Average Error: 30.8 → 0.4
Time: 2.0s
Precision: binary64
\[\sqrt{x \cdot x + x \cdot x}\]
↓
\[\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \left(\left|x\right| \cdot {\left(\sqrt{2}\right)}^{\frac{2}{3}}\right)\right)\]
\sqrt{x \cdot x + x \cdot x}
↓
\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \left(\left|x\right| \cdot {\left(\sqrt{2}\right)}^{\frac{2}{3}}\right)\right)
double code(double x) {
	return ((double) sqrt(((double) (((double) (x * x)) + ((double) (x * x))))));
}
↓
double code(double x) {
	return ((double) (((double) (((double) cbrt(((double) cbrt(((double) sqrt(2.0)))))) * ((double) cbrt(((double) cbrt(((double) sqrt(2.0)))))))) * ((double) (((double) cbrt(((double) cbrt(((double) sqrt(2.0)))))) * ((double) (((double) fabs(x)) * ((double) pow(((double) sqrt(2.0)), 0.6666666666666666))))))));
}

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

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

  3. Applied count-230.8

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

  4. Applied sqrt-prod31.0

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

  5. Simplified0.4

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

  7. Applied add-cube-cbrt0.4

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

  8. Applied associate-*l*0.4

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

  10. Applied associate-*l*0.6

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

  11. Simplified0.4

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

  13. Applied add-cube-cbrt0.4

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

  14. Applied associate-*l*0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2020169 
(FPCore (x)
  :name "sqrt A"
  :precision binary64
  (sqrt (+ (* x x) (* x x))))