Average Error: 30.8 → 0.5
Time: 1.9s
Precision: binary64
\[\sqrt{\left(2 \cdot x\right) \cdot x}\]
↓
\[\sqrt{\left|x \cdot \sqrt{2}\right|} \cdot \sqrt{\left|\sqrt{\sqrt{2}} \cdot \left(x \cdot \sqrt{\sqrt{2}}\right)\right|}\]
\sqrt{\left(2 \cdot x\right) \cdot x}
↓
\sqrt{\left|x \cdot \sqrt{2}\right|} \cdot \sqrt{\left|\sqrt{\sqrt{2}} \cdot \left(x \cdot \sqrt{\sqrt{2}}\right)\right|}
(FPCore (x) :precision binary64 (sqrt (* (* 2.0 x) x)))
↓
(FPCore (x)
 :precision binary64
 (*
  (sqrt (fabs (* x (sqrt 2.0))))
  (sqrt (fabs (* (sqrt (sqrt 2.0)) (* x (sqrt (sqrt 2.0))))))))
double code(double x) {
	return ((double) sqrt(((double) (((double) (2.0 * x)) * x))));
}
↓
double code(double x) {
	return ((double) (((double) sqrt(((double) fabs(((double) (x * ((double) sqrt(2.0)))))))) * ((double) sqrt(((double) fabs(((double) (((double) sqrt(((double) sqrt(2.0)))) * ((double) (x * ((double) sqrt(((double) 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 30.8

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

  3. Applied add-sqr-sqrt_binary6431.0

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

  4. Simplified31.0

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

  5. Simplified31.0

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

  7. Applied add-sqr-sqrt_binary6431.0

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

  8. Simplified31.0

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

  9. Simplified0.7

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

  11. Applied add-sqr-sqrt_binary640.7

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

  12. Applied sqrt-prod_binary640.5

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

  13. Applied associate-*r*_binary640.5

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

  14. Final simplification0.5

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

Reproduce

herbie shell --seed 2020205 
(FPCore (x)
  :name "sqrt B"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))