sqrt B (should all be same)

Average Error: 30.4 → 0.4

Time: 3.9s

Precision: binary64

Cost: 13252

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{2} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot x} \cdot \sqrt{x}\\ \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (* (* 2.0 x) x)))

↓

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310) (* (sqrt 2.0) (- x)) (* (sqrt (* 2.0 x)) (sqrt x))))

C

double code(double x) {
	return sqrt(((2.0 * x) * x));
}

↓

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = sqrt(2.0) * -x;
	} else {
		tmp = sqrt((2.0 * x)) * sqrt(x);
	}
	return tmp;
}

Fortran

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 <= (-5d-310)) then
        tmp = sqrt(2.0d0) * -x
    else
        tmp = sqrt((2.0d0 * x)) * sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	return Math.sqrt(((2.0 * x) * x));
}

↓

public static double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = Math.sqrt(2.0) * -x;
	} else {
		tmp = Math.sqrt((2.0 * x)) * Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x):
	return math.sqrt(((2.0 * x) * x))

↓

def code(x):
	tmp = 0
	if x <= -5e-310:
		tmp = math.sqrt(2.0) * -x
	else:
		tmp = math.sqrt((2.0 * x)) * math.sqrt(x)
	return tmp

Julia

function code(x)
	return sqrt(Float64(Float64(2.0 * x) * x))
end

↓

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(sqrt(2.0) * Float64(-x));
	else
		tmp = Float64(sqrt(Float64(2.0 * x)) * sqrt(x));
	end
	return tmp
end

MATLAB

function tmp = code(x)
	tmp = sqrt(((2.0 * x) * x));
end

↓

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -5e-310)
		tmp = sqrt(2.0) * -x;
	else
		tmp = sqrt((2.0 * x)) * sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := N[Sqrt[N[(N[(2.0 * x), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]

↓

code[x_] := If[LessEqual[x, -5e-310], N[(N[Sqrt[2.0], $MachinePrecision] * (-x)), $MachinePrecision], N[(N[Sqrt[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 30.7

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

   2. Taylor expanded in x around -inf 0.4

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

   3. Simplified 0.4

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

      [Start] 0.4	\[ -1 \cdot \left(\sqrt{2} \cdot x\right) \]
      mul-1-neg [=>] 0.4	\[ \color{blue}{-\sqrt{2} \cdot x} \]
      distribute-rgt-neg-in [=>] 0.4	\[ \color{blue}{\sqrt{2} \cdot \left(-x\right)} \]

   if -4.999999999999985e-310 < x

   1. Initial program 30.0

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

   2. Applied egg-rr 0.4

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{2} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot x} \cdot \sqrt{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.4
Cost	19584

\[{\left(\sqrt[3]{2 \cdot x} \cdot \sqrt[3]{x}\right)}^{1.5} \]

Alternative 2
Error	0.4
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{2} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{2}\\ \end{array} \]

Alternative 3
Error	31.0
Cost	6592

\[x \cdot \sqrt{2} \]

Reproduce

herbie shell --seed 2023041 
(FPCore (x)
  :name "sqrt B (should all be same)"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))