Optimisation.CirclePacking:place from circle-packing-0.1.0.4, C

Percentage Accurate: 100.0% → 100.0%

Time: 7.7s

Alternatives: 6

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \sqrt{\left|x - y\right|} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (sqrt (fabs (- x y))))

C

double code(double x, double y) {
	return sqrt(fabs((x - y)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt(abs((x - y)))
end function

Java

public static double code(double x, double y) {
	return Math.sqrt(Math.abs((x - y)));
}

Python

def code(x, y):
	return math.sqrt(math.fabs((x - y)))

Julia

function code(x, y)
	return sqrt(abs(Float64(x - y)))
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt(abs((x - y)));
end

Wolfram

code[x_, y_] := N[Sqrt[N[Abs[N[(x - y), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left|x - y\right|} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (sqrt (fabs (- x y))))

C

double code(double x, double y) {
	return sqrt(fabs((x - y)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt(abs((x - y)))
end function

Java

public static double code(double x, double y) {
	return Math.sqrt(Math.abs((x - y)));
}

Python

def code(x, y):
	return math.sqrt(math.fabs((x - y)))

Julia

function code(x, y)
	return sqrt(abs(Float64(x - y)))
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt(abs((x - y)));
end

Wolfram

code[x_, y_] := N[Sqrt[N[Abs[N[(x - y), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left|x - y\right|} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (sqrt (fabs (- x y))))

C

double code(double x, double y) {
	return sqrt(fabs((x - y)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt(abs((x - y)))
end function

Java

public static double code(double x, double y) {
	return Math.sqrt(Math.abs((x - y)));
}

Python

def code(x, y):
	return math.sqrt(math.fabs((x - y)))

Julia

function code(x, y)
	return sqrt(abs(Float64(x - y)))
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt(abs((x - y)));
end

Wolfram

code[x_, y_] := N[Sqrt[N[Abs[N[(x - y), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\sqrt{\left|x - y\right|} \]

2. Final simplification 100.0%

   \[\leadsto \sqrt{\left|x - y\right|} \]

Alternative 2: 74.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - y \leq -2 \cdot 10^{-243}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- x y) -2e-243) (sqrt y) (sqrt (- x y))))

C

double code(double x, double y) {
	double tmp;
	if ((x - y) <= -2e-243) {
		tmp = sqrt(y);
	} else {
		tmp = sqrt((x - y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x - y) <= (-2d-243)) then
        tmp = sqrt(y)
    else
        tmp = sqrt((x - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x - y) <= -2e-243) {
		tmp = Math.sqrt(y);
	} else {
		tmp = Math.sqrt((x - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x - y) <= -2e-243:
		tmp = math.sqrt(y)
	else:
		tmp = math.sqrt((x - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x - y) <= -2e-243)
		tmp = sqrt(y);
	else
		tmp = sqrt(Float64(x - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x - y) <= -2e-243)
		tmp = sqrt(y);
	else
		tmp = sqrt((x - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x - y), $MachinePrecision], -2e-243], N[Sqrt[y], $MachinePrecision], N[Sqrt[N[(x - y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x y) < -1.99999999999999999e-243

   1. Initial program 100.0%

      \[\sqrt{\left|x - y\right|} \]
   2. Step-by-step derivation

      1. fabs-sub 100.0%

         \[\leadsto \sqrt{\color{blue}{\left|y - x\right|}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sqrt{\left|y - x\right|}} \]

   4. Taylor expanded in y around -inf 100.0%

      \[\leadsto \color{blue}{\sqrt{\left|-\left(x + -1 \cdot y\right)\right|}} \]
   5. Step-by-step derivation

      1. fabs-neg 100.0%

         \[\leadsto \sqrt{\color{blue}{\left|x + -1 \cdot y\right|}} \]

      2. mul-1-neg 100.0%

         \[\leadsto \sqrt{\left|x + \color{blue}{\left(-y\right)}\right|} \]

      3. unsub-neg 100.0%

         \[\leadsto \sqrt{\left|\color{blue}{x - y}\right|} \]

      4. fabs-sub 100.0%

         \[\leadsto \sqrt{\color{blue}{\left|y - x\right|}} \]

      5. rem-square-sqrt 100.0%

         \[\leadsto \sqrt{\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right|} \]

      6. fabs-sqr 100.0%

         \[\leadsto \sqrt{\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}} \]

      7. rem-sqrt-square 100.0%

         \[\leadsto \color{blue}{\left|\sqrt{y - x}\right|} \]

      8. unpow1 100.0%

         \[\leadsto \left|\color{blue}{{\left(\sqrt{y - x}\right)}^{1}}\right| \]

      9. sqr-pow 99.4%

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

      10. fabs-sqr 99.4%

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

      11. sqr-pow 100.0%

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

      12. unpow1 100.0%

          \[\leadsto \color{blue}{\sqrt{y - x}} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\sqrt{y - x}} \]

   7. Taylor expanded in x around 0 41.1%

      \[\leadsto \color{blue}{\sqrt{y}} \]

   if -1.99999999999999999e-243 < (-.f64 x y)

   1. Initial program 100.0%

      \[\sqrt{\left|x - y\right|} \]
   2. Step-by-step derivation

      1. fabs-sub 100.0%

         \[\leadsto \sqrt{\color{blue}{\left|y - x\right|}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sqrt{\left|y - x\right|}} \]
   4. Step-by-step derivation

      1. pow1/2 100.0%

         \[\leadsto \color{blue}{{\left(\left|y - x\right|\right)}^{0.5}} \]

      2. add-cube-cbrt 98.8%

         \[\leadsto {\left(\left|\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \sqrt[3]{y - x}}\right|\right)}^{0.5} \]

      3. fabs-mul 98.8%

         \[\leadsto {\color{blue}{\left(\left|\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right| \cdot \left|\sqrt[3]{y - x}\right|\right)}}^{0.5} \]

      4. unpow-prod-down 98.8%

         \[\leadsto \color{blue}{{\left(\left|\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right|\right)}^{0.5} \cdot {\left(\left|\sqrt[3]{y - x}\right|\right)}^{0.5}} \]

      5. pow2 98.8%

         \[\leadsto {\left(\left|\color{blue}{{\left(\sqrt[3]{y - x}\right)}^{2}}\right|\right)}^{0.5} \cdot {\left(\left|\sqrt[3]{y - x}\right|\right)}^{0.5} \]

   5. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{{\left(\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right|\right)}^{0.5} \cdot {\left(\left|\sqrt[3]{y - x}\right|\right)}^{0.5}} \]

   6. Taylor expanded in y around 0 0.6%

      \[\leadsto \color{blue}{\sqrt{\left|{\left(y - x\right)}^{0.3333333333333333}\right| \cdot \left|{\left({\left(y - x\right)}^{0.3333333333333333}\right)}^{2}\right|}} \]
   7. Step-by-step derivation

      1. unpow1/3 0.6%

         \[\leadsto \sqrt{\left|\color{blue}{\sqrt[3]{y - x}}\right| \cdot \left|{\left({\left(y - x\right)}^{0.3333333333333333}\right)}^{2}\right|} \]

      2. sub-neg 0.6%

         \[\leadsto \sqrt{\left|\sqrt[3]{y - x}\right| \cdot \left|{\left({\color{blue}{\left(y + \left(-x\right)\right)}}^{0.3333333333333333}\right)}^{2}\right|} \]

      3. mul-1-neg 0.6%

         \[\leadsto \sqrt{\left|\sqrt[3]{y - x}\right| \cdot \left|{\left({\left(y + \color{blue}{-1 \cdot x}\right)}^{0.3333333333333333}\right)}^{2}\right|} \]

      4. mul-1-neg 0.6%

         \[\leadsto \sqrt{\left|\sqrt[3]{y - x}\right| \cdot \left|{\left({\left(y + \color{blue}{\left(-x\right)}\right)}^{0.3333333333333333}\right)}^{2}\right|} \]

      5. sub-neg 0.6%

         \[\leadsto \sqrt{\left|\sqrt[3]{y - x}\right| \cdot \left|{\left({\color{blue}{\left(y - x\right)}}^{0.3333333333333333}\right)}^{2}\right|} \]

      6. unpow1/3 98.7%

         \[\leadsto \sqrt{\left|\sqrt[3]{y - x}\right| \cdot \left|{\color{blue}{\left(\sqrt[3]{y - x}\right)}}^{2}\right|} \]

   8. Simplified 98.7%

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

      1. rem-cube-cbrt 98.3%

         \[\leadsto \sqrt{\left|\color{blue}{{\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{3}}\right| \cdot \left|{\left(\sqrt[3]{y - x}\right)}^{2}\right|} \]

      2. sqr-pow 0.7%

         \[\leadsto \sqrt{\left|\color{blue}{{\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)}}\right| \cdot \left|{\left(\sqrt[3]{y - x}\right)}^{2}\right|} \]

      3. fabs-sqr 0.7%

         \[\leadsto \sqrt{\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)}\right)} \cdot \left|{\left(\sqrt[3]{y - x}\right)}^{2}\right|} \]

      4. unpow2 0.7%

         \[\leadsto \sqrt{\left({\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)}\right) \cdot \left|\color{blue}{\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}}\right|} \]

      5. fabs-sqr 0.7%

         \[\leadsto \sqrt{\left({\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)}\right) \cdot \color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right)}} \]

      6. pow-prod-down 98.3%

         \[\leadsto \sqrt{\color{blue}{{\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right)} \]

      7. add-cube-cbrt 97.7%

         \[\leadsto \sqrt{{\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\sqrt[3]{y - x} \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right) \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)}\right)} \]

      8. add-cube-cbrt 97.3%

         \[\leadsto \sqrt{{\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right) \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)} \cdot \left(\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right) \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)\right)} \]

      9. swap-sqr 97.3%

         \[\leadsto \sqrt{{\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot \color{blue}{\left(\left(\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)\right)}} \]

   10. Applied egg-rr 97.3%

       \[\leadsto \sqrt{\color{blue}{{\left({\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{2}\right)}^{4.5}}} \]

   11. Taylor expanded in y around 0 99.3%

       \[\leadsto \sqrt{\color{blue}{x + -1 \cdot y}} \]
   12. Step-by-step derivation

       1. mul-1-neg 99.3%

          \[\leadsto \sqrt{x + \color{blue}{\left(-y\right)}} \]

       2. unsub-neg 99.3%

          \[\leadsto \sqrt{\color{blue}{x - y}} \]

   13. Simplified 99.3%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - y \leq -2 \cdot 10^{-243}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x - y}\\ \end{array} \]

Alternative 3: 99.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - y \leq -5 \cdot 10^{-251}:\\ \;\;\;\;\sqrt{y - x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- x y) -5e-251) (sqrt (- y x)) (sqrt (- x y))))

C

double code(double x, double y) {
	double tmp;
	if ((x - y) <= -5e-251) {
		tmp = sqrt((y - x));
	} else {
		tmp = sqrt((x - y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x - y) <= (-5d-251)) then
        tmp = sqrt((y - x))
    else
        tmp = sqrt((x - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x - y) <= -5e-251) {
		tmp = Math.sqrt((y - x));
	} else {
		tmp = Math.sqrt((x - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x - y) <= -5e-251:
		tmp = math.sqrt((y - x))
	else:
		tmp = math.sqrt((x - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x - y) <= -5e-251)
		tmp = sqrt(Float64(y - x));
	else
		tmp = sqrt(Float64(x - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x - y) <= -5e-251)
		tmp = sqrt((y - x));
	else
		tmp = sqrt((x - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x - y), $MachinePrecision], -5e-251], N[Sqrt[N[(y - x), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(x - y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x y) < -5.0000000000000003e-251

   1. Initial program 100.0%

      \[\sqrt{\left|x - y\right|} \]
   2. Step-by-step derivation

      1. fabs-sub 100.0%

         \[\leadsto \sqrt{\color{blue}{\left|y - x\right|}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sqrt{\left|y - x\right|}} \]

   4. Taylor expanded in y around -inf 100.0%

      \[\leadsto \color{blue}{\sqrt{\left|-\left(x + -1 \cdot y\right)\right|}} \]
   5. Step-by-step derivation

      1. fabs-neg 100.0%

         \[\leadsto \sqrt{\color{blue}{\left|x + -1 \cdot y\right|}} \]

      2. mul-1-neg 100.0%

         \[\leadsto \sqrt{\left|x + \color{blue}{\left(-y\right)}\right|} \]

      3. unsub-neg 100.0%

         \[\leadsto \sqrt{\left|\color{blue}{x - y}\right|} \]

      4. fabs-sub 100.0%

         \[\leadsto \sqrt{\color{blue}{\left|y - x\right|}} \]

      5. rem-square-sqrt 100.0%

         \[\leadsto \sqrt{\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right|} \]

      6. fabs-sqr 100.0%

         \[\leadsto \sqrt{\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}} \]

      7. rem-sqrt-square 100.0%

         \[\leadsto \color{blue}{\left|\sqrt{y - x}\right|} \]

      8. unpow1 100.0%

         \[\leadsto \left|\color{blue}{{\left(\sqrt{y - x}\right)}^{1}}\right| \]

      9. sqr-pow 99.4%

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

      10. fabs-sqr 99.4%

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

      11. sqr-pow 100.0%

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

      12. unpow1 100.0%

          \[\leadsto \color{blue}{\sqrt{y - x}} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\sqrt{y - x}} \]

   if -5.0000000000000003e-251 < (-.f64 x y)

   1. Initial program 100.0%

      \[\sqrt{\left|x - y\right|} \]
   2. Step-by-step derivation

      1. fabs-sub 100.0%

         \[\leadsto \sqrt{\color{blue}{\left|y - x\right|}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sqrt{\left|y - x\right|}} \]
   4. Step-by-step derivation

      1. pow1/2 100.0%

         \[\leadsto \color{blue}{{\left(\left|y - x\right|\right)}^{0.5}} \]

      2. add-cube-cbrt 98.8%

         \[\leadsto {\left(\left|\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \sqrt[3]{y - x}}\right|\right)}^{0.5} \]

      3. fabs-mul 98.8%

         \[\leadsto {\color{blue}{\left(\left|\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right| \cdot \left|\sqrt[3]{y - x}\right|\right)}}^{0.5} \]

      4. unpow-prod-down 98.8%

         \[\leadsto \color{blue}{{\left(\left|\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right|\right)}^{0.5} \cdot {\left(\left|\sqrt[3]{y - x}\right|\right)}^{0.5}} \]

      5. pow2 98.8%

         \[\leadsto {\left(\left|\color{blue}{{\left(\sqrt[3]{y - x}\right)}^{2}}\right|\right)}^{0.5} \cdot {\left(\left|\sqrt[3]{y - x}\right|\right)}^{0.5} \]

   5. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{{\left(\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right|\right)}^{0.5} \cdot {\left(\left|\sqrt[3]{y - x}\right|\right)}^{0.5}} \]

   6. Taylor expanded in y around 0 0.0%

      \[\leadsto \color{blue}{\sqrt{\left|{\left(y - x\right)}^{0.3333333333333333}\right| \cdot \left|{\left({\left(y - x\right)}^{0.3333333333333333}\right)}^{2}\right|}} \]
   7. Step-by-step derivation

      1. unpow1/3 0.0%

         \[\leadsto \sqrt{\left|\color{blue}{\sqrt[3]{y - x}}\right| \cdot \left|{\left({\left(y - x\right)}^{0.3333333333333333}\right)}^{2}\right|} \]

      2. sub-neg 0.0%

         \[\leadsto \sqrt{\left|\sqrt[3]{y - x}\right| \cdot \left|{\left({\color{blue}{\left(y + \left(-x\right)\right)}}^{0.3333333333333333}\right)}^{2}\right|} \]

      3. mul-1-neg 0.0%

         \[\leadsto \sqrt{\left|\sqrt[3]{y - x}\right| \cdot \left|{\left({\left(y + \color{blue}{-1 \cdot x}\right)}^{0.3333333333333333}\right)}^{2}\right|} \]

      4. mul-1-neg 0.0%

         \[\leadsto \sqrt{\left|\sqrt[3]{y - x}\right| \cdot \left|{\left({\left(y + \color{blue}{\left(-x\right)}\right)}^{0.3333333333333333}\right)}^{2}\right|} \]

      5. sub-neg 0.0%

         \[\leadsto \sqrt{\left|\sqrt[3]{y - x}\right| \cdot \left|{\left({\color{blue}{\left(y - x\right)}}^{0.3333333333333333}\right)}^{2}\right|} \]

      6. unpow1/3 98.8%

         \[\leadsto \sqrt{\left|\sqrt[3]{y - x}\right| \cdot \left|{\color{blue}{\left(\sqrt[3]{y - x}\right)}}^{2}\right|} \]

   8. Simplified 98.8%

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

      1. rem-cube-cbrt 98.3%

         \[\leadsto \sqrt{\left|\color{blue}{{\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{3}}\right| \cdot \left|{\left(\sqrt[3]{y - x}\right)}^{2}\right|} \]

      2. sqr-pow 0.0%

         \[\leadsto \sqrt{\left|\color{blue}{{\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)}}\right| \cdot \left|{\left(\sqrt[3]{y - x}\right)}^{2}\right|} \]

      3. fabs-sqr 0.0%

         \[\leadsto \sqrt{\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)}\right)} \cdot \left|{\left(\sqrt[3]{y - x}\right)}^{2}\right|} \]

      4. unpow2 0.0%

         \[\leadsto \sqrt{\left({\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)}\right) \cdot \left|\color{blue}{\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}}\right|} \]

      5. fabs-sqr 0.0%

         \[\leadsto \sqrt{\left({\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)}\right) \cdot \color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right)}} \]

      6. pow-prod-down 98.3%

         \[\leadsto \sqrt{\color{blue}{{\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right)} \]

      7. add-cube-cbrt 97.7%

         \[\leadsto \sqrt{{\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\sqrt[3]{y - x} \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right) \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)}\right)} \]

      8. add-cube-cbrt 97.3%

         \[\leadsto \sqrt{{\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right) \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)} \cdot \left(\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right) \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)\right)} \]

      9. swap-sqr 97.3%

         \[\leadsto \sqrt{{\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot \color{blue}{\left(\left(\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)\right)}} \]

   10. Applied egg-rr 97.3%

       \[\leadsto \sqrt{\color{blue}{{\left({\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{2}\right)}^{4.5}}} \]

   11. Taylor expanded in y around 0 100.0%

       \[\leadsto \sqrt{\color{blue}{x + -1 \cdot y}} \]
   12. Step-by-step derivation

       1. mul-1-neg 100.0%

          \[\leadsto \sqrt{x + \color{blue}{\left(-y\right)}} \]

       2. unsub-neg 100.0%

          \[\leadsto \sqrt{\color{blue}{x - y}} \]

   13. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - y \leq -5 \cdot 10^{-251}:\\ \;\;\;\;\sqrt{y - x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x - y}\\ \end{array} \]

Alternative 4: 54.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.25 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{-y}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.25e-45) (sqrt (- y)) (if (<= y 1.7e+97) (sqrt x) (sqrt y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.25e-45) {
		tmp = sqrt(-y);
	} else if (y <= 1.7e+97) {
		tmp = sqrt(x);
	} else {
		tmp = sqrt(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.25d-45)) then
        tmp = sqrt(-y)
    else if (y <= 1.7d+97) then
        tmp = sqrt(x)
    else
        tmp = sqrt(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.25e-45) {
		tmp = Math.sqrt(-y);
	} else if (y <= 1.7e+97) {
		tmp = Math.sqrt(x);
	} else {
		tmp = Math.sqrt(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.25e-45:
		tmp = math.sqrt(-y)
	elif y <= 1.7e+97:
		tmp = math.sqrt(x)
	else:
		tmp = math.sqrt(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.25e-45)
		tmp = sqrt(Float64(-y));
	elseif (y <= 1.7e+97)
		tmp = sqrt(x);
	else
		tmp = sqrt(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.25e-45)
		tmp = sqrt(-y);
	elseif (y <= 1.7e+97)
		tmp = sqrt(x);
	else
		tmp = sqrt(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.25e-45], N[Sqrt[(-y)], $MachinePrecision], If[LessEqual[y, 1.7e+97], N[Sqrt[x], $MachinePrecision], N[Sqrt[y], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.2499999999999999e-45

   1. Initial program 100.0%

      \[\sqrt{\left|x - y\right|} \]
   2. Step-by-step derivation

      1. fabs-sub 100.0%

         \[\leadsto \sqrt{\color{blue}{\left|y - x\right|}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sqrt{\left|y - x\right|}} \]
   4. Step-by-step derivation

      1. pow1/2 100.0%

         \[\leadsto \color{blue}{{\left(\left|y - x\right|\right)}^{0.5}} \]

      2. add-cube-cbrt 98.8%

         \[\leadsto {\left(\left|\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \sqrt[3]{y - x}}\right|\right)}^{0.5} \]

      3. fabs-mul 98.8%

         \[\leadsto {\color{blue}{\left(\left|\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right| \cdot \left|\sqrt[3]{y - x}\right|\right)}}^{0.5} \]

      4. unpow-prod-down 98.8%

         \[\leadsto \color{blue}{{\left(\left|\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right|\right)}^{0.5} \cdot {\left(\left|\sqrt[3]{y - x}\right|\right)}^{0.5}} \]

      5. pow2 98.8%

         \[\leadsto {\left(\left|\color{blue}{{\left(\sqrt[3]{y - x}\right)}^{2}}\right|\right)}^{0.5} \cdot {\left(\left|\sqrt[3]{y - x}\right|\right)}^{0.5} \]

   5. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{{\left(\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right|\right)}^{0.5} \cdot {\left(\left|\sqrt[3]{y - x}\right|\right)}^{0.5}} \]

   6. Taylor expanded in y around 0 18.6%

      \[\leadsto \color{blue}{\sqrt{\left|{\left(y - x\right)}^{0.3333333333333333}\right| \cdot \left|{\left({\left(y - x\right)}^{0.3333333333333333}\right)}^{2}\right|}} \]
   7. Step-by-step derivation

      1. unpow1/3 18.8%

         \[\leadsto \sqrt{\left|\color{blue}{\sqrt[3]{y - x}}\right| \cdot \left|{\left({\left(y - x\right)}^{0.3333333333333333}\right)}^{2}\right|} \]

      2. sub-neg 18.8%

         \[\leadsto \sqrt{\left|\sqrt[3]{y - x}\right| \cdot \left|{\left({\color{blue}{\left(y + \left(-x\right)\right)}}^{0.3333333333333333}\right)}^{2}\right|} \]

      3. mul-1-neg 18.8%

         \[\leadsto \sqrt{\left|\sqrt[3]{y - x}\right| \cdot \left|{\left({\left(y + \color{blue}{-1 \cdot x}\right)}^{0.3333333333333333}\right)}^{2}\right|} \]

      4. mul-1-neg 18.8%

         \[\leadsto \sqrt{\left|\sqrt[3]{y - x}\right| \cdot \left|{\left({\left(y + \color{blue}{\left(-x\right)}\right)}^{0.3333333333333333}\right)}^{2}\right|} \]

      5. sub-neg 18.8%

         \[\leadsto \sqrt{\left|\sqrt[3]{y - x}\right| \cdot \left|{\left({\color{blue}{\left(y - x\right)}}^{0.3333333333333333}\right)}^{2}\right|} \]

      6. unpow1/3 98.8%

         \[\leadsto \sqrt{\left|\sqrt[3]{y - x}\right| \cdot \left|{\color{blue}{\left(\sqrt[3]{y - x}\right)}}^{2}\right|} \]

   8. Simplified 98.8%

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

      1. rem-cube-cbrt 98.5%

         \[\leadsto \sqrt{\left|\color{blue}{{\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{3}}\right| \cdot \left|{\left(\sqrt[3]{y - x}\right)}^{2}\right|} \]

      2. sqr-pow 20.3%

         \[\leadsto \sqrt{\left|\color{blue}{{\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)}}\right| \cdot \left|{\left(\sqrt[3]{y - x}\right)}^{2}\right|} \]

      3. fabs-sqr 20.3%

         \[\leadsto \sqrt{\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)}\right)} \cdot \left|{\left(\sqrt[3]{y - x}\right)}^{2}\right|} \]

      4. unpow2 20.3%

         \[\leadsto \sqrt{\left({\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)}\right) \cdot \left|\color{blue}{\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}}\right|} \]

      5. fabs-sqr 20.3%

         \[\leadsto \sqrt{\left({\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)}\right) \cdot \color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right)}} \]

      6. pow-prod-down 98.5%

         \[\leadsto \sqrt{\color{blue}{{\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right)} \]

      7. add-cube-cbrt 97.8%

         \[\leadsto \sqrt{{\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\sqrt[3]{y - x} \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right) \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)}\right)} \]

      8. add-cube-cbrt 97.3%

         \[\leadsto \sqrt{{\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right) \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)} \cdot \left(\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right) \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)\right)} \]

      9. swap-sqr 97.2%

         \[\leadsto \sqrt{{\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot \color{blue}{\left(\left(\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)\right)}} \]

   10. Applied egg-rr 97.3%

       \[\leadsto \sqrt{\color{blue}{{\left({\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{2}\right)}^{4.5}}} \]

   11. Taylor expanded in y around -inf 66.9%

       \[\leadsto \sqrt{\color{blue}{-1 \cdot y}} \]
   12. Step-by-step derivation

       1. mul-1-neg 66.9%

          \[\leadsto \sqrt{\color{blue}{-y}} \]

   13. Simplified 66.9%

       \[\leadsto \sqrt{\color{blue}{-y}} \]

   if -2.2499999999999999e-45 < y < 1.70000000000000005e97

   1. Initial program 100.0%

      \[\sqrt{\left|x - y\right|} \]
   2. Step-by-step derivation

      1. fabs-sub 100.0%

         \[\leadsto \sqrt{\color{blue}{\left|y - x\right|}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sqrt{\left|y - x\right|}} \]
   4. Step-by-step derivation

      1. pow1/2 100.0%

         \[\leadsto \color{blue}{{\left(\left|y - x\right|\right)}^{0.5}} \]

      2. add-cube-cbrt 98.7%

         \[\leadsto {\left(\left|\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \sqrt[3]{y - x}}\right|\right)}^{0.5} \]

      3. fabs-mul 98.7%

         \[\leadsto {\color{blue}{\left(\left|\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right| \cdot \left|\sqrt[3]{y - x}\right|\right)}}^{0.5} \]

      4. unpow-prod-down 98.7%

         \[\leadsto \color{blue}{{\left(\left|\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right|\right)}^{0.5} \cdot {\left(\left|\sqrt[3]{y - x}\right|\right)}^{0.5}} \]

      5. pow2 98.7%

         \[\leadsto {\left(\left|\color{blue}{{\left(\sqrt[3]{y - x}\right)}^{2}}\right|\right)}^{0.5} \cdot {\left(\left|\sqrt[3]{y - x}\right|\right)}^{0.5} \]

   5. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{{\left(\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right|\right)}^{0.5} \cdot {\left(\left|\sqrt[3]{y - x}\right|\right)}^{0.5}} \]

   6. Taylor expanded in y around 0 42.1%

      \[\leadsto \color{blue}{\sqrt{\left|{\left(y - x\right)}^{0.3333333333333333}\right| \cdot \left|{\left({\left(y - x\right)}^{0.3333333333333333}\right)}^{2}\right|}} \]
   7. Step-by-step derivation

      1. unpow1/3 42.4%

         \[\leadsto \sqrt{\left|\color{blue}{\sqrt[3]{y - x}}\right| \cdot \left|{\left({\left(y - x\right)}^{0.3333333333333333}\right)}^{2}\right|} \]

      2. sub-neg 42.4%

         \[\leadsto \sqrt{\left|\sqrt[3]{y - x}\right| \cdot \left|{\left({\color{blue}{\left(y + \left(-x\right)\right)}}^{0.3333333333333333}\right)}^{2}\right|} \]

      3. mul-1-neg 42.4%

         \[\leadsto \sqrt{\left|\sqrt[3]{y - x}\right| \cdot \left|{\left({\left(y + \color{blue}{-1 \cdot x}\right)}^{0.3333333333333333}\right)}^{2}\right|} \]

      4. mul-1-neg 42.4%

         \[\leadsto \sqrt{\left|\sqrt[3]{y - x}\right| \cdot \left|{\left({\left(y + \color{blue}{\left(-x\right)}\right)}^{0.3333333333333333}\right)}^{2}\right|} \]

      5. sub-neg 42.4%

         \[\leadsto \sqrt{\left|\sqrt[3]{y - x}\right| \cdot \left|{\left({\color{blue}{\left(y - x\right)}}^{0.3333333333333333}\right)}^{2}\right|} \]

      6. unpow1/3 98.7%

         \[\leadsto \sqrt{\left|\sqrt[3]{y - x}\right| \cdot \left|{\color{blue}{\left(\sqrt[3]{y - x}\right)}}^{2}\right|} \]

   8. Simplified 98.7%

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

      1. rem-cube-cbrt 98.3%

         \[\leadsto \sqrt{\left|\color{blue}{{\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{3}}\right| \cdot \left|{\left(\sqrt[3]{y - x}\right)}^{2}\right|} \]

      2. sqr-pow 45.1%

         \[\leadsto \sqrt{\left|\color{blue}{{\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)}}\right| \cdot \left|{\left(\sqrt[3]{y - x}\right)}^{2}\right|} \]

      3. fabs-sqr 45.1%

         \[\leadsto \sqrt{\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)}\right)} \cdot \left|{\left(\sqrt[3]{y - x}\right)}^{2}\right|} \]

      4. unpow2 45.1%

         \[\leadsto \sqrt{\left({\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)}\right) \cdot \left|\color{blue}{\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}}\right|} \]

      5. fabs-sqr 45.1%

         \[\leadsto \sqrt{\left({\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)}\right) \cdot \color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right)}} \]

      6. pow-prod-down 98.2%

         \[\leadsto \sqrt{\color{blue}{{\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right)} \]

      7. add-cube-cbrt 97.7%

         \[\leadsto \sqrt{{\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\sqrt[3]{y - x} \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right) \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)}\right)} \]

      8. add-cube-cbrt 97.3%

         \[\leadsto \sqrt{{\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right) \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)} \cdot \left(\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right) \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)\right)} \]

      9. swap-sqr 97.3%

         \[\leadsto \sqrt{{\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot \color{blue}{\left(\left(\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)\right)}} \]

   10. Applied egg-rr 97.3%

       \[\leadsto \sqrt{\color{blue}{{\left({\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{2}\right)}^{4.5}}} \]

   11. Taylor expanded in y around 0 48.4%

       \[\leadsto \sqrt{\color{blue}{x}} \]

   if 1.70000000000000005e97 < y

   1. Initial program 100.0%

      \[\sqrt{\left|x - y\right|} \]
   2. Step-by-step derivation

      1. fabs-sub 100.0%

         \[\leadsto \sqrt{\color{blue}{\left|y - x\right|}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sqrt{\left|y - x\right|}} \]

   4. Taylor expanded in y around -inf 100.0%

      \[\leadsto \color{blue}{\sqrt{\left|-\left(x + -1 \cdot y\right)\right|}} \]
   5. Step-by-step derivation

      1. fabs-neg 100.0%

         \[\leadsto \sqrt{\color{blue}{\left|x + -1 \cdot y\right|}} \]

      2. mul-1-neg 100.0%

         \[\leadsto \sqrt{\left|x + \color{blue}{\left(-y\right)}\right|} \]

      3. unsub-neg 100.0%

         \[\leadsto \sqrt{\left|\color{blue}{x - y}\right|} \]

      4. fabs-sub 100.0%

         \[\leadsto \sqrt{\color{blue}{\left|y - x\right|}} \]

      5. rem-square-sqrt 90.2%

         \[\leadsto \sqrt{\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right|} \]

      6. fabs-sqr 90.2%

         \[\leadsto \sqrt{\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}} \]

      7. rem-sqrt-square 90.2%

         \[\leadsto \color{blue}{\left|\sqrt{y - x}\right|} \]

      8. unpow1 90.2%

         \[\leadsto \left|\color{blue}{{\left(\sqrt{y - x}\right)}^{1}}\right| \]

      9. sqr-pow 89.7%

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

      10. fabs-sqr 89.7%

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

      11. sqr-pow 90.2%

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

      12. unpow1 90.2%

          \[\leadsto \color{blue}{\sqrt{y - x}} \]

   6. Simplified 90.2%

      \[\leadsto \color{blue}{\sqrt{y - x}} \]

   7. Taylor expanded in x around 0 73.7%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.25 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{-y}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y}\\ \end{array} \]

Alternative 5: 41.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.55 \cdot 10^{-248}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 2.55e-248) (sqrt y) (sqrt x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.55e-248) {
		tmp = sqrt(y);
	} else {
		tmp = sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 2.55d-248) then
        tmp = sqrt(y)
    else
        tmp = sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2.55e-248) {
		tmp = Math.sqrt(y);
	} else {
		tmp = Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.55e-248:
		tmp = math.sqrt(y)
	else:
		tmp = math.sqrt(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.55e-248)
		tmp = sqrt(y);
	else
		tmp = sqrt(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.55e-248)
		tmp = sqrt(y);
	else
		tmp = sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.55e-248], N[Sqrt[y], $MachinePrecision], N[Sqrt[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.54999999999999986e-248

   1. Initial program 100.0%

      \[\sqrt{\left|x - y\right|} \]
   2. Step-by-step derivation

      1. fabs-sub 100.0%

         \[\leadsto \sqrt{\color{blue}{\left|y - x\right|}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sqrt{\left|y - x\right|}} \]

   4. Taylor expanded in y around -inf 100.0%

      \[\leadsto \color{blue}{\sqrt{\left|-\left(x + -1 \cdot y\right)\right|}} \]
   5. Step-by-step derivation

      1. fabs-neg 100.0%

         \[\leadsto \sqrt{\color{blue}{\left|x + -1 \cdot y\right|}} \]

      2. mul-1-neg 100.0%

         \[\leadsto \sqrt{\left|x + \color{blue}{\left(-y\right)}\right|} \]

      3. unsub-neg 100.0%

         \[\leadsto \sqrt{\left|\color{blue}{x - y}\right|} \]

      4. fabs-sub 100.0%

         \[\leadsto \sqrt{\color{blue}{\left|y - x\right|}} \]

      5. rem-square-sqrt 77.7%

         \[\leadsto \sqrt{\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right|} \]

      6. fabs-sqr 77.7%

         \[\leadsto \sqrt{\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}} \]

      7. rem-sqrt-square 77.7%

         \[\leadsto \color{blue}{\left|\sqrt{y - x}\right|} \]

      8. unpow1 77.7%

         \[\leadsto \left|\color{blue}{{\left(\sqrt{y - x}\right)}^{1}}\right| \]

      9. sqr-pow 77.2%

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

      10. fabs-sqr 77.2%

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

      11. sqr-pow 77.7%

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

      12. unpow1 77.7%

          \[\leadsto \color{blue}{\sqrt{y - x}} \]

   6. Simplified 77.7%

      \[\leadsto \color{blue}{\sqrt{y - x}} \]

   7. Taylor expanded in x around 0 24.7%

      \[\leadsto \color{blue}{\sqrt{y}} \]

   if 2.54999999999999986e-248 < x

   1. Initial program 100.0%

      \[\sqrt{\left|x - y\right|} \]
   2. Step-by-step derivation

      1. fabs-sub 100.0%

         \[\leadsto \sqrt{\color{blue}{\left|y - x\right|}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sqrt{\left|y - x\right|}} \]
   4. Step-by-step derivation

      1. pow1/2 100.0%

         \[\leadsto \color{blue}{{\left(\left|y - x\right|\right)}^{0.5}} \]

      2. add-cube-cbrt 98.8%

         \[\leadsto {\left(\left|\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \sqrt[3]{y - x}}\right|\right)}^{0.5} \]

      3. fabs-mul 98.8%

         \[\leadsto {\color{blue}{\left(\left|\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right| \cdot \left|\sqrt[3]{y - x}\right|\right)}}^{0.5} \]

      4. unpow-prod-down 98.8%

         \[\leadsto \color{blue}{{\left(\left|\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right|\right)}^{0.5} \cdot {\left(\left|\sqrt[3]{y - x}\right|\right)}^{0.5}} \]

      5. pow2 98.8%

         \[\leadsto {\left(\left|\color{blue}{{\left(\sqrt[3]{y - x}\right)}^{2}}\right|\right)}^{0.5} \cdot {\left(\left|\sqrt[3]{y - x}\right|\right)}^{0.5} \]

   5. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{{\left(\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right|\right)}^{0.5} \cdot {\left(\left|\sqrt[3]{y - x}\right|\right)}^{0.5}} \]

   6. Taylor expanded in y around 0 11.5%

      \[\leadsto \color{blue}{\sqrt{\left|{\left(y - x\right)}^{0.3333333333333333}\right| \cdot \left|{\left({\left(y - x\right)}^{0.3333333333333333}\right)}^{2}\right|}} \]
   7. Step-by-step derivation

      1. unpow1/3 11.6%

         \[\leadsto \sqrt{\left|\color{blue}{\sqrt[3]{y - x}}\right| \cdot \left|{\left({\left(y - x\right)}^{0.3333333333333333}\right)}^{2}\right|} \]

      2. sub-neg 11.6%

         \[\leadsto \sqrt{\left|\sqrt[3]{y - x}\right| \cdot \left|{\left({\color{blue}{\left(y + \left(-x\right)\right)}}^{0.3333333333333333}\right)}^{2}\right|} \]

      3. mul-1-neg 11.6%

         \[\leadsto \sqrt{\left|\sqrt[3]{y - x}\right| \cdot \left|{\left({\left(y + \color{blue}{-1 \cdot x}\right)}^{0.3333333333333333}\right)}^{2}\right|} \]

      4. mul-1-neg 11.6%

         \[\leadsto \sqrt{\left|\sqrt[3]{y - x}\right| \cdot \left|{\left({\left(y + \color{blue}{\left(-x\right)}\right)}^{0.3333333333333333}\right)}^{2}\right|} \]

      5. sub-neg 11.6%

         \[\leadsto \sqrt{\left|\sqrt[3]{y - x}\right| \cdot \left|{\left({\color{blue}{\left(y - x\right)}}^{0.3333333333333333}\right)}^{2}\right|} \]

      6. unpow1/3 98.8%

         \[\leadsto \sqrt{\left|\sqrt[3]{y - x}\right| \cdot \left|{\color{blue}{\left(\sqrt[3]{y - x}\right)}}^{2}\right|} \]

   8. Simplified 98.8%

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

      1. rem-cube-cbrt 98.4%

         \[\leadsto \sqrt{\left|\color{blue}{{\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{3}}\right| \cdot \left|{\left(\sqrt[3]{y - x}\right)}^{2}\right|} \]

      2. sqr-pow 12.5%

         \[\leadsto \sqrt{\left|\color{blue}{{\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)}}\right| \cdot \left|{\left(\sqrt[3]{y - x}\right)}^{2}\right|} \]

      3. fabs-sqr 12.5%

         \[\leadsto \sqrt{\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)}\right)} \cdot \left|{\left(\sqrt[3]{y - x}\right)}^{2}\right|} \]

      4. unpow2 12.5%

         \[\leadsto \sqrt{\left({\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)}\right) \cdot \left|\color{blue}{\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}}\right|} \]

      5. fabs-sqr 12.5%

         \[\leadsto \sqrt{\left({\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)}\right) \cdot \color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right)}} \]

      6. pow-prod-down 98.3%

         \[\leadsto \sqrt{\color{blue}{{\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right)} \]

      7. add-cube-cbrt 97.8%

         \[\leadsto \sqrt{{\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\sqrt[3]{y - x} \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right) \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)}\right)} \]

      8. add-cube-cbrt 97.3%

         \[\leadsto \sqrt{{\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right) \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)} \cdot \left(\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right) \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)\right)} \]

      9. swap-sqr 97.3%

         \[\leadsto \sqrt{{\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot \color{blue}{\left(\left(\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)\right)}} \]

   10. Applied egg-rr 97.3%

       \[\leadsto \sqrt{\color{blue}{{\left({\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{2}\right)}^{4.5}}} \]

   11. Taylor expanded in y around 0 66.6%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.55 \cdot 10^{-248}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x}\\ \end{array} \]

Alternative 6: 26.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (sqrt x))

C

double code(double x, double y) {
	return sqrt(x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt(x)
end function

Java

public static double code(double x, double y) {
	return Math.sqrt(x);
}

Python

def code(x, y):
	return math.sqrt(x)

Julia

function code(x, y)
	return sqrt(x)
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt(x);
end

Wolfram

code[x_, y_] := N[Sqrt[x], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\sqrt{\left|x - y\right|} \]
2. Step-by-step derivation

   1. fabs-sub 100.0%

      \[\leadsto \sqrt{\color{blue}{\left|y - x\right|}} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\sqrt{\left|y - x\right|}} \]
4. Step-by-step derivation

   1. pow1/2 100.0%

      \[\leadsto \color{blue}{{\left(\left|y - x\right|\right)}^{0.5}} \]

   2. add-cube-cbrt 98.7%

      \[\leadsto {\left(\left|\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \sqrt[3]{y - x}}\right|\right)}^{0.5} \]

   3. fabs-mul 98.7%

      \[\leadsto {\color{blue}{\left(\left|\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right| \cdot \left|\sqrt[3]{y - x}\right|\right)}}^{0.5} \]

   4. unpow-prod-down 98.8%

      \[\leadsto \color{blue}{{\left(\left|\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right|\right)}^{0.5} \cdot {\left(\left|\sqrt[3]{y - x}\right|\right)}^{0.5}} \]

   5. pow2 98.8%

      \[\leadsto {\left(\left|\color{blue}{{\left(\sqrt[3]{y - x}\right)}^{2}}\right|\right)}^{0.5} \cdot {\left(\left|\sqrt[3]{y - x}\right|\right)}^{0.5} \]

5. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{{\left(\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right|\right)}^{0.5} \cdot {\left(\left|\sqrt[3]{y - x}\right|\right)}^{0.5}} \]

6. Taylor expanded in y around 0 41.7%

   \[\leadsto \color{blue}{\sqrt{\left|{\left(y - x\right)}^{0.3333333333333333}\right| \cdot \left|{\left({\left(y - x\right)}^{0.3333333333333333}\right)}^{2}\right|}} \]
7. Step-by-step derivation

   1. unpow1/3 42.1%

      \[\leadsto \sqrt{\left|\color{blue}{\sqrt[3]{y - x}}\right| \cdot \left|{\left({\left(y - x\right)}^{0.3333333333333333}\right)}^{2}\right|} \]

   2. sub-neg 42.1%

      \[\leadsto \sqrt{\left|\sqrt[3]{y - x}\right| \cdot \left|{\left({\color{blue}{\left(y + \left(-x\right)\right)}}^{0.3333333333333333}\right)}^{2}\right|} \]

   3. mul-1-neg 42.1%

      \[\leadsto \sqrt{\left|\sqrt[3]{y - x}\right| \cdot \left|{\left({\left(y + \color{blue}{-1 \cdot x}\right)}^{0.3333333333333333}\right)}^{2}\right|} \]

   4. mul-1-neg 42.1%

      \[\leadsto \sqrt{\left|\sqrt[3]{y - x}\right| \cdot \left|{\left({\left(y + \color{blue}{\left(-x\right)}\right)}^{0.3333333333333333}\right)}^{2}\right|} \]

   5. sub-neg 42.1%

      \[\leadsto \sqrt{\left|\sqrt[3]{y - x}\right| \cdot \left|{\left({\color{blue}{\left(y - x\right)}}^{0.3333333333333333}\right)}^{2}\right|} \]

   6. unpow1/3 98.7%

      \[\leadsto \sqrt{\left|\sqrt[3]{y - x}\right| \cdot \left|{\color{blue}{\left(\sqrt[3]{y - x}\right)}}^{2}\right|} \]

8. Simplified 98.7%

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

   1. rem-cube-cbrt 98.4%

      \[\leadsto \sqrt{\left|\color{blue}{{\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{3}}\right| \cdot \left|{\left(\sqrt[3]{y - x}\right)}^{2}\right|} \]

   2. sqr-pow 45.0%

      \[\leadsto \sqrt{\left|\color{blue}{{\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)}}\right| \cdot \left|{\left(\sqrt[3]{y - x}\right)}^{2}\right|} \]

   3. fabs-sqr 45.0%

      \[\leadsto \sqrt{\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)}\right)} \cdot \left|{\left(\sqrt[3]{y - x}\right)}^{2}\right|} \]

   4. unpow2 45.0%

      \[\leadsto \sqrt{\left({\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)}\right) \cdot \left|\color{blue}{\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}}\right|} \]

   5. fabs-sqr 45.0%

      \[\leadsto \sqrt{\left({\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)}\right) \cdot \color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right)}} \]

   6. pow-prod-down 98.3%

      \[\leadsto \sqrt{\color{blue}{{\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right)} \]

   7. add-cube-cbrt 97.8%

      \[\leadsto \sqrt{{\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\sqrt[3]{y - x} \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right) \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)}\right)} \]

   8. add-cube-cbrt 97.3%

      \[\leadsto \sqrt{{\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right) \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)} \cdot \left(\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right) \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)\right)} \]

   9. swap-sqr 97.3%

      \[\leadsto \sqrt{{\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)}^{\left(\frac{3}{2}\right)} \cdot \color{blue}{\left(\left(\left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{y - x}} \cdot \sqrt[3]{\sqrt[3]{y - x}}\right)\right)}} \]

10. Applied egg-rr 97.3%

    \[\leadsto \sqrt{\color{blue}{{\left({\left(\sqrt[3]{\sqrt[3]{y - x}}\right)}^{2}\right)}^{4.5}}} \]

11. Taylor expanded in y around 0 33.0%

    \[\leadsto \sqrt{\color{blue}{x}} \]

12. Final simplification 33.0%

    \[\leadsto \sqrt{x} \]

Reproduce

herbie shell --seed 2023318 
(FPCore (x y)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, C"
  :precision binary64
  (sqrt (fabs (- x y))))