Average Error: 30.3 → 0.4
Time: 1.7s
Precision: binary64
\[\sqrt{2 \cdot \left(x \cdot x\right)}\]
↓
\[\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \left(\left|x\right| \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)\right)\right)\]
\sqrt{2 \cdot \left(x \cdot x\right)}
↓
\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \left(\left|x\right| \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)\right)\right)
(FPCore (x) :precision binary64 (sqrt (* 2.0 (* x x))))
↓
(FPCore (x)
 :precision binary64
 (*
  (* (cbrt (sqrt 2.0)) (cbrt (sqrt 2.0)))
  (*
   (cbrt (cbrt (sqrt 2.0)))
   (* (fabs x) (* (cbrt (cbrt (sqrt 2.0))) (cbrt (cbrt (sqrt 2.0))))))))
double code(double x) {
	return sqrt(2.0 * (x * x));
}
↓
double code(double x) {
	return (cbrt(sqrt(2.0)) * cbrt(sqrt(2.0))) * (cbrt(cbrt(sqrt(2.0))) * (fabs(x) * (cbrt(cbrt(sqrt(2.0))) * cbrt(cbrt(sqrt(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

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

  3. Applied sqrt-prod_binary6430.4

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

  4. Simplified0.4

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

  6. Applied add-cube-cbrt_binary640.4

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

  7. Applied associate-*l*_binary640.4

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

  8. Simplified0.4

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

  10. Applied add-cube-cbrt_binary640.4

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

  11. Applied associate-*r*_binary640.4

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

  12. Final simplification0.4

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

Alternatives

Reproduce

herbie shell --seed 2021118 
(FPCore (x)
  :name "sqrt C"
  :precision binary64
  (sqrt (* 2.0 (* x x))))