Average Error: 30.4 → 0.3
Time: 1.9s
Precision: binary64
\[\sqrt{x \cdot x + x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -2.8491681377424 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot {\left(\sqrt{\sqrt{2}}\right)}^{1.5}\right) \cdot \sqrt{\sqrt{\sqrt{2}}}\\ \end{array}\]
\sqrt{x \cdot x + x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -2.8491681377424 \cdot 10^{-310}:\\
\;\;\;\;x \cdot \left(-\sqrt{2}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(x \cdot {\left(\sqrt{\sqrt{2}}\right)}^{1.5}\right) \cdot \sqrt{\sqrt{\sqrt{2}}}\\

\end{array}
(FPCore (x) :precision binary64 (sqrt (+ (* x x) (* x x))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -2.8491681377424e-310)
   (* x (- (sqrt 2.0)))
   (* (* x (pow (sqrt (sqrt 2.0)) 1.5)) (sqrt (sqrt (sqrt 2.0))))))
double code(double x) {
	return ((double) sqrt(((double) (((double) (x * x)) + ((double) (x * x))))));
}
↓
double code(double x) {
	double tmp;
	if ((x <= -2.8491681377424e-310)) {
		tmp = ((double) (x * ((double) -(((double) sqrt(2.0))))));
	} else {
		tmp = ((double) (((double) (x * ((double) pow(((double) sqrt(((double) sqrt(2.0)))), 1.5)))) * ((double) sqrt(((double) sqrt(((double) 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 < -2.849168137742421e-310

    1. Initial program Error: 30.5 bits

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

    2. SimplifiedError: 30.5 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

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

    if -2.849168137742421e-310 < x

    1. Initial program Error: 30.2 bits

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

    2. SimplifiedError: 30.2 bits

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

    3. Taylor expanded around 0 Error: 0.4 bits

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

    5. Applied add-sqr-sqrtError: 0.4 bits

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

    6. Applied sqrt-prodError: 0.6 bits

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

    7. Applied associate-*r*Error: 0.4 bits

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

    9. Applied add-sqr-sqrtError: 0.4 bits

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

    10. Applied sqrt-prodError: 0.4 bits

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

    11. Applied sqrt-prodError: 0.4 bits

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

    12. Applied associate-*r*Error: 0.4 bits

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

    13. SimplifiedError: 0.3 bits

      \[\leadsto \color{blue}{\left(x \cdot {\left(\sqrt{\sqrt{2}}\right)}^{1.5}\right)} \cdot \sqrt{\sqrt{\sqrt{2}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplificationError: 0.3 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8491681377424 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot {\left(\sqrt{\sqrt{2}}\right)}^{1.5}\right) \cdot \sqrt{\sqrt{\sqrt{2}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(FPCore (x)
  :name "sqrt A"
  :precision binary64
  (sqrt (+ (* x x) (* x x))))