?

Average Error: 30.5 → 0.4
Time: 4.2s
Precision: binary64
Cost: 6788

?

\[\sqrt{2 \cdot \left(x \cdot x\right)} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{2} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot x\\ \end{array} \]
(FPCore (x) :precision binary64 (sqrt (* 2.0 (* x x))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -4e-310) (* (sqrt 2.0) (- x)) (* (sqrt 2.0) x)))
double code(double x) {
	return sqrt((2.0 * (x * x)));
}
↓
double code(double x) {
	double tmp;
	if (x <= -4e-310) {
		tmp = sqrt(2.0) * -x;
	} else {
		tmp = sqrt(2.0) * 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 <= (-4d-310)) then
        tmp = sqrt(2.0d0) * -x
    else
        tmp = sqrt(2.0d0) * 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 <= -4e-310) {
		tmp = Math.sqrt(2.0) * -x;
	} else {
		tmp = Math.sqrt(2.0) * x;
	}
	return tmp;
}

def code(x):
	return math.sqrt((2.0 * (x * x)))
↓
def code(x):
	tmp = 0
	if x <= -4e-310:
		tmp = math.sqrt(2.0) * -x
	else:
		tmp = math.sqrt(2.0) * x
	return tmp

function code(x)
	return sqrt(Float64(2.0 * Float64(x * x)))
end
↓
function code(x)
	tmp = 0.0
	if (x <= -4e-310)
		tmp = Float64(sqrt(2.0) * Float64(-x));
	else
		tmp = Float64(sqrt(2.0) * 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 <= -4e-310)
		tmp = sqrt(2.0) * -x;
	else
		tmp = sqrt(2.0) * x;
	end
	tmp_2 = tmp;
end

code[x_] := N[Sqrt[N[(2.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]
↓
code[x_] := If[LessEqual[x, -4e-310], N[(N[Sqrt[2.0], $MachinePrecision] * (-x)), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]]

\sqrt{2 \cdot \left(x \cdot x\right)}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\
\;\;\;\;\sqrt{2} \cdot \left(-x\right)\\

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


\end{array}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Split input into 2 regimes

  2. if x < -3.999999999999988e-310

    1. Initial program 30.9

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

    2. Taylor expanded in x around -inf 0.4

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

    3. Simplified0.4

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

      [Start]0.4

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

      rational_best-simplify-44 [=>]0.4

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

      rational_best-simplify-2 [=>]0.4

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

      rational_best-simplify-13 [<=]0.4

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

    if -3.999999999999988e-310 < x

    1. Initial program 30.2

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

    2. Taylor expanded in x around 0 0.4

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{2} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot x\\ \end{array} \]

Alternatives

Alternative 1
Error31.9
Cost6592
\[\sqrt{2} \cdot x \]

Error

Reproduce?

herbie shell --seed 2023094 
(FPCore (x)
  :name "sqrt C (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (* x x))))