Average Error: 30.4 → 14.9
Time: 3.3s
Precision: binary64
\[\sqrt{2 \cdot {x}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 3.8265285002463 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{2 \cdot {x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\sqrt{2}}} \cdot \left(\sqrt{\sqrt{\sqrt{2}}} \cdot \left(\sqrt{\sqrt{2}} \cdot {\left({x}^{1}\right)}^{1}\right)\right)\\ \end{array}\]
\sqrt{2 \cdot {x}^{2}}
↓
\begin{array}{l}
\mathbf{if}\;x \leq 3.8265285002463 \cdot 10^{-311}:\\
\;\;\;\;\sqrt{2 \cdot {x}^{2}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\sqrt{\sqrt{2}}} \cdot \left(\sqrt{\sqrt{\sqrt{2}}} \cdot \left(\sqrt{\sqrt{2}} \cdot {\left({x}^{1}\right)}^{1}\right)\right)\\

\end{array}
(FPCore (x) :precision binary64 (sqrt (* 2.0 (pow x 2.0))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x 3.8265285002463e-311)
   (sqrt (* 2.0 (pow x 2.0)))
   (*
    (sqrt (sqrt (sqrt 2.0)))
    (* (sqrt (sqrt (sqrt 2.0))) (* (sqrt (sqrt 2.0)) (pow (pow x 1.0) 1.0))))))
double code(double x) {
	return ((double) sqrt(((double) (2.0 * ((double) pow(x, 2.0))))));
}
↓
double code(double x) {
	double tmp;
	if ((x <= 3.8265285002463e-311)) {
		tmp = ((double) sqrt(((double) (2.0 * ((double) pow(x, 2.0))))));
	} else {
		tmp = ((double) (((double) sqrt(((double) sqrt(((double) sqrt(2.0)))))) * ((double) (((double) sqrt(((double) sqrt(((double) sqrt(2.0)))))) * ((double) (((double) sqrt(((double) sqrt(2.0)))) * ((double) pow(((double) pow(x, 1.0)), 1.0))))))));
	}
	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 < 3.82652850024632e-311

    1. Initial program 30.2

      \[\sqrt{2 \cdot {x}^{2}}\]

    if 3.82652850024632e-311 < x

    1. Initial program 30.6

      \[\sqrt{2 \cdot {x}^{2}}\]

    2. Taylor expanded around 0 0.4

      \[\leadsto \color{blue}{\sqrt{2} \cdot {\left({x}^{1}\right)}^{1}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt_binary640.4

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

    5. Applied sqrt-prod_binary640.6

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

    6. Applied associate-*l*_binary640.4

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

    7. Simplified0.4

      \[\leadsto \sqrt{\sqrt{2}} \cdot \color{blue}{\left({\left({x}^{1}\right)}^{1} \cdot \sqrt{\sqrt{2}}\right)}\]
    8. Using strategy rm

    9. Applied add-sqr-sqrt_binary640.4

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

    10. Applied sqrt-prod_binary640.4

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

    11. Applied sqrt-prod_binary640.4

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

    12. Applied associate-*l*_binary640.4

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

    13. Simplified0.4

      \[\leadsto \sqrt{\sqrt{\sqrt{2}}} \cdot \color{blue}{\left(\left(\sqrt{\sqrt{2}} \cdot {\left({x}^{1}\right)}^{1}\right) \cdot \sqrt{\sqrt{\sqrt{2}}}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification14.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.8265285002463 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{2 \cdot {x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\sqrt{2}}} \cdot \left(\sqrt{\sqrt{\sqrt{2}}} \cdot \left(\sqrt{\sqrt{2}} \cdot {\left({x}^{1}\right)}^{1}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020210 
(FPCore (x)
  :name "sqrt D"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))