Average Error: 30.3 → 1.4
Time: 4.3s
Precision: binary64
\[\sqrt{{x}^{2} + {x}^{2}}\]
↓
\[\left|{\left(\sqrt[3]{x}\right)}^{2}\right| \cdot \sqrt{{\left(\sqrt[3]{x}\right)}^{2} \cdot 2}\]
\sqrt{{x}^{2} + {x}^{2}}
↓
\left|{\left(\sqrt[3]{x}\right)}^{2}\right| \cdot \sqrt{{\left(\sqrt[3]{x}\right)}^{2} \cdot 2}
double code(double x) {
	return ((double) sqrt(((double) (((double) pow(x, 2.0)) + ((double) pow(x, 2.0))))));
}
↓
double code(double x) {
	return ((double) (((double) fabs(((double) pow(((double) cbrt(x)), 2.0)))) * ((double) sqrt(((double) (((double) pow(((double) cbrt(x)), 2.0)) * 2.0))))));
}

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

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

  2. Simplified30.3

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

  4. Applied add-cube-cbrt30.9

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

  5. Applied unpow-prod-down30.9

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

  6. Applied associate-*l*30.9

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

  7. Simplified30.9

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

  9. Applied sqrt-prod16.1

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

  11. Applied unpow-prod-down16.1

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

  12. Applied rem-sqrt-square1.4

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

  13. Final simplification1.4

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

Reproduce

herbie shell --seed 2020178 
(FPCore (x)
  :name "sqrt E"
  :precision binary64
  (sqrt (+ (pow x 2.0) (pow x 2.0))))