Average Error: 30.3 → 0.4
Time: 2.3s
Precision: binary64
\[\sqrt{\left(2 \cdot x\right) \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -3.2216398410802 \cdot 10^{-310}:\\ \;\;\;\;-\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \left(x \cdot {\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)}^{8}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 2} \cdot \sqrt{x}\\ \end{array}\]
\sqrt{\left(2 \cdot x\right) \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -3.2216398410802 \cdot 10^{-310}:\\
\;\;\;\;-\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \left(x \cdot {\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)}^{8}\right)\\

\mathbf{else}:\\
\;\;\;\;\sqrt{x \cdot 2} \cdot \sqrt{x}\\

\end{array}
(FPCore (x) :precision binary64 (sqrt (* (* 2.0 x) x)))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -3.2216398410802e-310)
   (- (* (cbrt (cbrt (sqrt 2.0))) (* x (pow (cbrt (cbrt (sqrt 2.0))) 8.0))))
   (* (sqrt (* x 2.0)) (sqrt x))))
double code(double x) {
	return sqrt((2.0 * x) * x);
}
↓
double code(double x) {
	double tmp;
	if (x <= -3.2216398410802e-310) {
		tmp = -(cbrt(cbrt(sqrt(2.0))) * (x * pow(cbrt(cbrt(sqrt(2.0))), 8.0)));
	} else {
		tmp = sqrt(x * 2.0) * sqrt(x);
	}
	return tmp;
}

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 < -3.22163984108021e-310

    1. Initial program 30.1

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

    2. Taylor expanded around -inf 0.4

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

    3. Simplified0.4

      \[\leadsto \color{blue}{-x \cdot \sqrt{2}}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt_binary64_1130.4

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

    6. Applied associate-*r*_binary64_180.4

      \[\leadsto -\color{blue}{\left(x \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right) \cdot \sqrt[3]{\sqrt{2}}}\]
    7. Using strategy rm

    8. Applied add-cube-cbrt_binary64_1130.4

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

    9. Applied associate-*r*_binary64_180.4

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

    10. Simplified0.4

      \[\leadsto -\color{blue}{\left(\left(x \cdot {\left(\sqrt[3]{\sqrt{2}}\right)}^{2}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)\right)} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\]
    11. Using strategy rm

    12. Applied associate-*l*_binary64_190.4

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

    13. Simplified0.4

      \[\leadsto -\left(x \cdot \color{blue}{{\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)}^{8}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\]

    if -3.22163984108021e-310 < x

    1. Initial program 30.4

      \[\sqrt{\left(2 \cdot x\right) \cdot x}\]
    2. Using strategy rm

    3. Applied sqrt-prod_binary64_940.3

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2216398410802 \cdot 10^{-310}:\\ \;\;\;\;-\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \left(x \cdot {\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)}^{8}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 2} \cdot \sqrt{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020355 
(FPCore (x)
  :name "sqrt B"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))