Average Error: 29.8 → 0.4
Time: 2.9s
Precision: binary64
\[\sqrt{2 \cdot {x}^{2}}\]
↓
\[\sqrt[3]{\sqrt{2}} \cdot \left(\left|{x}^{\left(\frac{2}{2}\right)}\right| \cdot {\left(\sqrt{2}\right)}^{\frac{2}{3}}\right)\]
\sqrt{2 \cdot {x}^{2}}
↓
\sqrt[3]{\sqrt{2}} \cdot \left(\left|{x}^{\left(\frac{2}{2}\right)}\right| \cdot {\left(\sqrt{2}\right)}^{\frac{2}{3}}\right)
double code(double x) {
	return ((double) sqrt(((double) (2.0 * ((double) pow(x, 2.0))))));
}
↓
double code(double x) {
	return ((double) (((double) cbrt(((double) sqrt(2.0)))) * ((double) (((double) fabs(((double) pow(x, ((double) (2.0 / 2.0)))))) * ((double) pow(((double) sqrt(2.0)), 0.6666666666666666))))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 29.8

    \[\sqrt{2 \cdot {x}^{2}}\]
  2. Using strategy rm

  3. Applied sqrt-prod29.9

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

  5. Applied sqr-pow29.9

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

  6. Applied rem-sqrt-square0.4

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

  8. Applied add-cube-cbrt0.4

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

  9. Applied associate-*l*0.4

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

  11. Applied associate-*l*0.6

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

  12. Simplified0.4

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

  13. Final simplification0.4

    \[\leadsto \sqrt[3]{\sqrt{2}} \cdot \left(\left|{x}^{\left(\frac{2}{2}\right)}\right| \cdot {\left(\sqrt{2}\right)}^{\frac{2}{3}}\right)\]

Reproduce

herbie shell --seed 2020157 
(FPCore (x)
  :name "sqrt D"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))