Average Error: 30.6 → 0.3
Time: 22.7s
Precision: binary64
Cost: 64512
\[\sqrt{x \cdot x + x \cdot x}\]
↓
\[\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right) \cdot \left(\left|x\right| \cdot {\left(\sqrt[3]{\sqrt{2}}\right)}^{2.3333333333333335}\right)\]
\sqrt{x \cdot x + x \cdot x}
↓
\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right) \cdot \left(\left|x\right| \cdot {\left(\sqrt[3]{\sqrt{2}}\right)}^{2.3333333333333335}\right)
(FPCore (x) :precision binary64 (sqrt (+ (* x x) (* x x))))
↓
(FPCore (x)
 :precision binary64
 (*
  (* (cbrt (cbrt (sqrt 2.0))) (cbrt (cbrt (sqrt 2.0))))
  (* (fabs x) (pow (cbrt (sqrt 2.0)) 2.3333333333333335))))
double code(double x) {
	return sqrt((x * x) + (x * x));
}
↓
double code(double x) {
	return (cbrt(cbrt(sqrt(2.0))) * cbrt(cbrt(sqrt(2.0)))) * (fabs(x) * pow(cbrt(sqrt(2.0)), 2.3333333333333335));
}

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
Error0.4
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
\[\begin{array}{l} \mathbf{if}\;x \leq -3.1772341154775 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{2} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot x\\ \end{array}\]
Alternative 5
Error30.1
Cost6913
\[\begin{array}{l} \mathbf{if}\;x \leq -3.559042902657881 \cdot 10^{-206}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot x\\ \end{array}\]
Alternative 6
Error60.6
Cost64
\[1\]
Alternative 7
Error61.6
Cost64
\[0\]

Error

Derivation

  1. Initial program 30.6

    \[\sqrt{x \cdot x + x \cdot x}\]
  2. Using strategy rm

  3. Applied count-2_binary64_36730.6

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

  4. Applied sqrt-prod_binary64_43530.8

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

  5. Simplified0.4

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

  7. Applied add-cube-cbrt_binary64_4540.4

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

  8. Applied associate-*l*_binary64_3600.4

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

  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-*l*_binary64_3600.6

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

  12. Simplified0.4

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

  14. Applied add-cube-cbrt_binary64_4540.4

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

  15. Applied associate-*l*_binary64_3600.4

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

  16. Simplified0.3

    \[\leadsto \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right) \cdot \color{blue}{\left(\left|x\right| \cdot {\left(\sqrt[3]{\sqrt{2}}\right)}^{2.3333333333333335}\right)}\]

  17. Simplified0.3

    \[\leadsto \color{blue}{\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right) \cdot \left(\left|x\right| \cdot {\left(\sqrt[3]{\sqrt{2}}\right)}^{2.3333333333333335}\right)}\]

  18. Final simplification0.3

    \[\leadsto \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right) \cdot \left(\left|x\right| \cdot {\left(\sqrt[3]{\sqrt{2}}\right)}^{2.3333333333333335}\right)\]

Reproduce

herbie shell --seed 2021014 
(FPCore (x)
  :name "sqrt A"
  :precision binary64
  (sqrt (+ (* x x) (* x x))))