Average Error: 30.5 → 0.4
Time: 1.7s
Precision: binary64
\[\sqrt{x \cdot x + x \cdot x} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 2.49609637258684 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} t_0 := \sqrt{\sqrt{2}}\\ t_1 := \sqrt{t_0}\\ -t_1 \cdot \left(t_1 \cdot \left(x \cdot t_0\right)\right) \end{array}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \sqrt{x + x}\\ \end{array} \]
\sqrt{x \cdot x + x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq 2.49609637258684 \cdot 10^{-310}:\\
\;\;\;\;\begin{array}{l}
t_0 := \sqrt{\sqrt{2}}\\
t_1 := \sqrt{t_0}\\
-t_1 \cdot \left(t_1 \cdot \left(x \cdot t_0\right)\right)
\end{array}\\

\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 2.49609637258684e-310)
   (let* ((t_0 (sqrt (sqrt 2.0))) (t_1 (sqrt t_0)))
     (- (* t_1 (* t_1 (* x t_0)))))
   (* (sqrt x) (sqrt (+ x x)))))
double code(double x) {
	return sqrt((x * x) + (x * x));
}
↓
double code(double x) {
	double tmp;
	if (x <= 2.49609637258684e-310) {
		double t_0_1 = sqrt(sqrt(2.0));
		double t_1_2 = sqrt(t_0_1);
		tmp = -(t_1_2 * (t_1_2 * (x * t_0_1)));
	} 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 < 2.496096372586836e-310

    1. Initial program 30.4

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

    2. Simplified30.4

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

    3. Taylor expanded around -inf 0.4

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

    4. Simplified0.4

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

    6. Applied add-sqr-sqrt_binary640.6

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

    7. Applied associate-*l*_binary640.4

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

    8. Simplified0.4

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

    10. Applied add-sqr-sqrt_binary640.4

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

    11. Applied associate-*l*_binary640.4

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

    12. Simplified0.4

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

    if 2.496096372586836e-310 < x

    1. Initial program 30.7

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

    2. Simplified30.7

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

    4. Applied sqrt-prod_binary640.3

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.49609637258684 \cdot 10^{-310}:\\ \;\;\;\;-\sqrt{\sqrt{\sqrt{2}}} \cdot \left(\sqrt{\sqrt{\sqrt{2}}} \cdot \left(x \cdot \sqrt{\sqrt{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \sqrt{x + x}\\ \end{array} \]

Reproduce

herbie shell --seed 2021205 
(FPCore (x)
  :name "sqrt A"
  :precision binary64
  (sqrt (+ (* x x) (* x x))))