Average Error: 30.3 → 0.5
Time: 2.7s
Precision: binary64
\[\sqrt{\left(2 \cdot x\right) \cdot x}\]
↓
\[\left(\sqrt{\sqrt{2}} \cdot \sqrt{\left|\sqrt{2} \cdot x\right|}\right) \cdot \sqrt{\left|x\right|}\]
\sqrt{\left(2 \cdot x\right) \cdot x}
↓
\left(\sqrt{\sqrt{2}} \cdot \sqrt{\left|\sqrt{2} \cdot x\right|}\right) \cdot \sqrt{\left|x\right|}
double code(double x) {
	return ((double) sqrt(((double) (((double) (2.0 * x)) * x))));
}
↓
double code(double x) {
	return ((double) (((double) (((double) sqrt(((double) sqrt(2.0)))) * ((double) sqrt(((double) fabs(((double) (((double) sqrt(2.0)) * x)))))))) * ((double) sqrt(((double) fabs(x))))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.3

    \[\sqrt{\left(2 \cdot x\right) \cdot x}\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt30.5

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

  4. Simplified30.5

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

  5. Simplified30.5

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

  7. Applied sqrt-prod30.5

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

  8. Applied sqrt-prod30.5

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

  9. Applied associate-*l*30.5

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

  10. Simplified0.5

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

  12. Applied associate-*r*0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2020196 
(FPCore (x)
  :name "sqrt B"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))