Average Error: 30.4 → 0.4
Time: 1.7s
Precision: binary64
\[\sqrt{x \cdot x + x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -5.45676362572406 \cdot 10^{-310}:\\ \;\;\;\;-\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \left(\left(x \cdot {\left(\sqrt[3]{\sqrt{2}}\right)}^{2}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \sqrt{x + x}\\ \end{array}\]
\sqrt{x \cdot x + x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -5.45676362572406 \cdot 10^{-310}:\\
\;\;\;\;-\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \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)\\

\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 -5.45676362572406e-310)
   (-
    (*
     (cbrt (cbrt (sqrt 2.0)))
     (*
      (* x (pow (cbrt (sqrt 2.0)) 2.0))
      (* (cbrt (cbrt (sqrt 2.0))) (cbrt (cbrt (sqrt 2.0)))))))
   (* (sqrt x) (sqrt (+ x x)))))
double code(double x) {
	return sqrt((x * x) + (x * x));
}
↓
double code(double x) {
	double tmp;
	if (x <= -5.45676362572406e-310) {
		tmp = -(cbrt(cbrt(sqrt(2.0))) * ((x * pow(cbrt(sqrt(2.0)), 2.0)) * (cbrt(cbrt(sqrt(2.0))) * cbrt(cbrt(sqrt(2.0))))));
	} 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 < -5.45676362572406e-310

    1. Initial program 29.9

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

    2. Simplified29.9

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

    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

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

    9. Applied add-cube-cbrt_binary640.4

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

    10. Applied associate-*r*_binary640.4

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

    11. Simplified0.4

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

    if -5.45676362572406e-310 < x

    1. Initial program 30.9

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

    2. Simplified30.9

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

    4. Applied sqrt-prod_binary640.3

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.45676362572406 \cdot 10^{-310}:\\ \;\;\;\;-\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \left(\left(x \cdot {\left(\sqrt[3]{\sqrt{2}}\right)}^{2}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \sqrt{x + x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020232 
(FPCore (x)
  :name "sqrt A"
  :precision binary64
  (sqrt (+ (* x x) (* x x))))