Average Error: 30.6 → 0.4
Time: 2.7s
Precision: binary64
\[\sqrt{{x}^{2} + {x}^{2}}\]
↓
\[\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \left(\left|x\right| \cdot {\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)}^{8}\right)\]
\sqrt{{x}^{2} + {x}^{2}}
↓
\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \left(\left|x\right| \cdot {\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)}^{8}\right)
(FPCore (x) :precision binary64 (sqrt (+ (pow x 2.0) (pow x 2.0))))
↓
(FPCore (x)
 :precision binary64
 (* (cbrt (cbrt (sqrt 2.0))) (* (fabs x) (pow (cbrt (cbrt (sqrt 2.0))) 8.0))))
double code(double x) {
	return sqrt(pow(x, 2.0) + pow(x, 2.0));
}
↓
double code(double x) {
	return cbrt(cbrt(sqrt(2.0))) * (fabs(x) * pow(cbrt(cbrt(sqrt(2.0))), 8.0));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.6

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

  2. Simplified30.6

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

  4. Applied sqrt-prod_binary6430.7

    \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{x \cdot x}}\]

  5. Simplified0.4

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

  7. Applied add-cube-cbrt_binary640.4

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

  8. Applied associate-*l*_binary640.4

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

  9. Simplified0.4

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

  11. Applied add-cube-cbrt_binary640.4

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

  12. Applied associate-*r*_binary640.4

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

  14. Applied associate-*r*_binary640.4

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

  15. Simplified0.4

    \[\leadsto \color{blue}{\left(\left|x\right| \cdot {\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)}^{8}\right)} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\]

  16. Final simplification0.4

    \[\leadsto \sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \left(\left|x\right| \cdot {\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)}^{8}\right)\]

Reproduce

herbie shell --seed 2020356 
(FPCore (x)
  :name "sqrt E"
  :precision binary64
  (sqrt (+ (pow x 2.0) (pow x 2.0))))