Average Error: 31.0 → 0.3
Time: 2.0s
Precision: binary64
\[\sqrt{x \cdot x + x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -6.8685384582662 \cdot 10^{-311}:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \left(x \cdot \left({\left(\sqrt[3]{\sqrt{2}}\right)}^{2} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \left(-\sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \sqrt{x + x}\\ \end{array}\]
\sqrt{x \cdot x + x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -6.8685384582662 \cdot 10^{-311}:\\
\;\;\;\;\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \left(x \cdot \left({\left(\sqrt[3]{\sqrt{2}}\right)}^{2} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \left(-\sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)\right)\right)\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 -6.8685384582662e-311)
   (*
    (cbrt (cbrt (sqrt 2.0)))
    (*
     x
     (*
      (pow (cbrt (sqrt 2.0)) 2.0)
      (* (cbrt (cbrt (sqrt 2.0))) (- (cbrt (cbrt (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 <= -6.8685384582662e-311)) {
		tmp = ((double) (((double) cbrt(((double) cbrt(((double) sqrt(2.0)))))) * ((double) (x * ((double) (((double) pow(((double) cbrt(((double) sqrt(2.0)))), 2.0)) * ((double) (((double) cbrt(((double) cbrt(((double) sqrt(2.0)))))) * ((double) -(((double) cbrt(((double) cbrt(((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 < -6.8685384582662e-311

    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. Taylor expanded around -inf Error: 0.4 bits

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

    4. SimplifiedError: 0.4 bits

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

    6. Applied add-cube-cbrtError: 0.4 bits

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

    7. Applied distribute-lft-neg-inError: 0.4 bits

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

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

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

    9. SimplifiedError: 0.4 bits

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

    11. Applied add-cube-cbrtError: 0.4 bits

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

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

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

    13. SimplifiedError: 0.4 bits

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

    if -6.8685384582662e-311 < x

    1. Initial program Error: 31.1 bits

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

    2. SimplifiedError: 31.0 bits

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

    4. Applied sqrt-prodError: 0.3 bits

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

  4. Final simplificationError: 0.3 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8685384582662 \cdot 10^{-311}:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \left(x \cdot \left({\left(\sqrt[3]{\sqrt{2}}\right)}^{2} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \left(-\sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \sqrt{x + x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(FPCore (x)
  :name "sqrt A"
  :precision binary64
  (sqrt (+ (* x x) (* x x))))