Average Error: 30.8 → 0.4
Time: 1.9s
Precision: binary64
\[\sqrt{x \cdot x + x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -3.9001145064361 \cdot 10^{-310}:\\ \;\;\;\;-x \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \sqrt{x + x}\\ \end{array}\]
\sqrt{x \cdot x + x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -3.9001145064361 \cdot 10^{-310}:\\
\;\;\;\;-x \cdot \sqrt{2}\\

\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 -3.9001145064361e-310)
   (- (* x (sqrt 2.0)))
   (* (sqrt x) (sqrt (+ x x)))))
double code(double x) {
	return ((double) sqrt(((double) (((double) (x * x)) + ((double) (x * x))))));
}
↓
double code(double x) {
	double tmp;
	if ((x <= -3.9001145064361e-310)) {
		tmp = ((double) -(((double) (x * ((double) sqrt(2.0))))));
	} else {
		tmp = ((double) (((double) sqrt(x)) * ((double) sqrt(((double) (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 < -3.90011450643612e-310

    1. Initial program Error: 30.8 bits

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

    2. SimplifiedError: 30.8 bits

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

    3. Taylor expanded around -inf Error: 0.4 bits

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

    4. SimplifiedError: 0.4 bits

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

    if -3.90011450643612e-310 < x

    1. Initial program Error: 30.9 bits

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

    2. SimplifiedError: 30.9 bits

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

    4. Applied sqrt-prodError: 0.4 bits

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

  4. Final simplificationError: 0.4 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9001145064361 \cdot 10^{-310}:\\ \;\;\;\;-x \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \sqrt{x + x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020205 
(FPCore (x)
  :name "sqrt A"
  :precision binary64
  (sqrt (+ (* x x) (* x x))))