Average Error: 19.4 → 8.4
Time: 55.8s
Precision: 64
Internal Precision: 384
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \le -2.709787285643665 \cdot 10^{+84}:\\ \;\;\;\;c0 \cdot \frac{\left|\sqrt[3]{\frac{A}{V}}\right|}{\sqrt{\frac{\ell}{\left(\sqrt[3]{\sqrt[3]{\frac{A}{V}}} \cdot \sqrt[3]{\sqrt[3]{\frac{A}{V}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{A}{V}}}}}}\\ \mathbf{if}\;V \cdot \ell \le -1.703813546141723 \cdot 10^{-132}:\\ \;\;\;\;\left(c0 \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right) \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\\ \mathbf{if}\;V \cdot \ell \le 3.9678772388993 \cdot 10^{-315}:\\ \;\;\;\;c0 \cdot \frac{\left|\sqrt[3]{A} \cdot \sqrt[3]{\frac{1}{V}}\right|}{\sqrt{\frac{\ell}{\sqrt[3]{\frac{A}{V}}}}}\\ \mathbf{if}\;V \cdot \ell \le 1.244520295797865 \cdot 10^{+271}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\left|\sqrt[3]{A} \cdot \sqrt[3]{\frac{1}{V}}\right|}{\sqrt{\frac{\ell}{\sqrt[3]{\frac{A}{V}}}}}\\ \end{array}\]

Error

Bits error versus c0

Bits error versus A

Bits error versus V

Bits error versus l

Derivation

  1. Split input into 4 regimes

  2. if (* V l) < -2.709787285643665e+84

    1. Initial program 22.7

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

    3. Applied associate-/r*18.6

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

    5. Applied add-cube-cbrt18.9

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

    6. Applied associate-/l*18.9

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

    8. Applied sqrt-div13.5

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

    9. Applied simplify13.5

      \[\leadsto c0 \cdot \frac{\color{blue}{\left|\sqrt[3]{\frac{A}{V}}\right|}}{\sqrt{\frac{\ell}{\sqrt[3]{\frac{A}{V}}}}}\]
    10. Using strategy rm

    11. Applied add-cube-cbrt13.7

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

    if -2.709787285643665e+84 < (* V l) < -1.703813546141723e-132

    1. Initial program 4.2

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

    3. Applied add-sqr-sqrt4.2

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

    4. Applied sqrt-prod4.5

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

    5. Applied associate-*r*4.4

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

    if -1.703813546141723e-132 < (* V l) < 3.9678772388993e-315 or 1.244520295797865e+271 < (* V l)

    1. Initial program 38.1

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

    3. Applied associate-/r*26.0

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

    5. Applied add-cube-cbrt26.3

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

    6. Applied associate-/l*26.3

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

    8. Applied sqrt-div17.1

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

    9. Applied simplify17.1

      \[\leadsto c0 \cdot \frac{\color{blue}{\left|\sqrt[3]{\frac{A}{V}}\right|}}{\sqrt{\frac{\ell}{\sqrt[3]{\frac{A}{V}}}}}\]
    10. Using strategy rm

    11. Applied div-inv17.1

      \[\leadsto c0 \cdot \frac{\left|\sqrt[3]{\color{blue}{A \cdot \frac{1}{V}}}\right|}{\sqrt{\frac{\ell}{\sqrt[3]{\frac{A}{V}}}}}\]

    12. Applied cbrt-prod17.2

      \[\leadsto c0 \cdot \frac{\left|\color{blue}{\sqrt[3]{A} \cdot \sqrt[3]{\frac{1}{V}}}\right|}{\sqrt{\frac{\ell}{\sqrt[3]{\frac{A}{V}}}}}\]

    if 3.9678772388993e-315 < (* V l) < 1.244520295797865e+271

    1. Initial program 10.1

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

    3. Applied sqrt-div0.5

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

Runtime

Time bar (total: 55.8s)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))