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

Percentage Accurate: 99.9% → 99.9%

Time: 7.6s

Alternatives: 9

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

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

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]

Derivation

1. Initial program 99.9%

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

Alternative 2: 83.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-48}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{x + y}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1e-8)
   (* 0.5 (+ x y))
   (if (<= x 2.1e-48) (+ x (/ (fabs y) 2.0)) (+ x (/ (+ x y) 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1e-8) {
		tmp = 0.5 * (x + y);
	} else if (x <= 2.1e-48) {
		tmp = x + (fabs(y) / 2.0);
	} else {
		tmp = x + ((x + y) / 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1d-8)) then
        tmp = 0.5d0 * (x + y)
    else if (x <= 2.1d-48) then
        tmp = x + (abs(y) / 2.0d0)
    else
        tmp = x + ((x + y) / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1e-8) {
		tmp = 0.5 * (x + y);
	} else if (x <= 2.1e-48) {
		tmp = x + (Math.abs(y) / 2.0);
	} else {
		tmp = x + ((x + y) / 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1e-8:
		tmp = 0.5 * (x + y)
	elif x <= 2.1e-48:
		tmp = x + (math.fabs(y) / 2.0)
	else:
		tmp = x + ((x + y) / 2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1e-8)
		tmp = Float64(0.5 * Float64(x + y));
	elseif (x <= 2.1e-48)
		tmp = Float64(x + Float64(abs(y) / 2.0));
	else
		tmp = Float64(x + Float64(Float64(x + y) / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1e-8)
		tmp = 0.5 * (x + y);
	elseif (x <= 2.1e-48)
		tmp = x + (abs(y) / 2.0);
	else
		tmp = x + ((x + y) / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1e-8], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.1e-48], N[(x + N[(N[Abs[y], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(x + y), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1e-8

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 97.5%

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

      2. fabs-sqr 97.5%

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

      3. add-sqr-sqrt 98.3%

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

      4. sub-neg 98.3%

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

      5. flip-+ 42.7%

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

      6. pow2 42.7%

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

   4. Applied egg-rr 42.7%

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

   5. Taylor expanded in x around 0 98.3%

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

      1. +-commutative 98.3%

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

      2. distribute-lft-in 98.3%

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

   7. Simplified 98.3%

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

   if -1e-8 < x < 2.09999999999999989e-48

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 88.8%

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

   if 2.09999999999999989e-48 < 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 98.8%

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

      2. fabs-mul 98.8%

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

      3. pow2 98.8%

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

   4. Applied egg-rr 98.8%

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

   6. Applied egg-rr 85.1%

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

      1. fma-undefine 85.1%

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

      2. associate-*r/ 85.1%

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

      3. unpow2 85.1%

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

      4. rem-3cbrt-lft 85.9%

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

   8. Simplified 85.9%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-48}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{x + y}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-45}:\\ \;\;\;\;0.5 \cdot \left|y\right|\\ \mathbf{else}:\\ \;\;\;\;x + \frac{x + y}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6e-8)
   (* 0.5 (+ x y))
   (if (<= x 5.8e-45) (* 0.5 (fabs y)) (+ x (/ (+ x y) 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6e-8) {
		tmp = 0.5 * (x + y);
	} else if (x <= 5.8e-45) {
		tmp = 0.5 * fabs(y);
	} else {
		tmp = x + ((x + y) / 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-6d-8)) then
        tmp = 0.5d0 * (x + y)
    else if (x <= 5.8d-45) then
        tmp = 0.5d0 * abs(y)
    else
        tmp = x + ((x + y) / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -6e-8) {
		tmp = 0.5 * (x + y);
	} else if (x <= 5.8e-45) {
		tmp = 0.5 * Math.abs(y);
	} else {
		tmp = x + ((x + y) / 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6e-8:
		tmp = 0.5 * (x + y)
	elif x <= 5.8e-45:
		tmp = 0.5 * math.fabs(y)
	else:
		tmp = x + ((x + y) / 2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6e-8)
		tmp = Float64(0.5 * Float64(x + y));
	elseif (x <= 5.8e-45)
		tmp = Float64(0.5 * abs(y));
	else
		tmp = Float64(x + Float64(Float64(x + y) / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6e-8)
		tmp = 0.5 * (x + y);
	elseif (x <= 5.8e-45)
		tmp = 0.5 * abs(y);
	else
		tmp = x + ((x + y) / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6e-8], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.8e-45], N[(0.5 * N[Abs[y], $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(x + y), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.99999999999999946e-8

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 97.5%

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

      2. fabs-sqr 97.5%

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

      3. add-sqr-sqrt 98.3%

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

      4. sub-neg 98.3%

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

      5. flip-+ 42.7%

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

      6. pow2 42.7%

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

   4. Applied egg-rr 42.7%

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

   5. Taylor expanded in x around 0 98.3%

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

      1. +-commutative 98.3%

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

      2. distribute-lft-in 98.3%

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

   7. Simplified 98.3%

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

   if -5.99999999999999946e-8 < x < 5.8e-45

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 88.8%

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

   4. Taylor expanded in x around 0 87.0%

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

   if 5.8e-45 < 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 98.8%

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

      2. fabs-mul 98.8%

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

      3. pow2 98.8%

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

   4. Applied egg-rr 98.8%

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

   6. Applied egg-rr 85.1%

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

      1. fma-undefine 85.1%

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

      2. associate-*r/ 85.1%

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

      3. unpow2 85.1%

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

      4. rem-3cbrt-lft 85.9%

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

   8. Simplified 85.9%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-45}:\\ \;\;\;\;0.5 \cdot \left|y\right|\\ \mathbf{else}:\\ \;\;\;\;x + \frac{x + y}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.0% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.85:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-125}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.85) (* x 0.5) (if (<= x 1.1e-125) (* y 0.5) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.85) {
		tmp = x * 0.5;
	} else if (x <= 1.1e-125) {
		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 <= (-0.85d0)) then
        tmp = x * 0.5d0
    else if (x <= 1.1d-125) 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 <= -0.85) {
		tmp = x * 0.5;
	} else if (x <= 1.1e-125) {
		tmp = y * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.85:
		tmp = x * 0.5
	elif x <= 1.1e-125:
		tmp = y * 0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.85)
		tmp = Float64(x * 0.5);
	elseif (x <= 1.1e-125)
		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 <= -0.85)
		tmp = x * 0.5;
	elseif (x <= 1.1e-125)
		tmp = y * 0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.85], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, 1.1e-125], N[(y * 0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.849999999999999978

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 97.4%

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

      2. fabs-sqr 97.4%

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

      3. add-sqr-sqrt 98.2%

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

      4. sub-neg 98.2%

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

      5. flip-+ 43.0%

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

      6. pow2 43.0%

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

   4. Applied egg-rr 43.0%

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

   5. Taylor expanded in x around inf 94.4%

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

   if -0.849999999999999978 < x < 1.09999999999999997e-125

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 90.9%

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

   4. Taylor expanded in x around 0 89.4%

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

      1. rem-square-sqrt 48.2%

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

      2. fabs-sqr 48.2%

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

      3. rem-square-sqrt 49.6%

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

   6. Simplified 49.6%

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

   if 1.09999999999999997e-125 < 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. mul-fabs 98.7%

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

      2. unpow2 98.7%

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

      3. add-cube-cbrt 99.8%

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

      4. add-sqr-sqrt 15.9%

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

      5. fabs-sqr 15.9%

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

      6. add-sqr-sqrt 27.4%

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

      7. add-sqr-sqrt 27.4%

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

      8. sqrt-prod 20.7%

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

      9. sqr-neg 20.7%

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

      10. pow2 20.7%

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

      11. sqrt-pow1 77.5%

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

      12. metadata-eval 77.5%

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

      13. pow1 77.5%

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

      14. add-cube-cbrt 76.8%

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

      15. pow3 76.9%

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

   6. Applied egg-rr 76.9%

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

   7. Taylor expanded in x around inf 63.8%

      \[\leadsto \color{blue}{1.5 \cdot x} \]
   8. Step-by-step derivation

      1. *-commutative 63.8%

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

   9. Simplified 63.8%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.85:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-125}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.0% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-266}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{x + y}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5e-266) (* 0.5 (+ x y)) (+ x (/ (+ x y) 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5e-266) {
		tmp = 0.5 * (x + y);
	} else {
		tmp = x + ((x + y) / 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5d-266)) then
        tmp = 0.5d0 * (x + y)
    else
        tmp = x + ((x + y) / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5e-266) {
		tmp = 0.5 * (x + y);
	} else {
		tmp = x + ((x + y) / 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5e-266:
		tmp = 0.5 * (x + y)
	else:
		tmp = x + ((x + y) / 2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5e-266)
		tmp = Float64(0.5 * Float64(x + y));
	else
		tmp = Float64(x + Float64(Float64(x + y) / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5e-266)
		tmp = 0.5 * (x + y);
	else
		tmp = x + ((x + y) / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5e-266], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(x + y), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.99999999999999992e-266

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 78.8%

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

      2. fabs-sqr 78.8%

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

      3. add-sqr-sqrt 79.7%

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

      4. sub-neg 79.7%

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

      5. flip-+ 41.5%

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

      6. pow2 41.5%

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

   4. Applied egg-rr 41.5%

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

   5. Taylor expanded in x around 0 79.7%

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

      1. +-commutative 79.7%

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

      2. distribute-lft-in 79.7%

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

   7. Simplified 79.7%

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

   if -4.99999999999999992e-266 < 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 98.3%

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

      2. fabs-mul 98.3%

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

      3. pow2 98.3%

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

   4. Applied egg-rr 98.3%

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

   6. Applied egg-rr 68.3%

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

      1. fma-undefine 68.3%

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

      2. associate-*r/ 68.3%

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

      3. unpow2 68.3%

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

      4. rem-3cbrt-lft 69.2%

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

   8. Simplified 69.2%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-266}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{x + y}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.0% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{-126}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 6.5e-126) (* 0.5 (+ x y)) (* x 1.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 6.5e-126) {
		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 <= 6.5d-126) 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 <= 6.5e-126) {
		tmp = 0.5 * (x + y);
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 6.5e-126:
		tmp = 0.5 * (x + y)
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 6.5e-126)
		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 <= 6.5e-126)
		tmp = 0.5 * (x + y);
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 6.5e-126], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.50000000000000014e-126

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 69.3%

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

      2. fabs-sqr 69.3%

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

      3. add-sqr-sqrt 70.6%

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

      4. sub-neg 70.6%

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

      5. flip-+ 37.6%

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

      6. pow2 37.6%

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

   4. Applied egg-rr 37.6%

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

   5. Taylor expanded in x around 0 70.6%

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

      1. +-commutative 70.6%

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

      2. distribute-lft-in 70.6%

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

   7. Simplified 70.6%

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

   if 6.50000000000000014e-126 < 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. mul-fabs 98.7%

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

      2. unpow2 98.7%

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

      3. add-cube-cbrt 99.8%

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

      4. add-sqr-sqrt 15.9%

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

      5. fabs-sqr 15.9%

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

      6. add-sqr-sqrt 27.4%

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

      7. add-sqr-sqrt 27.4%

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

      8. sqrt-prod 20.7%

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

      9. sqr-neg 20.7%

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

      10. pow2 20.7%

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

      11. sqrt-pow1 77.5%

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

      12. metadata-eval 77.5%

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

      13. pow1 77.5%

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

      14. add-cube-cbrt 76.8%

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

      15. pow3 76.9%

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

   6. Applied egg-rr 76.9%

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

   7. Taylor expanded in x around inf 63.8%

      \[\leadsto \color{blue}{1.5 \cdot x} \]
   8. Step-by-step derivation

      1. *-commutative 63.8%

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

   9. Simplified 63.8%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{-126}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 41.8% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.45:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -0.45) (* x 0.5) (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.45) {
		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 (x <= (-0.45d0)) 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 (x <= -0.45) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.45:
		tmp = x * 0.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.45)
		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 (x <= -0.45)
		tmp = x * 0.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.45], N[(x * 0.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.450000000000000011

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 97.4%

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

      2. fabs-sqr 97.4%

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

      3. add-sqr-sqrt 98.2%

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

      4. sub-neg 98.2%

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

      5. flip-+ 43.0%

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

      6. pow2 43.0%

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

   4. Applied egg-rr 43.0%

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

   5. Taylor expanded in x around inf 94.4%

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

   if -0.450000000000000011 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 72.4%

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

   4. Taylor expanded in x around 0 67.4%

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

      1. rem-square-sqrt 34.9%

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

      2. fabs-sqr 34.9%

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

      3. rem-square-sqrt 35.9%

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

   6. Simplified 35.9%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.45:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 30.4% accurate, 35.7× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.5 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x 0.5))

C

double code(double x, double y) {
	return x * 0.5;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * 0.5d0
end function

Java

public static double code(double x, double y) {
	return x * 0.5;
}

Python

def code(x, y):
	return x * 0.5

Julia

function code(x, y)
	return Float64(x * 0.5)
end

MATLAB

function tmp = code(x, y)
	tmp = x * 0.5;
end

Wolfram

code[x_, y_] := N[(x * 0.5), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. add-sqr-sqrt 51.2%

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

   2. fabs-sqr 51.2%

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

   3. add-sqr-sqrt 55.9%

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

   4. sub-neg 55.9%

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

   5. flip-+ 29.0%

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

   6. pow2 29.0%

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

4. Applied egg-rr 29.0%

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

5. Taylor expanded in x around inf 29.3%

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

6. Final simplification 29.3%

   \[\leadsto x \cdot 0.5 \]
7. Add Preprocessing

Alternative 9: 11.4% 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 10.9%

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

Reproduce

herbie shell --seed 2024165 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))