Average Error: 29.7 → 0.3
Time: 3.6s
Precision: binary64
Cost: 64512
\[\sqrt{{x}^{2} + {x}^{2}}\]
↓
\[\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right) \cdot \left(\left|x\right| \cdot {\left(\sqrt[3]{\sqrt{2}}\right)}^{2.3333333333333335}\right)\]
\sqrt{{x}^{2} + {x}^{2}}
↓
\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 (+ (pow x 2.0) (pow x 2.0))))
↓
(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(pow(x, 2.0) + pow(x, 2.0));
}
↓
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
Error0.6
Cost77504
\[\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt{\left|x\right| \cdot \sqrt[3]{\sqrt{2}}} \cdot \sqrt{\left|x\right| \cdot \sqrt[3]{\sqrt{2}}}\right)\]
Alternative 2
Error0.5
Cost71104
\[\sqrt[3]{\sqrt{2}} \cdot \left({\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}^{0.6666666666666666} \cdot \left(\left|x\right| \cdot {\left(\sqrt[3]{\sqrt{2}}\right)}^{0.6666666666666666}\right)\right)\]
Alternative 3
Error1.4
Cost58304
\[\sqrt[3]{\sqrt{2} \cdot \left|x\right|} \cdot \left(\sqrt[3]{\sqrt{2} \cdot \left|x\right|} \cdot \sqrt[3]{\sqrt{2} \cdot \left|x\right|}\right)\]
Alternative 4
Error0.7
Cost51648
\[\left(\sqrt{\left|x\right|} \cdot \sqrt{\sqrt{2}}\right) \cdot \left(\sqrt{\left|x\right|} \cdot \sqrt{\sqrt{2}}\right)\]
Alternative 5
Error0.4
Cost45248
\[\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\left|x\right| \cdot \sqrt[3]{\sqrt{2}}\right)\]
Alternative 6
Error1.4
Cost45248
\[\sqrt[3]{\left|x\right|} \cdot \left(\sqrt{2} \cdot \left(\sqrt[3]{\left|x\right|} \cdot \sqrt[3]{\left|x\right|}\right)\right)\]
Alternative 7
Error30.4
Cost39488
\[\sqrt[3]{\sqrt{2 \cdot \left(x \cdot x\right)}} \cdot \left(\sqrt[3]{\sqrt{2 \cdot \left(x \cdot x\right)}} \cdot \sqrt[3]{\sqrt{2 \cdot \left(x \cdot x\right)}}\right)\]
Alternative 8
Error0.7
Cost38848
\[\sqrt{\sqrt{2} \cdot \left|x\right|} \cdot \sqrt{\sqrt{2} \cdot \left|x\right|}\]
Alternative 9
Error0.4
Cost32384
\[\sqrt[3]{\sqrt{2}} \cdot \left(\left|x\right| \cdot {\left(\sqrt{2}\right)}^{0.6666666666666666}\right)\]
Alternative 10
Error0.6
Cost32320
\[\sqrt{\left|x\right|} \cdot \left(\sqrt{2} \cdot \sqrt{\left|x\right|}\right)\]
Alternative 11
Error0.5
Cost32320
\[\sqrt{\sqrt{2}} \cdot \left(\left|x\right| \cdot \sqrt{\sqrt{2}}\right)\]
Alternative 12
Error30.1
Cost26688
\[\sqrt{\sqrt[3]{2 \cdot \left(x \cdot x\right)} \cdot \left(\sqrt[3]{2 \cdot \left(x \cdot x\right)} \cdot \sqrt[3]{2 \cdot \left(x \cdot x\right)}\right)}\]
Alternative 13
Error29.9
Cost26304
\[\sqrt{\sqrt{2 \cdot \left(x \cdot x\right)}} \cdot \sqrt{\sqrt{2 \cdot \left(x \cdot x\right)}}\]
Alternative 14
Error40.4
Cost19584
\[\sqrt[3]{{\left(\sqrt{2 \cdot \left(x \cdot x\right)}\right)}^{3}}\]
Alternative 15
Error32.1
Cost19520
\[e^{\log \left(\sqrt{2 \cdot \left(x \cdot x\right)}\right)}\]
Alternative 16
Error29.7
Cost19520
\[\sqrt{{x}^{2} + {x}^{2}}\]
Alternative 17
Error51.7
Cost19456
\[\sqrt{\sqrt[3]{8 \cdot {x}^{6}}}\]
Alternative 18
Error0.4
Cost12992
\[\sqrt{2} \cdot \left|x\right|\]
Alternative 19
Error29.7
Cost6720
\[\sqrt{2 \cdot \left(x \cdot x\right)}\]
Alternative 20
Error32.5
Cost6656
\[-x \cdot \sqrt{2}\]
Alternative 21
Error30.5
Cost6592
\[x \cdot \sqrt{2}\]
Alternative 22
Error60.5
Cost64
\[1\]
Alternative 23
Error61.6
Cost64
\[0\]
Alternative 24
Error62.9
Cost64
\[-1\]

Error

Derivation

  1. Initial program 29.7

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

  2. Simplified29.7

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

  4. Applied sqrt-prod_binary6429.9

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

  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-*l*_binary640.6

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

  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_binary640.4

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

  15. Applied associate-*l*_binary640.4

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

  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 2021042 
(FPCore (x)
  :name "sqrt E"
  :precision binary64
  (sqrt (+ (pow x 2.0) (pow x 2.0))))