sqrt C (should all be same)

Percentage Accurate: 54.5% → 99.5%

Time: 10.5s

Precision: binary64

Cost: 13380

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;{2}^{0.125} \cdot \left(x \cdot {8}^{0.125}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -2e-310)
   (* x (- (sqrt 2.0)))
   (* (pow 2.0 0.125) (* x (pow 8.0 0.125)))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = x * -sqrt(2.0);
	} else {
		tmp = pow(2.0, 0.125) * (x * pow(8.0, 0.125));
	}
	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 = x * -sqrt(2.0d0)
    else
        tmp = (2.0d0 ** 0.125d0) * (x * (8.0d0 ** 0.125d0))
    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 = x * -Math.sqrt(2.0);
	} else {
		tmp = Math.pow(2.0, 0.125) * (x * Math.pow(8.0, 0.125));
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -2e-310:
		tmp = x * -math.sqrt(2.0)
	else:
		tmp = math.pow(2.0, 0.125) * (x * math.pow(8.0, 0.125))
	return tmp

Julia

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

↓

function code(x)
	tmp = 0.0
	if (x <= -2e-310)
		tmp = Float64(x * Float64(-sqrt(2.0)));
	else
		tmp = Float64((2.0 ^ 0.125) * Float64(x * (8.0 ^ 0.125)));
	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 = x * -sqrt(2.0);
	else
		tmp = (2.0 ^ 0.125) * (x * (8.0 ^ 0.125));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_] := If[LessEqual[x, -2e-310], N[(x * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[Power[2.0, 0.125], $MachinePrecision] * N[(x * N[Power[8.0, 0.125], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 50.0%

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

   2. Taylor expanded in x around -inf 99.4%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(-x\right)} \]
      Step-by-step derivation

      [Start] 99.4%	\[ -1 \cdot \left(\sqrt{2} \cdot x\right) \]
      mul-1-neg [=>] 99.4%	\[ \color{blue}{-\sqrt{2} \cdot x} \]
      distribute-rgt-neg-in [=>] 99.4%	\[ \color{blue}{\sqrt{2} \cdot \left(-x\right)} \]

   if -1.999999999999994e-310 < x

   1. Initial program 50.6%

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

   2. Taylor expanded in x around -inf 2.4%

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

   3. Simplified 2.4%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(-x\right)} \]
      Step-by-step derivation

      [Start] 2.4%	\[ -1 \cdot \left(\sqrt{2} \cdot x\right) \]
      mul-1-neg [=>] 2.4%	\[ \color{blue}{-\sqrt{2} \cdot x} \]
      distribute-rgt-neg-in [=>] 2.4%	\[ \color{blue}{\sqrt{2} \cdot \left(-x\right)} \]

   4. Applied egg-rr 50.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{2 \cdot \left(x \cdot x\right)}\right)}^{1.5}} \]
      Step-by-step derivation

      [Start] 2.4%	\[ \sqrt{2} \cdot \left(-x\right) \]
      add-sqr-sqrt [=>] 0.0%	\[ \sqrt{2} \cdot \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)} \]
      sqrt-unprod [=>] 50.2%	\[ \sqrt{2} \cdot \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} \]
      sqr-neg [=>] 50.2%	\[ \sqrt{2} \cdot \sqrt{\color{blue}{x \cdot x}} \]
      sqrt-unprod [<=] 98.9%	\[ \sqrt{2} \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \]
      add-sqr-sqrt [<=] 99.1%	\[ \sqrt{2} \cdot \color{blue}{x} \]
      *-commutative [=>] 99.1%	\[ \color{blue}{x \cdot \sqrt{2}} \]
      rem-cube-cbrt [<=] 97.9%	\[ \color{blue}{{\left(\sqrt[3]{x \cdot \sqrt{2}}\right)}^{3}} \]
      sqr-pow [=>] 97.9%	\[ \color{blue}{{\left(\sqrt[3]{x \cdot \sqrt{2}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{x \cdot \sqrt{2}}\right)}^{\left(\frac{3}{2}\right)}} \]
      pow-prod-down [=>] 97.9%	\[ \color{blue}{{\left(\sqrt[3]{x \cdot \sqrt{2}} \cdot \sqrt[3]{x \cdot \sqrt{2}}\right)}^{\left(\frac{3}{2}\right)}} \]
      cbrt-unprod [=>] 49.9%	\[ {\color{blue}{\left(\sqrt[3]{\left(x \cdot \sqrt{2}\right) \cdot \left(x \cdot \sqrt{2}\right)}\right)}}^{\left(\frac{3}{2}\right)} \]
      *-commutative [<=] 49.9%	\[ {\left(\sqrt[3]{\color{blue}{\left(\sqrt{2} \cdot x\right)} \cdot \left(x \cdot \sqrt{2}\right)}\right)}^{\left(\frac{3}{2}\right)} \]
      *-commutative [<=] 49.9%	\[ {\left(\sqrt[3]{\left(\sqrt{2} \cdot x\right) \cdot \color{blue}{\left(\sqrt{2} \cdot x\right)}}\right)}^{\left(\frac{3}{2}\right)} \]
      swap-sqr [=>] 49.9%	\[ {\left(\sqrt[3]{\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \left(x \cdot x\right)}}\right)}^{\left(\frac{3}{2}\right)} \]
      add-sqr-sqrt [<=] 50.0%	\[ {\left(\sqrt[3]{\color{blue}{2} \cdot \left(x \cdot x\right)}\right)}^{\left(\frac{3}{2}\right)} \]
      metadata-eval [=>] 50.0%	\[ {\left(\sqrt[3]{2 \cdot \left(x \cdot x\right)}\right)}^{\color{blue}{1.5}} \]

   5. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{{2}^{0.125} \cdot \left({2}^{0.125} \cdot \left({2}^{0.25} \cdot x\right)\right)} \]
      Step-by-step derivation

      [Start] 50.0%	\[ {\left(\sqrt[3]{2 \cdot \left(x \cdot x\right)}\right)}^{1.5} \]
      pow1/3 [=>] 46.6%	\[ {\color{blue}{\left({\left(2 \cdot \left(x \cdot x\right)\right)}^{0.3333333333333333}\right)}}^{1.5} \]
      pow-pow [=>] 50.6%	\[ \color{blue}{{\left(2 \cdot \left(x \cdot x\right)\right)}^{\left(0.3333333333333333 \cdot 1.5\right)}} \]
      metadata-eval [=>] 50.6%	\[ {\left(2 \cdot \left(x \cdot x\right)\right)}^{\color{blue}{0.5}} \]
      pow1/2 [<=] 50.6%	\[ \color{blue}{\sqrt{2 \cdot \left(x \cdot x\right)}} \]
      sqrt-prod [=>] 50.2%	\[ \color{blue}{\sqrt{2} \cdot \sqrt{x \cdot x}} \]
      sqrt-prod [=>] 98.9%	\[ \sqrt{2} \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \]
      add-sqr-sqrt [<=] 99.1%	\[ \sqrt{2} \cdot \color{blue}{x} \]
      pow1/2 [=>] 99.1%	\[ \color{blue}{{2}^{0.5}} \cdot x \]
      metadata-eval [<=] 99.1%	\[ {2}^{\color{blue}{\left(0.25 + 0.25\right)}} \cdot x \]
      pow-prod-up [<=] 99.2%	\[ \color{blue}{\left({2}^{0.25} \cdot {2}^{0.25}\right)} \cdot x \]
      associate-*r* [<=] 99.4%	\[ \color{blue}{{2}^{0.25} \cdot \left({2}^{0.25} \cdot x\right)} \]
      add-sqr-sqrt [=>] 99.4%	\[ \color{blue}{\left(\sqrt{{2}^{0.25}} \cdot \sqrt{{2}^{0.25}}\right)} \cdot \left({2}^{0.25} \cdot x\right) \]
      pow2 [=>] 99.4%	\[ \color{blue}{{\left(\sqrt{{2}^{0.25}}\right)}^{2}} \cdot \left({2}^{0.25} \cdot x\right) \]
      sqrt-pow1 [=>] 99.4%	\[ {\color{blue}{\left({2}^{\left(\frac{0.25}{2}\right)}\right)}}^{2} \cdot \left({2}^{0.25} \cdot x\right) \]
      sqrt-pow2 [<=] 99.4%	\[ {\color{blue}{\left({\left(\sqrt{2}\right)}^{0.25}\right)}}^{2} \cdot \left({2}^{0.25} \cdot x\right) \]
      pow2 [<=] 99.4%	\[ \color{blue}{\left({\left(\sqrt{2}\right)}^{0.25} \cdot {\left(\sqrt{2}\right)}^{0.25}\right)} \cdot \left({2}^{0.25} \cdot x\right) \]
      associate-*l* [=>] 99.4%	\[ \color{blue}{{\left(\sqrt{2}\right)}^{0.25} \cdot \left({\left(\sqrt{2}\right)}^{0.25} \cdot \left({2}^{0.25} \cdot x\right)\right)} \]
      sqrt-pow2 [=>] 99.4%	\[ \color{blue}{{2}^{\left(\frac{0.25}{2}\right)}} \cdot \left({\left(\sqrt{2}\right)}^{0.25} \cdot \left({2}^{0.25} \cdot x\right)\right) \]
      metadata-eval [=>] 99.4%	\[ {2}^{\color{blue}{0.125}} \cdot \left({\left(\sqrt{2}\right)}^{0.25} \cdot \left({2}^{0.25} \cdot x\right)\right) \]
      sqrt-pow2 [=>] 99.4%	\[ {2}^{0.125} \cdot \left(\color{blue}{{2}^{\left(\frac{0.25}{2}\right)}} \cdot \left({2}^{0.25} \cdot x\right)\right) \]
      metadata-eval [=>] 99.4%	\[ {2}^{0.125} \cdot \left({2}^{\color{blue}{0.125}} \cdot \left({2}^{0.25} \cdot x\right)\right) \]

   6. Taylor expanded in x around 0 99.7%

      \[\leadsto {2}^{0.125} \cdot \color{blue}{\left({8}^{0.125} \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;{2}^{0.125} \cdot \left(x \cdot {8}^{0.125}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.5%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;{2}^{0.125} \cdot \left(x \cdot {8}^{0.125}\right)\\ \end{array} \]

Alternative 2
Accuracy	97.9%
Cost	19584

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

Alternative 3
Accuracy	99.4%
Cost	13252

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

Alternative 4
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 5
Accuracy	50.5%
Cost	6592

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

Reproduce

herbie shell --seed 2023256 
(FPCore (x)
  :name "sqrt C (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (* x x))))