Average Error: 30.8 → 1.4
Time: 6.4s
Precision: binary64
\[\sqrt{{x}^{2} + {x}^{2}}\]
↓
\[\left|{\left(\sqrt[3]{x}\right)}^{2}\right| \cdot \sqrt{{\left(\sqrt[3]{x}\right)}^{2} + \left(\sqrt[3]{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \sqrt[3]{{\left(\sqrt[3]{x}\right)}^{2}}\right) \cdot \sqrt[3]{{\left(\sqrt[3]{x}\right)}^{2}}}\]
\sqrt{{x}^{2} + {x}^{2}}
↓
\left|{\left(\sqrt[3]{x}\right)}^{2}\right| \cdot \sqrt{{\left(\sqrt[3]{x}\right)}^{2} + \left(\sqrt[3]{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \sqrt[3]{{\left(\sqrt[3]{x}\right)}^{2}}\right) \cdot \sqrt[3]{{\left(\sqrt[3]{x}\right)}^{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)) + ((double) (((double) (((double) cbrt(((double) pow(((double) cbrt(x)), 2.0)))) * ((double) cbrt(((double) pow(((double) cbrt(x)), 2.0)))))) * ((double) cbrt(((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 30.8

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

  3. Applied add-cube-cbrt31.2

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

  4. Applied unpow-prod-down31.2

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

  5. Applied add-cube-cbrt31.4

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

  6. Applied unpow-prod-down31.4

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

  7. Applied distribute-lft-out31.4

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

  8. Applied sqrt-prod16.3

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

  10. Applied unpow-prod-down16.3

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

  11. Applied rem-sqrt-square1.4

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

  13. Applied add-cube-cbrt1.4

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

  14. Final simplification1.4

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

Reproduce

herbie shell --seed 2020169 
(FPCore (x)
  :name "sqrt E"
  :precision binary64
  (sqrt (+ (pow x 2.0) (pow x 2.0))))