sqrt D (should all be same)

Average Error: 47.86% → 0.67%

Time: 6.7s

Precision: binary64

Cost: 6788

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{2} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{2}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -1e-310) (* (sqrt 2.0) (- x)) (* x (sqrt 2.0))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -1e-310) {
		tmp = 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 ** 2.0d0)))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1d-310)) then
        tmp = 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 * Math.pow(x, 2.0)));
}

↓

public static double code(double x) {
	double tmp;
	if (x <= -1e-310) {
		tmp = Math.sqrt(2.0) * -x;
	} else {
		tmp = x * Math.sqrt(2.0);
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -1e-310:
		tmp = math.sqrt(2.0) * -x
	else:
		tmp = x * math.sqrt(2.0)
	return tmp

Julia

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

↓

function code(x)
	tmp = 0.0
	if (x <= -1e-310)
		tmp = Float64(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 ^ 2.0)));
end

↓

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1e-310)
		tmp = sqrt(2.0) * -x;
	else
		tmp = x * sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_] := If[LessEqual[x, -1e-310], N[(N[Sqrt[2.0], $MachinePrecision] * (-x)), $MachinePrecision], N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.999999999999969e-311

   1. Initial program 47.99

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

   2. Simplified 47.99

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

      [Start] 47.99	\[ \sqrt{2 \cdot {x}^{2}} \]
      unpow2 [=>] 47.99	\[ \sqrt{2 \cdot \color{blue}{\left(x \cdot x\right)}} \]

   3. Taylor expanded in x around -inf 0.67

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

   4. Simplified 0.67

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

      [Start] 0.67	\[ -1 \cdot \left(\sqrt{2} \cdot x\right) \]
      *-commutative [<=] 0.67	\[ -1 \cdot \color{blue}{\left(x \cdot \sqrt{2}\right)} \]
      mul-1-neg [=>] 0.67	\[ \color{blue}{-x \cdot \sqrt{2}} \]
      *-commutative [=>] 0.67	\[ -\color{blue}{\sqrt{2} \cdot x} \]
      distribute-rgt-neg-in [=>] 0.67	\[ \color{blue}{\sqrt{2} \cdot \left(-x\right)} \]

   if -9.999999999999969e-311 < x

   1. Initial program 47.73

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

   2. Simplified 47.73

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

      [Start] 47.73	\[ \sqrt{2 \cdot {x}^{2}} \]
      unpow2 [=>] 47.73	\[ \sqrt{2 \cdot \color{blue}{\left(x \cdot x\right)}} \]

   3. Taylor expanded in x around 0 0.68

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

4. Final simplification 0.67

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{2} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.11%
Cost	25920

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

Alternative 2
Error	2.04%
Cost	19584

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

Alternative 3
Error	48.82%
Cost	6592

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

Reproduce

herbie shell --seed 2023103 
(FPCore (x)
  :name "sqrt D (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))