Average Error: 19.0 → 11.6
Time: 43.3s
Precision: 64
Internal Precision: 576
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{A}{V} \le -7.937351850074292 \cdot 10^{+283}:\\ \;\;\;\;\left(\sqrt{\sqrt{\frac{\frac{A}{\ell}}{V}}} \cdot c0\right) \cdot \sqrt{\sqrt{A \cdot \frac{\frac{1}{V}}{\ell}}}\\ \mathbf{if}\;\frac{A}{V} \le -9.300712454372876 \cdot 10^{-162}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{if}\;\frac{A}{V} \le 0.0:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{\frac{1}{V}}{\ell}}\\ \mathbf{if}\;\frac{A}{V} \le 2.7709069046512633 \cdot 10^{+266}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(c0 \cdot \sqrt{A}\right) \cdot \sqrt{\frac{1}{V \cdot \ell}}\\ \end{array}\]

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 (/ A V) < -7.937351850074292e+283

    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 associate-/r*33.2

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

    7. Applied add-sqr-sqrt33.4

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

    8. Applied associate-*r*33.4

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

    9. Applied simplify33.3

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

    if -7.937351850074292e+283 < (/ A V) < -9.300712454372876e-162

    1. Initial program 16.5

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

    3. Applied associate-/r*8.5

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

    if -9.300712454372876e-162 < (/ A V) < 0.0

    1. Initial program 20.7

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

    3. Applied div-inv20.8

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

    5. Applied associate-/r*20.7

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

    if 0.0 < (/ A V) < 2.7709069046512633e+266

    1. Initial program 15.6

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

    3. Applied div-inv15.9

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

    5. Applied associate-/r*15.6

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

    7. Applied associate-*r/9.6

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

    8. Applied sqrt-div0.7

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

    9. Applied simplify0.7

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

    if 2.7709069046512633e+266 < (/ A V)

    1. Initial program 31.3

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

    3. Applied div-inv31.3

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

    4. Applied sqrt-prod37.6

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

    5. Applied associate-*r*38.0

      \[\leadsto \color{blue}{\left(c0 \cdot \sqrt{A}\right) \cdot \sqrt{\frac{1}{V \cdot \ell}}}\]
  3. Recombined 5 regimes into one program.

Runtime

Time bar (total: 43.3s)Debug logProfile

herbie shell --seed '#(1071501266 3581234924 1086666455 2685055582 1243441566 1802958749)' +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))