Average Error: 30.5 → 0.3
Time: 1.8s
Precision: binary64
\[\sqrt{x \cdot x + x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 2.5365652372159 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{\sqrt{\sqrt{2}}} \cdot \left(x \cdot \left(\sqrt{\sqrt{2}} \cdot \left(-\sqrt{\sqrt{\sqrt{2}}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \sqrt{x + x}\\ \end{array}\]
\sqrt{x \cdot x + x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq 2.5365652372159 \cdot 10^{-311}:\\
\;\;\;\;\sqrt{\sqrt{\sqrt{2}}} \cdot \left(x \cdot \left(\sqrt{\sqrt{2}} \cdot \left(-\sqrt{\sqrt{\sqrt{2}}}\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\sqrt{x} \cdot \sqrt{x + x}\\

\end{array}
double code(double x) {
	return ((double) sqrt(((double) (((double) (x * x)) + ((double) (x * x))))));
}
↓
double code(double x) {
	double VAR;
	if ((x <= 2.5365652372159e-311)) {
		VAR = ((double) (((double) sqrt(((double) sqrt(((double) sqrt(2.0)))))) * ((double) (x * ((double) (((double) sqrt(((double) sqrt(2.0)))) * ((double) -(((double) sqrt(((double) sqrt(((double) sqrt(2.0))))))))))))));
	} else {
		VAR = ((double) (((double) sqrt(x)) * ((double) sqrt(((double) (x + x))))));
	}
	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 < 2.53656523721594e-311

    1. Initial program 30.6

      \[\sqrt{x \cdot x + x \cdot x}\]

    2. Simplified30.6

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

    3. Taylor expanded around -inf 0.4

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

    4. Simplified0.4

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

    6. Applied add-sqr-sqrt0.4

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

    7. Applied sqrt-prod0.6

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

    8. Applied distribute-lft-neg-in0.6

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

    9. Applied associate-*r*0.4

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

    11. Applied add-sqr-sqrt0.4

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

    12. Applied sqrt-prod0.4

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

    13. Applied sqrt-prod0.4

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

    14. Applied associate-*r*0.4

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

    15. Simplified0.2

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

    if 2.53656523721594e-311 < x

    1. Initial program 30.3

      \[\sqrt{x \cdot x + x \cdot x}\]

    2. Simplified30.3

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

    4. Applied sqrt-prod0.4

      \[\leadsto \color{blue}{\sqrt{x} \cdot \sqrt{x + x}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5365652372159 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{\sqrt{\sqrt{2}}} \cdot \left(x \cdot \left(\sqrt{\sqrt{2}} \cdot \left(-\sqrt{\sqrt{\sqrt{2}}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \sqrt{x + x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(FPCore (x)
  :name "sqrt A"
  :precision binary64
  (sqrt (+ (* x x) (* x x))))