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

Percentage Accurate: 100.0% → 100.0%

Time: 8.5s

Alternatives: 9

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. Add Preprocessing
3. Add Preprocessing

Alternative 2: 66.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{x - y}\\ \mathbf{if}\;y \leq 1.35 \cdot 10^{-151}:\\ \;\;\;\;\sqrt{\left|x\right|}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+48}:\\ \;\;\;\;\frac{1}{{\left(t\_0 \cdot t\_0\right)}^{0.25}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 1.0 (- x y))))
   (if (<= y 1.35e-151)
     (sqrt (fabs x))
     (if (<= y 2.6e+48) (/ 1.0 (pow (* t_0 t_0) 0.25)) (sqrt y)))))

C

double code(double x, double y) {
	double t_0 = 1.0 / (x - y);
	double tmp;
	if (y <= 1.35e-151) {
		tmp = sqrt(fabs(x));
	} else if (y <= 2.6e+48) {
		tmp = 1.0 / pow((t_0 * t_0), 0.25);
	} 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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / (x - y)
    if (y <= 1.35d-151) then
        tmp = sqrt(abs(x))
    else if (y <= 2.6d+48) then
        tmp = 1.0d0 / ((t_0 * t_0) ** 0.25d0)
    else
        tmp = sqrt(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 / (x - y);
	double tmp;
	if (y <= 1.35e-151) {
		tmp = Math.sqrt(Math.abs(x));
	} else if (y <= 2.6e+48) {
		tmp = 1.0 / Math.pow((t_0 * t_0), 0.25);
	} else {
		tmp = Math.sqrt(y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 / (x - y)
	tmp = 0
	if y <= 1.35e-151:
		tmp = math.sqrt(math.fabs(x))
	elif y <= 2.6e+48:
		tmp = 1.0 / math.pow((t_0 * t_0), 0.25)
	else:
		tmp = math.sqrt(y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 / Float64(x - y))
	tmp = 0.0
	if (y <= 1.35e-151)
		tmp = sqrt(abs(x));
	elseif (y <= 2.6e+48)
		tmp = Float64(1.0 / (Float64(t_0 * t_0) ^ 0.25));
	else
		tmp = sqrt(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 / (x - y);
	tmp = 0.0;
	if (y <= 1.35e-151)
		tmp = sqrt(abs(x));
	elseif (y <= 2.6e+48)
		tmp = 1.0 / ((t_0 * t_0) ^ 0.25);
	else
		tmp = sqrt(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 / N[(x - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.35e-151], N[Sqrt[N[Abs[x], $MachinePrecision]], $MachinePrecision], If[LessEqual[y, 2.6e+48], N[(1.0 / N[Power[N[(t$95$0 * t$95$0), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision], N[Sqrt[y], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.35000000000000004e-151

   1. Initial program 100.0%

      \[\sqrt{\left|x - y\right|} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\color{blue}{x}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 59.8%

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

   if 1.35000000000000004e-151 < y < 2.59999999999999995e48

   1. Initial program 100.0%

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

      1. flip-- N/A

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

      2. fabs-div N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. div-fabs N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt{\left|\frac{x + y}{x \cdot x - y \cdot y}\right|}\right)}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\left|\frac{x + y}{x \cdot x - y \cdot y}\right|\right)\right)\right) \]

      9. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\left(\frac{x + y}{x \cdot x - y \cdot y}\right)\right)\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\left(\frac{1}{\frac{x \cdot x - y \cdot y}{x + y}}\right)\right)\right)\right) \]

      11. flip-- N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\left(\frac{1}{x - y}\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\mathsf{/.f64}\left(1, \left(x - y\right)\right)\right)\right)\right) \]

      13. --lowering--.f64 99.6%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(x, y\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.6%

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

      1. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({\left(\left|\frac{1}{x - y}\right|\right)}^{\color{blue}{\frac{1}{2}}}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({\left(\left|\frac{1}{x - y}\right|\right)}^{\left(\frac{1}{4} + \color{blue}{\frac{1}{4}}\right)}\right)\right) \]

      3. pow-prod-up N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({\left(\left|\frac{1}{x - y}\right|\right)}^{\frac{1}{4}} \cdot \color{blue}{{\left(\left|\frac{1}{x - y}\right|\right)}^{\frac{1}{4}}}\right)\right) \]

      4. pow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({\left(\left|\frac{1}{x - y}\right| \cdot \left|\frac{1}{x - y}\right|\right)}^{\color{blue}{\frac{1}{4}}}\right)\right) \]

      5. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({\left(\left|{\left(x - y\right)}^{-1}\right| \cdot \left|\frac{1}{x - y}\right|\right)}^{\frac{1}{4}}\right)\right) \]

      6. sqr-pow N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({\left(\left|{\left(x - y\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(x - y\right)}^{\left(\frac{-1}{2}\right)}\right| \cdot \left|\frac{1}{x - y}\right|\right)}^{\frac{1}{4}}\right)\right) \]

      7. fabs-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({\left(\left({\left(x - y\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(x - y\right)}^{\left(\frac{-1}{2}\right)}\right) \cdot \left|\frac{1}{x - y}\right|\right)}^{\frac{1}{4}}\right)\right) \]

      8. sqr-pow N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({\left({\left(x - y\right)}^{-1} \cdot \left|\frac{1}{x - y}\right|\right)}^{\frac{1}{4}}\right)\right) \]

      9. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({\left(\frac{1}{x - y} \cdot \left|\frac{1}{x - y}\right|\right)}^{\frac{1}{4}}\right)\right) \]

      10. inv-pow N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left({\left(\frac{1}{x - y} \cdot \left|{\left(x - y\right)}^{-1}\right|\right)}^{\frac{1}{4}}\right)\right) \]

      11. sqr-pow N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left({\left(\frac{1}{x - y} \cdot \left|{\left(x - y\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(x - y\right)}^{\left(\frac{-1}{2}\right)}\right|\right)}^{\frac{1}{4}}\right)\right) \]

      12. fabs-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left({\left(\frac{1}{x - y} \cdot \left({\left(x - y\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(x - y\right)}^{\left(\frac{-1}{2}\right)}\right)\right)}^{\frac{1}{4}}\right)\right) \]

      13. sqr-pow N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left({\left(\frac{1}{x - y} \cdot {\left(x - y\right)}^{-1}\right)}^{\frac{1}{4}}\right)\right) \]

      14. inv-pow N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left({\left(\frac{1}{x - y} \cdot \frac{1}{x - y}\right)}^{\frac{1}{4}}\right)\right) \]

      15. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(\left(\frac{1}{x - y} \cdot \frac{1}{x - y}\right), \color{blue}{\frac{1}{4}}\right)\right) \]

   6. Applied egg-rr 81.1%

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

   if 2.59999999999999995e48 < y

   1. Initial program 100.0%

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

      1. pow1/2 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-down N/A

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

      4. sqr-abs N/A

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

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(x - y\right) \cdot \left(x - y\right)\right), \color{blue}{\left(\frac{\frac{1}{2}}{2}\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(x - y\right), \left(x - y\right)\right), \left(\frac{\color{blue}{\frac{1}{2}}}{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(x - y\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{\_.f64}\left(x, y\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

      9. metadata-eval 25.4%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{\_.f64}\left(x, y\right)\right), \frac{1}{4}\right) \]

   4. Applied egg-rr 25.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{pow.f64}\left(\color{blue}{\left({y}^{2}\right)}, \frac{1}{4}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(y \cdot y\right), \frac{1}{4}\right) \]

      2. *-lowering-*.f64 24.1%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{4}\right) \]

   7. Simplified 24.1%

      \[\leadsto {\color{blue}{\left(y \cdot y\right)}}^{0.25} \]
   8. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto {\left({y}^{2}\right)}^{\frac{1}{4}} \]

      2. pow-pow N/A

         \[\leadsto {y}^{\color{blue}{\left(2 \cdot \frac{1}{4}\right)}} \]

      3. metadata-eval N/A

         \[\leadsto {y}^{\frac{1}{2}} \]

      4. unpow1/2 N/A

         \[\leadsto \sqrt{y} \]

      5. sqrt-lowering-sqrt.f64 76.3%

         \[\leadsto \mathsf{sqrt.f64}\left(y\right) \]

   9. Applied egg-rr 76.3%

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

Alternative 3: 62.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{x - y}\\ \mathbf{if}\;y \leq 2.6 \cdot 10^{+48}:\\ \;\;\;\;\frac{1}{{\left(t\_0 \cdot t\_0\right)}^{0.25}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 1.0 (- x y))))
   (if (<= y 2.6e+48) (/ 1.0 (pow (* t_0 t_0) 0.25)) (sqrt y))))

C

double code(double x, double y) {
	double t_0 = 1.0 / (x - y);
	double tmp;
	if (y <= 2.6e+48) {
		tmp = 1.0 / pow((t_0 * t_0), 0.25);
	} 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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / (x - y)
    if (y <= 2.6d+48) then
        tmp = 1.0d0 / ((t_0 * t_0) ** 0.25d0)
    else
        tmp = sqrt(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 / (x - y);
	double tmp;
	if (y <= 2.6e+48) {
		tmp = 1.0 / Math.pow((t_0 * t_0), 0.25);
	} else {
		tmp = Math.sqrt(y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 / (x - y)
	tmp = 0
	if y <= 2.6e+48:
		tmp = 1.0 / math.pow((t_0 * t_0), 0.25)
	else:
		tmp = math.sqrt(y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 / Float64(x - y))
	tmp = 0.0
	if (y <= 2.6e+48)
		tmp = Float64(1.0 / (Float64(t_0 * t_0) ^ 0.25));
	else
		tmp = sqrt(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 / (x - y);
	tmp = 0.0;
	if (y <= 2.6e+48)
		tmp = 1.0 / ((t_0 * t_0) ^ 0.25);
	else
		tmp = sqrt(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 / N[(x - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 2.6e+48], N[(1.0 / N[Power[N[(t$95$0 * t$95$0), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision], N[Sqrt[y], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 2.59999999999999995e48

   1. Initial program 100.0%

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

      1. flip-- N/A

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

      2. fabs-div N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. div-fabs N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt{\left|\frac{x + y}{x \cdot x - y \cdot y}\right|}\right)}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\left|\frac{x + y}{x \cdot x - y \cdot y}\right|\right)\right)\right) \]

      9. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\left(\frac{x + y}{x \cdot x - y \cdot y}\right)\right)\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\left(\frac{1}{\frac{x \cdot x - y \cdot y}{x + y}}\right)\right)\right)\right) \]

      11. flip-- N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\left(\frac{1}{x - y}\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\mathsf{/.f64}\left(1, \left(x - y\right)\right)\right)\right)\right) \]

      13. --lowering--.f64 99.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{fabs.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(x, y\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.5%

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

      1. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({\left(\left|\frac{1}{x - y}\right|\right)}^{\color{blue}{\frac{1}{2}}}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({\left(\left|\frac{1}{x - y}\right|\right)}^{\left(\frac{1}{4} + \color{blue}{\frac{1}{4}}\right)}\right)\right) \]

      3. pow-prod-up N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({\left(\left|\frac{1}{x - y}\right|\right)}^{\frac{1}{4}} \cdot \color{blue}{{\left(\left|\frac{1}{x - y}\right|\right)}^{\frac{1}{4}}}\right)\right) \]

      4. pow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({\left(\left|\frac{1}{x - y}\right| \cdot \left|\frac{1}{x - y}\right|\right)}^{\color{blue}{\frac{1}{4}}}\right)\right) \]

      5. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({\left(\left|{\left(x - y\right)}^{-1}\right| \cdot \left|\frac{1}{x - y}\right|\right)}^{\frac{1}{4}}\right)\right) \]

      6. sqr-pow N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({\left(\left|{\left(x - y\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(x - y\right)}^{\left(\frac{-1}{2}\right)}\right| \cdot \left|\frac{1}{x - y}\right|\right)}^{\frac{1}{4}}\right)\right) \]

      7. fabs-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({\left(\left({\left(x - y\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(x - y\right)}^{\left(\frac{-1}{2}\right)}\right) \cdot \left|\frac{1}{x - y}\right|\right)}^{\frac{1}{4}}\right)\right) \]

      8. sqr-pow N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({\left({\left(x - y\right)}^{-1} \cdot \left|\frac{1}{x - y}\right|\right)}^{\frac{1}{4}}\right)\right) \]

      9. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({\left(\frac{1}{x - y} \cdot \left|\frac{1}{x - y}\right|\right)}^{\frac{1}{4}}\right)\right) \]

      10. inv-pow N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left({\left(\frac{1}{x - y} \cdot \left|{\left(x - y\right)}^{-1}\right|\right)}^{\frac{1}{4}}\right)\right) \]

      11. sqr-pow N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left({\left(\frac{1}{x - y} \cdot \left|{\left(x - y\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(x - y\right)}^{\left(\frac{-1}{2}\right)}\right|\right)}^{\frac{1}{4}}\right)\right) \]

      12. fabs-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left({\left(\frac{1}{x - y} \cdot \left({\left(x - y\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(x - y\right)}^{\left(\frac{-1}{2}\right)}\right)\right)}^{\frac{1}{4}}\right)\right) \]

      13. sqr-pow N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left({\left(\frac{1}{x - y} \cdot {\left(x - y\right)}^{-1}\right)}^{\frac{1}{4}}\right)\right) \]

      14. inv-pow N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left({\left(\frac{1}{x - y} \cdot \frac{1}{x - y}\right)}^{\frac{1}{4}}\right)\right) \]

      15. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(\left(\frac{1}{x - y} \cdot \frac{1}{x - y}\right), \color{blue}{\frac{1}{4}}\right)\right) \]

   6. Applied egg-rr 60.5%

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

   if 2.59999999999999995e48 < y

   1. Initial program 100.0%

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

      1. pow1/2 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-down N/A

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

      4. sqr-abs N/A

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

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(x - y\right) \cdot \left(x - y\right)\right), \color{blue}{\left(\frac{\frac{1}{2}}{2}\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(x - y\right), \left(x - y\right)\right), \left(\frac{\color{blue}{\frac{1}{2}}}{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(x - y\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{\_.f64}\left(x, y\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

      9. metadata-eval 25.4%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{\_.f64}\left(x, y\right)\right), \frac{1}{4}\right) \]

   4. Applied egg-rr 25.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{pow.f64}\left(\color{blue}{\left({y}^{2}\right)}, \frac{1}{4}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(y \cdot y\right), \frac{1}{4}\right) \]

      2. *-lowering-*.f64 24.1%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{4}\right) \]

   7. Simplified 24.1%

      \[\leadsto {\color{blue}{\left(y \cdot y\right)}}^{0.25} \]
   8. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto {\left({y}^{2}\right)}^{\frac{1}{4}} \]

      2. pow-pow N/A

         \[\leadsto {y}^{\color{blue}{\left(2 \cdot \frac{1}{4}\right)}} \]

      3. metadata-eval N/A

         \[\leadsto {y}^{\frac{1}{2}} \]

      4. unpow1/2 N/A

         \[\leadsto \sqrt{y} \]

      5. sqrt-lowering-sqrt.f64 76.3%

         \[\leadsto \mathsf{sqrt.f64}\left(y\right) \]

   9. Applied egg-rr 76.3%

      \[\leadsto \color{blue}{\sqrt{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 61.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{+48}:\\ \;\;\;\;{\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{0.25}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.6e+48) (pow (* (- x y) (- x y)) 0.25) (sqrt y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.6e+48) {
		tmp = pow(((x - y) * (x - y)), 0.25);
	} 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 <= 2.6d+48) then
        tmp = ((x - y) * (x - y)) ** 0.25d0
    else
        tmp = sqrt(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.6e+48) {
		tmp = Math.pow(((x - y) * (x - y)), 0.25);
	} else {
		tmp = Math.sqrt(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.6e+48:
		tmp = math.pow(((x - y) * (x - y)), 0.25)
	else:
		tmp = math.sqrt(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.6e+48)
		tmp = Float64(Float64(x - y) * Float64(x - y)) ^ 0.25;
	else
		tmp = sqrt(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.6e+48)
		tmp = ((x - y) * (x - y)) ^ 0.25;
	else
		tmp = sqrt(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.6e+48], N[Power[N[(N[(x - y), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], N[Sqrt[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.59999999999999995e48

   1. Initial program 100.0%

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

      1. pow1/2 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-down N/A

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

      4. sqr-abs N/A

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

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(x - y\right) \cdot \left(x - y\right)\right), \color{blue}{\left(\frac{\frac{1}{2}}{2}\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(x - y\right), \left(x - y\right)\right), \left(\frac{\color{blue}{\frac{1}{2}}}{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(x - y\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{\_.f64}\left(x, y\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

      9. metadata-eval 60.4%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{\_.f64}\left(x, y\right)\right), \frac{1}{4}\right) \]

   4. Applied egg-rr 60.4%

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

   if 2.59999999999999995e48 < y

   1. Initial program 100.0%

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

      1. pow1/2 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-down N/A

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

      4. sqr-abs N/A

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

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(x - y\right) \cdot \left(x - y\right)\right), \color{blue}{\left(\frac{\frac{1}{2}}{2}\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(x - y\right), \left(x - y\right)\right), \left(\frac{\color{blue}{\frac{1}{2}}}{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(x - y\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{\_.f64}\left(x, y\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

      9. metadata-eval 25.4%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{\_.f64}\left(x, y\right)\right), \frac{1}{4}\right) \]

   4. Applied egg-rr 25.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{pow.f64}\left(\color{blue}{\left({y}^{2}\right)}, \frac{1}{4}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(y \cdot y\right), \frac{1}{4}\right) \]

      2. *-lowering-*.f64 24.1%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{4}\right) \]

   7. Simplified 24.1%

      \[\leadsto {\color{blue}{\left(y \cdot y\right)}}^{0.25} \]
   8. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto {\left({y}^{2}\right)}^{\frac{1}{4}} \]

      2. pow-pow N/A

         \[\leadsto {y}^{\color{blue}{\left(2 \cdot \frac{1}{4}\right)}} \]

      3. metadata-eval N/A

         \[\leadsto {y}^{\frac{1}{2}} \]

      4. unpow1/2 N/A

         \[\leadsto \sqrt{y} \]

      5. sqrt-lowering-sqrt.f64 76.3%

         \[\leadsto \mathsf{sqrt.f64}\left(y\right) \]

   9. Applied egg-rr 76.3%

      \[\leadsto \color{blue}{\sqrt{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 43.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{-174}:\\ \;\;\;\;{\left(x \cdot \left(x + y \cdot -2\right)\right)}^{0.25}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.3e-174) (pow (* x (+ x (* y -2.0))) 0.25) (sqrt y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.3e-174) {
		tmp = pow((x * (x + (y * -2.0))), 0.25);
	} 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 <= 3.3d-174) then
        tmp = (x * (x + (y * (-2.0d0)))) ** 0.25d0
    else
        tmp = sqrt(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.3e-174) {
		tmp = Math.pow((x * (x + (y * -2.0))), 0.25);
	} else {
		tmp = Math.sqrt(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.3e-174:
		tmp = math.pow((x * (x + (y * -2.0))), 0.25)
	else:
		tmp = math.sqrt(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.3e-174)
		tmp = Float64(x * Float64(x + Float64(y * -2.0))) ^ 0.25;
	else
		tmp = sqrt(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.3e-174)
		tmp = (x * (x + (y * -2.0))) ^ 0.25;
	else
		tmp = sqrt(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.3e-174], N[Power[N[(x * N[(x + N[(y * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], N[Sqrt[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.3000000000000001e-174

   1. Initial program 100.0%

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

      1. pow1/2 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-down N/A

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

      4. sqr-abs N/A

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

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(x - y\right) \cdot \left(x - y\right)\right), \color{blue}{\left(\frac{\frac{1}{2}}{2}\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(x - y\right), \left(x - y\right)\right), \left(\frac{\color{blue}{\frac{1}{2}}}{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(x - y\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{\_.f64}\left(x, y\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

      9. metadata-eval 55.0%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{\_.f64}\left(x, y\right)\right), \frac{1}{4}\right) \]

   4. Applied egg-rr 55.0%

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

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{pow.f64}\left(\color{blue}{\left(-2 \cdot \left(x \cdot y\right) + {x}^{2}\right)}, \frac{1}{4}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(x \cdot y\right) \cdot -2 + {x}^{2}\right), \frac{1}{4}\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(x \cdot \left(y \cdot -2\right) + {x}^{2}\right), \frac{1}{4}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(x \cdot \left(-2 \cdot y\right) + {x}^{2}\right), \frac{1}{4}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(x \cdot \left(-2 \cdot y\right) + x \cdot x\right), \frac{1}{4}\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(x \cdot \left(-2 \cdot y + x\right)\right), \frac{1}{4}\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(x \cdot \left(x + -2 \cdot y\right)\right), \frac{1}{4}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(x, \left(x + -2 \cdot y\right)\right), \frac{1}{4}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, \left(-2 \cdot y\right)\right)\right), \frac{1}{4}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, \left(y \cdot -2\right)\right)\right), \frac{1}{4}\right) \]

      10. *-lowering-*.f64 32.4%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, -2\right)\right)\right), \frac{1}{4}\right) \]

   7. Simplified 32.4%

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

   if 3.3000000000000001e-174 < y

   1. Initial program 100.0%

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

      1. pow1/2 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-down N/A

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

      4. sqr-abs N/A

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

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(x - y\right) \cdot \left(x - y\right)\right), \color{blue}{\left(\frac{\frac{1}{2}}{2}\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(x - y\right), \left(x - y\right)\right), \left(\frac{\color{blue}{\frac{1}{2}}}{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(x - y\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{\_.f64}\left(x, y\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

      9. metadata-eval 48.5%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{\_.f64}\left(x, y\right)\right), \frac{1}{4}\right) \]

   4. Applied egg-rr 48.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{pow.f64}\left(\color{blue}{\left({y}^{2}\right)}, \frac{1}{4}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(y \cdot y\right), \frac{1}{4}\right) \]

      2. *-lowering-*.f64 30.5%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{4}\right) \]

   7. Simplified 30.5%

      \[\leadsto {\color{blue}{\left(y \cdot y\right)}}^{0.25} \]
   8. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto {\left({y}^{2}\right)}^{\frac{1}{4}} \]

      2. pow-pow N/A

         \[\leadsto {y}^{\color{blue}{\left(2 \cdot \frac{1}{4}\right)}} \]

      3. metadata-eval N/A

         \[\leadsto {y}^{\frac{1}{2}} \]

      4. unpow1/2 N/A

         \[\leadsto \sqrt{y} \]

      5. sqrt-lowering-sqrt.f64 59.1%

         \[\leadsto \mathsf{sqrt.f64}\left(y\right) \]

   9. Applied egg-rr 59.1%

      \[\leadsto \color{blue}{\sqrt{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 43.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{-174}:\\ \;\;\;\;{\left(x \cdot \left(x - y\right)\right)}^{0.25}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.3e-174) (pow (* x (- x y)) 0.25) (sqrt y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.3e-174) {
		tmp = pow((x * (x - y)), 0.25);
	} 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 <= 3.3d-174) then
        tmp = (x * (x - y)) ** 0.25d0
    else
        tmp = sqrt(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.3e-174) {
		tmp = Math.pow((x * (x - y)), 0.25);
	} else {
		tmp = Math.sqrt(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.3e-174:
		tmp = math.pow((x * (x - y)), 0.25)
	else:
		tmp = math.sqrt(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.3e-174)
		tmp = Float64(x * Float64(x - y)) ^ 0.25;
	else
		tmp = sqrt(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.3e-174)
		tmp = (x * (x - y)) ^ 0.25;
	else
		tmp = sqrt(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.3e-174], N[Power[N[(x * N[(x - y), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], N[Sqrt[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.3000000000000001e-174

   1. Initial program 100.0%

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

      1. pow1/2 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-down N/A

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

      4. sqr-abs N/A

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

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(x - y\right) \cdot \left(x - y\right)\right), \color{blue}{\left(\frac{\frac{1}{2}}{2}\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(x - y\right), \left(x - y\right)\right), \left(\frac{\color{blue}{\frac{1}{2}}}{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(x - y\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{\_.f64}\left(x, y\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

      9. metadata-eval 55.0%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{\_.f64}\left(x, y\right)\right), \frac{1}{4}\right) \]

   4. Applied egg-rr 55.0%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \color{blue}{x}\right), \frac{1}{4}\right) \]
   6. Step-by-step derivation

   7. Simplified 32.1%

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

   if 3.3000000000000001e-174 < y

   1. Initial program 100.0%

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

      1. pow1/2 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-down N/A

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

      4. sqr-abs N/A

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

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(x - y\right) \cdot \left(x - y\right)\right), \color{blue}{\left(\frac{\frac{1}{2}}{2}\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(x - y\right), \left(x - y\right)\right), \left(\frac{\color{blue}{\frac{1}{2}}}{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(x - y\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{\_.f64}\left(x, y\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

      9. metadata-eval 48.5%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{\_.f64}\left(x, y\right)\right), \frac{1}{4}\right) \]

   4. Applied egg-rr 48.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{pow.f64}\left(\color{blue}{\left({y}^{2}\right)}, \frac{1}{4}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(y \cdot y\right), \frac{1}{4}\right) \]

      2. *-lowering-*.f64 30.5%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{4}\right) \]

   7. Simplified 30.5%

      \[\leadsto {\color{blue}{\left(y \cdot y\right)}}^{0.25} \]
   8. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto {\left({y}^{2}\right)}^{\frac{1}{4}} \]

      2. pow-pow N/A

         \[\leadsto {y}^{\color{blue}{\left(2 \cdot \frac{1}{4}\right)}} \]

      3. metadata-eval N/A

         \[\leadsto {y}^{\frac{1}{2}} \]

      4. unpow1/2 N/A

         \[\leadsto \sqrt{y} \]

      5. sqrt-lowering-sqrt.f64 59.1%

         \[\leadsto \mathsf{sqrt.f64}\left(y\right) \]

   9. Applied egg-rr 59.1%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{-174}:\\ \;\;\;\;{\left(x \cdot \left(x - y\right)\right)}^{0.25}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 44.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{-174}:\\ \;\;\;\;{\left(x \cdot x\right)}^{0.25}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.3e-174) (pow (* x x) 0.25) (sqrt y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.3e-174) {
		tmp = pow((x * x), 0.25);
	} 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 <= 3.3d-174) then
        tmp = (x * x) ** 0.25d0
    else
        tmp = sqrt(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.3e-174) {
		tmp = Math.pow((x * x), 0.25);
	} else {
		tmp = Math.sqrt(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.3e-174:
		tmp = math.pow((x * x), 0.25)
	else:
		tmp = math.sqrt(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.3e-174)
		tmp = Float64(x * x) ^ 0.25;
	else
		tmp = sqrt(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.3e-174)
		tmp = (x * x) ^ 0.25;
	else
		tmp = sqrt(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.3e-174], N[Power[N[(x * x), $MachinePrecision], 0.25], $MachinePrecision], N[Sqrt[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.3000000000000001e-174

   1. Initial program 100.0%

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

      1. pow1/2 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-down N/A

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

      4. sqr-abs N/A

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

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(x - y\right) \cdot \left(x - y\right)\right), \color{blue}{\left(\frac{\frac{1}{2}}{2}\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(x - y\right), \left(x - y\right)\right), \left(\frac{\color{blue}{\frac{1}{2}}}{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(x - y\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{\_.f64}\left(x, y\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

      9. metadata-eval 55.0%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{\_.f64}\left(x, y\right)\right), \frac{1}{4}\right) \]

   4. Applied egg-rr 55.0%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{pow.f64}\left(\color{blue}{\left({x}^{2}\right)}, \frac{1}{4}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(x \cdot x\right), \frac{1}{4}\right) \]

      2. *-lowering-*.f64 32.7%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{4}\right) \]

   7. Simplified 32.7%

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

   if 3.3000000000000001e-174 < y

   1. Initial program 100.0%

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

      1. pow1/2 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-down N/A

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

      4. sqr-abs N/A

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

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(x - y\right) \cdot \left(x - y\right)\right), \color{blue}{\left(\frac{\frac{1}{2}}{2}\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(x - y\right), \left(x - y\right)\right), \left(\frac{\color{blue}{\frac{1}{2}}}{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(x - y\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{\_.f64}\left(x, y\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

      9. metadata-eval 48.5%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{\_.f64}\left(x, y\right)\right), \frac{1}{4}\right) \]

   4. Applied egg-rr 48.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{pow.f64}\left(\color{blue}{\left({y}^{2}\right)}, \frac{1}{4}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(y \cdot y\right), \frac{1}{4}\right) \]

      2. *-lowering-*.f64 30.5%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{4}\right) \]

   7. Simplified 30.5%

      \[\leadsto {\color{blue}{\left(y \cdot y\right)}}^{0.25} \]
   8. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto {\left({y}^{2}\right)}^{\frac{1}{4}} \]

      2. pow-pow N/A

         \[\leadsto {y}^{\color{blue}{\left(2 \cdot \frac{1}{4}\right)}} \]

      3. metadata-eval N/A

         \[\leadsto {y}^{\frac{1}{2}} \]

      4. unpow1/2 N/A

         \[\leadsto \sqrt{y} \]

      5. sqrt-lowering-sqrt.f64 59.1%

         \[\leadsto \mathsf{sqrt.f64}\left(y\right) \]

   9. Applied egg-rr 59.1%

      \[\leadsto \color{blue}{\sqrt{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 26.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. pow1/2 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-down N/A

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

   4. sqr-abs N/A

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

   5. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(\left(x - y\right) \cdot \left(x - y\right)\right), \color{blue}{\left(\frac{\frac{1}{2}}{2}\right)}\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(x - y\right), \left(x - y\right)\right), \left(\frac{\color{blue}{\frac{1}{2}}}{2}\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(x - y\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{\_.f64}\left(x, y\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

   9. metadata-eval 52.3%

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{\_.f64}\left(x, y\right)\right), \frac{1}{4}\right) \]

4. Applied egg-rr 52.3%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{pow.f64}\left(\color{blue}{\left({y}^{2}\right)}, \frac{1}{4}\right) \]
6. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(y \cdot y\right), \frac{1}{4}\right) \]

   2. *-lowering-*.f64 28.0%

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{4}\right) \]

7. Simplified 28.0%

   \[\leadsto {\color{blue}{\left(y \cdot y\right)}}^{0.25} \]
8. Step-by-step derivation

   1. pow2 N/A

      \[\leadsto {\left({y}^{2}\right)}^{\frac{1}{4}} \]

   2. pow-pow N/A

      \[\leadsto {y}^{\color{blue}{\left(2 \cdot \frac{1}{4}\right)}} \]

   3. metadata-eval N/A

      \[\leadsto {y}^{\frac{1}{2}} \]

   4. unpow1/2 N/A

      \[\leadsto \sqrt{y} \]

   5. sqrt-lowering-sqrt.f64 25.6%

      \[\leadsto \mathsf{sqrt.f64}\left(y\right) \]

9. Applied egg-rr 25.6%

   \[\leadsto \color{blue}{\sqrt{y}} \]
10. Add Preprocessing

Alternative 9: 26.8% 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. Add Preprocessing
3. Step-by-step derivation

   1. pow1/2 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-down N/A

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

   4. sqr-abs N/A

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

   5. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(\left(x - y\right) \cdot \left(x - y\right)\right), \color{blue}{\left(\frac{\frac{1}{2}}{2}\right)}\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(x - y\right), \left(x - y\right)\right), \left(\frac{\color{blue}{\frac{1}{2}}}{2}\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(x - y\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{\_.f64}\left(x, y\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

   9. metadata-eval 52.3%

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{\_.f64}\left(x, y\right)\right), \frac{1}{4}\right) \]

4. Applied egg-rr 52.3%

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

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\sqrt{x}} \]
6. Step-by-step derivation

   1. sqrt-lowering-sqrt.f64 25.3%

      \[\leadsto \mathsf{sqrt.f64}\left(x\right) \]

7. Simplified 25.3%

   \[\leadsto \color{blue}{\sqrt{x}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024138 
(FPCore (x y)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, C"
  :precision binary64
  (sqrt (fabs (- x y))))