Average Error: 30.6 → 0.4
Time: 3.5s
Precision: binary64
\[\sqrt{{x}^{2} + {x}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 2.6083400758351 \cdot 10^{-310}:\\ \;\;\;\;-x \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{2}} \cdot \left(x \cdot \sqrt{\sqrt{2}}\right)\\ \end{array}\]
\sqrt{{x}^{2} + {x}^{2}}
↓
\begin{array}{l}
\mathbf{if}\;x \leq 2.6083400758351 \cdot 10^{-310}:\\
\;\;\;\;-x \cdot \sqrt{2}\\

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

\end{array}
(FPCore (x) :precision binary64 (sqrt (+ (pow x 2.0) (pow x 2.0))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x 2.6083400758351e-310)
   (- (* x (sqrt 2.0)))
   (* (sqrt (sqrt 2.0)) (* x (sqrt (sqrt 2.0))))))
double code(double x) {
	return sqrt(pow(x, 2.0) + pow(x, 2.0));
}
↓
double code(double x) {
	double tmp;
	if (x <= 2.6083400758351e-310) {
		tmp = -(x * sqrt(2.0));
	} else {
		tmp = sqrt(sqrt(2.0)) * (x * sqrt(sqrt(2.0)));
	}
	return tmp;
}

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 < 2.608340075835121e-310

    1. Initial program 30.7

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

    2. Simplified30.7

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

    3. Taylor expanded around -inf 0.4

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

    4. Simplified0.4

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

    if 2.608340075835121e-310 < x

    1. Initial program 30.5

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

    2. Simplified30.5

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

    3. Taylor expanded around 0 0.4

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

    5. Applied add-sqr-sqrt_binary640.6

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

    6. Applied associate-*r*_binary640.4

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6083400758351 \cdot 10^{-310}:\\ \;\;\;\;-x \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{2}} \cdot \left(x \cdot \sqrt{\sqrt{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021024 
(FPCore (x)
  :name "sqrt E"
  :precision binary64
  (sqrt (+ (pow x 2.0) (pow x 2.0))))