Average Error: 30.4 → 0.4
Time: 4.7s
Precision: binary64
\[\sqrt{\left(2 \cdot x\right) \cdot x} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.85078885647426 \cdot 10^{-310}:\\ \;\;\;\;-x \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \sqrt{x + x}\\ \end{array} \]
(FPCore (x) :precision binary64 (sqrt (* (* 2.0 x) x)))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -1.85078885647426e-310)
   (- (* x (sqrt 2.0)))
   (* (sqrt x) (sqrt (+ x x)))))
double code(double x) {
	return sqrt(((2.0 * x) * x));
}
↓
double code(double x) {
	double tmp;
	if (x <= -1.85078885647426e-310) {
		tmp = -(x * sqrt(2.0));
	} else {
		tmp = sqrt(x) * sqrt((x + x));
	}
	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 <= (-1.85078885647426d-310)) then
        tmp = -(x * sqrt(2.0d0))
    else
        tmp = sqrt(x) * sqrt((x + x))
    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 <= -1.85078885647426e-310) {
		tmp = -(x * Math.sqrt(2.0));
	} else {
		tmp = Math.sqrt(x) * Math.sqrt((x + x));
	}
	return tmp;
}

def code(x):
	return math.sqrt(((2.0 * x) * x))
↓
def code(x):
	tmp = 0
	if x <= -1.85078885647426e-310:
		tmp = -(x * math.sqrt(2.0))
	else:
		tmp = math.sqrt(x) * math.sqrt((x + x))
	return tmp

function code(x)
	return sqrt(Float64(Float64(2.0 * x) * x))
end
↓
function code(x)
	tmp = 0.0
	if (x <= -1.85078885647426e-310)
		tmp = Float64(-Float64(x * sqrt(2.0)));
	else
		tmp = Float64(sqrt(x) * sqrt(Float64(x + x)));
	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 <= -1.85078885647426e-310)
		tmp = -(x * sqrt(2.0));
	else
		tmp = sqrt(x) * sqrt((x + x));
	end
	tmp_2 = tmp;
end

code[x_] := N[Sqrt[N[(N[(2.0 * x), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]
↓
code[x_] := If[LessEqual[x, -1.85078885647426e-310], (-N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), N[(N[Sqrt[x], $MachinePrecision] * N[Sqrt[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

\sqrt{\left(2 \cdot x\right) \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.85078885647426 \cdot 10^{-310}:\\
\;\;\;\;-x \cdot \sqrt{2}\\

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


\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 < -1.850788856474255e-310

    1. Initial program 30.8

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

    2. Taylor expanded in x around -inf 0.4

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

    3. Simplified0.4

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

    if -1.850788856474255e-310 < x

    1. Initial program 30.0

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

    2. Applied egg-rr0.3

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.85078885647426 \cdot 10^{-310}:\\ \;\;\;\;-x \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \sqrt{x + x}\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(FPCore (x)
  :name "sqrt B"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))