Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3

Percentage Accurate: 99.9% → 99.9%

Time: 10.3s

Alternatives: 10

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ x + \frac{\left|y - x\right|}{2} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x (/ (fabs (- y x)) 2.0)))

C

double code(double x, double y) {
	return x + (fabs((y - x)) / 2.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + (abs((y - x)) / 2.0d0)
end function

Java

public static double code(double x, double y) {
	return x + (Math.abs((y - x)) / 2.0);
}

Python

def code(x, y):
	return x + (math.fabs((y - x)) / 2.0)

Julia

function code(x, y)
	return Float64(x + Float64(abs(Float64(y - x)) / 2.0))
end

MATLAB

function tmp = code(x, y)
	tmp = x + (abs((y - x)) / 2.0);
end

Wolfram

code[x_, y_] := N[(x + N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{\left|y - x\right|}{2} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x (/ (fabs (- y x)) 2.0)))

C

double code(double x, double y) {
	return x + (fabs((y - x)) / 2.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + (abs((y - x)) / 2.0d0)
end function

Java

public static double code(double x, double y) {
	return x + (Math.abs((y - x)) / 2.0);
}

Python

def code(x, y):
	return x + (math.fabs((y - x)) / 2.0)

Julia

function code(x, y)
	return Float64(x + Float64(abs(Float64(y - x)) / 2.0))
end

MATLAB

function tmp = code(x, y)
	tmp = x + (abs((y - x)) / 2.0);
end

Wolfram

code[x_, y_] := N[(x + N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{\left|x - y\right|}{2} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x (/ (fabs (- x y)) 2.0)))

C

double code(double x, double y) {
	return x + (fabs((x - y)) / 2.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + (abs((x - y)) / 2.0d0)
end function

Java

public static double code(double x, double y) {
	return x + (Math.abs((x - y)) / 2.0);
}

Python

def code(x, y):
	return x + (math.fabs((x - y)) / 2.0)

Julia

function code(x, y)
	return Float64(x + Float64(abs(Float64(x - y)) / 2.0))
end

MATLAB

function tmp = code(x, y)
	tmp = x + (abs((x - y)) / 2.0);
end

Wolfram

code[x_, y_] := N[(x + N[(N[Abs[N[(x - y), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

   \[\leadsto x + \frac{\left|x - y\right|}{2} \]
4. Add Preprocessing

Alternative 2: 79.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-22}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-30}:\\ \;\;\;\;x + \frac{\left|x\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.8e-22)
   (+ x (/ (fabs y) 2.0))
   (if (<= y 5.5e-30) (+ x (/ (fabs x) 2.0)) (* 0.5 (+ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.8e-22) {
		tmp = x + (fabs(y) / 2.0);
	} else if (y <= 5.5e-30) {
		tmp = x + (fabs(x) / 2.0);
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.8d-22)) then
        tmp = x + (abs(y) / 2.0d0)
    else if (y <= 5.5d-30) then
        tmp = x + (abs(x) / 2.0d0)
    else
        tmp = 0.5d0 * (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.8e-22) {
		tmp = x + (Math.abs(y) / 2.0);
	} else if (y <= 5.5e-30) {
		tmp = x + (Math.abs(x) / 2.0);
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.8e-22:
		tmp = x + (math.fabs(y) / 2.0)
	elif y <= 5.5e-30:
		tmp = x + (math.fabs(x) / 2.0)
	else:
		tmp = 0.5 * (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.8e-22)
		tmp = Float64(x + Float64(abs(y) / 2.0));
	elseif (y <= 5.5e-30)
		tmp = Float64(x + Float64(abs(x) / 2.0));
	else
		tmp = Float64(0.5 * Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.8e-22)
		tmp = x + (abs(y) / 2.0);
	elseif (y <= 5.5e-30)
		tmp = x + (abs(x) / 2.0);
	else
		tmp = 0.5 * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.8e-22], N[(x + N[(N[Abs[y], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.5e-30], N[(x + N[(N[Abs[x], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.79999999999999995e-22

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 77.4%

      \[\leadsto x + \frac{\left|\color{blue}{y}\right|}{2} \]

   if -2.79999999999999995e-22 < y < 5.49999999999999976e-30

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 87.7%

      \[\leadsto x + \frac{\left|\color{blue}{-1 \cdot x}\right|}{2} \]
   4. Step-by-step derivation

      1. neg-mul-1 87.7%

         \[\leadsto x + \frac{\left|\color{blue}{-x}\right|}{2} \]

   5. Simplified 87.7%

      \[\leadsto x + \frac{\left|\color{blue}{-x}\right|}{2} \]

   if 5.49999999999999976e-30 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 82.4%

      \[\leadsto \color{blue}{x \cdot \left(1 + 0.5 \cdot \frac{\left|y - x\right|}{x}\right)} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 81.1%

         \[\leadsto x \cdot \left(1 + 0.5 \cdot \frac{\left|\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \sqrt[3]{y - x}}\right|}{x}\right) \]

      2. unpow2 81.1%

         \[\leadsto x \cdot \left(1 + 0.5 \cdot \frac{\left|\color{blue}{{\left(\sqrt[3]{y - x}\right)}^{2}} \cdot \sqrt[3]{y - x}\right|}{x}\right) \]

      3. mul-fabs 81.1%

         \[\leadsto x \cdot \left(1 + 0.5 \cdot \frac{\color{blue}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{x}\right) \]

      4. clear-num 81.1%

         \[\leadsto x \cdot \left(1 + 0.5 \cdot \color{blue}{\frac{1}{\frac{x}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}}\right) \]

      5. un-div-inv 81.1%

         \[\leadsto x \cdot \left(1 + \color{blue}{\frac{0.5}{\frac{x}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}}\right) \]

      6. unpow2 81.1%

         \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\left|\color{blue}{\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}}\right| \cdot \left|\sqrt[3]{y - x}\right|}}\right) \]

      7. fabs-sqr 81.1%

         \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right)} \cdot \left|\sqrt[3]{y - x}\right|}}\right) \]

      8. add-sqr-sqrt 69.5%

         \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \left|\color{blue}{\sqrt{\sqrt[3]{y - x}} \cdot \sqrt{\sqrt[3]{y - x}}}\right|}}\right) \]

      9. fabs-sqr 69.5%

         \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \color{blue}{\left(\sqrt{\sqrt[3]{y - x}} \cdot \sqrt{\sqrt[3]{y - x}}\right)}}}\right) \]

      10. add-sqr-sqrt 71.8%

          \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \color{blue}{\sqrt[3]{y - x}}}}\right) \]

      11. add-cube-cbrt 72.9%

          \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\color{blue}{y - x}}}\right) \]

   5. Applied egg-rr 72.9%

      \[\leadsto x \cdot \left(1 + \color{blue}{\frac{0.5}{\frac{x}{y - x}}}\right) \]
   6. Step-by-step derivation

      1. associate-/r/ 72.9%

         \[\leadsto x \cdot \left(1 + \color{blue}{\frac{0.5}{x} \cdot \left(y - x\right)}\right) \]

   7. Simplified 72.9%

      \[\leadsto x \cdot \left(1 + \color{blue}{\frac{0.5}{x} \cdot \left(y - x\right)}\right) \]

   8. Taylor expanded in x around 0 90.6%

      \[\leadsto \color{blue}{0.5 \cdot x + 0.5 \cdot y} \]
   9. Step-by-step derivation

      1. distribute-lft-out 90.6%

         \[\leadsto \color{blue}{0.5 \cdot \left(x + y\right)} \]

      2. +-commutative 90.6%

         \[\leadsto 0.5 \cdot \color{blue}{\left(y + x\right)} \]

   10. Simplified 90.6%

       \[\leadsto \color{blue}{0.5 \cdot \left(y + x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-22}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-30}:\\ \;\;\;\;x + \frac{\left|x\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 1.18 \cdot 10^{-120}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{-1}{\frac{2}{x + y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3e-21)
   (* 0.5 (+ x y))
   (if (<= x 1.18e-120)
     (+ x (/ (fabs y) 2.0))
     (- x (/ -1.0 (/ 2.0 (+ x y)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3e-21) {
		tmp = 0.5 * (x + y);
	} else if (x <= 1.18e-120) {
		tmp = x + (fabs(y) / 2.0);
	} else {
		tmp = x - (-1.0 / (2.0 / (x + y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3d-21)) then
        tmp = 0.5d0 * (x + y)
    else if (x <= 1.18d-120) then
        tmp = x + (abs(y) / 2.0d0)
    else
        tmp = x - ((-1.0d0) / (2.0d0 / (x + y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3e-21) {
		tmp = 0.5 * (x + y);
	} else if (x <= 1.18e-120) {
		tmp = x + (Math.abs(y) / 2.0);
	} else {
		tmp = x - (-1.0 / (2.0 / (x + y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3e-21:
		tmp = 0.5 * (x + y)
	elif x <= 1.18e-120:
		tmp = x + (math.fabs(y) / 2.0)
	else:
		tmp = x - (-1.0 / (2.0 / (x + y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3e-21)
		tmp = Float64(0.5 * Float64(x + y));
	elseif (x <= 1.18e-120)
		tmp = Float64(x + Float64(abs(y) / 2.0));
	else
		tmp = Float64(x - Float64(-1.0 / Float64(2.0 / Float64(x + y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3e-21)
		tmp = 0.5 * (x + y);
	elseif (x <= 1.18e-120)
		tmp = x + (abs(y) / 2.0);
	else
		tmp = x - (-1.0 / (2.0 / (x + y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3e-21], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.18e-120], N[(x + N[(N[Abs[y], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(x - N[(-1.0 / N[(2.0 / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.99999999999999991e-21

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 99.9%

      \[\leadsto \color{blue}{x \cdot \left(1 + 0.5 \cdot \frac{\left|y - x\right|}{x}\right)} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 98.0%

         \[\leadsto x \cdot \left(1 + 0.5 \cdot \frac{\left|\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \sqrt[3]{y - x}}\right|}{x}\right) \]

      2. unpow2 98.0%

         \[\leadsto x \cdot \left(1 + 0.5 \cdot \frac{\left|\color{blue}{{\left(\sqrt[3]{y - x}\right)}^{2}} \cdot \sqrt[3]{y - x}\right|}{x}\right) \]

      3. mul-fabs 98.0%

         \[\leadsto x \cdot \left(1 + 0.5 \cdot \frac{\color{blue}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{x}\right) \]

      4. clear-num 97.7%

         \[\leadsto x \cdot \left(1 + 0.5 \cdot \color{blue}{\frac{1}{\frac{x}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}}\right) \]

      5. un-div-inv 97.7%

         \[\leadsto x \cdot \left(1 + \color{blue}{\frac{0.5}{\frac{x}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}}\right) \]

      6. unpow2 97.7%

         \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\left|\color{blue}{\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}}\right| \cdot \left|\sqrt[3]{y - x}\right|}}\right) \]

      7. fabs-sqr 97.7%

         \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right)} \cdot \left|\sqrt[3]{y - x}\right|}}\right) \]

      8. add-sqr-sqrt 81.5%

         \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \left|\color{blue}{\sqrt{\sqrt[3]{y - x}} \cdot \sqrt{\sqrt[3]{y - x}}}\right|}}\right) \]

      9. fabs-sqr 81.5%

         \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \color{blue}{\left(\sqrt{\sqrt[3]{y - x}} \cdot \sqrt{\sqrt[3]{y - x}}\right)}}}\right) \]

      10. add-sqr-sqrt 81.7%

          \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \color{blue}{\sqrt[3]{y - x}}}}\right) \]

      11. add-cube-cbrt 83.6%

          \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\color{blue}{y - x}}}\right) \]

   5. Applied egg-rr 83.6%

      \[\leadsto x \cdot \left(1 + \color{blue}{\frac{0.5}{\frac{x}{y - x}}}\right) \]
   6. Step-by-step derivation

      1. associate-/r/ 83.6%

         \[\leadsto x \cdot \left(1 + \color{blue}{\frac{0.5}{x} \cdot \left(y - x\right)}\right) \]

   7. Simplified 83.6%

      \[\leadsto x \cdot \left(1 + \color{blue}{\frac{0.5}{x} \cdot \left(y - x\right)}\right) \]

   8. Taylor expanded in x around 0 83.6%

      \[\leadsto \color{blue}{0.5 \cdot x + 0.5 \cdot y} \]
   9. Step-by-step derivation

      1. distribute-lft-out 83.6%

         \[\leadsto \color{blue}{0.5 \cdot \left(x + y\right)} \]

      2. +-commutative 83.6%

         \[\leadsto 0.5 \cdot \color{blue}{\left(y + x\right)} \]

   10. Simplified 83.6%

       \[\leadsto \color{blue}{0.5 \cdot \left(y + x\right)} \]

   if -2.99999999999999991e-21 < x < 1.17999999999999999e-120

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 81.1%

      \[\leadsto x + \frac{\left|\color{blue}{y}\right|}{2} \]

   if 1.17999999999999999e-120 < x

   1. Initial program 99.8%

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

      1. add-cube-cbrt 98.7%

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

      2. fabs-mul 98.7%

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

      3. pow2 98.7%

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

   4. Applied egg-rr 98.7%

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

      1. add-sqr-sqrt 20.4%

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

      2. fabs-sqr 20.4%

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

      3. add-sqr-sqrt 98.7%

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

      4. flip-- 51.3%

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

      5. unpow2 51.3%

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

      6. unpow2 51.3%

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

      7. div-fabs 51.3%

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

      8. clear-num 51.3%

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

      9. cbrt-div 51.4%

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

      10. metadata-eval 51.4%

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

      11. add-sqr-sqrt 48.9%

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

      12. fabs-sqr 48.9%

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

      13. add-sqr-sqrt 51.4%

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

      14. add-sqr-sqrt 10.0%

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

      15. fabs-sqr 10.0%

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

      16. add-sqr-sqrt 51.4%

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

   6. Applied egg-rr 98.9%

      \[\leadsto x + \frac{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\color{blue}{\frac{1}{\sqrt[3]{\frac{1}{y - x}}}}\right|}{2} \]
   7. Step-by-step derivation

      1. clear-num 98.9%

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

      2. inv-pow 98.9%

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

      3. mul-fabs 98.9%

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

      4. cbrt-div 98.7%

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

      5. metadata-eval 98.7%

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

   8. Applied egg-rr 98.7%

      \[\leadsto x + \color{blue}{{\left(\frac{2}{\left|{\left(\sqrt[3]{y - x}\right)}^{2} \cdot \frac{1}{\frac{1}{\sqrt[3]{y - x}}}\right|}\right)}^{-1}} \]
   9. Step-by-step derivation

      1. fabs-mul 98.7%

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

      2. unpow2 98.7%

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

      3. fabs-sqr 98.7%

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

      4. unpow2 98.7%

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

      5. remove-double-div 98.7%

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

      6. unpow-1 98.7%

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

      7. unpow2 98.7%

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

      8. fabs-sqr 98.7%

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

      9. unpow2 98.7%

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

   10. Simplified 99.7%

       \[\leadsto x + \color{blue}{\frac{1}{\frac{2}{\left|y - x\right|}}} \]
   11. Step-by-step derivation

       1. *-un-lft-identity 99.7%

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

       2. add-sqr-sqrt 20.8%

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

       3. fabs-sqr 20.8%

          \[\leadsto x + \frac{1}{1 \cdot \frac{2}{\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}}} \]

       4. add-sqr-sqrt 33.7%

          \[\leadsto x + \frac{1}{1 \cdot \frac{2}{\color{blue}{y - x}}} \]

       5. sub-neg 33.7%

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

       6. add-sqr-sqrt 0.0%

          \[\leadsto x + \frac{1}{1 \cdot \frac{2}{y + \color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}} \]

       7. sqrt-unprod 60.2%

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

       8. sqr-neg 60.2%

          \[\leadsto x + \frac{1}{1 \cdot \frac{2}{y + \sqrt{\color{blue}{x \cdot x}}}} \]

       9. sqrt-prod 92.1%

          \[\leadsto x + \frac{1}{1 \cdot \frac{2}{y + \color{blue}{\sqrt{x} \cdot \sqrt{x}}}} \]

       10. add-sqr-sqrt 92.2%

           \[\leadsto x + \frac{1}{1 \cdot \frac{2}{y + \color{blue}{x}}} \]

   12. Applied egg-rr 92.2%

       \[\leadsto x + \frac{1}{\color{blue}{1 \cdot \frac{2}{y + x}}} \]
   13. Step-by-step derivation

       1. *-lft-identity 92.2%

          \[\leadsto x + \frac{1}{\color{blue}{\frac{2}{y + x}}} \]

   14. Simplified 92.2%

       \[\leadsto x + \frac{1}{\color{blue}{\frac{2}{y + x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 1.18 \cdot 10^{-120}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{-1}{\frac{2}{x + y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-190}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-114}:\\ \;\;\;\;\left|y\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x - \frac{-1}{\frac{2}{x + y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.3e-190)
   (* 0.5 (+ x y))
   (if (<= x 6e-114) (* (fabs y) 0.5) (- x (/ -1.0 (/ 2.0 (+ x y)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.3e-190) {
		tmp = 0.5 * (x + y);
	} else if (x <= 6e-114) {
		tmp = fabs(y) * 0.5;
	} else {
		tmp = x - (-1.0 / (2.0 / (x + y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.3d-190)) then
        tmp = 0.5d0 * (x + y)
    else if (x <= 6d-114) then
        tmp = abs(y) * 0.5d0
    else
        tmp = x - ((-1.0d0) / (2.0d0 / (x + y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.3e-190) {
		tmp = 0.5 * (x + y);
	} else if (x <= 6e-114) {
		tmp = Math.abs(y) * 0.5;
	} else {
		tmp = x - (-1.0 / (2.0 / (x + y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.3e-190:
		tmp = 0.5 * (x + y)
	elif x <= 6e-114:
		tmp = math.fabs(y) * 0.5
	else:
		tmp = x - (-1.0 / (2.0 / (x + y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.3e-190)
		tmp = Float64(0.5 * Float64(x + y));
	elseif (x <= 6e-114)
		tmp = Float64(abs(y) * 0.5);
	else
		tmp = Float64(x - Float64(-1.0 / Float64(2.0 / Float64(x + y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.3e-190)
		tmp = 0.5 * (x + y);
	elseif (x <= 6e-114)
		tmp = abs(y) * 0.5;
	else
		tmp = x - (-1.0 / (2.0 / (x + y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.3e-190], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6e-114], N[(N[Abs[y], $MachinePrecision] * 0.5), $MachinePrecision], N[(x - N[(-1.0 / N[(2.0 / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.30000000000000019e-190

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 98.4%

      \[\leadsto \color{blue}{x \cdot \left(1 + 0.5 \cdot \frac{\left|y - x\right|}{x}\right)} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 96.5%

         \[\leadsto x \cdot \left(1 + 0.5 \cdot \frac{\left|\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \sqrt[3]{y - x}}\right|}{x}\right) \]

      2. unpow2 96.5%

         \[\leadsto x \cdot \left(1 + 0.5 \cdot \frac{\left|\color{blue}{{\left(\sqrt[3]{y - x}\right)}^{2}} \cdot \sqrt[3]{y - x}\right|}{x}\right) \]

      3. mul-fabs 96.5%

         \[\leadsto x \cdot \left(1 + 0.5 \cdot \frac{\color{blue}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{x}\right) \]

      4. clear-num 96.3%

         \[\leadsto x \cdot \left(1 + 0.5 \cdot \color{blue}{\frac{1}{\frac{x}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}}\right) \]

      5. un-div-inv 96.3%

         \[\leadsto x \cdot \left(1 + \color{blue}{\frac{0.5}{\frac{x}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}}\right) \]

      6. unpow2 96.3%

         \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\left|\color{blue}{\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}}\right| \cdot \left|\sqrt[3]{y - x}\right|}}\right) \]

      7. fabs-sqr 96.3%

         \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right)} \cdot \left|\sqrt[3]{y - x}\right|}}\right) \]

      8. add-sqr-sqrt 76.2%

         \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \left|\color{blue}{\sqrt{\sqrt[3]{y - x}} \cdot \sqrt{\sqrt[3]{y - x}}}\right|}}\right) \]

      9. fabs-sqr 76.2%

         \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \color{blue}{\left(\sqrt{\sqrt[3]{y - x}} \cdot \sqrt{\sqrt[3]{y - x}}\right)}}}\right) \]

      10. add-sqr-sqrt 76.5%

          \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \color{blue}{\sqrt[3]{y - x}}}}\right) \]

      11. add-cube-cbrt 78.2%

          \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\color{blue}{y - x}}}\right) \]

   5. Applied egg-rr 78.2%

      \[\leadsto x \cdot \left(1 + \color{blue}{\frac{0.5}{\frac{x}{y - x}}}\right) \]
   6. Step-by-step derivation

      1. associate-/r/ 78.2%

         \[\leadsto x \cdot \left(1 + \color{blue}{\frac{0.5}{x} \cdot \left(y - x\right)}\right) \]

   7. Simplified 78.2%

      \[\leadsto x \cdot \left(1 + \color{blue}{\frac{0.5}{x} \cdot \left(y - x\right)}\right) \]

   8. Taylor expanded in x around 0 78.2%

      \[\leadsto \color{blue}{0.5 \cdot x + 0.5 \cdot y} \]
   9. Step-by-step derivation

      1. distribute-lft-out 78.2%

         \[\leadsto \color{blue}{0.5 \cdot \left(x + y\right)} \]

      2. +-commutative 78.2%

         \[\leadsto 0.5 \cdot \color{blue}{\left(y + x\right)} \]

   10. Simplified 78.2%

       \[\leadsto \color{blue}{0.5 \cdot \left(y + x\right)} \]

   if -3.30000000000000019e-190 < x < 6.0000000000000003e-114

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 88.1%

      \[\leadsto x + \frac{\left|\color{blue}{y}\right|}{2} \]

   4. Taylor expanded in x around 0 86.3%

      \[\leadsto \color{blue}{0.5 \cdot \left|y\right|} \]

   if 6.0000000000000003e-114 < x

   1. Initial program 99.8%

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

      1. add-cube-cbrt 98.7%

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

      2. fabs-mul 98.7%

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

      3. pow2 98.7%

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

   4. Applied egg-rr 98.7%

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

      1. add-sqr-sqrt 20.4%

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

      2. fabs-sqr 20.4%

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

      3. add-sqr-sqrt 98.7%

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

      4. flip-- 51.3%

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

      5. unpow2 51.3%

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

      6. unpow2 51.3%

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

      7. div-fabs 51.3%

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

      8. clear-num 51.3%

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

      9. cbrt-div 51.4%

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

      10. metadata-eval 51.4%

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

      11. add-sqr-sqrt 48.9%

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

      12. fabs-sqr 48.9%

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

      13. add-sqr-sqrt 51.4%

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

      14. add-sqr-sqrt 10.0%

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

      15. fabs-sqr 10.0%

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

      16. add-sqr-sqrt 51.4%

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

   6. Applied egg-rr 98.9%

      \[\leadsto x + \frac{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\color{blue}{\frac{1}{\sqrt[3]{\frac{1}{y - x}}}}\right|}{2} \]
   7. Step-by-step derivation

      1. clear-num 98.9%

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

      2. inv-pow 98.9%

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

      3. mul-fabs 98.9%

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

      4. cbrt-div 98.7%

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

      5. metadata-eval 98.7%

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

   8. Applied egg-rr 98.7%

      \[\leadsto x + \color{blue}{{\left(\frac{2}{\left|{\left(\sqrt[3]{y - x}\right)}^{2} \cdot \frac{1}{\frac{1}{\sqrt[3]{y - x}}}\right|}\right)}^{-1}} \]
   9. Step-by-step derivation

      1. fabs-mul 98.7%

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

      2. unpow2 98.7%

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

      3. fabs-sqr 98.7%

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

      4. unpow2 98.7%

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

      5. remove-double-div 98.7%

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

      6. unpow-1 98.7%

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

      7. unpow2 98.7%

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

      8. fabs-sqr 98.7%

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

      9. unpow2 98.7%

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

   10. Simplified 99.7%

       \[\leadsto x + \color{blue}{\frac{1}{\frac{2}{\left|y - x\right|}}} \]
   11. Step-by-step derivation

       1. *-un-lft-identity 99.7%

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

       2. add-sqr-sqrt 20.8%

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

       3. fabs-sqr 20.8%

          \[\leadsto x + \frac{1}{1 \cdot \frac{2}{\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}}} \]

       4. add-sqr-sqrt 33.7%

          \[\leadsto x + \frac{1}{1 \cdot \frac{2}{\color{blue}{y - x}}} \]

       5. sub-neg 33.7%

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

       6. add-sqr-sqrt 0.0%

          \[\leadsto x + \frac{1}{1 \cdot \frac{2}{y + \color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}} \]

       7. sqrt-unprod 60.2%

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

       8. sqr-neg 60.2%

          \[\leadsto x + \frac{1}{1 \cdot \frac{2}{y + \sqrt{\color{blue}{x \cdot x}}}} \]

       9. sqrt-prod 92.1%

          \[\leadsto x + \frac{1}{1 \cdot \frac{2}{y + \color{blue}{\sqrt{x} \cdot \sqrt{x}}}} \]

       10. add-sqr-sqrt 92.2%

           \[\leadsto x + \frac{1}{1 \cdot \frac{2}{y + \color{blue}{x}}} \]

   12. Applied egg-rr 92.2%

       \[\leadsto x + \frac{1}{\color{blue}{1 \cdot \frac{2}{y + x}}} \]
   13. Step-by-step derivation

       1. *-lft-identity 92.2%

          \[\leadsto x + \frac{1}{\color{blue}{\frac{2}{y + x}}} \]

   14. Simplified 92.2%

       \[\leadsto x + \frac{1}{\color{blue}{\frac{2}{y + x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-190}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-114}:\\ \;\;\;\;\left|y\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x - \frac{-1}{\frac{2}{x + y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.9% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.18 \cdot 10^{-268}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{-1}{\frac{2}{x + y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.18e-268) (* 0.5 (+ x y)) (- x (/ -1.0 (/ 2.0 (+ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.18e-268) {
		tmp = 0.5 * (x + y);
	} else {
		tmp = x - (-1.0 / (2.0 / (x + y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.18d-268)) then
        tmp = 0.5d0 * (x + y)
    else
        tmp = x - ((-1.0d0) / (2.0d0 / (x + y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.18e-268) {
		tmp = 0.5 * (x + y);
	} else {
		tmp = x - (-1.0 / (2.0 / (x + y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.18e-268:
		tmp = 0.5 * (x + y)
	else:
		tmp = x - (-1.0 / (2.0 / (x + y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.18e-268)
		tmp = Float64(0.5 * Float64(x + y));
	else
		tmp = Float64(x - Float64(-1.0 / Float64(2.0 / Float64(x + y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.18e-268)
		tmp = 0.5 * (x + y);
	else
		tmp = x - (-1.0 / (2.0 / (x + y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.18e-268], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], N[(x - N[(-1.0 / N[(2.0 / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.18e-268

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 94.3%

      \[\leadsto \color{blue}{x \cdot \left(1 + 0.5 \cdot \frac{\left|y - x\right|}{x}\right)} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 92.4%

         \[\leadsto x \cdot \left(1 + 0.5 \cdot \frac{\left|\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \sqrt[3]{y - x}}\right|}{x}\right) \]

      2. unpow2 92.4%

         \[\leadsto x \cdot \left(1 + 0.5 \cdot \frac{\left|\color{blue}{{\left(\sqrt[3]{y - x}\right)}^{2}} \cdot \sqrt[3]{y - x}\right|}{x}\right) \]

      3. mul-fabs 92.4%

         \[\leadsto x \cdot \left(1 + 0.5 \cdot \frac{\color{blue}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{x}\right) \]

      4. clear-num 92.2%

         \[\leadsto x \cdot \left(1 + 0.5 \cdot \color{blue}{\frac{1}{\frac{x}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}}\right) \]

      5. un-div-inv 92.2%

         \[\leadsto x \cdot \left(1 + \color{blue}{\frac{0.5}{\frac{x}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}}\right) \]

      6. unpow2 92.2%

         \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\left|\color{blue}{\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}}\right| \cdot \left|\sqrt[3]{y - x}\right|}}\right) \]

      7. fabs-sqr 92.2%

         \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right)} \cdot \left|\sqrt[3]{y - x}\right|}}\right) \]

      8. add-sqr-sqrt 70.8%

         \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \left|\color{blue}{\sqrt{\sqrt[3]{y - x}} \cdot \sqrt{\sqrt[3]{y - x}}}\right|}}\right) \]

      9. fabs-sqr 70.8%

         \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \color{blue}{\left(\sqrt{\sqrt[3]{y - x}} \cdot \sqrt{\sqrt[3]{y - x}}\right)}}}\right) \]

      10. add-sqr-sqrt 71.2%

          \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \color{blue}{\sqrt[3]{y - x}}}}\right) \]

      11. add-cube-cbrt 72.7%

          \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\color{blue}{y - x}}}\right) \]

   5. Applied egg-rr 72.7%

      \[\leadsto x \cdot \left(1 + \color{blue}{\frac{0.5}{\frac{x}{y - x}}}\right) \]
   6. Step-by-step derivation

      1. associate-/r/ 72.7%

         \[\leadsto x \cdot \left(1 + \color{blue}{\frac{0.5}{x} \cdot \left(y - x\right)}\right) \]

   7. Simplified 72.7%

      \[\leadsto x \cdot \left(1 + \color{blue}{\frac{0.5}{x} \cdot \left(y - x\right)}\right) \]

   8. Taylor expanded in x around 0 76.4%

      \[\leadsto \color{blue}{0.5 \cdot x + 0.5 \cdot y} \]
   9. Step-by-step derivation

      1. distribute-lft-out 76.4%

         \[\leadsto \color{blue}{0.5 \cdot \left(x + y\right)} \]

      2. +-commutative 76.4%

         \[\leadsto 0.5 \cdot \color{blue}{\left(y + x\right)} \]

   10. Simplified 76.4%

       \[\leadsto \color{blue}{0.5 \cdot \left(y + x\right)} \]

   if -1.18e-268 < x

   1. Initial program 99.8%

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

      1. add-cube-cbrt 98.4%

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

      2. fabs-mul 98.4%

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

      3. pow2 98.4%

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

   4. Applied egg-rr 98.4%

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

      1. add-sqr-sqrt 24.9%

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

      2. fabs-sqr 24.9%

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

      3. add-sqr-sqrt 98.4%

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

      4. flip-- 55.5%

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

      5. unpow2 55.5%

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

      6. unpow2 55.5%

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

      7. div-fabs 55.5%

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

      8. clear-num 55.5%

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

      9. cbrt-div 55.7%

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

      10. metadata-eval 55.7%

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

      11. add-sqr-sqrt 42.0%

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

      12. fabs-sqr 42.0%

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

      13. add-sqr-sqrt 55.7%

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

      14. add-sqr-sqrt 25.9%

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

      15. fabs-sqr 25.9%

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

      16. add-sqr-sqrt 55.7%

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

   6. Applied egg-rr 98.6%

      \[\leadsto x + \frac{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\color{blue}{\frac{1}{\sqrt[3]{\frac{1}{y - x}}}}\right|}{2} \]
   7. Step-by-step derivation

      1. clear-num 98.6%

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

      2. inv-pow 98.6%

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

      3. mul-fabs 98.6%

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

      4. cbrt-div 98.4%

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

      5. metadata-eval 98.4%

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

   8. Applied egg-rr 98.4%

      \[\leadsto x + \color{blue}{{\left(\frac{2}{\left|{\left(\sqrt[3]{y - x}\right)}^{2} \cdot \frac{1}{\frac{1}{\sqrt[3]{y - x}}}\right|}\right)}^{-1}} \]
   9. Step-by-step derivation

      1. fabs-mul 98.4%

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

      2. unpow2 98.4%

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

      3. fabs-sqr 98.4%

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

      4. unpow2 98.4%

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

      5. remove-double-div 98.4%

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

      6. unpow-1 98.4%

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

      7. unpow2 98.4%

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

      8. fabs-sqr 98.4%

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

      9. unpow2 98.4%

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

   10. Simplified 99.7%

       \[\leadsto x + \color{blue}{\frac{1}{\frac{2}{\left|y - x\right|}}} \]
   11. Step-by-step derivation

       1. *-un-lft-identity 99.7%

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

       2. add-sqr-sqrt 25.2%

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

       3. fabs-sqr 25.2%

          \[\leadsto x + \frac{1}{1 \cdot \frac{2}{\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}}} \]

       4. add-sqr-sqrt 35.1%

          \[\leadsto x + \frac{1}{1 \cdot \frac{2}{\color{blue}{y - x}}} \]

       5. sub-neg 35.1%

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

       6. add-sqr-sqrt 3.2%

          \[\leadsto x + \frac{1}{1 \cdot \frac{2}{y + \color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}} \]

       7. sqrt-unprod 54.5%

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

       8. sqr-neg 54.5%

          \[\leadsto x + \frac{1}{1 \cdot \frac{2}{y + \sqrt{\color{blue}{x \cdot x}}}} \]

       9. sqrt-prod 74.9%

          \[\leadsto x + \frac{1}{1 \cdot \frac{2}{y + \color{blue}{\sqrt{x} \cdot \sqrt{x}}}} \]

       10. add-sqr-sqrt 78.2%

           \[\leadsto x + \frac{1}{1 \cdot \frac{2}{y + \color{blue}{x}}} \]

   12. Applied egg-rr 78.2%

       \[\leadsto x + \frac{1}{\color{blue}{1 \cdot \frac{2}{y + x}}} \]
   13. Step-by-step derivation

       1. *-lft-identity 78.2%

          \[\leadsto x + \frac{1}{\color{blue}{\frac{2}{y + x}}} \]

   14. Simplified 78.2%

       \[\leadsto x + \frac{1}{\color{blue}{\frac{2}{y + x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.18 \cdot 10^{-268}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{-1}{\frac{2}{x + y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.0% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-192}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-129}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6.8e-192) (* x 0.5) (if (<= x 1.25e-129) (* y 0.5) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6.8e-192) {
		tmp = x * 0.5;
	} else if (x <= 1.25e-129) {
		tmp = y * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-6.8d-192)) then
        tmp = x * 0.5d0
    else if (x <= 1.25d-129) then
        tmp = y * 0.5d0
    else
        tmp = x * 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -6.8e-192) {
		tmp = x * 0.5;
	} else if (x <= 1.25e-129) {
		tmp = y * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6.8e-192:
		tmp = x * 0.5
	elif x <= 1.25e-129:
		tmp = y * 0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6.8e-192)
		tmp = Float64(x * 0.5);
	elseif (x <= 1.25e-129)
		tmp = Float64(y * 0.5);
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6.8e-192)
		tmp = x * 0.5;
	elseif (x <= 1.25e-129)
		tmp = y * 0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6.8e-192], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, 1.25e-129], N[(y * 0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.80000000000000003e-192

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 98.4%

      \[\leadsto \color{blue}{x \cdot \left(1 + 0.5 \cdot \frac{\left|y - x\right|}{x}\right)} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 96.5%

         \[\leadsto x \cdot \left(1 + 0.5 \cdot \frac{\left|\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \sqrt[3]{y - x}}\right|}{x}\right) \]

      2. unpow2 96.5%

         \[\leadsto x \cdot \left(1 + 0.5 \cdot \frac{\left|\color{blue}{{\left(\sqrt[3]{y - x}\right)}^{2}} \cdot \sqrt[3]{y - x}\right|}{x}\right) \]

      3. mul-fabs 96.5%

         \[\leadsto x \cdot \left(1 + 0.5 \cdot \frac{\color{blue}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{x}\right) \]

      4. clear-num 96.3%

         \[\leadsto x \cdot \left(1 + 0.5 \cdot \color{blue}{\frac{1}{\frac{x}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}}\right) \]

      5. un-div-inv 96.3%

         \[\leadsto x \cdot \left(1 + \color{blue}{\frac{0.5}{\frac{x}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}}\right) \]

      6. unpow2 96.3%

         \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\left|\color{blue}{\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}}\right| \cdot \left|\sqrt[3]{y - x}\right|}}\right) \]

      7. fabs-sqr 96.3%

         \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right)} \cdot \left|\sqrt[3]{y - x}\right|}}\right) \]

      8. add-sqr-sqrt 75.5%

         \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \left|\color{blue}{\sqrt{\sqrt[3]{y - x}} \cdot \sqrt{\sqrt[3]{y - x}}}\right|}}\right) \]

      9. fabs-sqr 75.5%

         \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \color{blue}{\left(\sqrt{\sqrt[3]{y - x}} \cdot \sqrt{\sqrt[3]{y - x}}\right)}}}\right) \]

      10. add-sqr-sqrt 75.9%

          \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \color{blue}{\sqrt[3]{y - x}}}}\right) \]

      11. add-cube-cbrt 77.5%

          \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\color{blue}{y - x}}}\right) \]

   5. Applied egg-rr 77.5%

      \[\leadsto x \cdot \left(1 + \color{blue}{\frac{0.5}{\frac{x}{y - x}}}\right) \]
   6. Step-by-step derivation

      1. associate-/r/ 77.5%

         \[\leadsto x \cdot \left(1 + \color{blue}{\frac{0.5}{x} \cdot \left(y - x\right)}\right) \]

   7. Simplified 77.5%

      \[\leadsto x \cdot \left(1 + \color{blue}{\frac{0.5}{x} \cdot \left(y - x\right)}\right) \]

   8. Taylor expanded in x around inf 61.5%

      \[\leadsto x \cdot \color{blue}{0.5} \]

   if -6.80000000000000003e-192 < x < 1.25000000000000007e-129

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 87.2%

      \[\leadsto x + \frac{\left|\color{blue}{y}\right|}{2} \]

   4. Taylor expanded in x around 0 85.2%

      \[\leadsto \color{blue}{0.5 \cdot \left|y\right|} \]
   5. Step-by-step derivation

      1. rem-square-sqrt 40.0%

         \[\leadsto 0.5 \cdot \left|\color{blue}{\sqrt{y} \cdot \sqrt{y}}\right| \]

      2. fabs-sqr 40.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \]

      3. rem-square-sqrt 41.8%

         \[\leadsto 0.5 \cdot \color{blue}{y} \]

   6. Simplified 41.8%

      \[\leadsto \color{blue}{0.5 \cdot y} \]

   if 1.25000000000000007e-129 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 71.1%

      \[\leadsto x + \frac{\left|\color{blue}{-1 \cdot x}\right|}{2} \]
   4. Step-by-step derivation

      1. neg-mul-1 71.1%

         \[\leadsto x + \frac{\left|\color{blue}{-x}\right|}{2} \]

   5. Simplified 71.1%

      \[\leadsto x + \frac{\left|\color{blue}{-x}\right|}{2} \]

   6. Taylor expanded in x around 0 71.1%

      \[\leadsto \color{blue}{x + 0.5 \cdot \left|-x\right|} \]
   7. Step-by-step derivation

      1. fabs-neg 71.1%

         \[\leadsto x + 0.5 \cdot \color{blue}{\left|x\right|} \]

      2. rem-square-sqrt 71.0%

         \[\leadsto x + 0.5 \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \]

      3. fabs-sqr 71.0%

         \[\leadsto x + 0.5 \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \]

      4. rem-square-sqrt 71.1%

         \[\leadsto x + 0.5 \cdot \color{blue}{x} \]

      5. *-commutative 71.1%

         \[\leadsto x + \color{blue}{x \cdot 0.5} \]

      6. *-rgt-identity 71.1%

         \[\leadsto \color{blue}{x \cdot 1} + x \cdot 0.5 \]

      7. distribute-lft-out 71.1%

         \[\leadsto \color{blue}{x \cdot \left(1 + 0.5\right)} \]

      8. metadata-eval 71.1%

         \[\leadsto x \cdot \color{blue}{1.5} \]

   8. Simplified 71.1%

      \[\leadsto \color{blue}{x \cdot 1.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-192}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-129}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.5% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{-129}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.25e-129) (* 0.5 (+ x y)) (* x 1.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.25e-129) {
		tmp = 0.5 * (x + y);
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.25d-129) then
        tmp = 0.5d0 * (x + y)
    else
        tmp = x * 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.25e-129) {
		tmp = 0.5 * (x + y);
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.25e-129:
		tmp = 0.5 * (x + y)
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.25e-129)
		tmp = Float64(0.5 * Float64(x + y));
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.25e-129)
		tmp = 0.5 * (x + y);
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.25e-129], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25000000000000007e-129

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 87.6%

      \[\leadsto \color{blue}{x \cdot \left(1 + 0.5 \cdot \frac{\left|y - x\right|}{x}\right)} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 86.0%

         \[\leadsto x \cdot \left(1 + 0.5 \cdot \frac{\left|\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \sqrt[3]{y - x}}\right|}{x}\right) \]

      2. unpow2 86.0%

         \[\leadsto x \cdot \left(1 + 0.5 \cdot \frac{\left|\color{blue}{{\left(\sqrt[3]{y - x}\right)}^{2}} \cdot \sqrt[3]{y - x}\right|}{x}\right) \]

      3. mul-fabs 86.0%

         \[\leadsto x \cdot \left(1 + 0.5 \cdot \frac{\color{blue}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{x}\right) \]

      4. clear-num 85.8%

         \[\leadsto x \cdot \left(1 + 0.5 \cdot \color{blue}{\frac{1}{\frac{x}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}}\right) \]

      5. un-div-inv 85.8%

         \[\leadsto x \cdot \left(1 + \color{blue}{\frac{0.5}{\frac{x}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}}\right) \]

      6. unpow2 85.8%

         \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\left|\color{blue}{\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}}\right| \cdot \left|\sqrt[3]{y - x}\right|}}\right) \]

      7. fabs-sqr 85.8%

         \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right)} \cdot \left|\sqrt[3]{y - x}\right|}}\right) \]

      8. add-sqr-sqrt 59.0%

         \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \left|\color{blue}{\sqrt{\sqrt[3]{y - x}} \cdot \sqrt{\sqrt[3]{y - x}}}\right|}}\right) \]

      9. fabs-sqr 59.0%

         \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \color{blue}{\left(\sqrt{\sqrt[3]{y - x}} \cdot \sqrt{\sqrt[3]{y - x}}\right)}}}\right) \]

      10. add-sqr-sqrt 60.0%

          \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \color{blue}{\sqrt[3]{y - x}}}}\right) \]

      11. add-cube-cbrt 61.2%

          \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\color{blue}{y - x}}}\right) \]

   5. Applied egg-rr 61.2%

      \[\leadsto x \cdot \left(1 + \color{blue}{\frac{0.5}{\frac{x}{y - x}}}\right) \]
   6. Step-by-step derivation

      1. associate-/r/ 61.2%

         \[\leadsto x \cdot \left(1 + \color{blue}{\frac{0.5}{x} \cdot \left(y - x\right)}\right) \]

   7. Simplified 61.2%

      \[\leadsto x \cdot \left(1 + \color{blue}{\frac{0.5}{x} \cdot \left(y - x\right)}\right) \]

   8. Taylor expanded in x around 0 68.6%

      \[\leadsto \color{blue}{0.5 \cdot x + 0.5 \cdot y} \]
   9. Step-by-step derivation

      1. distribute-lft-out 68.6%

         \[\leadsto \color{blue}{0.5 \cdot \left(x + y\right)} \]

      2. +-commutative 68.6%

         \[\leadsto 0.5 \cdot \color{blue}{\left(y + x\right)} \]

   10. Simplified 68.6%

       \[\leadsto \color{blue}{0.5 \cdot \left(y + x\right)} \]

   if 1.25000000000000007e-129 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 71.1%

      \[\leadsto x + \frac{\left|\color{blue}{-1 \cdot x}\right|}{2} \]
   4. Step-by-step derivation

      1. neg-mul-1 71.1%

         \[\leadsto x + \frac{\left|\color{blue}{-x}\right|}{2} \]

   5. Simplified 71.1%

      \[\leadsto x + \frac{\left|\color{blue}{-x}\right|}{2} \]

   6. Taylor expanded in x around 0 71.1%

      \[\leadsto \color{blue}{x + 0.5 \cdot \left|-x\right|} \]
   7. Step-by-step derivation

      1. fabs-neg 71.1%

         \[\leadsto x + 0.5 \cdot \color{blue}{\left|x\right|} \]

      2. rem-square-sqrt 71.0%

         \[\leadsto x + 0.5 \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \]

      3. fabs-sqr 71.0%

         \[\leadsto x + 0.5 \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \]

      4. rem-square-sqrt 71.1%

         \[\leadsto x + 0.5 \cdot \color{blue}{x} \]

      5. *-commutative 71.1%

         \[\leadsto x + \color{blue}{x \cdot 0.5} \]

      6. *-rgt-identity 71.1%

         \[\leadsto \color{blue}{x \cdot 1} + x \cdot 0.5 \]

      7. distribute-lft-out 71.1%

         \[\leadsto \color{blue}{x \cdot \left(1 + 0.5\right)} \]

      8. metadata-eval 71.1%

         \[\leadsto x \cdot \color{blue}{1.5} \]

   8. Simplified 71.1%

      \[\leadsto \color{blue}{x \cdot 1.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{-129}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 44.1% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{-10}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.7e-10) (* x 0.5) (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.7e-10) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.7d-10) then
        tmp = x * 0.5d0
    else
        tmp = y * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.7e-10) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.7e-10:
		tmp = x * 0.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.7e-10)
		tmp = Float64(x * 0.5);
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.7e-10)
		tmp = x * 0.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.7e-10], N[(x * 0.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.7e-10

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 94.9%

      \[\leadsto \color{blue}{x \cdot \left(1 + 0.5 \cdot \frac{\left|y - x\right|}{x}\right)} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 93.5%

         \[\leadsto x \cdot \left(1 + 0.5 \cdot \frac{\left|\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \sqrt[3]{y - x}}\right|}{x}\right) \]

      2. unpow2 93.5%

         \[\leadsto x \cdot \left(1 + 0.5 \cdot \frac{\left|\color{blue}{{\left(\sqrt[3]{y - x}\right)}^{2}} \cdot \sqrt[3]{y - x}\right|}{x}\right) \]

      3. mul-fabs 93.5%

         \[\leadsto x \cdot \left(1 + 0.5 \cdot \frac{\color{blue}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{x}\right) \]

      4. clear-num 93.3%

         \[\leadsto x \cdot \left(1 + 0.5 \cdot \color{blue}{\frac{1}{\frac{x}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}}\right) \]

      5. un-div-inv 93.3%

         \[\leadsto x \cdot \left(1 + \color{blue}{\frac{0.5}{\frac{x}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}}\right) \]

      6. unpow2 93.3%

         \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\left|\color{blue}{\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}}\right| \cdot \left|\sqrt[3]{y - x}\right|}}\right) \]

      7. fabs-sqr 93.3%

         \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right)} \cdot \left|\sqrt[3]{y - x}\right|}}\right) \]

      8. add-sqr-sqrt 37.2%

         \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \left|\color{blue}{\sqrt{\sqrt[3]{y - x}} \cdot \sqrt{\sqrt[3]{y - x}}}\right|}}\right) \]

      9. fabs-sqr 37.2%

         \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \color{blue}{\left(\sqrt{\sqrt[3]{y - x}} \cdot \sqrt{\sqrt[3]{y - x}}\right)}}}\right) \]

      10. add-sqr-sqrt 43.1%

          \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \color{blue}{\sqrt[3]{y - x}}}}\right) \]

      11. add-cube-cbrt 44.0%

          \[\leadsto x \cdot \left(1 + \frac{0.5}{\frac{x}{\color{blue}{y - x}}}\right) \]

   5. Applied egg-rr 44.0%

      \[\leadsto x \cdot \left(1 + \color{blue}{\frac{0.5}{\frac{x}{y - x}}}\right) \]
   6. Step-by-step derivation

      1. associate-/r/ 44.0%

         \[\leadsto x \cdot \left(1 + \color{blue}{\frac{0.5}{x} \cdot \left(y - x\right)}\right) \]

   7. Simplified 44.0%

      \[\leadsto x \cdot \left(1 + \color{blue}{\frac{0.5}{x} \cdot \left(y - x\right)}\right) \]

   8. Taylor expanded in x around inf 42.1%

      \[\leadsto x \cdot \color{blue}{0.5} \]

   if 2.7e-10 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 82.1%

      \[\leadsto x + \frac{\left|\color{blue}{y}\right|}{2} \]

   4. Taylor expanded in x around 0 78.7%

      \[\leadsto \color{blue}{0.5 \cdot \left|y\right|} \]
   5. Step-by-step derivation

      1. rem-square-sqrt 78.1%

         \[\leadsto 0.5 \cdot \left|\color{blue}{\sqrt{y} \cdot \sqrt{y}}\right| \]

      2. fabs-sqr 78.1%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \]

      3. rem-square-sqrt 78.7%

         \[\leadsto 0.5 \cdot \color{blue}{y} \]

   6. Simplified 78.7%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{-10}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 28.1% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-30}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 5.5e-30) x (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.5e-30) {
		tmp = x;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5.5d-30) then
        tmp = x
    else
        tmp = y * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5.5e-30) {
		tmp = x;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.5e-30:
		tmp = x
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.5e-30)
		tmp = x;
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.5e-30)
		tmp = x;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.5e-30], x, N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.49999999999999976e-30

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 14.0%

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

   if 5.49999999999999976e-30 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 80.5%

      \[\leadsto x + \frac{\left|\color{blue}{y}\right|}{2} \]

   4. Taylor expanded in x around 0 76.8%

      \[\leadsto \color{blue}{0.5 \cdot \left|y\right|} \]
   5. Step-by-step derivation

      1. rem-square-sqrt 76.2%

         \[\leadsto 0.5 \cdot \left|\color{blue}{\sqrt{y} \cdot \sqrt{y}}\right| \]

      2. fabs-sqr 76.2%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \]

      3. rem-square-sqrt 76.8%

         \[\leadsto 0.5 \cdot \color{blue}{y} \]

   6. Simplified 76.8%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-30}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 11.5% accurate, 107.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around inf 12.0%

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

Reproduce

herbie shell --seed 2024137 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))