Average Error: 30.5 → 15.5
Time: 4.2s
Precision: binary64
\[\sqrt{{x}^{2} + {x}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 6.0263395317301 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{\sqrt{{x}^{2} \cdot 2}} \cdot \sqrt{\sqrt{{x}^{2} \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot {x}^{1}\right)\\ \end{array}\]
\sqrt{{x}^{2} + {x}^{2}}
↓
\begin{array}{l}
\mathbf{if}\;x \leq 6.0263395317301 \cdot 10^{-311}:\\
\;\;\;\;\sqrt{\sqrt{{x}^{2} \cdot 2}} \cdot \sqrt{\sqrt{{x}^{2} \cdot 2}}\\

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

\end{array}
(FPCore (x) :precision binary64 (sqrt (+ (pow x 2.0) (pow x 2.0))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x 6.0263395317301e-311)
   (* (sqrt (sqrt (* (pow x 2.0) 2.0))) (sqrt (sqrt (* (pow x 2.0) 2.0))))
   (* (sqrt (sqrt 2.0)) (* (sqrt (sqrt 2.0)) (pow x 1.0)))))
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 <= 6.0263395317301e-311)) {
		VAR = ((double) (((double) sqrt(((double) sqrt(((double) (((double) pow(x, 2.0)) * 2.0)))))) * ((double) sqrt(((double) sqrt(((double) (((double) pow(x, 2.0)) * 2.0))))))));
	} else {
		VAR = ((double) (((double) sqrt(((double) sqrt(2.0)))) * ((double) (((double) sqrt(((double) sqrt(2.0)))) * ((double) pow(x, 1.0))))));
	}
	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 < 6.02633953173012e-311

    1. Initial program 30.6

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

    2. Simplified30.6

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

    4. Applied add-sqr-sqrt30.8

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

    if 6.02633953173012e-311 < x

    1. Initial program 30.3

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

    2. Simplified30.3

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

    3. Taylor expanded around 0 5.7

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

    4. Simplified0.4

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

    6. Applied add-sqr-sqrt0.4

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

    7. Applied sqrt-prod0.6

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

    8. Applied associate-*l*0.4

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

    9. Simplified0.4

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

  4. Final simplification15.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.0263395317301 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{\sqrt{{x}^{2} \cdot 2}} \cdot \sqrt{\sqrt{{x}^{2} \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot {x}^{1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020198 
(FPCore (x)
  :name "sqrt E"
  :precision binary64
  (sqrt (+ (pow x 2.0) (pow x 2.0))))