sqrt B (should all be same)

Average Accuracy: 52.1% → 99.3%

Time: 2.4s

Precision: binary64

Cost: 19780

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;{2}^{0.25} \cdot \left(\sqrt{\sqrt{2}} \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{2}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -2e-310)
   (* (pow 2.0 0.25) (* (sqrt (sqrt 2.0)) (- x)))
   (* x (sqrt 2.0))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = pow(2.0, 0.25) * (sqrt(sqrt(2.0)) * -x);
	} else {
		tmp = x * sqrt(2.0);
	}
	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 <= (-2d-310)) then
        tmp = (2.0d0 ** 0.25d0) * (sqrt(sqrt(2.0d0)) * -x)
    else
        tmp = x * sqrt(2.0d0)
    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 <= -2e-310) {
		tmp = Math.pow(2.0, 0.25) * (Math.sqrt(Math.sqrt(2.0)) * -x);
	} else {
		tmp = x * Math.sqrt(2.0);
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -2e-310:
		tmp = math.pow(2.0, 0.25) * (math.sqrt(math.sqrt(2.0)) * -x)
	else:
		tmp = x * math.sqrt(2.0)
	return tmp

Julia

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

↓

function code(x)
	tmp = 0.0
	if (x <= -2e-310)
		tmp = Float64((2.0 ^ 0.25) * Float64(sqrt(sqrt(2.0)) * Float64(-x)));
	else
		tmp = Float64(x * sqrt(2.0));
	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 <= -2e-310)
		tmp = (2.0 ^ 0.25) * (sqrt(sqrt(2.0)) * -x);
	else
		tmp = x * sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_] := If[LessEqual[x, -2e-310], N[(N[Power[2.0, 0.25], $MachinePrecision] * N[(N[Sqrt[N[Sqrt[2.0], $MachinePrecision]], $MachinePrecision] * (-x)), $MachinePrecision]), $MachinePrecision], N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 51.6%

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

   2. Applied egg-rr 0.0%

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

   3. Applied egg-rr 2.3%

      \[\leadsto \color{blue}{{2}^{0.25} \cdot \left({2}^{0.25} \cdot x\right)} \]

   4. Applied egg-rr 51.4%

      \[\leadsto {2}^{0.25} \cdot \color{blue}{\sqrt{x \cdot \left(x \cdot \sqrt{2}\right)}} \]

   5. Simplified 51.4%

      \[\leadsto {2}^{0.25} \cdot \color{blue}{\sqrt{\sqrt{2} \cdot \left(x \cdot x\right)}} \]
      Proof

      [Start] 51.4	\[ {2}^{0.25} \cdot \sqrt{x \cdot \left(x \cdot \sqrt{2}\right)} \]
      associate-*r* [=>] 51.4	\[ {2}^{0.25} \cdot \sqrt{\color{blue}{\left(x \cdot x\right) \cdot \sqrt{2}}} \]
      *-commutative [=>] 51.4	\[ {2}^{0.25} \cdot \sqrt{\color{blue}{\sqrt{2} \cdot \left(x \cdot x\right)}} \]

   6. Taylor expanded in x around -inf 99.3%

      \[\leadsto {2}^{0.25} \cdot \color{blue}{\left(-1 \cdot \left(\sqrt{\sqrt{2}} \cdot x\right)\right)} \]

   7. Simplified 99.3%

      \[\leadsto {2}^{0.25} \cdot \color{blue}{\left(\sqrt{\sqrt{2}} \cdot \left(-x\right)\right)} \]
      Proof

      [Start] 99.3	\[ {2}^{0.25} \cdot \left(-1 \cdot \left(\sqrt{\sqrt{2}} \cdot x\right)\right) \]
      mul-1-neg [=>] 99.3	\[ {2}^{0.25} \cdot \color{blue}{\left(-\sqrt{\sqrt{2}} \cdot x\right)} \]
      distribute-rgt-neg-in [=>] 99.3	\[ {2}^{0.25} \cdot \color{blue}{\left(\sqrt{\sqrt{2}} \cdot \left(-x\right)\right)} \]

   if -1.999999999999994e-310 < x

   1. Initial program 52.6%

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

   2. Taylor expanded in x around 0 99.3%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;{2}^{0.25} \cdot \left(\sqrt{\sqrt{2}} \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{2}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.3%
Cost	6788

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

Alternative 2
Accuracy	51.2%
Cost	6592

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

Reproduce

herbie shell --seed 2023129 
(FPCore (x)
  :name "sqrt B (should all be same)"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))