Average Error: 31.0 → 0.4
Time: 1.4s
Precision: binary64
\[\sqrt{2 \cdot \left(x \cdot x\right)} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 1.29087552084645 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \sqrt{x \cdot 2}\\ \end{array} \]
(FPCore (x) :precision binary64 (sqrt (* 2.0 (* x x))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x 1.29087552084645e-310)
   (* x (- (sqrt 2.0)))
   (* (sqrt x) (sqrt (* x 2.0)))))
double code(double x) {
	return sqrt((2.0 * (x * x)));
}
↓
double code(double x) {
	double tmp;
	if (x <= 1.29087552084645e-310) {
		tmp = x * -sqrt(2.0);
	} else {
		tmp = sqrt(x) * sqrt((x * 2.0));
	}
	return tmp;
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((2.0d0 * (x * x)))
end function
↓
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.29087552084645d-310) then
        tmp = x * -sqrt(2.0d0)
    else
        tmp = sqrt(x) * sqrt((x * 2.0d0))
    end if
    code = tmp
end function

public static double code(double x) {
	return Math.sqrt((2.0 * (x * x)));
}
↓
public static double code(double x) {
	double tmp;
	if (x <= 1.29087552084645e-310) {
		tmp = x * -Math.sqrt(2.0);
	} else {
		tmp = Math.sqrt(x) * Math.sqrt((x * 2.0));
	}
	return tmp;
}

def code(x):
	return math.sqrt((2.0 * (x * x)))
↓
def code(x):
	tmp = 0
	if x <= 1.29087552084645e-310:
		tmp = x * -math.sqrt(2.0)
	else:
		tmp = math.sqrt(x) * math.sqrt((x * 2.0))
	return tmp

function code(x)
	return sqrt(Float64(2.0 * Float64(x * x)))
end
↓
function code(x)
	tmp = 0.0
	if (x <= 1.29087552084645e-310)
		tmp = Float64(x * Float64(-sqrt(2.0)));
	else
		tmp = Float64(sqrt(x) * sqrt(Float64(x * 2.0)));
	end
	return tmp
end

function tmp = code(x)
	tmp = sqrt((2.0 * (x * x)));
end
↓
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.29087552084645e-310)
		tmp = x * -sqrt(2.0);
	else
		tmp = sqrt(x) * sqrt((x * 2.0));
	end
	tmp_2 = tmp;
end

code[x_] := N[Sqrt[N[(2.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]
↓
code[x_] := If[LessEqual[x, 1.29087552084645e-310], N[(x * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[Sqrt[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

\sqrt{2 \cdot \left(x \cdot x\right)}
↓
\begin{array}{l}
\mathbf{if}\;x \leq 1.29087552084645 \cdot 10^{-310}:\\
\;\;\;\;x \cdot \left(-\sqrt{2}\right)\\

\mathbf{else}:\\
\;\;\;\;\sqrt{x} \cdot \sqrt{x \cdot 2}\\


\end{array}

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 < 1.290875520846451e-310

    1. Initial program 31.3

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

    2. Taylor expanded in x around -inf 0.4

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

    3. Simplified0.4

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

    if 1.290875520846451e-310 < x

    1. Initial program 30.7

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

    2. Applied egg-rr0.4

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.29087552084645 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \sqrt{x \cdot 2}\\ \end{array} \]

Reproduce

herbie shell --seed 2022150 
(FPCore (x)
  :name "sqrt C"
  :precision binary64
  (sqrt (* 2.0 (* x x))))