*r/37.8

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

    10. Applied simplify37.8

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

    if -6.086047157823372e+243 < (/ 1 (* V l)) < -8.983466433386799e-150

    1. Initial program 7.0

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

    3. Applied add-cube-cbrt7.9

      \[\leadsto \color{blue}{\left(\sqrt[3]{c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}} \cdot \sqrt[3]{c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}}\right) \cdot \sqrt[3]{c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}}}\]

    if -8.983466433386799e-150 < (/ 1 (* V l)) < 2.670367123990877e-305

    1. Initial program 30.9

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

    3. Applied associate-/r*20.3

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

    5. Applied div-inv20.3

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

    7. Applied associate-*l/20.4

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

    8. Applied simplify20.4

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

    if 2.670367123990877e-305 < (/ 1 (* V l)) < 2.07818662014676e+291

    1. Initial program 9.6

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

    3. Applied div-inv9.6

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

    4. Applied sqrt-prod0.4

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

    5. Applied associate-*r*2.6

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

    if 2.07818662014676e+291 < (/ 1 (* V l))

    1. Initial program 55.3

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

    3. Applied associate-/r*34.0

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

    5. Applied div-inv34.0

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

    7. Applied associate-*l/34.0

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

    8. Applied simplify34.0

      \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{\ell}}}{V}}\]
  3. Recombined 5 regimes into one program.

Runtime

Time bar (total: 38.1s)Debug logProfile

herbie shell --seed '#(1072330854 3074818769 591214268 3603999196 3863745332 3332387116)' +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))