Average Error: 30.5 → 15.5
Time: 4.5s
Precision: binary64
\[\sqrt{{x}^{2} + {x}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 1.18320777133784 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{{x}^{2} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\sqrt{\sqrt{2}}}} \cdot \left(\sqrt{\sqrt{\sqrt{\sqrt{2}}}} \cdot \left({x}^{1} \cdot {\left(\sqrt{\sqrt{\sqrt{2}}}\right)}^{3}\right)\right)\\ \end{array}\]
\sqrt{{x}^{2} + {x}^{2}}
↓
\begin{array}{l}
\mathbf{if}\;x \leq 1.18320777133784 \cdot 10^{-310}:\\
\;\;\;\;\sqrt{{x}^{2} \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\sqrt{\sqrt{\sqrt{2}}}} \cdot \left(\sqrt{\sqrt{\sqrt{\sqrt{2}}}} \cdot \left({x}^{1} \cdot {\left(\sqrt{\sqrt{\sqrt{2}}}\right)}^{3}\right)\right)\\

\end{array}
double code(double x) {
	return ((double) sqrt(((double) (((double) pow(x, 2.0)) + ((double) pow(x, 2.0))))));
}
↓
double code(double x) {
	double VAR;
	if ((x <= 1.18320777133784e-310)) {
		VAR = ((double) sqrt(((double) (((double) pow(x, 2.0)) * 2.0))));
	} else {
		VAR = ((double) (((double) sqrt(((double) sqrt(((double) sqrt(((double) sqrt(2.0)))))))) * ((double) (((double) sqrt(((double) sqrt(((double) sqrt(((double) sqrt(2.0)))))))) * ((double) (((double) pow(x, 1.0)) * ((double) pow(((double) sqrt(((double) sqrt(((double) sqrt(2.0)))))), 3.0))))))));
	}
	return VAR;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < 1.18320777133784e-310

    1. Initial program Error: 30.7 bits

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

    2. SimplifiedError: 30.7 bits

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

    if 1.18320777133784e-310 < x

    1. Initial program Error: 30.3 bits

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

    2. SimplifiedError: 30.3 bits

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

    3. Taylor expanded around 0 Error: 5.7 bits

      \[\leadsto \color{blue}{\sqrt{2} \cdot e^{1 \cdot \left(\log 1 + \log x\right)}}\]

    4. SimplifiedError: 0.4 bits

      \[\leadsto \color{blue}{\sqrt{2} \cdot {x}^{1}}\]
    5. Using strategy rm

    6. Applied add-sqr-sqrtError: 0.4 bits

      \[\leadsto \sqrt{\color{blue}{\sqrt{2} \cdot \sqrt{2}}} \cdot {x}^{1}\]

    7. Applied sqrt-prodError: 0.6 bits

      \[\leadsto \color{blue}{\left(\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}\right)} \cdot {x}^{1}\]

    8. Applied associate-*l*Error: 0.4 bits

      \[\leadsto \color{blue}{\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot {x}^{1}\right)}\]

    9. SimplifiedError: 0.4 bits

      \[\leadsto \sqrt{\sqrt{2}} \cdot \color{blue}{\left({x}^{1} \cdot \sqrt{\sqrt{2}}\right)}\]
    10. Using strategy rm

    11. Applied add-sqr-sqrtError: 0.4 bits

      \[\leadsto \sqrt{\sqrt{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}} \cdot \left({x}^{1} \cdot \sqrt{\sqrt{2}}\right)\]

    12. Applied sqrt-prodError: 0.4 bits

      \[\leadsto \sqrt{\color{blue}{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}}} \cdot \left({x}^{1} \cdot \sqrt{\sqrt{2}}\right)\]

    13. Applied sqrt-prodError: 0.4 bits

      \[\leadsto \color{blue}{\left(\sqrt{\sqrt{\sqrt{2}}} \cdot \sqrt{\sqrt{\sqrt{2}}}\right)} \cdot \left({x}^{1} \cdot \sqrt{\sqrt{2}}\right)\]

    14. Applied associate-*l*Error: 0.4 bits

      \[\leadsto \color{blue}{\sqrt{\sqrt{\sqrt{2}}} \cdot \left(\sqrt{\sqrt{\sqrt{2}}} \cdot \left({x}^{1} \cdot \sqrt{\sqrt{2}}\right)\right)}\]

    15. SimplifiedError: 0.4 bits

      \[\leadsto \sqrt{\sqrt{\sqrt{2}}} \cdot \color{blue}{\left(\sqrt{\sqrt{2}} \cdot \left({x}^{1} \cdot \sqrt{\sqrt{\sqrt{2}}}\right)\right)}\]
    16. Using strategy rm

    17. Applied add-sqr-sqrtError: 0.4 bits

      \[\leadsto \sqrt{\sqrt{\sqrt{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left({x}^{1} \cdot \sqrt{\sqrt{\sqrt{2}}}\right)\right)\]

    18. Applied sqrt-prodError: 0.4 bits

      \[\leadsto \sqrt{\sqrt{\color{blue}{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}}}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left({x}^{1} \cdot \sqrt{\sqrt{\sqrt{2}}}\right)\right)\]

    19. Applied sqrt-prodError: 0.4 bits

      \[\leadsto \sqrt{\color{blue}{\sqrt{\sqrt{\sqrt{2}}} \cdot \sqrt{\sqrt{\sqrt{2}}}}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left({x}^{1} \cdot \sqrt{\sqrt{\sqrt{2}}}\right)\right)\]

    20. Applied sqrt-prodError: 1.0 bits

      \[\leadsto \color{blue}{\left(\sqrt{\sqrt{\sqrt{\sqrt{2}}}} \cdot \sqrt{\sqrt{\sqrt{\sqrt{2}}}}\right)} \cdot \left(\sqrt{\sqrt{2}} \cdot \left({x}^{1} \cdot \sqrt{\sqrt{\sqrt{2}}}\right)\right)\]

    21. Applied associate-*l*Error: 1.0 bits

      \[\leadsto \color{blue}{\sqrt{\sqrt{\sqrt{\sqrt{2}}}} \cdot \left(\sqrt{\sqrt{\sqrt{\sqrt{2}}}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left({x}^{1} \cdot \sqrt{\sqrt{\sqrt{2}}}\right)\right)\right)}\]

    22. SimplifiedError: 0.3 bits

      \[\leadsto \sqrt{\sqrt{\sqrt{\sqrt{2}}}} \cdot \color{blue}{\left(\left({x}^{1} \cdot {\left(\sqrt{\sqrt{\sqrt{2}}}\right)}^{3}\right) \cdot \sqrt{\sqrt{\sqrt{\sqrt{2}}}}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplificationError: 15.5 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.18320777133784 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{{x}^{2} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\sqrt{\sqrt{2}}}} \cdot \left(\sqrt{\sqrt{\sqrt{\sqrt{2}}}} \cdot \left({x}^{1} \cdot {\left(\sqrt{\sqrt{\sqrt{2}}}\right)}^{3}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020200 
(FPCore (x)
  :name "sqrt E"
  :precision binary64
  (sqrt (+ (pow x 2.0) (pow x 2.0))))