Average Error: 30.7 → 0.4
Time: 3.1s
Precision: binary64
\[\sqrt{2 \cdot {x}^{2}}\]
↓
\[\left|\sqrt[3]{\sqrt{2}} \cdot \left({x}^{\left(\frac{2}{2}\right)} \cdot {\left(\sqrt{2}\right)}^{\frac{2}{3}}\right)\right|\]
\sqrt{2 \cdot {x}^{2}}
↓
\left|\sqrt[3]{\sqrt{2}} \cdot \left({x}^{\left(\frac{2}{2}\right)} \cdot {\left(\sqrt{2}\right)}^{\frac{2}{3}}\right)\right|
double code(double x) {
	return ((double) sqrt(((double) (2.0 * ((double) pow(x, 2.0))))));
}
↓
double code(double x) {
	return ((double) fabs(((double) (((double) cbrt(((double) sqrt(2.0)))) * ((double) (((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 30.7

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

  3. Applied sqr-pow30.7

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

  4. Applied add-sqr-sqrt31.0

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

  5. Applied unswap-sqr30.9

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

  7. Applied rem-sqrt-square0.4

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

  9. Applied add-cube-cbrt0.4

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

  10. Applied associate-*l*0.4

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

  12. Applied associate-*l*0.6

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

  13. Simplified0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2020152 
(FPCore (x)
  :name "sqrt D"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))