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

Percentage Accurate: 100.0% → 100.0%

Time: 7.9s

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: 48.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\left|x\right|}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-142}:\\ \;\;\;\;{\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{0.25}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.12e+154)
   (sqrt (fabs x))
   (if (<= x -5e-142) (pow (* (- x y) (- x y)) 0.25) (sqrt y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.12e+154) {
		tmp = sqrt(fabs(x));
	} else if (x <= -5e-142) {
		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 (x <= (-1.12d+154)) then
        tmp = sqrt(abs(x))
    else if (x <= (-5d-142)) 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 (x <= -1.12e+154) {
		tmp = Math.sqrt(Math.abs(x));
	} else if (x <= -5e-142) {
		tmp = Math.pow(((x - y) * (x - y)), 0.25);
	} else {
		tmp = Math.sqrt(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.12e+154:
		tmp = math.sqrt(math.fabs(x))
	elif x <= -5e-142:
		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 (x <= -1.12e+154)
		tmp = sqrt(abs(x));
	elseif (x <= -5e-142)
		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 (x <= -1.12e+154)
		tmp = sqrt(abs(x));
	elseif (x <= -5e-142)
		tmp = ((x - y) * (x - y)) ^ 0.25;
	else
		tmp = sqrt(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.12e+154], N[Sqrt[N[Abs[x], $MachinePrecision]], $MachinePrecision], If[LessEqual[x, -5e-142], N[Power[N[(N[(x - y), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], N[Sqrt[y], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.11999999999999994e154

   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 86.9%

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

   if -1.11999999999999994e154 < x < -5.0000000000000002e-142

   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 79.5%

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

   4. Applied egg-rr 79.5%

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

   if -5.0000000000000002e-142 < x

   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 48.1%

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

   4. Applied egg-rr 48.1%

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

   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 30.2%

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

   7. Simplified 30.2%

      \[\leadsto {\color{blue}{\left(y \cdot y\right)}}^{0.25} \]
   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 29.3%

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

   9. Applied egg-rr 29.3%

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

Alternative 3: 38.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - y \leq -2 \cdot 10^{+160}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{elif}\;x - y \leq -1 \cdot 10^{-172}:\\ \;\;\;\;{\left(\frac{\frac{1}{x - y}}{x - y}\right)}^{-0.25}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- x y) -2e+160)
   (sqrt y)
   (if (<= (- x y) -1e-172) (pow (/ (/ 1.0 (- x y)) (- x y)) -0.25) (sqrt y))))

C

double code(double x, double y) {
	double tmp;
	if ((x - y) <= -2e+160) {
		tmp = sqrt(y);
	} else if ((x - y) <= -1e-172) {
		tmp = pow(((1.0 / (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 ((x - y) <= (-2d+160)) then
        tmp = sqrt(y)
    else if ((x - y) <= (-1d-172)) then
        tmp = ((1.0d0 / (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 ((x - y) <= -2e+160) {
		tmp = Math.sqrt(y);
	} else if ((x - y) <= -1e-172) {
		tmp = Math.pow(((1.0 / (x - y)) / (x - y)), -0.25);
	} else {
		tmp = Math.sqrt(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x - y) <= -2e+160:
		tmp = math.sqrt(y)
	elif (x - y) <= -1e-172:
		tmp = math.pow(((1.0 / (x - y)) / (x - y)), -0.25)
	else:
		tmp = math.sqrt(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x - y) <= -2e+160)
		tmp = sqrt(y);
	elseif (Float64(x - y) <= -1e-172)
		tmp = Float64(Float64(1.0 / 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 ((x - y) <= -2e+160)
		tmp = sqrt(y);
	elseif ((x - y) <= -1e-172)
		tmp = ((1.0 / (x - y)) / (x - y)) ^ -0.25;
	else
		tmp = sqrt(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x - y), $MachinePrecision], -2e+160], N[Sqrt[y], $MachinePrecision], If[LessEqual[N[(x - y), $MachinePrecision], -1e-172], N[Power[N[(N[(1.0 / N[(x - y), $MachinePrecision]), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision], -0.25], $MachinePrecision], N[Sqrt[y], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x y) < -2.00000000000000001e160 or -1e-172 < (-.f64 x 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 31.7%

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

   4. Applied egg-rr 31.7%

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

   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 17.1%

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

   7. Simplified 17.1%

      \[\leadsto {\color{blue}{\left(y \cdot y\right)}}^{0.25} \]
   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 15.4%

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

   9. Applied egg-rr 15.4%

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

   if -2.00000000000000001e160 < (-.f64 x y) < -1e-172

   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 97.1%

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

   4. Applied egg-rr 97.1%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

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

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

      6. clear-num N/A

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

      7. flip-- N/A

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

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

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

      9. --lowering--.f64 97.1%

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

   6. Applied egg-rr 97.1%

      \[\leadsto {\color{blue}{\left(\frac{x - y}{\frac{1}{x - y}}\right)}}^{0.25} \]
   7. Step-by-step derivation

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. pow-pow N/A

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

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

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

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

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

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

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

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

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

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

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

      9. metadata-eval 97.0%

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

   8. Applied egg-rr 97.0%

      \[\leadsto \color{blue}{{\left(\frac{\frac{1}{x - y}}{x - y}\right)}^{-0.25}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 38.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - y \leq -2 \cdot 10^{+155}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{elif}\;x - y \leq -1 \cdot 10^{-172}:\\ \;\;\;\;{\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{0.25}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- x y) -2e+155)
   (sqrt y)
   (if (<= (- x y) -1e-172) (pow (* (- x y) (- x y)) 0.25) (sqrt y))))

C

double code(double x, double y) {
	double tmp;
	if ((x - y) <= -2e+155) {
		tmp = sqrt(y);
	} else if ((x - y) <= -1e-172) {
		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 ((x - y) <= (-2d+155)) then
        tmp = sqrt(y)
    else if ((x - y) <= (-1d-172)) 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 ((x - y) <= -2e+155) {
		tmp = Math.sqrt(y);
	} else if ((x - y) <= -1e-172) {
		tmp = Math.pow(((x - y) * (x - y)), 0.25);
	} else {
		tmp = Math.sqrt(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x - y) <= -2e+155:
		tmp = math.sqrt(y)
	elif (x - y) <= -1e-172:
		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 (Float64(x - y) <= -2e+155)
		tmp = sqrt(y);
	elseif (Float64(x - y) <= -1e-172)
		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 ((x - y) <= -2e+155)
		tmp = sqrt(y);
	elseif ((x - y) <= -1e-172)
		tmp = ((x - y) * (x - y)) ^ 0.25;
	else
		tmp = sqrt(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x - y), $MachinePrecision], -2e+155], N[Sqrt[y], $MachinePrecision], If[LessEqual[N[(x - y), $MachinePrecision], -1e-172], 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 (-.f64 x y) < -2.00000000000000001e155 or -1e-172 < (-.f64 x 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 31.7%

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

   4. Applied egg-rr 31.7%

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

   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 17.1%

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

   7. Simplified 17.1%

      \[\leadsto {\color{blue}{\left(y \cdot y\right)}}^{0.25} \]
   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 15.4%

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

   9. Applied egg-rr 15.4%

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

   if -2.00000000000000001e155 < (-.f64 x y) < -1e-172

   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 97.1%

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

   4. Applied egg-rr 97.1%

      \[\leadsto \color{blue}{{\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{0.25}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 42.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{-233}:\\ \;\;\;\;{\left(x \cdot \left(x + y \cdot -2\right)\right)}^{0.25}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.9e-233) (pow (* x (+ x (* y -2.0))) 0.25) (sqrt y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.9e-233) {
		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 <= 1.9d-233) 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 <= 1.9e-233) {
		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 <= 1.9e-233:
		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 <= 1.9e-233)
		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 <= 1.9e-233)
		tmp = (x * (x + (y * -2.0))) ^ 0.25;
	else
		tmp = sqrt(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.9e-233], 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 < 1.9e-233

   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 46.2%

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

   4. Applied egg-rr 46.2%

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

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-rgt-identity N/A

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

      10. times-frac N/A

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

      11. *-rgt-identity N/A

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

      12. times-frac N/A

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

      13. *-inverses N/A

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

      14. /-rgt-identity N/A

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

      15. *-lft-identity N/A

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

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

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

      19. metadata-eval N/A

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

      20. distribute-lft-neg-in N/A

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

      21. mul-1-neg N/A

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

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

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

   7. Simplified 29.6%

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

   if 1.9e-233 < 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 53.0%

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

   4. Applied egg-rr 53.0%

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

   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 32.9%

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

   7. Simplified 32.9%

      \[\leadsto {\color{blue}{\left(y \cdot y\right)}}^{0.25} \]
   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 56.1%

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

   9. Applied egg-rr 56.1%

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

Alternative 6: 42.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.08 \cdot 10^{-234}:\\ \;\;\;\;{\left(x \cdot \left(x - y\right)\right)}^{0.25}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.08e-234) (pow (* x (- x y)) 0.25) (sqrt y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.08e-234) {
		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 <= 1.08d-234) 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 <= 1.08e-234) {
		tmp = Math.pow((x * (x - y)), 0.25);
	} else {
		tmp = Math.sqrt(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.08e-234:
		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 <= 1.08e-234)
		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 <= 1.08e-234)
		tmp = (x * (x - y)) ^ 0.25;
	else
		tmp = sqrt(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.08e-234], 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 < 1.0800000000000001e-234

   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 45.8%

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

   4. Applied egg-rr 45.8%

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

   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 29.2%

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

   if 1.0800000000000001e-234 < 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 53.4%

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

   4. Applied egg-rr 53.4%

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

   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 32.7%

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

   7. Simplified 32.7%

      \[\leadsto {\color{blue}{\left(y \cdot y\right)}}^{0.25} \]
   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 55.6%

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

   9. Applied egg-rr 55.6%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.08 \cdot 10^{-234}:\\ \;\;\;\;{\left(x \cdot \left(x - y\right)\right)}^{0.25}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{-233}:\\ \;\;\;\;{\left(x \cdot x\right)}^{0.25}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.9e-233) (pow (* x x) 0.25) (sqrt y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.9e-233) {
		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 <= 1.9d-233) 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 <= 1.9e-233) {
		tmp = Math.pow((x * x), 0.25);
	} else {
		tmp = Math.sqrt(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.9e-233:
		tmp = math.pow((x * x), 0.25)
	else:
		tmp = math.sqrt(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.9e-233)
		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 <= 1.9e-233)
		tmp = (x * x) ^ 0.25;
	else
		tmp = sqrt(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.9e-233], N[Power[N[(x * x), $MachinePrecision], 0.25], $MachinePrecision], N[Sqrt[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.9e-233

   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 46.2%

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

   4. Applied egg-rr 46.2%

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

   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 30.1%

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

   7. Simplified 30.1%

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

   if 1.9e-233 < 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 53.0%

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

   4. Applied egg-rr 53.0%

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

   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 32.9%

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

   7. Simplified 32.9%

      \[\leadsto {\color{blue}{\left(y \cdot y\right)}}^{0.25} \]
   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 56.1%

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

   9. Applied egg-rr 56.1%

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

Alternative 8: 26.1% 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 49.1%

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

4. Applied egg-rr 49.1%

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

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 26.0%

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

7. Simplified 26.0%

   \[\leadsto {\color{blue}{\left(y \cdot y\right)}}^{0.25} \]
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 24.4%

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

9. Applied egg-rr 24.4%

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

Alternative 9: 25.7% 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 49.1%

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

4. Applied egg-rr 49.1%

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

5. Taylor expanded in x around inf

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

   1. sqrt-lowering-sqrt.f64 24.7%

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

7. Simplified 24.7%

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

Reproduce

herbie shell --seed 2024152 
(FPCore (x y)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, C"
  :precision binary64
  (sqrt (fabs (- x y))))