Average Error: 30.8 → 0.4
Time: 2.2s
Precision: binary64
\[\sqrt{2 \cdot \left(x \cdot x\right)}\]
↓
\[\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \left|x\right|\right)\]
\sqrt{2 \cdot \left(x \cdot x\right)}
↓
\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \left|x\right|\right)
(FPCore (x) :precision binary64 (sqrt (* 2.0 (* x x))))
↓
(FPCore (x)
 :precision binary64
 (* (* (cbrt (sqrt 2.0)) (cbrt (sqrt 2.0))) (* (cbrt (sqrt 2.0)) (fabs x))))
double code(double x) {
	return ((double) sqrt(((double) (2.0 * ((double) (x * x))))));
}
↓
double code(double x) {
	return ((double) (((double) (((double) cbrt(((double) sqrt(2.0)))) * ((double) cbrt(((double) sqrt(2.0)))))) * ((double) (((double) cbrt(((double) sqrt(2.0)))) * ((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: 30.8 bits

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

  3. Applied sqrt-prodError: 31.0 bits

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

  4. SimplifiedError: 0.4 bits

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

  6. Applied add-cube-cbrtError: 0.4 bits

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

  7. Applied associate-*l*Error: 0.4 bits

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

  8. SimplifiedError: 0.4 bits

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

  9. Final simplificationError: 0.4 bits

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

Reproduce

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