Average Error: 30.6 → 0.4
Time: 2.6s
Precision: binary64
\[\sqrt{\left(2 \cdot x\right) \cdot x} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -3.85243106171524 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} t_0 := \sqrt[3]{\sqrt{2}}\\ t_1 := \sqrt[3]{t_0}\\ -\left(t_0 \cdot t_0\right) \cdot \left(t_1 \cdot \left(x \cdot {t_1}^{2}\right)\right) \end{array}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 2} \cdot \sqrt{x}\\ \end{array} \]
\sqrt{\left(2 \cdot x\right) \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -3.85243106171524 \cdot 10^{-310}:\\
\;\;\;\;\begin{array}{l}
t_0 := \sqrt[3]{\sqrt{2}}\\
t_1 := \sqrt[3]{t_0}\\
-\left(t_0 \cdot t_0\right) \cdot \left(t_1 \cdot \left(x \cdot {t_1}^{2}\right)\right)
\end{array}\\

\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 -3.85243106171524e-310)
   (let* ((t_0 (cbrt (sqrt 2.0))) (t_1 (cbrt t_0)))
     (- (* (* t_0 t_0) (* t_1 (* x (pow t_1 2.0))))))
   (* (sqrt (* x 2.0)) (sqrt x))))
double code(double x) {
	return sqrt((2.0 * x) * x);
}
↓
double code(double x) {
	double tmp;
	if (x <= -3.85243106171524e-310) {
		double t_0_1 = cbrt(sqrt(2.0));
		double t_1_2 = cbrt(t_0_1);
		tmp = -((t_0_1 * t_0_1) * (t_1_2 * (x * pow(t_1_2, 2.0))));
	} 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 < -3.85243106171524e-310

    1. Initial program 30.8

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

    2. Taylor expanded in x around -inf 0.4

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

    3. Simplified0.4

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

    4. Applied add-cube-cbrt_binary640.4

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

    5. Applied associate-*l*_binary640.4

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

    6. Applied add-cube-cbrt_binary640.4

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

    7. Applied associate-*l*_binary640.4

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

    8. Applied associate-*l*_binary640.4

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

    9. Simplified0.4

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

    if -3.85243106171524e-310 < x

    1. Initial program 30.4

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

    2. Applied sqrt-prod_binary640.3

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.85243106171524 \cdot 10^{-310}:\\ \;\;\;\;-\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \left(x \cdot {\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 2} \cdot \sqrt{x}\\ \end{array} \]

Reproduce

herbie shell --seed 2022077 
(FPCore (x)
  :name "sqrt B"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))