Average Error: 30.7 → 0.4
Time: 2.4s
Precision: binary64
\[\sqrt{x \cdot x + x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -2.98332070175644 \cdot 10^{-310}:\\ \;\;\;\;-x \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot {16}^{0.1111111111111111}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\\ \end{array}\]
\sqrt{x \cdot x + x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -2.98332070175644 \cdot 10^{-310}:\\
\;\;\;\;-x \cdot \sqrt{2}\\

\mathbf{else}:\\
\;\;\;\;\left(x \cdot {16}^{0.1111111111111111}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\\

\end{array}
(FPCore (x) :precision binary64 (sqrt (+ (* x x) (* x x))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -2.98332070175644e-310)
   (- (* x (sqrt 2.0)))
   (* (* x (pow 16.0 0.1111111111111111)) (cbrt (cbrt (sqrt 2.0))))))
double code(double x) {
	return sqrt((x * x) + (x * x));
}
↓
double code(double x) {
	double tmp;
	if (x <= -2.98332070175644e-310) {
		tmp = -(x * sqrt(2.0));
	} else {
		tmp = (x * pow(16.0, 0.1111111111111111)) * cbrt(cbrt(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 < -2.983320701756437e-310

    1. Initial program 31.3

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

    2. Simplified31.2

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

    3. Taylor expanded around -inf 0.4

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

    4. Simplified0.4

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

    if -2.983320701756437e-310 < x

    1. Initial program 30.2

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

    2. Simplified30.2

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

    3. Taylor expanded around 0 0.4

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

    5. Applied add-cube-cbrt_binary64_1090.4

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

    6. Applied associate-*r*_binary64_190.4

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

    8. Applied add-cube-cbrt_binary64_1090.4

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

    9. Applied associate-*r*_binary64_190.4

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

    10. Simplified0.4

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

    11. Taylor expanded around 0 0.4

      \[\leadsto \color{blue}{\left(x \cdot {\left({\left(\sqrt{2}\right)}^{8}\right)}^{0.1111111111111111}\right)} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\]

    12. Simplified0.4

      \[\leadsto \color{blue}{\left(x \cdot {16}^{0.1111111111111111}\right)} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.98332070175644 \cdot 10^{-310}:\\ \;\;\;\;-x \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot {16}^{0.1111111111111111}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020281 
(FPCore (x)
  :name "sqrt A"
  :precision binary64
  (sqrt (+ (* x x) (* x x))))