Average Error: 30.6 → 0.4
Time: 4.9s
Precision: binary64
\[\sqrt{\left(2 \cdot x\right) \cdot x}\]
↓
\[\left|\sqrt[3]{\sqrt{2}} \cdot \left(x \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right)\right|\]
\sqrt{\left(2 \cdot x\right) \cdot x}
↓
\left|\sqrt[3]{\sqrt{2}} \cdot \left(x \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right)\right|
(FPCore (x) :precision binary64 (sqrt (* (* 2.0 x) x)))
↓
(FPCore (x)
 :precision binary64
 (fabs (* (cbrt (sqrt 2.0)) (* x (* (cbrt (sqrt 2.0)) (cbrt (sqrt 2.0)))))))
double code(double x) {
	return sqrt((2.0 * x) * x);
}
↓
double code(double x) {
	return fabs(cbrt(sqrt(2.0)) * (x * (cbrt(sqrt(2.0)) * 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 30.6

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

  3. Applied add-sqr-sqrt_binary64_10030.8

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

  4. Simplified30.8

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

  5. Simplified0.7

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

  7. Applied pow1_binary64_1390.7

    \[\leadsto \sqrt{\left|x \cdot \sqrt{2}\right|} \cdot \color{blue}{{\left(\sqrt{\left|x \cdot \sqrt{2}\right|}\right)}^{1}}\]

  8. Applied pow1_binary64_1390.7

    \[\leadsto \color{blue}{{\left(\sqrt{\left|x \cdot \sqrt{2}\right|}\right)}^{1}} \cdot {\left(\sqrt{\left|x \cdot \sqrt{2}\right|}\right)}^{1}\]

  9. Applied pow-prod-down_binary64_1490.7

    \[\leadsto \color{blue}{{\left(\sqrt{\left|x \cdot \sqrt{2}\right|} \cdot \sqrt{\left|x \cdot \sqrt{2}\right|}\right)}^{1}}\]

  10. Simplified0.4

    \[\leadsto {\color{blue}{\left(\left|x \cdot \sqrt{2}\right|\right)}}^{1}\]
  11. Using strategy rm

  12. Applied add-cube-cbrt_binary64_1130.4

    \[\leadsto {\left(\left|x \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}\right)}\right|\right)}^{1}\]

  13. Applied associate-*r*_binary64_180.4

    \[\leadsto {\left(\left|\color{blue}{\left(x \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right) \cdot \sqrt[3]{\sqrt{2}}}\right|\right)}^{1}\]

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2021176 
(FPCore (x)
  :name "sqrt B"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))