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

↓

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \le 1.19787663429483 \cdot 10^{-310} \lor \neg \left(V \cdot \ell \le 2.1944054542886264 \cdot 10^{+306}\right):\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{1}{V \cdot \ell}} \cdot \sqrt{A}\right)\\ \end{array}\]

Derivation

Split input into 2 regimes
if (* V l) < 1.19787663429483e-310 or 2.1944054542886264e+306 < (* V l)
1. Initial program 24.4
  \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
2. Initial simplification20.3
  \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\]
if 1.19787663429483e-310 < (* V l) < 2.1944054542886264e+306
1. Initial program 9.5
  \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
2. Taylor expanded around inf 9.5
  \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{\ell \cdot V}}}\]
3. Using strategy rm
4. Applied div-inv9.6
  \[\leadsto c0 \cdot \sqrt{\color{blue}{A \cdot \frac{1}{\ell \cdot V}}}\]
5. Applied sqrt-prod0.5
  \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{A} \cdot \sqrt{\frac{1}{\ell \cdot V}}\right)}\]
Recombined 2 regimes into one program.
Final simplification12.7
\[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \le 1.19787663429483 \cdot 10^{-310} \lor \neg \left(V \cdot \ell \le 2.1944054542886264 \cdot 10^{+306}\right):\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{1}{V \cdot \ell}} \cdot \sqrt{A}\right)\\ \end{array}\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	19.1	12.7	10.1	9.0	70.9%

herbie shell --seed 2018351 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))

Error

Try it out

Derivation

`if (* V l) < 1.19787663429483e-310 or 2.1944054542886264e+306 < (* V l)`

`if 1.19787663429483e-310 < (* V l) < 2.1944054542886264e+306`

Runtime

In
Out