Average Error: 30.6 → 0.3
Time: 2.3s
Precision: binary64
\[\sqrt{2 \cdot {x}^{2}}\]
↓
\[\sqrt{\sqrt{\sqrt{2}}} \cdot \left(\left|x\right| \cdot {\left(2 \cdot \sqrt{2}\right)}^{0.25}\right)\]
\sqrt{2 \cdot {x}^{2}}
↓
\sqrt{\sqrt{\sqrt{2}}} \cdot \left(\left|x\right| \cdot {\left(2 \cdot \sqrt{2}\right)}^{0.25}\right)
(FPCore (x) :precision binary64 (sqrt (* 2.0 (pow x 2.0))))
↓
(FPCore (x)
 :precision binary64
 (* (sqrt (sqrt (sqrt 2.0))) (* (fabs x) (pow (* 2.0 (sqrt 2.0)) 0.25))))
double code(double x) {
	return sqrt(2.0 * pow(x, 2.0));
}
↓
double code(double x) {
	return sqrt(sqrt(sqrt(2.0))) * (fabs(x) * pow((2.0 * sqrt(2.0)), 0.25));
}

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

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

  3. Applied sqrt-prod_binary6430.8

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

  4. Simplified0.4

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

  6. Applied add-sqr-sqrt_binary640.6

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

  7. Applied associate-*l*_binary640.4

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

  8. Simplified0.4

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

  10. Applied add-sqr-sqrt_binary640.4

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

  11. Applied associate-*l*_binary640.4

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

  12. Simplified0.4

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

  13. Taylor expanded around 0 0.3

    \[\leadsto \sqrt{\sqrt{\sqrt{2}}} \cdot \color{blue}{\left(\left|x\right| \cdot {\left({\left(\sqrt{2}\right)}^{3}\right)}^{0.25}\right)}\]

  14. Simplified0.3

    \[\leadsto \sqrt{\sqrt{\sqrt{2}}} \cdot \color{blue}{\left(\left|x\right| \cdot {\left(2 \cdot \sqrt{2}\right)}^{0.25}\right)}\]

  15. Final simplification0.3

    \[\leadsto \sqrt{\sqrt{\sqrt{2}}} \cdot \left(\left|x\right| \cdot {\left(2 \cdot \sqrt{2}\right)}^{0.25}\right)\]

Reproduce

herbie shell --seed 2020233 
(FPCore (x)
  :name "sqrt D"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))