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

Percentage Accurate: 99.9% → 99.9%

Time: 8.5s

Alternatives: 12

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 y around -inf 99.9%

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

      1. fabs-neg 99.9%

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

      2. mul-1-neg 99.9%

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

      3. sub-neg 99.9%

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

      4. fabs-sub 99.9%

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

      5. rem-square-sqrt 87.9%

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

      6. fabs-sqr 87.9%

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

      7. rem-square-sqrt 90.5%

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

   5. Simplified 90.5%

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

   6. Taylor expanded in x around 0 90.6%

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

      1. distribute-lft-out 90.6%

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

   8. Simplified 90.6%

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

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

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 (/ (* (+ x y) (* (+ x y) (/ 1.0 (+ x y)))) 2.0)))))

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 + (((x + y) * ((x + y) * (1.0 / (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 <= (-3d-21)) then
        tmp = 0.5d0 * (x + y)
    else if (x <= 1.18d-120) then
        tmp = x + (abs(y) / 2.0d0)
    else
        tmp = x + (((x + y) * ((x + y) * (1.0d0 / (x + y)))) / 2.0d0)
    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 + (((x + y) * ((x + y) * (1.0 / (x + y)))) / 2.0);
	}
	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 + (((x + y) * ((x + y) * (1.0 / (x + y)))) / 2.0)
	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(Float64(Float64(x + y) * Float64(Float64(x + y) * Float64(1.0 / Float64(x + y)))) / 2.0));
	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 + (((x + y) * ((x + y) * (1.0 / (x + y)))) / 2.0);
	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[(N[(N[(x + y), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] * N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $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 y around -inf 100.0%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-sub 100.0%

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

      5. rem-square-sqrt 82.9%

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

      6. fabs-sqr 82.9%

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

      7. rem-square-sqrt 83.6%

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

   5. Simplified 83.6%

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

   6. Taylor expanded in x around 0 83.6%

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

      1. distribute-lft-out 83.6%

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

   8. Simplified 83.6%

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

   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. 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 20.8%

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

      5. fabs-sqr 20.8%

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

      6. add-sqr-sqrt 33.7%

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

      7. flip-- 14.7%

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

      8. difference-of-squares 14.9%

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

      9. add-sqr-sqrt 14.9%

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

      10. sqrt-unprod 15.0%

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

      11. sqr-neg 15.0%

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

      12. sqrt-unprod 0.0%

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

      13. add-sqr-sqrt 49.1%

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

      14. sub-neg 49.1%

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

      15. pow2 49.1%

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

      16. sub-neg 49.1%

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

      17. add-sqr-sqrt 0.0%

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

      18. sqrt-unprod 49.1%

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

      19. sqr-neg 49.1%

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

      20. sqrt-unprod 48.9%

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

      21. add-sqr-sqrt 49.1%

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

   6. Applied egg-rr 49.1%

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

      1. div-inv 49.1%

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

      2. inv-pow 49.1%

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

      3. metadata-eval 49.1%

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

      4. pow-div 45.9%

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

      5. pow1 45.9%

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

      6. unpow2 45.9%

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

      7. associate-*l* 53.3%

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

      8. +-commutative 53.3%

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

      9. +-commutative 53.3%

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

      10. pow1 53.3%

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

      11. pow-div 92.1%

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

      12. metadata-eval 92.1%

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

      13. inv-pow 92.1%

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

      14. +-commutative 92.1%

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

   8. Applied egg-rr 92.1%

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

Alternative 4: 81.7% accurate, 0.9× speedup

Math

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

FPCore

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

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 + (((x + y) * ((x + y) * (1.0 / (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 <= (-3.3d-190)) then
        tmp = 0.5d0 * (x + y)
    else if (x <= 6d-114) then
        tmp = abs(y) * 0.5d0
    else
        tmp = x + (((x + y) * ((x + y) * (1.0d0 / (x + y)))) / 2.0d0)
    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 + (((x + y) * ((x + y) * (1.0 / (x + y)))) / 2.0);
	}
	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 + (((x + y) * ((x + y) * (1.0 / (x + y)))) / 2.0)
	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(Float64(Float64(x + y) * Float64(Float64(x + y) * Float64(1.0 / Float64(x + y)))) / 2.0));
	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 + (((x + y) * ((x + y) * (1.0 / (x + y)))) / 2.0);
	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[(N[(N[(x + y), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] * N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $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 y around -inf 100.0%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-sub 100.0%

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

      5. rem-square-sqrt 77.4%

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

      6. fabs-sqr 77.4%

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

      7. rem-square-sqrt 78.2%

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

   5. Simplified 78.2%

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

   6. Taylor expanded in x around 0 78.2%

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

      1. distribute-lft-out 78.2%

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

   8. Simplified 78.2%

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

   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. 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 20.8%

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

      5. fabs-sqr 20.8%

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

      6. add-sqr-sqrt 33.7%

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

      7. flip-- 14.7%

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

      8. difference-of-squares 14.9%

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

      9. add-sqr-sqrt 14.9%

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

      10. sqrt-unprod 15.0%

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

      11. sqr-neg 15.0%

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

      12. sqrt-unprod 0.0%

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

      13. add-sqr-sqrt 49.1%

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

      14. sub-neg 49.1%

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

      15. pow2 49.1%

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

      16. sub-neg 49.1%

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

      17. add-sqr-sqrt 0.0%

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

      18. sqrt-unprod 49.1%

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

      19. sqr-neg 49.1%

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

      20. sqrt-unprod 48.9%

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

      21. add-sqr-sqrt 49.1%

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

   6. Applied egg-rr 49.1%

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

      1. div-inv 49.1%

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

      2. inv-pow 49.1%

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

      3. metadata-eval 49.1%

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

      4. pow-div 45.9%

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

      5. pow1 45.9%

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

      6. unpow2 45.9%

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

      7. associate-*l* 53.3%

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

      8. +-commutative 53.3%

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

      9. +-commutative 53.3%

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

      10. pow1 53.3%

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

      11. pow-div 92.1%

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

      12. metadata-eval 92.1%

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

      13. inv-pow 92.1%

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

      14. +-commutative 92.1%

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

   8. Applied egg-rr 92.1%

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

4. Final simplification 84.7%

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

Alternative 5: 71.7% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-279}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+137}:\\ \;\;\;\;x + \frac{-1}{\frac{\frac{2}{y}}{-1 - \frac{x}{y}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.4e-279)
   (* 0.5 (+ x y))
   (if (<= x 9.5e+137)
     (+ x (/ -1.0 (/ (/ 2.0 y) (- -1.0 (/ x y)))))
     (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.4e-279) {
		tmp = 0.5 * (x + y);
	} else if (x <= 9.5e+137) {
		tmp = x + (-1.0 / ((2.0 / y) / (-1.0 - (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 <= (-4.4d-279)) then
        tmp = 0.5d0 * (x + y)
    else if (x <= 9.5d+137) then
        tmp = x + ((-1.0d0) / ((2.0d0 / y) / ((-1.0d0) - (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 <= -4.4e-279) {
		tmp = 0.5 * (x + y);
	} else if (x <= 9.5e+137) {
		tmp = x + (-1.0 / ((2.0 / y) / (-1.0 - (x / y))));
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.4e-279:
		tmp = 0.5 * (x + y)
	elif x <= 9.5e+137:
		tmp = x + (-1.0 / ((2.0 / y) / (-1.0 - (x / y))))
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.4e-279)
		tmp = Float64(0.5 * Float64(x + y));
	elseif (x <= 9.5e+137)
		tmp = Float64(x + Float64(-1.0 / Float64(Float64(2.0 / y) / Float64(-1.0 - 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 <= -4.4e-279)
		tmp = 0.5 * (x + y);
	elseif (x <= 9.5e+137)
		tmp = x + (-1.0 / ((2.0 / y) / (-1.0 - (x / y))));
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.4e-279], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.5e+137], N[(x + N[(-1.0 / N[(N[(2.0 / y), $MachinePrecision] / N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.40000000000000001e-279

   1. Initial program 100.0%

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

   3. Taylor expanded in y around -inf 100.0%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-sub 100.0%

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

      5. rem-square-sqrt 74.6%

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

      6. fabs-sqr 74.6%

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

      7. rem-square-sqrt 75.4%

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

   5. Simplified 75.4%

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

   6. Taylor expanded in x around 0 75.5%

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

      1. distribute-lft-out 75.5%

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

   8. Simplified 75.5%

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

   if -4.40000000000000001e-279 < x < 9.50000000000000031e137

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 97.6%

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

      1. mul-1-neg 97.6%

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

      2. unsub-neg 97.6%

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

   5. Simplified 97.6%

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

      1. clear-num 97.5%

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

      2. inv-pow 97.5%

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

      3. add-sqr-sqrt 33.9%

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

      4. fabs-sqr 33.9%

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

      5. add-sqr-sqrt 40.8%

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

      6. associate-/r* 40.8%

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

      7. div-inv 40.8%

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

      8. cancel-sign-sub-inv 40.8%

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

      9. add-sqr-sqrt 3.6%

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

      10. sqrt-unprod 66.4%

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

      11. sqr-neg 66.4%

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

      12. sqrt-unprod 65.6%

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

      13. add-sqr-sqrt 69.2%

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

      14. div-inv 69.2%

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

   7. Applied egg-rr 69.2%

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

      1. unpow-1 69.2%

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

   9. Simplified 69.2%

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

   if 9.50000000000000031e137 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 92.2%

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

      1. neg-mul-1 92.2%

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

   5. Simplified 92.2%

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

      1. clear-num 92.1%

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

      2. inv-pow 92.1%

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

      3. fabs-neg 92.1%

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

      4. add-sqr-sqrt 92.0%

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

      5. fabs-sqr 92.0%

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

      6. add-sqr-sqrt 92.1%

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

   7. Applied egg-rr 92.1%

      \[\leadsto x + \color{blue}{{\left(\frac{2}{x}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 92.1%

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

   9. Simplified 92.1%

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

   10. Taylor expanded in x around 0 92.2%

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

       1. *-commutative 92.2%

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

   12. Simplified 92.2%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-279}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+137}:\\ \;\;\;\;x + \frac{-1}{\frac{\frac{2}{y}}{-1 - \frac{x}{y}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.0% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-296}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \frac{1}{x + y}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1e-296)
   (* 0.5 (+ x y))
   (+ x (/ (* (+ x y) (* (+ x y) (/ 1.0 (+ x y)))) 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1e-296) {
		tmp = 0.5 * (x + y);
	} else {
		tmp = x + (((x + y) * ((x + y) * (1.0 / (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-296)) then
        tmp = 0.5d0 * (x + y)
    else
        tmp = x + (((x + y) * ((x + y) * (1.0d0 / (x + y)))) / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1e-296) {
		tmp = 0.5 * (x + y);
	} else {
		tmp = x + (((x + y) * ((x + y) * (1.0 / (x + y)))) / 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1e-296:
		tmp = 0.5 * (x + y)
	else:
		tmp = x + (((x + y) * ((x + y) * (1.0 / (x + y)))) / 2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1e-296)
		tmp = Float64(0.5 * Float64(x + y));
	else
		tmp = Float64(x + Float64(Float64(Float64(x + y) * Float64(Float64(x + y) * Float64(1.0 / Float64(x + y)))) / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1e-296)
		tmp = 0.5 * (x + y);
	else
		tmp = x + (((x + y) * ((x + y) * (1.0 / (x + y)))) / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1e-296], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(x + y), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] * N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1e-296

   1. Initial program 100.0%

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

   3. Taylor expanded in y around -inf 100.0%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-sub 100.0%

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

      5. rem-square-sqrt 74.4%

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

      6. fabs-sqr 74.4%

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

      7. rem-square-sqrt 75.3%

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

   5. Simplified 75.3%

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

   6. Taylor expanded in x around 0 75.3%

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

      1. distribute-lft-out 75.3%

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

   8. Simplified 75.3%

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

   if -1e-296 < 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. mul-fabs 98.4%

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

      2. unpow2 98.4%

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

      3. add-cube-cbrt 99.8%

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

      4. add-sqr-sqrt 24.0%

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

      5. fabs-sqr 24.0%

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

      6. add-sqr-sqrt 34.3%

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

      7. flip-- 16.9%

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

      8. difference-of-squares 17.0%

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

      9. add-sqr-sqrt 16.2%

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

      10. sqrt-unprod 17.1%

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

      11. sqr-neg 17.1%

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

      12. sqrt-unprod 0.8%

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

      13. add-sqr-sqrt 42.9%

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

      14. sub-neg 42.9%

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

      15. pow2 42.9%

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

      16. sub-neg 42.9%

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

      17. add-sqr-sqrt 0.8%

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

      18. sqrt-unprod 42.9%

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

      19. sqr-neg 42.9%

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

      20. sqrt-unprod 41.9%

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

      21. add-sqr-sqrt 42.9%

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

   6. Applied egg-rr 42.9%

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

      1. div-inv 42.9%

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

      2. inv-pow 42.9%

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

      3. metadata-eval 42.9%

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

      4. pow-div 39.7%

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

      5. pow1 39.7%

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

      6. unpow2 39.7%

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

      7. associate-*l* 45.3%

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

      8. +-commutative 45.3%

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

      9. +-commutative 45.3%

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

      10. pow1 45.3%

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

      11. pow-div 79.6%

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

      12. metadata-eval 79.6%

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

      13. inv-pow 79.6%

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

      14. +-commutative 79.6%

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

   8. Applied egg-rr 79.6%

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

Alternative 7: 71.7% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-279}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+137}:\\ \;\;\;\;y \cdot \left(0.5 + \frac{x}{y} \cdot 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.5e-279)
   (* 0.5 (+ x y))
   (if (<= x 9.5e+137) (* y (+ 0.5 (* (/ x y) 1.5))) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.5e-279) {
		tmp = 0.5 * (x + y);
	} else if (x <= 9.5e+137) {
		tmp = y * (0.5 + ((x / y) * 1.5));
	} 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 <= (-4.5d-279)) then
        tmp = 0.5d0 * (x + y)
    else if (x <= 9.5d+137) then
        tmp = y * (0.5d0 + ((x / y) * 1.5d0))
    else
        tmp = x * 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4.5e-279) {
		tmp = 0.5 * (x + y);
	} else if (x <= 9.5e+137) {
		tmp = y * (0.5 + ((x / y) * 1.5));
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.5e-279:
		tmp = 0.5 * (x + y)
	elif x <= 9.5e+137:
		tmp = y * (0.5 + ((x / y) * 1.5))
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.5e-279)
		tmp = Float64(0.5 * Float64(x + y));
	elseif (x <= 9.5e+137)
		tmp = Float64(y * Float64(0.5 + Float64(Float64(x / y) * 1.5)));
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.5e-279)
		tmp = 0.5 * (x + y);
	elseif (x <= 9.5e+137)
		tmp = y * (0.5 + ((x / y) * 1.5));
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.5e-279], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.5e+137], N[(y * N[(0.5 + N[(N[(x / y), $MachinePrecision] * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.49999999999999995e-279

   1. Initial program 100.0%

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

   3. Taylor expanded in y around -inf 100.0%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-sub 100.0%

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

      5. rem-square-sqrt 74.6%

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

      6. fabs-sqr 74.6%

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

      7. rem-square-sqrt 75.4%

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

   5. Simplified 75.4%

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

   6. Taylor expanded in x around 0 75.5%

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

      1. distribute-lft-out 75.5%

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

   8. Simplified 75.5%

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

   if -4.49999999999999995e-279 < x < 9.50000000000000031e137

   1. Initial program 99.9%

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

      1. add-cube-cbrt 98.1%

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

      2. fabs-mul 98.1%

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

      3. pow2 98.1%

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

   4. Applied egg-rr 98.1%

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

      1. mul-fabs 98.1%

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

      2. unpow2 98.1%

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

      3. add-cube-cbrt 99.9%

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

      4. add-sqr-sqrt 33.9%

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

      5. fabs-sqr 33.9%

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

      6. add-sqr-sqrt 41.2%

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

      7. flip-- 24.7%

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

      8. difference-of-squares 24.7%

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

      9. add-sqr-sqrt 21.1%

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

      10. sqrt-unprod 24.7%

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

      11. sqr-neg 24.7%

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

      12. sqrt-unprod 3.5%

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

      13. add-sqr-sqrt 52.1%

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

      14. sub-neg 52.1%

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

      15. pow2 52.1%

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

      16. sub-neg 52.1%

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

      17. add-sqr-sqrt 3.5%

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

      18. sqrt-unprod 52.1%

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

      19. sqr-neg 52.1%

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

      20. sqrt-unprod 48.4%

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

      21. add-sqr-sqrt 52.1%

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

   6. Applied egg-rr 52.1%

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

   7. Taylor expanded in y around inf 69.2%

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

      1. distribute-lft1-in 69.2%

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

      2. metadata-eval 69.2%

         \[\leadsto y \cdot \left(0.5 + \color{blue}{1.5} \cdot \frac{x}{y}\right) \]

   9. Simplified 69.2%

      \[\leadsto \color{blue}{y \cdot \left(0.5 + 1.5 \cdot \frac{x}{y}\right)} \]

   if 9.50000000000000031e137 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 92.2%

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

      1. neg-mul-1 92.2%

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

   5. Simplified 92.2%

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

      1. clear-num 92.1%

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

      2. inv-pow 92.1%

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

      3. fabs-neg 92.1%

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

      4. add-sqr-sqrt 92.0%

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

      5. fabs-sqr 92.0%

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

      6. add-sqr-sqrt 92.1%

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

   7. Applied egg-rr 92.1%

      \[\leadsto x + \color{blue}{{\left(\frac{2}{x}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 92.1%

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

   9. Simplified 92.1%

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

   10. Taylor expanded in x around 0 92.2%

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

       1. *-commutative 92.2%

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

   12. Simplified 92.2%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-279}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+137}:\\ \;\;\;\;y \cdot \left(0.5 + \frac{x}{y} \cdot 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.9% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-191}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{-132}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.7e-191) (* x 0.5) (if (<= x 9.6e-132) (* y 0.5) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.7e-191) {
		tmp = x * 0.5;
	} else if (x <= 9.6e-132) {
		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 <= (-2.7d-191)) then
        tmp = x * 0.5d0
    else if (x <= 9.6d-132) 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 <= -2.7e-191) {
		tmp = x * 0.5;
	} else if (x <= 9.6e-132) {
		tmp = y * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.7e-191:
		tmp = x * 0.5
	elif x <= 9.6e-132:
		tmp = y * 0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.7e-191)
		tmp = Float64(x * 0.5);
	elseif (x <= 9.6e-132)
		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 <= -2.7e-191)
		tmp = x * 0.5;
	elseif (x <= 9.6e-132)
		tmp = y * 0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.7e-191], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, 9.6e-132], N[(y * 0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.69999999999999999e-191

   1. Initial program 100.0%

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

   3. Taylor expanded in y around -inf 100.0%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-sub 100.0%

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

      5. rem-square-sqrt 76.8%

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

      6. fabs-sqr 76.8%

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

      7. rem-square-sqrt 77.6%

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

   5. Simplified 77.6%

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

   6. Taylor expanded in x around inf 61.5%

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

      1. *-commutative 61.5%

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

   8. Simplified 61.5%

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

   if -2.69999999999999999e-191 < x < 9.60000000000000062e-132

   1. Initial program 99.9%

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

   3. Taylor expanded in y around -inf 99.9%

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

      1. fabs-neg 99.9%

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

      2. mul-1-neg 99.9%

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

      3. sub-neg 99.9%

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

      4. fabs-sub 99.9%

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

      5. rem-square-sqrt 45.7%

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

      6. fabs-sqr 45.7%

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

      7. rem-square-sqrt 48.7%

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

   5. Simplified 48.7%

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

   6. Taylor expanded in x around 0 41.8%

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

      1. *-commutative 41.8%

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

   8. Simplified 41.8%

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

   if 9.60000000000000062e-132 < 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. Step-by-step derivation

      1. clear-num 71.1%

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

      2. inv-pow 71.1%

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

      3. fabs-neg 71.1%

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

      4. add-sqr-sqrt 71.0%

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

      5. fabs-sqr 71.0%

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

      6. add-sqr-sqrt 71.1%

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

   7. Applied egg-rr 71.1%

      \[\leadsto x + \color{blue}{{\left(\frac{2}{x}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 71.1%

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

   9. Simplified 71.1%

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

   10. Taylor expanded in x around 0 71.1%

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

       1. *-commutative 71.1%

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

   12. Simplified 71.1%

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

Alternative 9: 67.6% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-126}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 6e-126) (* 0.5 (+ x y)) (* x 1.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 6e-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 <= 6d-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 <= 6e-126) {
		tmp = 0.5 * (x + y);
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 6e-126:
		tmp = 0.5 * (x + y)
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 6e-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 <= 6e-126)
		tmp = 0.5 * (x + y);
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 6e-126], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.0000000000000003e-126

   1. Initial program 100.0%

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

   3. Taylor expanded in y around -inf 100.0%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-sub 100.0%

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

      5. rem-square-sqrt 67.1%

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

      6. fabs-sqr 67.1%

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

      7. rem-square-sqrt 68.6%

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

   5. Simplified 68.6%

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

   6. Taylor expanded in x around 0 68.6%

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

      1. distribute-lft-out 68.6%

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

   8. Simplified 68.6%

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

   if 6.0000000000000003e-126 < 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. Step-by-step derivation

      1. clear-num 71.1%

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

      2. inv-pow 71.1%

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

      3. fabs-neg 71.1%

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

      4. add-sqr-sqrt 71.0%

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

      5. fabs-sqr 71.0%

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

      6. add-sqr-sqrt 71.1%

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

   7. Applied egg-rr 71.1%

      \[\leadsto x + \color{blue}{{\left(\frac{2}{x}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 71.1%

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

   9. Simplified 71.1%

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

   10. Taylor expanded in x around 0 71.1%

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

       1. *-commutative 71.1%

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

   12. Simplified 71.1%

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

Alternative 10: 51.1% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -5e-310) (* x 0.5) (* x 1.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5e-310) {
		tmp = x * 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 <= (-5d-310)) then
        tmp = x * 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 <= -5e-310) {
		tmp = x * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5e-310:
		tmp = x * 0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(x * 0.5);
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5e-310)
		tmp = x * 0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5e-310], N[(x * 0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 100.0%

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

   3. Taylor expanded in y around -inf 100.0%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-sub 100.0%

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

      5. rem-square-sqrt 74.0%

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

      6. fabs-sqr 74.0%

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

      7. rem-square-sqrt 74.9%

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

   5. Simplified 74.9%

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

   6. Taylor expanded in x around inf 53.8%

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

      1. *-commutative 53.8%

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

   8. Simplified 53.8%

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

   if -4.999999999999985e-310 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 58.7%

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

      1. neg-mul-1 58.7%

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

   5. Simplified 58.7%

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

      1. clear-num 58.6%

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

      2. inv-pow 58.6%

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

      3. fabs-neg 58.6%

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

      4. add-sqr-sqrt 58.6%

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

      5. fabs-sqr 58.6%

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

      6. add-sqr-sqrt 58.6%

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

   7. Applied egg-rr 58.6%

      \[\leadsto x + \color{blue}{{\left(\frac{2}{x}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 58.6%

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

   9. Simplified 58.6%

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

   10. Taylor expanded in x around 0 58.7%

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

       1. *-commutative 58.7%

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

   12. Simplified 58.7%

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

Alternative 11: 30.3% 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. Taylor expanded in y around -inf 99.9%

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

   1. fabs-neg 99.9%

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

   2. mul-1-neg 99.9%

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

   3. sub-neg 99.9%

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

   4. fabs-sub 99.9%

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

   5. rem-square-sqrt 50.8%

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

   6. fabs-sqr 50.8%

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

   7. rem-square-sqrt 56.1%

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

5. Simplified 56.1%

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

6. Taylor expanded in x around inf 34.6%

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

   1. *-commutative 34.6%

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

8. Simplified 34.6%

   \[\leadsto \color{blue}{x \cdot 0.5} \]
9. Add Preprocessing

Alternative 12: 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)))