Average Error: 31.2 → 0.4
Time: 5.7s
Precision: binary64
\[\sqrt{2 \cdot {x}^{2}}\]
↓
\[\sqrt{\sqrt{\sqrt{2}}} \cdot \left(\sqrt{\sqrt{\sqrt{2}}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left|x\right|\right)\right)\]
\sqrt{2 \cdot {x}^{2}}
↓
\sqrt{\sqrt{\sqrt{2}}} \cdot \left(\sqrt{\sqrt{\sqrt{2}}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left|x\right|\right)\right)
(FPCore (x) :precision binary64 (sqrt (* 2.0 (pow x 2.0))))
↓
(FPCore (x)
 :precision binary64
 (*
  (sqrt (sqrt (sqrt 2.0)))
  (* (sqrt (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(sqrt(2.0))) * (sqrt(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 31.2

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

  3. Applied add-sqr-sqrt_binary6431.4

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

  4. Simplified31.4

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

  5. Simplified31.4

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

  7. Applied sqrt-prod_binary6431.4

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

  8. Applied sqrt-prod_binary6431.4

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

  9. Applied associate-*l*_binary6431.4

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

  10. Simplified31.4

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

  12. Applied sqrt-prod_binary6431.3

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

  13. Applied sqrt-prod_binary6431.4

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

  14. Applied associate-*l*_binary6431.4

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

  15. Simplified0.4

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

  17. Applied add-sqr-sqrt_binary640.4

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

  18. Applied associate-*l*_binary640.4

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

  19. Final simplification0.4

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

Reproduce

herbie shell --seed 2021090 
(FPCore (x)
  :name "sqrt D"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))