Average Error: 30.5 → 0.4
Time: 26.8s
Precision: binary64
Cost: 58240
\[\sqrt{x \cdot x + x \cdot x}\]
↓
\[\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \left(\left|x\right| \cdot {2}^{0.1111111111111111}\right)\right)\]
\sqrt{x \cdot x + x \cdot x}
↓
\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \left(\left|x\right| \cdot {2}^{0.1111111111111111}\right)\right)
(FPCore (x) :precision binary64 (sqrt (+ (* x x) (* x x))))
↓
(FPCore (x)
 :precision binary64
 (*
  (* (cbrt (sqrt 2.0)) (cbrt (sqrt 2.0)))
  (* (cbrt (cbrt (sqrt 2.0))) (* (fabs x) (pow 2.0 0.1111111111111111)))))
double code(double x) {
	return sqrt((x * x) + (x * x));
}
↓
double code(double x) {
	return (cbrt(sqrt(2.0)) * cbrt(sqrt(2.0))) * (cbrt(cbrt(sqrt(2.0))) * (fabs(x) * pow(2.0, 0.1111111111111111)));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error0.4
Cost51584
\[\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \left(\left|x\right| \cdot {\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)}^{8}\right)\]
Alternative 2
Error0.4
Cost45248
\[\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \left|x\right|\right)\]
Alternative 3
Error0.4
Cost32384
\[\sqrt[3]{\sqrt{2}} \cdot \left(\left|x\right| \cdot {\left(\sqrt{2}\right)}^{0.6666666666666666}\right)\]
Alternative 4
Error0.4
Cost32320
\[\sqrt{\sqrt{2}} \cdot \left(\left|x\right| \cdot \sqrt{\sqrt{2}}\right)\]
Alternative 5
Error0.4
Cost12992
\[\sqrt{2} \cdot \left|x\right|\]
Alternative 6
Error0.4
Cost6977
\[\begin{array}{l} \mathbf{if}\;x \leq -1.43388241055396 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{2} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot x\\ \end{array}\]
Alternative 7
Error30.2
Cost6913
\[\begin{array}{l} \mathbf{if}\;x \leq -3.489520080313717 \cdot 10^{-206}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot x\\ \end{array}\]
Alternative 8
Error60.6
Cost64
\[1\]

Error

Time

Derivation

  1. Initial program 30.5

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

  3. Applied count-2_binary64_2630.5

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

  4. Applied sqrt-prod_binary64_9430.7

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

  5. Simplified0.4

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

  7. Applied add-cube-cbrt_binary64_1130.4

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

  8. Applied associate-*l*_binary64_190.4

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

  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 add-cube-cbrt_binary64_1130.4

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

  12. Applied associate-*r*_binary64_180.4

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

  13. Taylor expanded around 0 0.4

    \[\leadsto \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\color{blue}{\left(\left|x\right| \cdot {\left({\left(\sqrt{2}\right)}^{2}\right)}^{0.1111111111111111}\right)} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)\]

  14. Simplified0.4

    \[\leadsto \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\color{blue}{\left(\left|x\right| \cdot {2}^{0.1111111111111111}\right)} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)\]

  15. Simplified0.4

    \[\leadsto \color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \left(\left|x\right| \cdot {2}^{0.1111111111111111}\right)\right)}\]

  16. Final simplification0.4

    \[\leadsto \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \left(\left|x\right| \cdot {2}^{0.1111111111111111}\right)\right)\]

Reproduce

herbie shell --seed 2021065 
(FPCore (x)
  :name "sqrt A"
  :precision binary64
  (sqrt (+ (* x x) (* x x))))