Average Error: 30.4 → 0.4
Time: 2.2s
Precision: binary64
\[\sqrt{{x}^{2} + {x}^{2}}\]
↓
\[\left({\left(\sqrt{2}\right)}^{0.6666666666666666} \cdot \left|x\right|\right) \cdot \sqrt[3]{\sqrt{2}}\]
\sqrt{{x}^{2} + {x}^{2}}
↓
\left({\left(\sqrt{2}\right)}^{0.6666666666666666} \cdot \left|x\right|\right) \cdot \sqrt[3]{\sqrt{2}}
(FPCore (x) :precision binary64 (sqrt (+ (pow x 2.0) (pow x 2.0))))
↓
(FPCore (x)
 :precision binary64
 (* (* (pow (sqrt 2.0) 0.6666666666666666) (fabs x)) (cbrt (sqrt 2.0))))
double code(double x) {
	return ((double) sqrt(((double) (((double) pow(x, 2.0)) + ((double) pow(x, 2.0))))));
}
↓
double code(double x) {
	return ((double) (((double) (((double) pow(((double) sqrt(2.0)), 0.6666666666666666)) * ((double) fabs(x)))) * ((double) cbrt(((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 30.4

    \[\sqrt{{x}^{2} + {x}^{2}}\]

  2. Simplified30.4

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

  4. Applied sqrt-prod_binary6430.5

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

  5. Simplified0.4

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

  7. Applied add-cube-cbrt_binary640.4

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

  8. Applied associate-*l*_binary640.4

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

  9. Simplified0.4

    \[\leadsto \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \color{blue}{\left(\left|x\right| \cdot \sqrt[3]{\sqrt{2}}\right)}\]
  10. Using strategy rm

  11. Applied associate-*r*_binary640.4

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

  12. Simplified0.4

    \[\leadsto \color{blue}{\left({\left(\sqrt{2}\right)}^{0.6666666666666666} \cdot \left|x\right|\right)} \cdot \sqrt[3]{\sqrt{2}}\]

  13. Final simplification0.4

    \[\leadsto \left({\left(\sqrt{2}\right)}^{0.6666666666666666} \cdot \left|x\right|\right) \cdot \sqrt[3]{\sqrt{2}}\]

Reproduce

herbie shell --seed 2020210 
(FPCore (x)
  :name "sqrt E"
  :precision binary64
  (sqrt (+ (pow x 2.0) (pow x 2.0))))