Average Error: 30.9 → 0.4
Time: 2.1s
Precision: binary64
\[\sqrt{x \cdot x + x \cdot x} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -3.90929137846693 \cdot 10^{-310}:\\ \;\;\;\;-\sqrt[3]{\sqrt{2}} \cdot \left(x \cdot {\left(\sqrt{2}\right)}^{0.6666666666666666}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \sqrt{x + x}\\ \end{array} \]
\sqrt{x \cdot x + x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -3.90929137846693 \cdot 10^{-310}:\\
\;\;\;\;-\sqrt[3]{\sqrt{2}} \cdot \left(x \cdot {\left(\sqrt{2}\right)}^{0.6666666666666666}\right)\\

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


\end{array}

(FPCore (x) :precision binary64 (sqrt (+ (* x x) (* x x))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -3.90929137846693e-310)
   (- (* (cbrt (sqrt 2.0)) (* x (pow (sqrt 2.0) 0.6666666666666666))))
   (* (sqrt x) (sqrt (+ x x)))))
double code(double x) {
	return sqrt((x * x) + (x * x));
}
↓
double code(double x) {
	double tmp;
	if (x <= -3.90929137846693e-310) {
		tmp = -(cbrt(sqrt(2.0)) * (x * pow(sqrt(2.0), 0.6666666666666666)));
	} else {
		tmp = sqrt(x) * sqrt(x + 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 < -3.909291378466935e-310

    1. Initial program 31.6

      \[\sqrt{x \cdot x + x \cdot x} \]

    2. Simplified31.6

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

    3. Taylor expanded in x around -inf 0.4

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

    4. Simplified0.4

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

    6. Applied add-cube-cbrt_binary640.4

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

    7. Applied associate-*l*_binary640.4

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

    8. Simplified0.4

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

    10. Applied associate-*l*_binary640.6

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

    11. Simplified0.4

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

    if -3.909291378466935e-310 < x

    1. Initial program 30.3

      \[\sqrt{x \cdot x + x \cdot x} \]

    2. Simplified30.3

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

    4. Applied sqrt-prod_binary640.4

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.90929137846693 \cdot 10^{-310}:\\ \;\;\;\;-\sqrt[3]{\sqrt{2}} \cdot \left(x \cdot {\left(\sqrt{2}\right)}^{0.6666666666666666}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \sqrt{x + x}\\ \end{array} \]

Reproduce

herbie shell --seed 2021206 
(FPCore (x)
  :name "sqrt A"
  :precision binary64
  (sqrt (+ (* x x) (* x x))))