Average Error: 31.0 → 0.5
Time: 2.6s
Precision: binary64
\[\sqrt{\left(2 \cdot x\right) \cdot x}\]
↓
\[\sqrt{\sqrt{2}} \cdot \left(\sqrt{\left|\sqrt[3]{\sqrt{2}} \cdot \left(x \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right)\right|} \cdot \sqrt{\left|x\right|}\right)\]
\sqrt{\left(2 \cdot x\right) \cdot x}
↓
\sqrt{\sqrt{2}} \cdot \left(\sqrt{\left|\sqrt[3]{\sqrt{2}} \cdot \left(x \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right)\right|} \cdot \sqrt{\left|x\right|}\right)
(FPCore (x) :precision binary64 (sqrt (* (* 2.0 x) x)))
↓
(FPCore (x)
 :precision binary64
 (*
  (sqrt (sqrt 2.0))
  (*
   (sqrt
    (fabs (* (cbrt (sqrt 2.0)) (* x (* (cbrt (sqrt 2.0)) (cbrt (sqrt 2.0)))))))
   (sqrt (fabs x)))))
double code(double x) {
	return ((double) sqrt(((double) (((double) (2.0 * x)) * x))));
}
↓
double code(double x) {
	return ((double) (((double) sqrt(((double) sqrt(2.0)))) * ((double) (((double) sqrt(((double) fabs(((double) (((double) cbrt(((double) sqrt(2.0)))) * ((double) (x * ((double) (((double) cbrt(((double) sqrt(2.0)))) * ((double) cbrt(((double) sqrt(2.0)))))))))))))) * ((double) sqrt(((double) 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 Error: 31.0 bits

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

  3. Applied add-sqr-sqrtError: 31.2 bits

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

  4. SimplifiedError: 31.2 bits

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

  5. SimplifiedError: 31.2 bits

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

  7. Applied sqrt-prodError: 31.2 bits

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

  8. Applied sqrt-prodError: 31.2 bits

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

  9. Applied associate-*l*Error: 31.1 bits

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

  10. SimplifiedError: 0.5 bits

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

  12. Applied add-cube-cbrtError: 0.5 bits

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

  13. Applied associate-*r*Error: 0.5 bits

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

  14. Final simplificationError: 0.5 bits

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

Reproduce

herbie shell --seed 2020203 
(FPCore (x)
  :name "sqrt B"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))