Average Error: 30.5 → 0.4
Time: 3.0s
Precision: binary64
\[\sqrt{2 \cdot \left(x \cdot x\right)}\]
↓
\[\sqrt{\sqrt{\sqrt{2}}} \cdot \left(\sqrt{\sqrt{\sqrt{2}}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left|x\right|\right)\right)\]
\sqrt{2 \cdot \left(x \cdot x\right)}
↓
\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 (* x x))))
↓
(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 * (x * x));
}
↓
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 30.5

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

  3. Applied add-sqr-sqrt_binary64_10030.7

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

  5. Applied sqrt-prod_binary64_9430.7

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

  6. Applied sqrt-prod_binary64_9430.6

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

  7. Applied associate-*l*_binary64_1930.6

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

  8. Simplified30.6

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

  10. Applied sqrt-prod_binary64_9430.6

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

  11. Applied sqrt-prod_binary64_9430.7

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

  12. Applied associate-*l*_binary64_1930.7

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

  13. Simplified0.4

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

  15. Applied add-sqr-sqrt_binary64_1000.4

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

  16. Applied associate-*l*_binary64_190.4

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

  17. Simplified0.4

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

  18. 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 2021045 
(FPCore (x)
  :name "sqrt C"
  :precision binary64
  (sqrt (* 2.0 (* x x))))