Average Error: 30.7 → 0.4
Time: 2.9s
Precision: binary64
Cost: 45248
\[\sqrt{{x}^{2} + {x}^{2}}\]
↓
\[\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \left|x\right|\right)\]
\sqrt{{x}^{2} + {x}^{2}}
↓
\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 (+ (pow x 2.0) (pow x 2.0))))
↓
(FPCore (x)
 :precision binary64
 (* (* (cbrt (sqrt 2.0)) (cbrt (sqrt 2.0))) (* (cbrt (sqrt 2.0)) (fabs x))))
double code(double x) {
	return sqrt(pow(x, 2.0) + pow(x, 2.0));
}
↓
double code(double x) {
	return (cbrt(sqrt(2.0)) * cbrt(sqrt(2.0))) * (cbrt(sqrt(2.0)) * fabs(x));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error0.4
Cost32384
\[\sqrt[3]{\sqrt{2}} \cdot \left(\left|x\right| \cdot {\left(\sqrt{2}\right)}^{0.6666666666666666}\right)\]
Alternative 2
Error0.5
Cost32320
\[\sqrt{\sqrt{2}} \cdot \left(\left|x\right| \cdot \sqrt{\sqrt{2}}\right)\]
Alternative 3
Error0.4
Cost12992
\[\sqrt{2} \cdot \left|x\right|\]
Alternative 4
Error0.4
Cost6977
\[\begin{array}{l} \mathbf{if}\;x \leq -3.89140349991603 \cdot 10^{-310}:\\ \;\;\;\;-\sqrt{2} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot x\\ \end{array}\]
Alternative 5
Error30.2
Cost6913
\[\begin{array}{l} \mathbf{if}\;x \leq -3.571696274009956 \cdot 10^{-206}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot x\\ \end{array}\]
Alternative 6
Error60.6
Cost64
\[1\]

Error

Derivation

  1. Initial program 30.7

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

  2. Simplified30.7

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

  4. Applied sqrt-prod_binary6430.8

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

  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. Simplified0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2021044 
(FPCore (x)
  :name "sqrt E"
  :precision binary64
  (sqrt (+ (pow x 2.0) (pow x 2.0))))