Average Error: 30.3 → 0.4
Time: 2.0s
Precision: binary64
\[\sqrt{x \cdot x + x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 3.34772934547096 \cdot 10^{-310}:\\ \;\;\;\;-\sqrt[3]{\sqrt{2}} \cdot \left(x \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{2}\\ \end{array}\]
\sqrt{x \cdot x + x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq 3.34772934547096 \cdot 10^{-310}:\\
\;\;\;\;-\sqrt[3]{\sqrt{2}} \cdot \left(x \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right)\\

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

\end{array}
(FPCore (x) :precision binary64 (sqrt (+ (* x x) (* x x))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x 3.34772934547096e-310)
   (- (* (cbrt (sqrt 2.0)) (* x (* (cbrt (sqrt 2.0)) (cbrt (sqrt 2.0))))))
   (* x (sqrt 2.0))))
double code(double x) {
	return sqrt((x * x) + (x * x));
}
↓
double code(double x) {
	double tmp;
	if (x <= 3.34772934547096e-310) {
		tmp = -(cbrt(sqrt(2.0)) * (x * (cbrt(sqrt(2.0)) * cbrt(sqrt(2.0)))));
	} else {
		tmp = x * sqrt(2.0);
	}
	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 < 3.347729345470958e-310

    1. Initial program 30.5

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

    2. Simplified30.4

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

    3. Taylor expanded around -inf 0.4

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

    4. Simplified0.4

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

    6. Applied add-cube-cbrt_binary640.4

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

    7. Applied associate-*r*_binary640.4

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

    if 3.347729345470958e-310 < x

    1. Initial program 30.1

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

    2. Simplified30.1

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

    3. Taylor expanded around 0 0.4

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.34772934547096 \cdot 10^{-310}:\\ \;\;\;\;-\sqrt[3]{\sqrt{2}} \cdot \left(x \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{2}\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021118 
(FPCore (x)
  :name "sqrt A"
  :precision binary64
  (sqrt (+ (* x x) (* x x))))