Average Error: 29.9 → 0.4
Time: 2.2s
Precision: binary64
\[\sqrt{\left(2 \cdot x\right) \cdot x} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 5.8146655483607 \cdot 10^{-310}:\\ \;\;\;\;-x \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \sqrt[3]{\sqrt{2}}\right) \cdot {\left(\sqrt{2}\right)}^{0.6666666666666666}\\ \end{array} \]
(FPCore (x) :precision binary64 (sqrt (* (* 2.0 x) x)))
↓
(FPCore (x)
 :precision binary64
 (if (<= x 5.8146655483607e-310)
   (- (* x (sqrt 2.0)))
   (* (* x (cbrt (sqrt 2.0))) (pow (sqrt 2.0) 0.6666666666666666))))
double code(double x) {
	return sqrt(((2.0 * x) * x));
}
↓
double code(double x) {
	double tmp;
	if (x <= 5.8146655483607e-310) {
		tmp = -(x * sqrt(2.0));
	} else {
		tmp = (x * cbrt(sqrt(2.0))) * pow(sqrt(2.0), 0.6666666666666666);
	}
	return tmp;
}

public static double code(double x) {
	return Math.sqrt(((2.0 * x) * x));
}
↓
public static double code(double x) {
	double tmp;
	if (x <= 5.8146655483607e-310) {
		tmp = -(x * Math.sqrt(2.0));
	} else {
		tmp = (x * Math.cbrt(Math.sqrt(2.0))) * Math.pow(Math.sqrt(2.0), 0.6666666666666666);
	}
	return tmp;
}

function code(x)
	return sqrt(Float64(Float64(2.0 * x) * x))
end
↓
function code(x)
	tmp = 0.0
	if (x <= 5.8146655483607e-310)
		tmp = Float64(-Float64(x * sqrt(2.0)));
	else
		tmp = Float64(Float64(x * cbrt(sqrt(2.0))) * (sqrt(2.0) ^ 0.6666666666666666));
	end
	return tmp
end

code[x_] := N[Sqrt[N[(N[(2.0 * x), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]
↓
code[x_] := If[LessEqual[x, 5.8146655483607e-310], (-N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), N[(N[(x * N[Power[N[Sqrt[2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sqrt[2.0], $MachinePrecision], 0.6666666666666666], $MachinePrecision]), $MachinePrecision]]

\sqrt{\left(2 \cdot x\right) \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq 5.8146655483607 \cdot 10^{-310}:\\
\;\;\;\;-x \cdot \sqrt{2}\\

\mathbf{else}:\\
\;\;\;\;\left(x \cdot \sqrt[3]{\sqrt{2}}\right) \cdot {\left(\sqrt{2}\right)}^{0.6666666666666666}\\


\end{array}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < 5.814665548360677e-310

    1. Initial program 29.6

      \[\sqrt{\left(2 \cdot x\right) \cdot x} \]

    2. Taylor expanded in x around -inf 0.4

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{2} \cdot x\right)} \]

    3. Simplified0.4

      \[\leadsto \color{blue}{-\sqrt{2} \cdot x} \]

    if 5.814665548360677e-310 < x

    1. Initial program 30.2

      \[\sqrt{\left(2 \cdot x\right) \cdot x} \]

    2. Taylor expanded in x around 0 0.4

      \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]

    3. Applied add-cube-cbrt_binary640.4

      \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}\right)} \cdot x \]

    4. Applied associate-*l*_binary640.4

      \[\leadsto \color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot x\right)} \]

    5. Applied associate-*l*_binary640.6

      \[\leadsto \color{blue}{\sqrt[3]{\sqrt{2}} \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \left(\sqrt[3]{\sqrt{2}} \cdot x\right)\right)} \]

    6. Simplified0.4

      \[\leadsto \sqrt[3]{\sqrt{2}} \cdot \color{blue}{\left(x \cdot {\left(\sqrt{2}\right)}^{0.6666666666666666}\right)} \]

    7. Applied associate-*r*_binary640.4

      \[\leadsto \color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot x\right) \cdot {\left(\sqrt{2}\right)}^{0.6666666666666666}} \]

    8. Simplified0.4

      \[\leadsto \color{blue}{\left(x \cdot \sqrt[3]{\sqrt{2}}\right)} \cdot {\left(\sqrt{2}\right)}^{0.6666666666666666} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.8146655483607 \cdot 10^{-310}:\\ \;\;\;\;-x \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \sqrt[3]{\sqrt{2}}\right) \cdot {\left(\sqrt{2}\right)}^{0.6666666666666666}\\ \end{array} \]

Reproduce

herbie shell --seed 2022129 
(FPCore (x)
  :name "sqrt B"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))