Henrywood and Agarwal, Equation (3) Percentage Accurate: 74.4% → 95.6%
Time: 9.1s
Alternatives: 13
Speedup: TODO×
Unsound rule application detected in e-graph. Results may not be sound. (more) Specification ? \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
Enter valid numbers for all inputs
Local Percentage Accuracy vs ?
The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples. Accuracy vs Speed? The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs. Alternative 1? \[\begin{array}{l}
\mathbf{if}\;V \cdot \ell \leq -4 \cdot 10^{-312}:\\
\;\;\;\;c0 \cdot \frac{\frac{\sqrt{-A}}{\sqrt{-V}}}{\sqrt{\ell}}\\
\mathbf{elif}\;V \cdot \ell \leq 0:\\
\;\;\;\;c0 \cdot {\left(\frac{\sqrt[3]{\frac{A}{V}}}{\sqrt[3]{\ell}}\right)}^{1.5}\\
\mathbf{elif}\;V \cdot \ell \leq 10^{+278}:\\
\;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\
\mathbf{else}:\\
\;\;\;\;c0 \cdot {\left(\frac{\sqrt[3]{\frac{A}{\ell}}}{\sqrt[3]{V}}\right)}^{1.5}\\
\end{array}
\]
Derivation Split input into 4 regimes if (*.f64 V l) < -3.9999999999988e-312 Initial program 81.2%
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
Step-by-step derivation associate-/r*79.7%
\[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}}
\]
sqrt-div44.4%
\[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}}
\]
Applied egg-rr 44.4%
\[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}}
\]
Step-by-step derivation frac-2neg44.4%
\[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{-A}{-V}}}}{\sqrt{\ell}}
\]
sqrt-div46.7%
\[\leadsto c0 \cdot \frac{\color{blue}{\frac{\sqrt{-A}}{\sqrt{-V}}}}{\sqrt{\ell}}
\]
Applied egg-rr 46.7%
\[\leadsto c0 \cdot \frac{\color{blue}{\frac{\sqrt{-A}}{\sqrt{-V}}}}{\sqrt{\ell}}
\]
if -3.9999999999988e-312 < (*.f64 V l) < 0.0 Initial program 30.6%
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
Step-by-step derivation pow1/230.6%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}}
\]
add-cube-cbrt30.6%
\[\leadsto c0 \cdot {\color{blue}{\left(\left(\sqrt[3]{\frac{A}{V \cdot \ell}} \cdot \sqrt[3]{\frac{A}{V \cdot \ell}}\right) \cdot \sqrt[3]{\frac{A}{V \cdot \ell}}\right)}}^{0.5}
\]
pow330.6%
\[\leadsto c0 \cdot {\color{blue}{\left({\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{3}\right)}}^{0.5}
\]
pow-pow30.6%
\[\leadsto c0 \cdot \color{blue}{{\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{\left(3 \cdot 0.5\right)}}
\]
metadata-eval30.6%
\[\leadsto c0 \cdot {\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{\color{blue}{1.5}}
\]
Applied egg-rr 30.6%
\[\leadsto c0 \cdot \color{blue}{{\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{1.5}}
\]
Step-by-step derivation associate-/r*58.1%
\[\leadsto c0 \cdot {\left(\sqrt[3]{\color{blue}{\frac{\frac{A}{V}}{\ell}}}\right)}^{1.5}
\]
cbrt-div79.6%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{\sqrt[3]{\frac{A}{V}}}{\sqrt[3]{\ell}}\right)}}^{1.5}
\]
Applied egg-rr 79.6%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{\sqrt[3]{\frac{A}{V}}}{\sqrt[3]{\ell}}\right)}}^{1.5}
\]
if 0.0 < (*.f64 V l) < 9.99999999999999964e277 Initial program 86.5%
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
Step-by-step derivation sqrt-div99.6%
\[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}}
\]
associate-*r/95.3%
\[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}}
\]
Applied egg-rr 95.3%
\[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}}
\]
Step-by-step derivation associate-/l*99.5%
\[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}}
\]
Simplified99.5%
\[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}}
\]
if 9.99999999999999964e277 < (*.f64 V l) Initial program 53.6%
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
Step-by-step derivation pow1/253.6%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}}
\]
add-cube-cbrt53.6%
\[\leadsto c0 \cdot {\color{blue}{\left(\left(\sqrt[3]{\frac{A}{V \cdot \ell}} \cdot \sqrt[3]{\frac{A}{V \cdot \ell}}\right) \cdot \sqrt[3]{\frac{A}{V \cdot \ell}}\right)}}^{0.5}
\]
pow353.6%
\[\leadsto c0 \cdot {\color{blue}{\left({\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{3}\right)}}^{0.5}
\]
pow-pow53.6%
\[\leadsto c0 \cdot \color{blue}{{\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{\left(3 \cdot 0.5\right)}}
\]
metadata-eval53.6%
\[\leadsto c0 \cdot {\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{\color{blue}{1.5}}
\]
Applied egg-rr 53.6%
\[\leadsto c0 \cdot \color{blue}{{\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{1.5}}
\]
Step-by-step derivation *-un-lft-identity53.6%
\[\leadsto c0 \cdot {\left(\sqrt[3]{\frac{\color{blue}{1 \cdot A}}{V \cdot \ell}}\right)}^{1.5}
\]
frac-times66.9%
\[\leadsto c0 \cdot {\left(\sqrt[3]{\color{blue}{\frac{1}{V} \cdot \frac{A}{\ell}}}\right)}^{1.5}
\]
cbrt-prod92.7%
\[\leadsto c0 \cdot {\color{blue}{\left(\sqrt[3]{\frac{1}{V}} \cdot \sqrt[3]{\frac{A}{\ell}}\right)}}^{1.5}
\]
cbrt-div92.7%
\[\leadsto c0 \cdot {\left(\color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{V}}} \cdot \sqrt[3]{\frac{A}{\ell}}\right)}^{1.5}
\]
metadata-eval92.7%
\[\leadsto c0 \cdot {\left(\frac{\color{blue}{1}}{\sqrt[3]{V}} \cdot \sqrt[3]{\frac{A}{\ell}}\right)}^{1.5}
\]
Applied egg-rr 92.7%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{1}{\sqrt[3]{V}} \cdot \sqrt[3]{\frac{A}{\ell}}\right)}}^{1.5}
\]
Step-by-step derivation associate-*l/92.5%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{1 \cdot \sqrt[3]{\frac{A}{\ell}}}{\sqrt[3]{V}}\right)}}^{1.5}
\]
*-lft-identity92.5%
\[\leadsto c0 \cdot {\left(\frac{\color{blue}{\sqrt[3]{\frac{A}{\ell}}}}{\sqrt[3]{V}}\right)}^{1.5}
\]
Simplified92.5%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{\sqrt[3]{\frac{A}{\ell}}}{\sqrt[3]{V}}\right)}}^{1.5}
\]
Recombined 4 regimes into one program. Final simplification72.2%
\[\leadsto \begin{array}{l}
\mathbf{if}\;V \cdot \ell \leq -4 \cdot 10^{-312}:\\
\;\;\;\;c0 \cdot \frac{\frac{\sqrt{-A}}{\sqrt{-V}}}{\sqrt{\ell}}\\
\mathbf{elif}\;V \cdot \ell \leq 0:\\
\;\;\;\;c0 \cdot {\left(\frac{\sqrt[3]{\frac{A}{V}}}{\sqrt[3]{\ell}}\right)}^{1.5}\\
\mathbf{elif}\;V \cdot \ell \leq 10^{+278}:\\
\;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\
\mathbf{else}:\\
\;\;\;\;c0 \cdot {\left(\frac{\sqrt[3]{\frac{A}{\ell}}}{\sqrt[3]{V}}\right)}^{1.5}\\
\end{array}
\]
Alternative 2? \[\begin{array}{l}
\mathbf{if}\;V \cdot \ell \leq -\infty:\\
\;\;\;\;c0 \cdot \left({\ell}^{-0.5} \cdot \sqrt{\frac{A}{V}}\right)\\
\mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-290}:\\
\;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\
\mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 5 \cdot 10^{+291}\right):\\
\;\;\;\;c0 \cdot {\left(\frac{\sqrt[3]{\frac{A}{V}}}{\sqrt[3]{\ell}}\right)}^{1.5}\\
\mathbf{else}:\\
\;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\
\end{array}
\]
Derivation Split input into 4 regimes if (*.f64 V l) < -inf.0 Initial program 53.3%
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
Step-by-step derivation pow1/253.3%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}}
\]
clear-num53.3%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)}}^{0.5}
\]
inv-pow53.3%
\[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{V \cdot \ell}{A}\right)}^{-1}\right)}}^{0.5}
\]
pow-pow53.3%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{\left(-1 \cdot 0.5\right)}}
\]
associate-/l*74.1%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{\frac{A}{\ell}}\right)}}^{\left(-1 \cdot 0.5\right)}
\]
metadata-eval74.1%
\[\leadsto c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{\color{blue}{-0.5}}
\]
Applied egg-rr 74.1%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5}}
\]
Step-by-step derivation associate-/l*53.3%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{V \cdot \ell}{A}\right)}}^{-0.5}
\]
Simplified53.3%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}}
\]
Step-by-step derivation add-sqr-sqrt53.3%
\[\leadsto c0 \cdot \color{blue}{\left(\sqrt{{\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}} \cdot \sqrt{{\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}}\right)}
\]
sqrt-unprod53.3%
\[\leadsto c0 \cdot \color{blue}{\sqrt{{\left(\frac{V \cdot \ell}{A}\right)}^{-0.5} \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}}}
\]
pow-prod-up53.3%
\[\leadsto c0 \cdot \sqrt{\color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{\left(-0.5 + -0.5\right)}}}
\]
metadata-eval53.3%
\[\leadsto c0 \cdot \sqrt{{\left(\frac{V \cdot \ell}{A}\right)}^{\color{blue}{-1}}}
\]
inv-pow53.3%
\[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\frac{V \cdot \ell}{A}}}}
\]
clear-num53.3%
\[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}}
\]
associate-/r*74.1%
\[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}}
\]
sqrt-undiv40.0%
\[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}}
\]
clear-num40.0%
\[\leadsto c0 \cdot \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}}
\]
associate-/r/40.0%
\[\leadsto c0 \cdot \color{blue}{\left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\right)}
\]
pow1/240.0%
\[\leadsto c0 \cdot \left(\frac{1}{\color{blue}{{\ell}^{0.5}}} \cdot \sqrt{\frac{A}{V}}\right)
\]
pow-flip40.0%
\[\leadsto c0 \cdot \left(\color{blue}{{\ell}^{\left(-0.5\right)}} \cdot \sqrt{\frac{A}{V}}\right)
\]
metadata-eval40.0%
\[\leadsto c0 \cdot \left({\ell}^{\color{blue}{-0.5}} \cdot \sqrt{\frac{A}{V}}\right)
\]
Applied egg-rr 40.0%
\[\leadsto c0 \cdot \color{blue}{\left({\ell}^{-0.5} \cdot \sqrt{\frac{A}{V}}\right)}
\]
if -inf.0 < (*.f64 V l) < -1.0000000000000001e-290 Initial program 85.8%
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
Step-by-step derivation frac-2neg85.8%
\[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{-A}{-V \cdot \ell}}}
\]
sqrt-div99.5%
\[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{-A}}{\sqrt{-V \cdot \ell}}}
\]
*-commutative99.5%
\[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{-\color{blue}{\ell \cdot V}}}
\]
distribute-rgt-neg-in99.5%
\[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\color{blue}{\ell \cdot \left(-V\right)}}}
\]
Applied egg-rr 99.5%
\[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}}
\]
if -1.0000000000000001e-290 < (*.f64 V l) < 0.0 or 5.0000000000000001e291 < (*.f64 V l) Initial program 39.7%
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
Step-by-step derivation pow1/239.7%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}}
\]
add-cube-cbrt39.7%
\[\leadsto c0 \cdot {\color{blue}{\left(\left(\sqrt[3]{\frac{A}{V \cdot \ell}} \cdot \sqrt[3]{\frac{A}{V \cdot \ell}}\right) \cdot \sqrt[3]{\frac{A}{V \cdot \ell}}\right)}}^{0.5}
\]
pow339.7%
\[\leadsto c0 \cdot {\color{blue}{\left({\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{3}\right)}}^{0.5}
\]
pow-pow39.7%
\[\leadsto c0 \cdot \color{blue}{{\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{\left(3 \cdot 0.5\right)}}
\]
metadata-eval39.7%
\[\leadsto c0 \cdot {\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{\color{blue}{1.5}}
\]
Applied egg-rr 39.7%
\[\leadsto c0 \cdot \color{blue}{{\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{1.5}}
\]
Step-by-step derivation associate-/r*61.8%
\[\leadsto c0 \cdot {\left(\sqrt[3]{\color{blue}{\frac{\frac{A}{V}}{\ell}}}\right)}^{1.5}
\]
cbrt-div83.5%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{\sqrt[3]{\frac{A}{V}}}{\sqrt[3]{\ell}}\right)}}^{1.5}
\]
Applied egg-rr 83.5%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{\sqrt[3]{\frac{A}{V}}}{\sqrt[3]{\ell}}\right)}}^{1.5}
\]
if 0.0 < (*.f64 V l) < 5.0000000000000001e291 Initial program 85.7%
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
Step-by-step derivation sqrt-div99.5%
\[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}}
\]
associate-*r/95.4%
\[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}}
\]
Applied egg-rr 95.4%
\[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}}
\]
Step-by-step derivation associate-/l*99.4%
\[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}}
\]
Simplified99.4%
\[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}}
\]
Recombined 4 regimes into one program. Final simplification92.7%
\[\leadsto \begin{array}{l}
\mathbf{if}\;V \cdot \ell \leq -\infty:\\
\;\;\;\;c0 \cdot \left({\ell}^{-0.5} \cdot \sqrt{\frac{A}{V}}\right)\\
\mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-290}:\\
\;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\
\mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 5 \cdot 10^{+291}\right):\\
\;\;\;\;c0 \cdot {\left(\frac{\sqrt[3]{\frac{A}{V}}}{\sqrt[3]{\ell}}\right)}^{1.5}\\
\mathbf{else}:\\
\;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\
\end{array}
\]
Alternative 3? \[\begin{array}{l}
\mathbf{if}\;V \cdot \ell \leq -\infty:\\
\;\;\;\;c0 \cdot \left({\ell}^{-0.5} \cdot \sqrt{\frac{A}{V}}\right)\\
\mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-290}:\\
\;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\
\mathbf{elif}\;V \cdot \ell \leq 0:\\
\;\;\;\;c0 \cdot {\left(\frac{\sqrt[3]{\frac{A}{V}}}{\sqrt[3]{\ell}}\right)}^{1.5}\\
\mathbf{elif}\;V \cdot \ell \leq 10^{+278}:\\
\;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\
\mathbf{else}:\\
\;\;\;\;c0 \cdot {\left(\frac{\sqrt[3]{\frac{A}{\ell}}}{\sqrt[3]{V}}\right)}^{1.5}\\
\end{array}
\]
Derivation Split input into 5 regimes if (*.f64 V l) < -inf.0 Initial program 53.3%
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
Step-by-step derivation pow1/253.3%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}}
\]
clear-num53.3%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)}}^{0.5}
\]
inv-pow53.3%
\[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{V \cdot \ell}{A}\right)}^{-1}\right)}}^{0.5}
\]
pow-pow53.3%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{\left(-1 \cdot 0.5\right)}}
\]
associate-/l*74.1%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{\frac{A}{\ell}}\right)}}^{\left(-1 \cdot 0.5\right)}
\]
metadata-eval74.1%
\[\leadsto c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{\color{blue}{-0.5}}
\]
Applied egg-rr 74.1%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5}}
\]
Step-by-step derivation associate-/l*53.3%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{V \cdot \ell}{A}\right)}}^{-0.5}
\]
Simplified53.3%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}}
\]
Step-by-step derivation add-sqr-sqrt53.3%
\[\leadsto c0 \cdot \color{blue}{\left(\sqrt{{\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}} \cdot \sqrt{{\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}}\right)}
\]
sqrt-unprod53.3%
\[\leadsto c0 \cdot \color{blue}{\sqrt{{\left(\frac{V \cdot \ell}{A}\right)}^{-0.5} \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}}}
\]
pow-prod-up53.3%
\[\leadsto c0 \cdot \sqrt{\color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{\left(-0.5 + -0.5\right)}}}
\]
metadata-eval53.3%
\[\leadsto c0 \cdot \sqrt{{\left(\frac{V \cdot \ell}{A}\right)}^{\color{blue}{-1}}}
\]
inv-pow53.3%
\[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\frac{V \cdot \ell}{A}}}}
\]
clear-num53.3%
\[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}}
\]
associate-/r*74.1%
\[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}}
\]
sqrt-undiv40.0%
\[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}}
\]
clear-num40.0%
\[\leadsto c0 \cdot \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}}
\]
associate-/r/40.0%
\[\leadsto c0 \cdot \color{blue}{\left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\right)}
\]
pow1/240.0%
\[\leadsto c0 \cdot \left(\frac{1}{\color{blue}{{\ell}^{0.5}}} \cdot \sqrt{\frac{A}{V}}\right)
\]
pow-flip40.0%
\[\leadsto c0 \cdot \left(\color{blue}{{\ell}^{\left(-0.5\right)}} \cdot \sqrt{\frac{A}{V}}\right)
\]
metadata-eval40.0%
\[\leadsto c0 \cdot \left({\ell}^{\color{blue}{-0.5}} \cdot \sqrt{\frac{A}{V}}\right)
\]
Applied egg-rr 40.0%
\[\leadsto c0 \cdot \color{blue}{\left({\ell}^{-0.5} \cdot \sqrt{\frac{A}{V}}\right)}
\]
if -inf.0 < (*.f64 V l) < -1.0000000000000001e-290 Initial program 85.8%
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
Step-by-step derivation frac-2neg85.8%
\[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{-A}{-V \cdot \ell}}}
\]
sqrt-div99.5%
\[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{-A}}{\sqrt{-V \cdot \ell}}}
\]
*-commutative99.5%
\[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{-\color{blue}{\ell \cdot V}}}
\]
distribute-rgt-neg-in99.5%
\[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\color{blue}{\ell \cdot \left(-V\right)}}}
\]
Applied egg-rr 99.5%
\[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}}
\]
if -1.0000000000000001e-290 < (*.f64 V l) < 0.0 Initial program 33.6%
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
Step-by-step derivation pow1/233.6%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}}
\]
add-cube-cbrt33.6%
\[\leadsto c0 \cdot {\color{blue}{\left(\left(\sqrt[3]{\frac{A}{V \cdot \ell}} \cdot \sqrt[3]{\frac{A}{V \cdot \ell}}\right) \cdot \sqrt[3]{\frac{A}{V \cdot \ell}}\right)}}^{0.5}
\]
pow333.5%
\[\leadsto c0 \cdot {\color{blue}{\left({\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{3}\right)}}^{0.5}
\]
pow-pow33.5%
\[\leadsto c0 \cdot \color{blue}{{\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{\left(3 \cdot 0.5\right)}}
\]
metadata-eval33.5%
\[\leadsto c0 \cdot {\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{\color{blue}{1.5}}
\]
Applied egg-rr 33.5%
\[\leadsto c0 \cdot \color{blue}{{\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{1.5}}
\]
Step-by-step derivation associate-/r*58.8%
\[\leadsto c0 \cdot {\left(\sqrt[3]{\color{blue}{\frac{\frac{A}{V}}{\ell}}}\right)}^{1.5}
\]
cbrt-div81.2%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{\sqrt[3]{\frac{A}{V}}}{\sqrt[3]{\ell}}\right)}}^{1.5}
\]
Applied egg-rr 81.2%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{\sqrt[3]{\frac{A}{V}}}{\sqrt[3]{\ell}}\right)}}^{1.5}
\]
if 0.0 < (*.f64 V l) < 9.99999999999999964e277 Initial program 86.5%
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
Step-by-step derivation sqrt-div99.6%
\[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}}
\]
associate-*r/95.3%
\[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}}
\]
Applied egg-rr 95.3%
\[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}}
\]
Step-by-step derivation associate-/l*99.5%
\[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}}
\]
Simplified99.5%
\[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}}
\]
if 9.99999999999999964e277 < (*.f64 V l) Initial program 53.6%
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
Step-by-step derivation pow1/253.6%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}}
\]
add-cube-cbrt53.6%
\[\leadsto c0 \cdot {\color{blue}{\left(\left(\sqrt[3]{\frac{A}{V \cdot \ell}} \cdot \sqrt[3]{\frac{A}{V \cdot \ell}}\right) \cdot \sqrt[3]{\frac{A}{V \cdot \ell}}\right)}}^{0.5}
\]
pow353.6%
\[\leadsto c0 \cdot {\color{blue}{\left({\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{3}\right)}}^{0.5}
\]
pow-pow53.6%
\[\leadsto c0 \cdot \color{blue}{{\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{\left(3 \cdot 0.5\right)}}
\]
metadata-eval53.6%
\[\leadsto c0 \cdot {\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{\color{blue}{1.5}}
\]
Applied egg-rr 53.6%
\[\leadsto c0 \cdot \color{blue}{{\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{1.5}}
\]
Step-by-step derivation *-un-lft-identity53.6%
\[\leadsto c0 \cdot {\left(\sqrt[3]{\frac{\color{blue}{1 \cdot A}}{V \cdot \ell}}\right)}^{1.5}
\]
frac-times66.9%
\[\leadsto c0 \cdot {\left(\sqrt[3]{\color{blue}{\frac{1}{V} \cdot \frac{A}{\ell}}}\right)}^{1.5}
\]
cbrt-prod92.7%
\[\leadsto c0 \cdot {\color{blue}{\left(\sqrt[3]{\frac{1}{V}} \cdot \sqrt[3]{\frac{A}{\ell}}\right)}}^{1.5}
\]
cbrt-div92.7%
\[\leadsto c0 \cdot {\left(\color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{V}}} \cdot \sqrt[3]{\frac{A}{\ell}}\right)}^{1.5}
\]
metadata-eval92.7%
\[\leadsto c0 \cdot {\left(\frac{\color{blue}{1}}{\sqrt[3]{V}} \cdot \sqrt[3]{\frac{A}{\ell}}\right)}^{1.5}
\]
Applied egg-rr 92.7%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{1}{\sqrt[3]{V}} \cdot \sqrt[3]{\frac{A}{\ell}}\right)}}^{1.5}
\]
Step-by-step derivation associate-*l/92.5%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{1 \cdot \sqrt[3]{\frac{A}{\ell}}}{\sqrt[3]{V}}\right)}}^{1.5}
\]
*-lft-identity92.5%
\[\leadsto c0 \cdot {\left(\frac{\color{blue}{\sqrt[3]{\frac{A}{\ell}}}}{\sqrt[3]{V}}\right)}^{1.5}
\]
Simplified92.5%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{\sqrt[3]{\frac{A}{\ell}}}{\sqrt[3]{V}}\right)}}^{1.5}
\]
Recombined 5 regimes into one program. Final simplification92.9%
\[\leadsto \begin{array}{l}
\mathbf{if}\;V \cdot \ell \leq -\infty:\\
\;\;\;\;c0 \cdot \left({\ell}^{-0.5} \cdot \sqrt{\frac{A}{V}}\right)\\
\mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-290}:\\
\;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\
\mathbf{elif}\;V \cdot \ell \leq 0:\\
\;\;\;\;c0 \cdot {\left(\frac{\sqrt[3]{\frac{A}{V}}}{\sqrt[3]{\ell}}\right)}^{1.5}\\
\mathbf{elif}\;V \cdot \ell \leq 10^{+278}:\\
\;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\
\mathbf{else}:\\
\;\;\;\;c0 \cdot {\left(\frac{\sqrt[3]{\frac{A}{\ell}}}{\sqrt[3]{V}}\right)}^{1.5}\\
\end{array}
\]
Alternative 4? \[\begin{array}{l}
\mathbf{if}\;\ell \leq -1 \cdot 10^{-309}:\\
\;\;\;\;\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}\\
\mathbf{else}:\\
\;\;\;\;c0 \cdot \left({\ell}^{-0.5} \cdot \sqrt{\frac{A}{V}}\right)\\
\end{array}
\]
Derivation Split input into 2 regimes if l < -1.000000000000002e-309 Initial program 70.8%
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
Step-by-step derivation *-commutative70.8%
\[\leadsto \color{blue}{\sqrt{\frac{A}{V \cdot \ell}} \cdot c0}
\]
sqrt-div36.3%
\[\leadsto \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \cdot c0
\]
associate-*l/36.4%
\[\leadsto \color{blue}{\frac{\sqrt{A} \cdot c0}{\sqrt{V \cdot \ell}}}
\]
Applied egg-rr 36.4%
\[\leadsto \color{blue}{\frac{\sqrt{A} \cdot c0}{\sqrt{V \cdot \ell}}}
\]
if -1.000000000000002e-309 < l Initial program 77.6%
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
Step-by-step derivation pow1/277.6%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}}
\]
clear-num77.1%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)}}^{0.5}
\]
inv-pow77.1%
\[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{V \cdot \ell}{A}\right)}^{-1}\right)}}^{0.5}
\]
pow-pow77.8%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{\left(-1 \cdot 0.5\right)}}
\]
associate-/l*74.4%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{\frac{A}{\ell}}\right)}}^{\left(-1 \cdot 0.5\right)}
\]
metadata-eval74.4%
\[\leadsto c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{\color{blue}{-0.5}}
\]
Applied egg-rr 74.4%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5}}
\]
Step-by-step derivation associate-/l*77.8%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{V \cdot \ell}{A}\right)}}^{-0.5}
\]
Simplified77.8%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}}
\]
Step-by-step derivation add-sqr-sqrt77.5%
\[\leadsto c0 \cdot \color{blue}{\left(\sqrt{{\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}} \cdot \sqrt{{\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}}\right)}
\]
sqrt-unprod77.1%
\[\leadsto c0 \cdot \color{blue}{\sqrt{{\left(\frac{V \cdot \ell}{A}\right)}^{-0.5} \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}}}
\]
pow-prod-up77.1%
\[\leadsto c0 \cdot \sqrt{\color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{\left(-0.5 + -0.5\right)}}}
\]
metadata-eval77.1%
\[\leadsto c0 \cdot \sqrt{{\left(\frac{V \cdot \ell}{A}\right)}^{\color{blue}{-1}}}
\]
inv-pow77.1%
\[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\frac{V \cdot \ell}{A}}}}
\]
clear-num77.6%
\[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}}
\]
associate-/r*81.7%
\[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}}
\]
sqrt-undiv90.0%
\[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}}
\]
clear-num90.0%
\[\leadsto c0 \cdot \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}}
\]
associate-/r/90.0%
\[\leadsto c0 \cdot \color{blue}{\left(\frac{1}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\right)}
\]
pow1/290.0%
\[\leadsto c0 \cdot \left(\frac{1}{\color{blue}{{\ell}^{0.5}}} \cdot \sqrt{\frac{A}{V}}\right)
\]
pow-flip90.0%
\[\leadsto c0 \cdot \left(\color{blue}{{\ell}^{\left(-0.5\right)}} \cdot \sqrt{\frac{A}{V}}\right)
\]
metadata-eval90.0%
\[\leadsto c0 \cdot \left({\ell}^{\color{blue}{-0.5}} \cdot \sqrt{\frac{A}{V}}\right)
\]
Applied egg-rr 90.0%
\[\leadsto c0 \cdot \color{blue}{\left({\ell}^{-0.5} \cdot \sqrt{\frac{A}{V}}\right)}
\]
Recombined 2 regimes into one program. Final simplification64.2%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\ell \leq -1 \cdot 10^{-309}:\\
\;\;\;\;\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}\\
\mathbf{else}:\\
\;\;\;\;c0 \cdot \left({\ell}^{-0.5} \cdot \sqrt{\frac{A}{V}}\right)\\
\end{array}
\]
Alternative 5? \[\begin{array}{l}
\mathbf{if}\;\ell \leq 7.2 \cdot 10^{-307}:\\
\;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\
\mathbf{else}:\\
\;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\
\end{array}
\]
Derivation Split input into 2 regimes if l < 7.20000000000000014e-307 Initial program 70.8%
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
Step-by-step derivation pow1/270.8%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}}
\]
clear-num70.7%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)}}^{0.5}
\]
inv-pow70.7%
\[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{V \cdot \ell}{A}\right)}^{-1}\right)}}^{0.5}
\]
pow-pow71.6%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{\left(-1 \cdot 0.5\right)}}
\]
associate-/l*71.7%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{\frac{A}{\ell}}\right)}}^{\left(-1 \cdot 0.5\right)}
\]
metadata-eval71.7%
\[\leadsto c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{\color{blue}{-0.5}}
\]
Applied egg-rr 71.7%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5}}
\]
Step-by-step derivation associate-/l*71.6%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{V \cdot \ell}{A}\right)}}^{-0.5}
\]
*-lft-identity71.6%
\[\leadsto c0 \cdot {\left(\frac{V \cdot \ell}{\color{blue}{1 \cdot A}}\right)}^{-0.5}
\]
times-frac71.8%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{1} \cdot \frac{\ell}{A}\right)}}^{-0.5}
\]
/-rgt-identity71.8%
\[\leadsto c0 \cdot {\left(\color{blue}{V} \cdot \frac{\ell}{A}\right)}^{-0.5}
\]
Simplified71.8%
\[\leadsto c0 \cdot \color{blue}{{\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}}
\]
if 7.20000000000000014e-307 < l Initial program 77.6%
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
Step-by-step derivation associate-/r*81.7%
\[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}}
\]
sqrt-div90.0%
\[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}}
\]
Applied egg-rr 90.0%
\[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}}
\]
Recombined 2 regimes into one program. Final simplification81.2%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\ell \leq 7.2 \cdot 10^{-307}:\\
\;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\
\mathbf{else}:\\
\;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\
\end{array}
\]
Alternative 6? \[\begin{array}{l}
\mathbf{if}\;\ell \leq -1 \cdot 10^{-309}:\\
\;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\
\mathbf{else}:\\
\;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\
\end{array}
\]
Derivation Split input into 2 regimes if l < -1.000000000000002e-309 Initial program 70.8%
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
Step-by-step derivation sqrt-div36.3%
\[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}}
\]
associate-*r/36.4%
\[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}}
\]
Applied egg-rr 36.4%
\[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}}
\]
Step-by-step derivation associate-/l*36.3%
\[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}}
\]
Simplified36.3%
\[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}}
\]
if -1.000000000000002e-309 < l Initial program 77.6%
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
Step-by-step derivation associate-/r*81.7%
\[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}}
\]
sqrt-div90.0%
\[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}}
\]
Applied egg-rr 90.0%
\[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}}
\]
Recombined 2 regimes into one program. Final simplification64.2%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\ell \leq -1 \cdot 10^{-309}:\\
\;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\
\mathbf{else}:\\
\;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\
\end{array}
\]
Alternative 7? \[\begin{array}{l}
\mathbf{if}\;\ell \leq -1 \cdot 10^{-309}:\\
\;\;\;\;\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}\\
\mathbf{else}:\\
\;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\
\end{array}
\]
Derivation Split input into 2 regimes if l < -1.000000000000002e-309 Initial program 70.8%
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
Step-by-step derivation *-commutative70.8%
\[\leadsto \color{blue}{\sqrt{\frac{A}{V \cdot \ell}} \cdot c0}
\]
sqrt-div36.3%
\[\leadsto \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \cdot c0
\]
associate-*l/36.4%
\[\leadsto \color{blue}{\frac{\sqrt{A} \cdot c0}{\sqrt{V \cdot \ell}}}
\]
Applied egg-rr 36.4%
\[\leadsto \color{blue}{\frac{\sqrt{A} \cdot c0}{\sqrt{V \cdot \ell}}}
\]
if -1.000000000000002e-309 < l Initial program 77.6%
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
Step-by-step derivation associate-/r*81.7%
\[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}}
\]
sqrt-div90.0%
\[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}}
\]
Applied egg-rr 90.0%
\[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}}
\]
Recombined 2 regimes into one program. Final simplification64.2%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\ell \leq -1 \cdot 10^{-309}:\\
\;\;\;\;\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}\\
\mathbf{else}:\\
\;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\
\end{array}
\]
Alternative 8? \[\begin{array}{l}
t_0 := \frac{A}{V \cdot \ell}\\
\mathbf{if}\;t_0 \leq 0:\\
\;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\
\mathbf{elif}\;t_0 \leq 2 \cdot 10^{+271}:\\
\;\;\;\;c0 \cdot \sqrt{t_0}\\
\mathbf{else}:\\
\;\;\;\;c0 \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{-0.5}\\
\end{array}
\]
Derivation Split input into 3 regimes if (/.f64 A (*.f64 V l)) < 0.0 Initial program 42.8%
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
Step-by-step derivation *-un-lft-identity42.8%
\[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{1 \cdot A}}{V \cdot \ell}}
\]
times-frac54.0%
\[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V} \cdot \frac{A}{\ell}}}
\]
Applied egg-rr 54.0%
\[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V} \cdot \frac{A}{\ell}}}
\]
Step-by-step derivation associate-*l/54.0%
\[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1 \cdot \frac{A}{\ell}}{V}}}
\]
*-un-lft-identity54.0%
\[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{\ell}}}{V}}
\]
Applied egg-rr 54.0%
\[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{\ell}}{V}}}
\]
if 0.0 < (/.f64 A (*.f64 V l)) < 1.99999999999999991e271 Initial program 99.5%
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
if 1.99999999999999991e271 < (/.f64 A (*.f64 V l)) Initial program 34.0%
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
Step-by-step derivation pow1/234.0%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}}
\]
clear-num34.0%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)}}^{0.5}
\]
inv-pow34.0%
\[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{V \cdot \ell}{A}\right)}^{-1}\right)}}^{0.5}
\]
pow-pow37.6%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{\left(-1 \cdot 0.5\right)}}
\]
associate-/l*51.1%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{\frac{A}{\ell}}\right)}}^{\left(-1 \cdot 0.5\right)}
\]
metadata-eval51.1%
\[\leadsto c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{\color{blue}{-0.5}}
\]
Applied egg-rr 51.1%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5}}
\]
Step-by-step derivation associate-/l*37.6%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{V \cdot \ell}{A}\right)}}^{-0.5}
\]
Simplified37.6%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}}
\]
Step-by-step derivation associate-/l*51.1%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{\frac{A}{\ell}}\right)}}^{-0.5}
\]
associate-/r/54.1%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{A} \cdot \ell\right)}}^{-0.5}
\]
Applied egg-rr 54.1%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{A} \cdot \ell\right)}}^{-0.5}
\]
Recombined 3 regimes into one program. Final simplification80.9%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\
\;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\
\mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+271}:\\
\;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\
\mathbf{else}:\\
\;\;\;\;c0 \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{-0.5}\\
\end{array}
\]
Alternative 9? \[\begin{array}{l}
t_0 := \frac{A}{V \cdot \ell}\\
\mathbf{if}\;t_0 \leq 10^{-312}:\\
\;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\
\mathbf{elif}\;t_0 \leq 2 \cdot 10^{+271}:\\
\;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\
\mathbf{else}:\\
\;\;\;\;c0 \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{-0.5}\\
\end{array}
\]
Derivation Split input into 3 regimes if (/.f64 A (*.f64 V l)) < 9.9999999999847e-313 Initial program 43.5%
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
Step-by-step derivation *-un-lft-identity43.5%
\[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{1 \cdot A}}{V \cdot \ell}}
\]
times-frac54.6%
\[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V} \cdot \frac{A}{\ell}}}
\]
Applied egg-rr 54.6%
\[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V} \cdot \frac{A}{\ell}}}
\]
Step-by-step derivation associate-*l/54.5%
\[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1 \cdot \frac{A}{\ell}}{V}}}
\]
*-un-lft-identity54.5%
\[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{\ell}}}{V}}
\]
Applied egg-rr 54.5%
\[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{\ell}}{V}}}
\]
if 9.9999999999847e-313 < (/.f64 A (*.f64 V l)) < 1.99999999999999991e271 Initial program 99.7%
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
Step-by-step derivation pow1/299.7%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}}
\]
clear-num99.7%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)}}^{0.5}
\]
inv-pow99.7%
\[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{V \cdot \ell}{A}\right)}^{-1}\right)}}^{0.5}
\]
pow-pow99.7%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{\left(-1 \cdot 0.5\right)}}
\]
associate-/l*88.4%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{\frac{A}{\ell}}\right)}}^{\left(-1 \cdot 0.5\right)}
\]
metadata-eval88.4%
\[\leadsto c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{\color{blue}{-0.5}}
\]
Applied egg-rr 88.4%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5}}
\]
Step-by-step derivation associate-/l*99.7%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{V \cdot \ell}{A}\right)}}^{-0.5}
\]
Simplified99.7%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}}
\]
if 1.99999999999999991e271 < (/.f64 A (*.f64 V l)) Initial program 34.0%
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
Step-by-step derivation pow1/234.0%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}}
\]
clear-num34.0%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)}}^{0.5}
\]
inv-pow34.0%
\[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{V \cdot \ell}{A}\right)}^{-1}\right)}}^{0.5}
\]
pow-pow37.6%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{\left(-1 \cdot 0.5\right)}}
\]
associate-/l*51.1%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{\frac{A}{\ell}}\right)}}^{\left(-1 \cdot 0.5\right)}
\]
metadata-eval51.1%
\[\leadsto c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{\color{blue}{-0.5}}
\]
Applied egg-rr 51.1%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5}}
\]
Step-by-step derivation associate-/l*37.6%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{V \cdot \ell}{A}\right)}}^{-0.5}
\]
Simplified37.6%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}}
\]
Step-by-step derivation associate-/l*51.1%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{\frac{A}{\ell}}\right)}}^{-0.5}
\]
associate-/r/54.1%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{A} \cdot \ell\right)}}^{-0.5}
\]
Applied egg-rr 54.1%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{A} \cdot \ell\right)}}^{-0.5}
\]
Recombined 3 regimes into one program. Final simplification80.9%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\frac{A}{V \cdot \ell} \leq 10^{-312}:\\
\;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\
\mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+271}:\\
\;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\
\mathbf{else}:\\
\;\;\;\;c0 \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{-0.5}\\
\end{array}
\]
Alternative 10? \[\begin{array}{l}
t_0 := \frac{A}{V \cdot \ell}\\
\mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 2 \cdot 10^{+271}\right):\\
\;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\
\mathbf{else}:\\
\;\;\;\;c0 \cdot \sqrt{t_0}\\
\end{array}
\]
Derivation Split input into 2 regimes if (/.f64 A (*.f64 V l)) < 0.0 or 1.99999999999999991e271 < (/.f64 A (*.f64 V l)) Initial program 38.1%
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
Step-by-step derivation pow1/238.1%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}}
\]
clear-num38.1%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)}}^{0.5}
\]
inv-pow38.1%
\[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{V \cdot \ell}{A}\right)}^{-1}\right)}}^{0.5}
\]
pow-pow40.0%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{\left(-1 \cdot 0.5\right)}}
\]
associate-/l*52.0%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{\frac{A}{\ell}}\right)}}^{\left(-1 \cdot 0.5\right)}
\]
metadata-eval52.0%
\[\leadsto c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{\color{blue}{-0.5}}
\]
Applied egg-rr 52.0%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5}}
\]
Step-by-step derivation associate-/l*40.0%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{V \cdot \ell}{A}\right)}}^{-0.5}
\]
Simplified40.0%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}}
\]
Step-by-step derivation add-sqr-sqrt40.0%
\[\leadsto c0 \cdot \color{blue}{\left(\sqrt{{\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}} \cdot \sqrt{{\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}}\right)}
\]
sqrt-unprod38.1%
\[\leadsto c0 \cdot \color{blue}{\sqrt{{\left(\frac{V \cdot \ell}{A}\right)}^{-0.5} \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}}}
\]
pow-prod-up38.1%
\[\leadsto c0 \cdot \sqrt{\color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{\left(-0.5 + -0.5\right)}}}
\]
metadata-eval38.1%
\[\leadsto c0 \cdot \sqrt{{\left(\frac{V \cdot \ell}{A}\right)}^{\color{blue}{-1}}}
\]
inv-pow38.1%
\[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\frac{V \cdot \ell}{A}}}}
\]
clear-num38.1%
\[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}}
\]
associate-/r*52.3%
\[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}}
\]
Applied egg-rr 52.3%
\[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{\frac{A}{V}}{\ell}}}
\]
if 0.0 < (/.f64 A (*.f64 V l)) < 1.99999999999999991e271 Initial program 99.5%
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
Recombined 2 regimes into one program. Final simplification80.2%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+271}\right):\\
\;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\
\mathbf{else}:\\
\;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\
\end{array}
\]
Alternative 11? \[\begin{array}{l}
t_0 := \frac{A}{V \cdot \ell}\\
\mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 2 \cdot 10^{+306}\right):\\
\;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\
\mathbf{else}:\\
\;\;\;\;c0 \cdot \sqrt{t_0}\\
\end{array}
\]
Derivation Split input into 2 regimes if (/.f64 A (*.f64 V l)) < 0.0 or 2.00000000000000003e306 < (/.f64 A (*.f64 V l)) Initial program 36.3%
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
Step-by-step derivation *-un-lft-identity36.3%
\[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{1 \cdot A}}{V \cdot \ell}}
\]
times-frac50.9%
\[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V} \cdot \frac{A}{\ell}}}
\]
Applied egg-rr 50.9%
\[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V} \cdot \frac{A}{\ell}}}
\]
Step-by-step derivation associate-*l/50.9%
\[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1 \cdot \frac{A}{\ell}}{V}}}
\]
*-un-lft-identity50.9%
\[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{\ell}}}{V}}
\]
Applied egg-rr 50.9%
\[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{\ell}}{V}}}
\]
if 0.0 < (/.f64 A (*.f64 V l)) < 2.00000000000000003e306 Initial program 99.5%
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
Recombined 2 regimes into one program. Final simplification80.2%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+306}\right):\\
\;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\
\mathbf{else}:\\
\;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\
\end{array}
\]
Alternative 12? \[\begin{array}{l}
t_0 := \frac{A}{V \cdot \ell}\\
\mathbf{if}\;t_0 \leq 0:\\
\;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\
\mathbf{elif}\;t_0 \leq 2 \cdot 10^{+306}:\\
\;\;\;\;c0 \cdot \sqrt{t_0}\\
\mathbf{else}:\\
\;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\
\end{array}
\]
Derivation Split input into 3 regimes if (/.f64 A (*.f64 V l)) < 0.0 Initial program 42.8%
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
Step-by-step derivation *-un-lft-identity42.8%
\[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{1 \cdot A}}{V \cdot \ell}}
\]
times-frac54.0%
\[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V} \cdot \frac{A}{\ell}}}
\]
Applied egg-rr 54.0%
\[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V} \cdot \frac{A}{\ell}}}
\]
Step-by-step derivation associate-*l/54.0%
\[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1 \cdot \frac{A}{\ell}}{V}}}
\]
*-un-lft-identity54.0%
\[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{\ell}}}{V}}
\]
Applied egg-rr 54.0%
\[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{\ell}}{V}}}
\]
if 0.0 < (/.f64 A (*.f64 V l)) < 2.00000000000000003e306 Initial program 99.5%
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
if 2.00000000000000003e306 < (/.f64 A (*.f64 V l)) Initial program 30.3%
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
Step-by-step derivation pow1/230.3%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}}
\]
clear-num30.3%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)}}^{0.5}
\]
inv-pow30.3%
\[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{V \cdot \ell}{A}\right)}^{-1}\right)}}^{0.5}
\]
pow-pow34.0%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{\left(-1 \cdot 0.5\right)}}
\]
associate-/l*51.9%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{\frac{A}{\ell}}\right)}}^{\left(-1 \cdot 0.5\right)}
\]
metadata-eval51.9%
\[\leadsto c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{\color{blue}{-0.5}}
\]
Applied egg-rr 51.9%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5}}
\]
Step-by-step derivation associate-/l*34.0%
\[\leadsto c0 \cdot {\color{blue}{\left(\frac{V \cdot \ell}{A}\right)}}^{-0.5}
\]
Simplified34.0%
\[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}}
\]
Step-by-step derivation add-sqr-sqrt34.0%
\[\leadsto c0 \cdot \color{blue}{\left(\sqrt{{\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}} \cdot \sqrt{{\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}}\right)}
\]
sqrt-unprod30.3%
\[\leadsto c0 \cdot \color{blue}{\sqrt{{\left(\frac{V \cdot \ell}{A}\right)}^{-0.5} \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}}}
\]
pow-prod-up30.3%
\[\leadsto c0 \cdot \sqrt{\color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{\left(-0.5 + -0.5\right)}}}
\]
metadata-eval30.3%
\[\leadsto c0 \cdot \sqrt{{\left(\frac{V \cdot \ell}{A}\right)}^{\color{blue}{-1}}}
\]
inv-pow30.3%
\[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\frac{V \cdot \ell}{A}}}}
\]
clear-num30.3%
\[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}}
\]
associate-/r*48.0%
\[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}}
\]
sqrt-undiv40.3%
\[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}}
\]
associate-*r/38.6%
\[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}}
\]
associate-/l*40.4%
\[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}}
\]
sqrt-undiv51.5%
\[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{\ell}{\frac{A}{V}}}}}
\]
associate-/l*34.0%
\[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{\ell \cdot V}{A}}}}
\]
*-commutative34.0%
\[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{V \cdot \ell}}{A}}}
\]
associate-*r/51.8%
\[\leadsto \frac{c0}{\sqrt{\color{blue}{V \cdot \frac{\ell}{A}}}}
\]
Applied egg-rr 51.8%
\[\leadsto \color{blue}{\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}}
\]
Recombined 3 regimes into one program. Final simplification80.9%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\
\;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\
\mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+306}:\\
\;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\
\mathbf{else}:\\
\;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\
\end{array}
\]
Alternative 13? \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
Derivation Initial program 74.3%
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
Final simplification74.3%
\[\leadsto c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
\]
Reproduce ? herbie shell --seed 2023166
(FPCore (c0 A V l)
:name "Henrywood and Agarwal, Equation (3)"
:precision binary64
(* c0 (sqrt (/ A (* V l)))))
