Average Error: 29.9 → 0.4
Time: 2.2s
Precision: binary64
\[\sqrt{2 \cdot \left(x \cdot x\right)} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{\sqrt{2}}\\ \mathbf{if}\;x \leq 5.8146655483607 \cdot 10^{-310}:\\ \;\;\;\;-x \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 \cdot t_0\right) \cdot \left(x \cdot t_0\right)\\ \end{array} \]
(FPCore (x) :precision binary64 (sqrt (* 2.0 (* x x))))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (sqrt 2.0))))
   (if (<= x 5.8146655483607e-310)
     (- (* x (sqrt 2.0)))
     (* (* t_0 t_0) (* x t_0)))))
double code(double x) {
	return sqrt((2.0 * (x * x)));
}
↓
double code(double x) {
	double t_0 = cbrt(sqrt(2.0));
	double tmp;
	if (x <= 5.8146655483607e-310) {
		tmp = -(x * sqrt(2.0));
	} else {
		tmp = (t_0 * t_0) * (x * t_0);
	}
	return tmp;
}

public static double code(double x) {
	return Math.sqrt((2.0 * (x * x)));
}
↓
public static double code(double x) {
	double t_0 = Math.cbrt(Math.sqrt(2.0));
	double tmp;
	if (x <= 5.8146655483607e-310) {
		tmp = -(x * Math.sqrt(2.0));
	} else {
		tmp = (t_0 * t_0) * (x * t_0);
	}
	return tmp;
}

function code(x)
	return sqrt(Float64(2.0 * Float64(x * x)))
end
↓
function code(x)
	t_0 = cbrt(sqrt(2.0))
	tmp = 0.0
	if (x <= 5.8146655483607e-310)
		tmp = Float64(-Float64(x * sqrt(2.0)));
	else
		tmp = Float64(Float64(t_0 * t_0) * Float64(x * t_0));
	end
	return tmp
end

code[x_] := N[Sqrt[N[(2.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]
↓
code[x_] := Block[{t$95$0 = N[Power[N[Sqrt[2.0], $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[x, 5.8146655483607e-310], (-N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]]]

\sqrt{2 \cdot \left(x \cdot x\right)}
↓
\begin{array}{l}
t_0 := \sqrt[3]{\sqrt{2}}\\
\mathbf{if}\;x \leq 5.8146655483607 \cdot 10^{-310}:\\
\;\;\;\;-x \cdot \sqrt{2}\\

\mathbf{else}:\\
\;\;\;\;\left(t_0 \cdot t_0\right) \cdot \left(x \cdot t_0\right)\\


\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

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

    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

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

    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

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

  4. Final simplification0.4

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

Reproduce

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