Average Error: 30.1 → 0.4
Time: 2.1s
Precision: binary64
\[\sqrt{{x}^{2} + {x}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 1.25277216594145 \cdot 10^{-310}:\\ \;\;\;\;-\sqrt[3]{\sqrt{2}} \cdot \left(x \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{2}} \cdot \left(x \cdot \sqrt{\sqrt{2}}\right)\\ \end{array}\]
\sqrt{{x}^{2} + {x}^{2}}
↓
\begin{array}{l}
\mathbf{if}\;x \leq 1.25277216594145 \cdot 10^{-310}:\\
\;\;\;\;-\sqrt[3]{\sqrt{2}} \cdot \left(x \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right)\\

\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 1.25277216594145e-310)
   (- (* (cbrt (sqrt 2.0)) (* x (* (cbrt (sqrt 2.0)) (cbrt (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 <= 1.25277216594145e-310) {
		tmp = -(cbrt(sqrt(2.0)) * (x * (cbrt(sqrt(2.0)) * cbrt(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 < 1.252772165941452e-310

    1. Initial program 30.0

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

    2. Simplified30.0

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

    3. Taylor expanded around -inf 0.4

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

    4. Simplified0.4

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

    6. Applied add-cube-cbrt_binary640.4

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

    7. Applied associate-*r*_binary640.4

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

    if 1.252772165941452e-310 < x

    1. Initial program 30.2

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

    2. Simplified30.2

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

    4. Applied sqrt-prod_binary6430.3

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

    5. Simplified0.4

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

    7. Applied add-sqr-sqrt_binary640.6

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

    8. Applied associate-*l*_binary640.4

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

    9. Simplified0.4

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25277216594145 \cdot 10^{-310}:\\ \;\;\;\;-\sqrt[3]{\sqrt{2}} \cdot \left(x \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{2}} \cdot \left(x \cdot \sqrt{\sqrt{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021173 
(FPCore (x)
  :name "sqrt E"
  :precision binary64
  (sqrt (+ (pow x 2.0) (pow x 2.0))))