Average Error: 30.5 → 0.4
Time: 3.8s
Precision: binary64
\[\sqrt{2 \cdot {x}^{2}}\]
↓
\[\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left|x\right|\right)\]
\sqrt{2 \cdot {x}^{2}}
↓
\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left|x\right|\right)
(FPCore (x) :precision binary64 (sqrt (* 2.0 (pow x 2.0))))
↓
(FPCore (x)
 :precision binary64
 (* (sqrt (sqrt 2.0)) (* (sqrt (sqrt 2.0)) (fabs x))))
double code(double x) {
	return sqrt(2.0 * pow(x, 2.0));
}
↓
double code(double x) {
	return sqrt(sqrt(2.0)) * (sqrt(sqrt(2.0)) * fabs(x));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.5

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

  3. Applied add-sqr-sqrt_binary6430.7

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

  4. Simplified30.7

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

  5. Simplified30.7

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

  7. Applied sqrt-prod_binary6430.7

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

  8. Applied sqrt-prod_binary6430.6

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

  9. Simplified30.6

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

  11. Applied sqrt-prod_binary6430.6

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

  12. Applied sqrt-prod_binary6430.7

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

  13. Applied associate-*l*_binary6430.7

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

  14. Simplified0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2021045 
(FPCore (x)
  :name "sqrt D"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))