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: 65.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{-201}:\\ \;\;\;\;\sqrt{\left|x\right|}\\ \mathbf{elif}\;y \leq 6.7 \cdot 10^{-178}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+124}:\\ \;\;\;\;{\left(\frac{x - y}{\frac{1}{x - y}}\right)}^{0.25}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.45e-201)
   (sqrt (fabs x))
   (if (<= y 6.7e-178)
     (sqrt y)
     (if (<= y 3.8e+124) (pow (/ (- x y) (/ 1.0 (- x y))) 0.25) (sqrt y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.45e-201) {
		tmp = sqrt(fabs(x));
	} else if (y <= 6.7e-178) {
		tmp = sqrt(y);
	} else if (y <= 3.8e+124) {
		tmp = pow(((x - y) / (1.0 / (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 <= 1.45d-201) then
        tmp = sqrt(abs(x))
    else if (y <= 6.7d-178) then
        tmp = sqrt(y)
    else if (y <= 3.8d+124) then
        tmp = ((x - y) / (1.0d0 / (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 <= 1.45e-201) {
		tmp = Math.sqrt(Math.abs(x));
	} else if (y <= 6.7e-178) {
		tmp = Math.sqrt(y);
	} else if (y <= 3.8e+124) {
		tmp = Math.pow(((x - y) / (1.0 / (x - y))), 0.25);
	} else {
		tmp = Math.sqrt(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.45e-201:
		tmp = math.sqrt(math.fabs(x))
	elif y <= 6.7e-178:
		tmp = math.sqrt(y)
	elif y <= 3.8e+124:
		tmp = math.pow(((x - y) / (1.0 / (x - y))), 0.25)
	else:
		tmp = math.sqrt(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.45e-201)
		tmp = sqrt(abs(x));
	elseif (y <= 6.7e-178)
		tmp = sqrt(y);
	elseif (y <= 3.8e+124)
		tmp = Float64(Float64(x - y) / Float64(1.0 / 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 <= 1.45e-201)
		tmp = sqrt(abs(x));
	elseif (y <= 6.7e-178)
		tmp = sqrt(y);
	elseif (y <= 3.8e+124)
		tmp = ((x - y) / (1.0 / (x - y))) ^ 0.25;
	else
		tmp = sqrt(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.45e-201], N[Sqrt[N[Abs[x], $MachinePrecision]], $MachinePrecision], If[LessEqual[y, 6.7e-178], N[Sqrt[y], $MachinePrecision], If[LessEqual[y, 3.8e+124], N[Power[N[(N[(x - y), $MachinePrecision] / N[(1.0 / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], N[Sqrt[y], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.4500000000000001e-201

   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 60.1%

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

   if 1.4500000000000001e-201 < y < 6.7000000000000004e-178 or 3.7999999999999998e124 < 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 16.9%

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

   4. Applied egg-rr 16.9%

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

   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 13.9%

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

   7. Simplified 13.9%

      \[\leadsto {\color{blue}{\left(y \cdot y\right)}}^{0.25} \]
   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 80.8%

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

   9. Applied egg-rr 80.8%

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

   if 6.7000000000000004e-178 < y < 3.7999999999999998e124

   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 72.3%

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

   4. Applied egg-rr 72.3%

      \[\leadsto \color{blue}{{\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{0.25}} \]
   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 72.3%

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

   6. Applied egg-rr 72.3%

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

Alternative 3: 39.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - y \leq -1.6 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{elif}\;x - y \leq -2 \cdot 10^{-161}:\\ \;\;\;\;{\left(\frac{x - y}{\frac{1}{x - y}}\right)}^{0.25}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- x y) -1.6e+154)
   (sqrt y)
   (if (<= (- x y) -2e-161) (pow (/ (- x y) (/ 1.0 (- x y))) 0.25) (sqrt y))))

C

double code(double x, double y) {
	double tmp;
	if ((x - y) <= -1.6e+154) {
		tmp = sqrt(y);
	} else if ((x - y) <= -2e-161) {
		tmp = pow(((x - y) / (1.0 / (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) <= (-1.6d+154)) then
        tmp = sqrt(y)
    else if ((x - y) <= (-2d-161)) then
        tmp = ((x - y) / (1.0d0 / (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) <= -1.6e+154) {
		tmp = Math.sqrt(y);
	} else if ((x - y) <= -2e-161) {
		tmp = Math.pow(((x - y) / (1.0 / (x - y))), 0.25);
	} else {
		tmp = Math.sqrt(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x - y) <= -1.6e+154:
		tmp = math.sqrt(y)
	elif (x - y) <= -2e-161:
		tmp = math.pow(((x - y) / (1.0 / (x - y))), 0.25)
	else:
		tmp = math.sqrt(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x - y) <= -1.6e+154)
		tmp = sqrt(y);
	elseif (Float64(x - y) <= -2e-161)
		tmp = Float64(Float64(x - y) / Float64(1.0 / 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) <= -1.6e+154)
		tmp = sqrt(y);
	elseif ((x - y) <= -2e-161)
		tmp = ((x - y) / (1.0 / (x - y))) ^ 0.25;
	else
		tmp = sqrt(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x - y), $MachinePrecision], -1.6e+154], N[Sqrt[y], $MachinePrecision], If[LessEqual[N[(x - y), $MachinePrecision], -2e-161], N[Power[N[(N[(x - y), $MachinePrecision] / N[(1.0 / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], N[Sqrt[y], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x y) < -1.6e154 or -2.00000000000000006e-161 < (-.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 36.1%

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

   4. Applied egg-rr 36.1%

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

   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 19.9%

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

   7. Simplified 19.9%

      \[\leadsto {\color{blue}{\left(y \cdot y\right)}}^{0.25} \]
   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 12.4%

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

   9. Applied egg-rr 12.4%

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

   if -1.6e154 < (-.f64 x y) < -2.00000000000000006e-161

   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 98.5%

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

   4. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{{\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{0.25}} \]
   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 98.5%

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

   6. Applied egg-rr 98.5%

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

Alternative 4: 39.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - y \leq -1.6 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{elif}\;x - y \leq -2 \cdot 10^{-161}:\\ \;\;\;\;{\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{0.25}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- x y) -1.6e+154)
   (sqrt y)
   (if (<= (- x y) -2e-161) (pow (* (- x y) (- x y)) 0.25) (sqrt y))))

C

double code(double x, double y) {
	double tmp;
	if ((x - y) <= -1.6e+154) {
		tmp = sqrt(y);
	} else if ((x - y) <= -2e-161) {
		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) <= (-1.6d+154)) then
        tmp = sqrt(y)
    else if ((x - y) <= (-2d-161)) 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) <= -1.6e+154) {
		tmp = Math.sqrt(y);
	} else if ((x - y) <= -2e-161) {
		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) <= -1.6e+154:
		tmp = math.sqrt(y)
	elif (x - y) <= -2e-161:
		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) <= -1.6e+154)
		tmp = sqrt(y);
	elseif (Float64(x - y) <= -2e-161)
		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) <= -1.6e+154)
		tmp = sqrt(y);
	elseif ((x - y) <= -2e-161)
		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], -1.6e+154], N[Sqrt[y], $MachinePrecision], If[LessEqual[N[(x - y), $MachinePrecision], -2e-161], 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) < -1.6e154 or -2.00000000000000006e-161 < (-.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 36.1%

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

   4. Applied egg-rr 36.1%

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

   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 19.9%

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

   7. Simplified 19.9%

      \[\leadsto {\color{blue}{\left(y \cdot y\right)}}^{0.25} \]
   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 12.4%

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

   9. Applied egg-rr 12.4%

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

   if -1.6e154 < (-.f64 x y) < -2.00000000000000006e-161

   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 98.5%

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

   4. Applied egg-rr 98.5%

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

Alternative 5: 44.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-202}:\\ \;\;\;\;{\left(x \cdot \left(x + y \cdot -2\right)\right)}^{0.25}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 8e-202) (pow (* x (+ x (* y -2.0))) 0.25) (sqrt y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8e-202) {
		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 <= 8d-202) 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 <= 8e-202) {
		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 <= 8e-202:
		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 <= 8e-202)
		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 <= 8e-202)
		tmp = (x * (x + (y * -2.0))) ^ 0.25;
	else
		tmp = sqrt(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8e-202], 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 < 8.0000000000000003e-202

   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 52.5%

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

   4. Applied egg-rr 52.5%

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

   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 33.2%

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

   if 8.0000000000000003e-202 < 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 53.6%

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

   4. Applied egg-rr 53.6%

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

   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 34.5%

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

   7. Simplified 34.5%

      \[\leadsto {\color{blue}{\left(y \cdot y\right)}}^{0.25} \]
   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 57.2%

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

   9. Applied egg-rr 57.2%

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

Alternative 6: 43.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{-201}:\\ \;\;\;\;{\left(x \cdot \left(x - y\right)\right)}^{0.25}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.45e-201) (pow (* x (- x y)) 0.25) (sqrt y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.45e-201) {
		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 <= 1.45d-201) 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 <= 1.45e-201) {
		tmp = Math.pow((x * (x - y)), 0.25);
	} else {
		tmp = Math.sqrt(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.45e-201:
		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 <= 1.45e-201)
		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 <= 1.45e-201)
		tmp = (x * (x - y)) ^ 0.25;
	else
		tmp = sqrt(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.45e-201], 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 < 1.4500000000000001e-201

   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 52.5%

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

   4. Applied egg-rr 52.5%

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

   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 32.6%

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

   if 1.4500000000000001e-201 < 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 53.6%

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

   4. Applied egg-rr 53.6%

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

   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 34.5%

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

   7. Simplified 34.5%

      \[\leadsto {\color{blue}{\left(y \cdot y\right)}}^{0.25} \]
   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 57.2%

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

   9. Applied egg-rr 57.2%

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

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{-201}:\\ \;\;\;\;{\left(x \cdot \left(x - y\right)\right)}^{0.25}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 44.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{-201}:\\ \;\;\;\;{\left(x \cdot x\right)}^{0.25}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.45e-201) (pow (* x x) 0.25) (sqrt y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.45e-201) {
		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 <= 1.45d-201) 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 <= 1.45e-201) {
		tmp = Math.pow((x * x), 0.25);
	} else {
		tmp = Math.sqrt(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.45e-201:
		tmp = math.pow((x * x), 0.25)
	else:
		tmp = math.sqrt(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.45e-201)
		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 <= 1.45e-201)
		tmp = (x * x) ^ 0.25;
	else
		tmp = sqrt(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.45e-201], N[Power[N[(x * x), $MachinePrecision], 0.25], $MachinePrecision], N[Sqrt[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.4500000000000001e-201

   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 52.5%

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

   4. Applied egg-rr 52.5%

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

   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 33.2%

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

   7. Simplified 33.2%

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

   if 1.4500000000000001e-201 < 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 53.6%

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

   4. Applied egg-rr 53.6%

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

   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 34.5%

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

   7. Simplified 34.5%

      \[\leadsto {\color{blue}{\left(y \cdot y\right)}}^{0.25} \]
   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 57.2%

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

   9. Applied egg-rr 57.2%

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

Alternative 8: 26.8% 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 52.9%

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

4. Applied egg-rr 52.9%

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

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 26.1%

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

7. Simplified 26.1%

   \[\leadsto {\color{blue}{\left(y \cdot y\right)}}^{0.25} \]
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 20.3%

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

9. Applied egg-rr 20.3%

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

Alternative 9: 25.5% 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 52.9%

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

4. Applied egg-rr 52.9%

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

5. Taylor expanded in x around inf

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

   1. sqrt-lowering-sqrt.f64 25.5%

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

7. Simplified 25.5%

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

Reproduce

herbie shell --seed 2024161 
(FPCore (x y)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, C"
  :precision binary64
  (sqrt (fabs (- x y))))