Average Error: 31.1 → 1.4
Time: 5.7s
Precision: binary64
\[\sqrt{{x}^{2} + {x}^{2}}\]
↓
\[\sqrt{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(\sqrt{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \sqrt{{\left(\sqrt[3]{x}\right)}^{2} + {\left(\sqrt[3]{x}\right)}^{2}}\right)\]
\sqrt{{x}^{2} + {x}^{2}}
↓
\sqrt{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(\sqrt{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \sqrt{{\left(\sqrt[3]{x}\right)}^{2} + {\left(\sqrt[3]{x}\right)}^{2}}\right)
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) sqrt(((double) pow(((double) cbrt(x)), 2.0)))) * ((double) (((double) sqrt(((double) pow(((double) cbrt(x)), 2.0)))) * ((double) sqrt(((double) (((double) pow(((double) cbrt(x)), 2.0)) + ((double) pow(((double) cbrt(x)), 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 31.1

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

  3. Applied add-cube-cbrt31.4

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

  4. Applied unpow-prod-down31.4

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

  5. Applied add-cube-cbrt31.7

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

  6. Applied unpow-prod-down31.7

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

  7. Applied distribute-rgt-out31.6

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

  8. Applied sqrt-prod15.6

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

  10. Applied unpow-prod-down15.6

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

  11. Applied unpow-prod-down15.6

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

  12. Applied distribute-lft-out15.6

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

  13. Applied sqrt-prod1.4

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

  14. Final simplification1.4

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

Reproduce

herbie shell --seed 2020171 
(FPCore (x)
  :name "sqrt E"
  :precision binary64
  (sqrt (+ (pow x 2.0) (pow x 2.0))))