Average Error: 30.5 → 15.4
Time: 12.9s
Precision: binary64
\[\sqrt{x \cdot x + x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -2.1487218123004 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{x \cdot \left(x + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \sqrt{x + x}\\ \end{array}\]
\sqrt{x \cdot x + x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -2.1487218123004 \cdot 10^{-310}:\\
\;\;\;\;\sqrt{x \cdot \left(x + x\right)}\\

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

\end{array}
(FPCore (x) :precision binary64 (sqrt (+ (* x x) (* x x))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -2.1487218123004e-310)
   (sqrt (* x (+ x x)))
   (* (sqrt x) (sqrt (+ x x)))))
double code(double x) {
	return sqrt((x * x) + (x * x));
}
↓
double code(double x) {
	double tmp;
	if (x <= -2.1487218123004e-310) {
		tmp = sqrt(x * (x + x));
	} else {
		tmp = sqrt(x) * sqrt(x + x);
	}
	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 < -2.148721812300404e-310

    1. Initial program 30.4

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

    2. Simplified30.4

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

    if -2.148721812300404e-310 < x

    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. Using strategy rm

    4. Applied sqrt-prod_binary640.4

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

  4. Final simplification15.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1487218123004 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{x \cdot \left(x + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \sqrt{x + x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020288 
(FPCore (x)
  :name "sqrt A"
  :precision binary64
  (sqrt (+ (* x x) (* x x))))