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

Percentage Accurate: 99.9% → 99.9%

Time: 13.6s

Alternatives: 13

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: 83.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-195}:\\ \;\;\;\;x - 0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-88}:\\ \;\;\;\;x + \frac{\left|x\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;x + \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \frac{1}{2 \cdot \left(x + y\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.1e-195)
   (- x (* 0.5 (+ x y)))
   (if (<= y 1.6e-88)
     (+ x (/ (fabs x) 2.0))
     (+ x (* (+ x y) (* (+ x y) (/ 1.0 (* 2.0 (+ x y)))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.1e-195) {
		tmp = x - (0.5 * (x + y));
	} else if (y <= 1.6e-88) {
		tmp = x + (fabs(x) / 2.0);
	} else {
		tmp = x + ((x + y) * ((x + y) * (1.0 / (2.0 * (x + y)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.1d-195)) then
        tmp = x - (0.5d0 * (x + y))
    else if (y <= 1.6d-88) then
        tmp = x + (abs(x) / 2.0d0)
    else
        tmp = x + ((x + y) * ((x + y) * (1.0d0 / (2.0d0 * (x + y)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.1e-195) {
		tmp = x - (0.5 * (x + y));
	} else if (y <= 1.6e-88) {
		tmp = x + (Math.abs(x) / 2.0);
	} else {
		tmp = x + ((x + y) * ((x + y) * (1.0 / (2.0 * (x + y)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.1e-195:
		tmp = x - (0.5 * (x + y))
	elif y <= 1.6e-88:
		tmp = x + (math.fabs(x) / 2.0)
	else:
		tmp = x + ((x + y) * ((x + y) * (1.0 / (2.0 * (x + y)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.1e-195)
		tmp = Float64(x - Float64(0.5 * Float64(x + y)));
	elseif (y <= 1.6e-88)
		tmp = Float64(x + Float64(abs(x) / 2.0));
	else
		tmp = Float64(x + Float64(Float64(x + y) * Float64(Float64(x + y) * Float64(1.0 / Float64(2.0 * Float64(x + y))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.1e-195)
		tmp = x - (0.5 * (x + y));
	elseif (y <= 1.6e-88)
		tmp = x + (abs(x) / 2.0);
	else
		tmp = x + ((x + y) * ((x + y) * (1.0 / (2.0 * (x + y)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.1e-195], N[(x - N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.6e-88], N[(x + N[(N[Abs[x], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(x + y), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] * N[(1.0 / N[(2.0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.10000000000000003e-195

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 85.2%

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

      1. *-inverses 85.2%

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

      2. div-sub 85.2%

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

      3. add-sqr-sqrt 42.3%

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

      4. associate-/r* 42.4%

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

   5. Applied egg-rr 42.4%

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

      1. div-inv 42.4%

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

      2. metadata-eval 42.4%

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

   7. Applied egg-rr 70.6%

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

      1. associate-*r* 70.6%

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

      2. associate-*r/ 55.9%

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

      3. *-commutative 55.9%

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

      4. associate-/l* 85.3%

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

      5. *-inverses 85.3%

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

      6. *-rgt-identity 85.3%

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

      7. *-commutative 85.3%

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

      8. +-commutative 85.3%

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

   9. Simplified 85.3%

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

   if -1.10000000000000003e-195 < y < 1.60000000000000006e-88

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 93.6%

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

      1. neg-mul-1 93.6%

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

   5. Simplified 93.6%

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

   if 1.60000000000000006e-88 < y

   1. Initial program 99.9%

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

      1. add-cube-cbrt 98.2%

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

      2. fabs-mul 98.2%

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

      3. pow2 98.2%

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

   4. Applied egg-rr 98.2%

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

      1. unpow2 98.2%

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

      2. fabs-sqr 98.2%

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

   6. Applied egg-rr 98.2%

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

      1. unpow2 98.2%

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

   8. Simplified 98.2%

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

      1. add-sqr-sqrt 82.2%

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

      2. fabs-sqr 82.2%

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

      3. unpow2 82.2%

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

      4. add-sqr-sqrt 85.1%

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

      5. add-cube-cbrt 86.7%

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

      6. sub-neg 86.7%

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

      7. add-sqr-sqrt 44.7%

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

      8. sqrt-unprod 79.0%

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

      9. sqr-neg 79.0%

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

      10. sqrt-unprod 53.8%

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

      11. add-sqr-sqrt 88.8%

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

      12. flip-+ 52.6%

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

      13. clear-num 52.5%

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

      14. sub-neg 52.5%

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

      15. add-sqr-sqrt 22.5%

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

      16. sqrt-unprod 50.8%

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

      17. sqr-neg 50.8%

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

      18. sqrt-unprod 28.7%

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

      19. add-sqr-sqrt 54.8%

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

      20. difference-of-squares 54.9%

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

      21. sub-neg 54.9%

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

   10. Applied egg-rr 52.9%

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

       1. associate-/r/ 52.9%

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

       2. *-commutative 52.9%

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

       3. +-commutative 52.9%

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

       4. +-commutative 52.9%

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

   12. Simplified 52.9%

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

       1. associate-/l* 52.9%

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

       2. unpow2 52.9%

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

       3. associate-*l* 88.7%

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

       4. +-commutative 88.7%

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

       5. +-commutative 88.7%

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

       6. associate-/l/ 88.7%

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

       7. +-commutative 88.7%

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

   14. Applied egg-rr 88.7%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-195}:\\ \;\;\;\;x - 0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-88}:\\ \;\;\;\;x + \frac{\left|x\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;x + \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \frac{1}{2 \cdot \left(x + y\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+14}:\\ \;\;\;\;x - 0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-34}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;x + \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \frac{1}{2 \cdot \left(x + y\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.75e+14)
   (- x (* 0.5 (+ x y)))
   (if (<= x 6.2e-34)
     (+ x (/ (fabs y) 2.0))
     (+ x (* (+ x y) (* (+ x y) (/ 1.0 (* 2.0 (+ x y)))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.75e+14) {
		tmp = x - (0.5 * (x + y));
	} else if (x <= 6.2e-34) {
		tmp = x + (fabs(y) / 2.0);
	} else {
		tmp = x + ((x + y) * ((x + y) * (1.0 / (2.0 * (x + y)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.75d+14)) then
        tmp = x - (0.5d0 * (x + y))
    else if (x <= 6.2d-34) then
        tmp = x + (abs(y) / 2.0d0)
    else
        tmp = x + ((x + y) * ((x + y) * (1.0d0 / (2.0d0 * (x + y)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.75e+14) {
		tmp = x - (0.5 * (x + y));
	} else if (x <= 6.2e-34) {
		tmp = x + (Math.abs(y) / 2.0);
	} else {
		tmp = x + ((x + y) * ((x + y) * (1.0 / (2.0 * (x + y)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.75e+14:
		tmp = x - (0.5 * (x + y))
	elif x <= 6.2e-34:
		tmp = x + (math.fabs(y) / 2.0)
	else:
		tmp = x + ((x + y) * ((x + y) * (1.0 / (2.0 * (x + y)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.75e+14)
		tmp = Float64(x - Float64(0.5 * Float64(x + y)));
	elseif (x <= 6.2e-34)
		tmp = Float64(x + Float64(abs(y) / 2.0));
	else
		tmp = Float64(x + Float64(Float64(x + y) * Float64(Float64(x + y) * Float64(1.0 / Float64(2.0 * Float64(x + y))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.75e+14)
		tmp = x - (0.5 * (x + y));
	elseif (x <= 6.2e-34)
		tmp = x + (abs(y) / 2.0);
	else
		tmp = x + ((x + y) * ((x + y) * (1.0 / (2.0 * (x + y)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.75e+14], N[(x - N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.2e-34], N[(x + N[(N[Abs[y], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(x + y), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] * N[(1.0 / N[(2.0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.75e14

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 99.9%

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

      1. *-inverses 99.9%

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

      2. div-sub 99.9%

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

      3. add-sqr-sqrt 0.0%

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

      4. associate-/r* 0.0%

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

   5. Applied egg-rr 0.0%

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

      1. div-inv 0.0%

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

      2. metadata-eval 0.0%

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

   7. Applied egg-rr 92.1%

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

      1. associate-*r* 92.1%

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

      2. associate-*r/ 36.3%

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

      3. *-commutative 36.3%

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

      4. associate-/l* 92.1%

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

      5. *-inverses 92.1%

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

      6. *-rgt-identity 92.1%

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

      7. *-commutative 92.1%

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

      8. +-commutative 92.1%

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

   9. Simplified 92.1%

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

   if -1.75e14 < x < 6.1999999999999996e-34

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 84.1%

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

   if 6.1999999999999996e-34 < x

   1. Initial program 99.7%

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

      1. add-cube-cbrt 99.0%

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

      2. fabs-mul 99.0%

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

      3. pow2 99.0%

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

   4. Applied egg-rr 99.0%

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

      1. unpow2 99.0%

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

      2. fabs-sqr 99.0%

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

   6. Applied egg-rr 99.0%

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

      1. unpow2 99.0%

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

   8. Simplified 99.0%

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

      1. add-sqr-sqrt 15.5%

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

      2. fabs-sqr 15.5%

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

      3. unpow2 15.5%

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

      4. add-sqr-sqrt 28.0%

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

      5. add-cube-cbrt 28.2%

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

      6. sub-neg 28.2%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 48.9%

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

      9. sqr-neg 48.9%

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

      10. sqrt-unprod 82.4%

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

      11. add-sqr-sqrt 82.4%

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

      12. flip-+ 40.7%

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

      13. clear-num 40.7%

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

      14. sub-neg 40.7%

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

      15. add-sqr-sqrt 0.0%

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

      16. sqrt-unprod 14.2%

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

      17. sqr-neg 14.2%

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

      18. sqrt-unprod 14.2%

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

      19. add-sqr-sqrt 14.2%

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

      20. difference-of-squares 14.2%

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

      21. sub-neg 14.2%

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

   10. Applied egg-rr 41.1%

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

       1. associate-/r/ 41.1%

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

       2. *-commutative 41.1%

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

       3. +-commutative 41.1%

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

       4. +-commutative 41.1%

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

   12. Simplified 41.1%

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

       1. associate-/l* 41.1%

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

       2. unpow2 41.1%

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

       3. associate-*l* 82.4%

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

       4. +-commutative 82.4%

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

       5. +-commutative 82.4%

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

       6. associate-/l/ 82.4%

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

       7. +-commutative 82.4%

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

   14. Applied egg-rr 82.4%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+14}:\\ \;\;\;\;x - 0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-34}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;x + \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \frac{1}{2 \cdot \left(x + y\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-70}:\\ \;\;\;\;x - 0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-31}:\\ \;\;\;\;0.5 \cdot \left|y\right|\\ \mathbf{else}:\\ \;\;\;\;x + \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \frac{1}{2 \cdot \left(x + y\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.45e-70)
   (- x (* 0.5 (+ x y)))
   (if (<= x 1.35e-31)
     (* 0.5 (fabs y))
     (+ x (* (+ x y) (* (+ x y) (/ 1.0 (* 2.0 (+ x y)))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.45e-70) {
		tmp = x - (0.5 * (x + y));
	} else if (x <= 1.35e-31) {
		tmp = 0.5 * fabs(y);
	} else {
		tmp = x + ((x + y) * ((x + y) * (1.0 / (2.0 * (x + y)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.45d-70)) then
        tmp = x - (0.5d0 * (x + y))
    else if (x <= 1.35d-31) then
        tmp = 0.5d0 * abs(y)
    else
        tmp = x + ((x + y) * ((x + y) * (1.0d0 / (2.0d0 * (x + y)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.45e-70) {
		tmp = x - (0.5 * (x + y));
	} else if (x <= 1.35e-31) {
		tmp = 0.5 * Math.abs(y);
	} else {
		tmp = x + ((x + y) * ((x + y) * (1.0 / (2.0 * (x + y)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.45e-70:
		tmp = x - (0.5 * (x + y))
	elif x <= 1.35e-31:
		tmp = 0.5 * math.fabs(y)
	else:
		tmp = x + ((x + y) * ((x + y) * (1.0 / (2.0 * (x + y)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.45e-70)
		tmp = Float64(x - Float64(0.5 * Float64(x + y)));
	elseif (x <= 1.35e-31)
		tmp = Float64(0.5 * abs(y));
	else
		tmp = Float64(x + Float64(Float64(x + y) * Float64(Float64(x + y) * Float64(1.0 / Float64(2.0 * Float64(x + y))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.45e-70)
		tmp = x - (0.5 * (x + y));
	elseif (x <= 1.35e-31)
		tmp = 0.5 * abs(y);
	else
		tmp = x + ((x + y) * ((x + y) * (1.0 / (2.0 * (x + y)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.45e-70], N[(x - N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.35e-31], N[(0.5 * N[Abs[y], $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(x + y), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] * N[(1.0 / N[(2.0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.44999999999999986e-70

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 98.7%

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

      1. *-inverses 98.7%

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

      2. div-sub 98.7%

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

      3. add-sqr-sqrt 0.0%

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

      4. associate-/r* 0.0%

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

   5. Applied egg-rr 0.0%

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

      1. div-inv 0.0%

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

      2. metadata-eval 0.0%

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

   7. Applied egg-rr 86.7%

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

      1. associate-*r* 86.7%

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

      2. associate-*r/ 42.1%

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

      3. *-commutative 42.1%

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

      4. associate-/l* 86.7%

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

      5. *-inverses 86.7%

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

      6. *-rgt-identity 86.7%

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

      7. *-commutative 86.7%

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

      8. +-commutative 86.7%

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

   9. Simplified 86.7%

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

   if -1.44999999999999986e-70 < x < 1.35000000000000007e-31

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 86.6%

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

   4. Taylor expanded in x around 0 84.4%

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

   if 1.35000000000000007e-31 < x

   1. Initial program 99.7%

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

      1. add-cube-cbrt 99.0%

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

      2. fabs-mul 99.0%

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

      3. pow2 99.0%

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

   4. Applied egg-rr 99.0%

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

      1. unpow2 99.0%

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

      2. fabs-sqr 99.0%

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

   6. Applied egg-rr 99.0%

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

      1. unpow2 99.0%

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

   8. Simplified 99.0%

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

      1. add-sqr-sqrt 15.5%

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

      2. fabs-sqr 15.5%

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

      3. unpow2 15.5%

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

      4. add-sqr-sqrt 28.0%

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

      5. add-cube-cbrt 28.2%

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

      6. sub-neg 28.2%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 48.9%

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

      9. sqr-neg 48.9%

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

      10. sqrt-unprod 82.4%

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

      11. add-sqr-sqrt 82.4%

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

      12. flip-+ 40.7%

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

      13. clear-num 40.7%

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

      14. sub-neg 40.7%

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

      15. add-sqr-sqrt 0.0%

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

      16. sqrt-unprod 14.2%

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

      17. sqr-neg 14.2%

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

      18. sqrt-unprod 14.2%

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

      19. add-sqr-sqrt 14.2%

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

      20. difference-of-squares 14.2%

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

      21. sub-neg 14.2%

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

   10. Applied egg-rr 41.1%

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

       1. associate-/r/ 41.1%

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

       2. *-commutative 41.1%

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

       3. +-commutative 41.1%

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

       4. +-commutative 41.1%

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

   12. Simplified 41.1%

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

       1. associate-/l* 41.1%

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

       2. unpow2 41.1%

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

       3. associate-*l* 82.4%

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

       4. +-commutative 82.4%

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

       5. +-commutative 82.4%

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

       6. associate-/l/ 82.4%

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

       7. +-commutative 82.4%

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

   14. Applied egg-rr 82.4%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-70}:\\ \;\;\;\;x - 0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-31}:\\ \;\;\;\;0.5 \cdot \left|y\right|\\ \mathbf{else}:\\ \;\;\;\;x + \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \frac{1}{2 \cdot \left(x + y\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.1% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-208}:\\ \;\;\;\;x - 0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \frac{1}{2 \cdot \left(x + y\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9e-208)
   (- x (* 0.5 (+ x y)))
   (+ x (* (+ x y) (* (+ x y) (/ 1.0 (* 2.0 (+ x y))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9e-208) {
		tmp = x - (0.5 * (x + y));
	} else {
		tmp = x + ((x + y) * ((x + y) * (1.0 / (2.0 * (x + y)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-9d-208)) then
        tmp = x - (0.5d0 * (x + y))
    else
        tmp = x + ((x + y) * ((x + y) * (1.0d0 / (2.0d0 * (x + y)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -9e-208) {
		tmp = x - (0.5 * (x + y));
	} else {
		tmp = x + ((x + y) * ((x + y) * (1.0 / (2.0 * (x + y)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -9e-208:
		tmp = x - (0.5 * (x + y))
	else:
		tmp = x + ((x + y) * ((x + y) * (1.0 / (2.0 * (x + y)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9e-208)
		tmp = Float64(x - Float64(0.5 * Float64(x + y)));
	else
		tmp = Float64(x + Float64(Float64(x + y) * Float64(Float64(x + y) * Float64(1.0 / Float64(2.0 * Float64(x + y))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -9e-208)
		tmp = x - (0.5 * (x + y));
	else
		tmp = x + ((x + y) * ((x + y) * (1.0 / (2.0 * (x + y)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9e-208], N[(x - N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(x + y), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] * N[(1.0 / N[(2.0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.9999999999999992e-208

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 85.5%

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

      1. *-inverses 85.5%

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

      2. div-sub 85.5%

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

      3. add-sqr-sqrt 41.5%

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

      4. associate-/r* 41.6%

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

   5. Applied egg-rr 41.6%

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

      1. div-inv 41.6%

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

      2. metadata-eval 41.6%

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

   7. Applied egg-rr 71.1%

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

      1. associate-*r* 71.1%

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

      2. associate-*r/ 56.8%

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

      3. *-commutative 56.8%

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

      4. associate-/l* 85.6%

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

      5. *-inverses 85.6%

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

      6. *-rgt-identity 85.6%

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

      7. *-commutative 85.6%

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

      8. +-commutative 85.6%

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

   9. Simplified 85.6%

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

   if -8.9999999999999992e-208 < y

   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%

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

      1. unpow2 98.3%

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

      2. fabs-sqr 98.3%

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

   6. Applied egg-rr 98.3%

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

      1. unpow2 98.3%

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

   8. Simplified 98.3%

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

      1. add-sqr-sqrt 65.7%

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

      2. fabs-sqr 65.7%

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

      3. unpow2 65.7%

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

      4. add-sqr-sqrt 71.4%

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

      5. add-cube-cbrt 72.7%

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

      6. sub-neg 72.7%

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

      7. add-sqr-sqrt 43.9%

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

      8. sqrt-unprod 70.3%

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

      9. sqr-neg 70.3%

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

      10. sqrt-unprod 53.4%

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

      11. add-sqr-sqrt 77.3%

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

      12. flip-+ 48.2%

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

      13. clear-num 48.1%

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

      14. sub-neg 48.1%

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

      15. add-sqr-sqrt 15.3%

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

      16. sqrt-unprod 34.9%

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

      17. sqr-neg 34.9%

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

      18. sqrt-unprod 20.4%

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

      19. add-sqr-sqrt 44.2%

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

      20. difference-of-squares 44.2%

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

      21. sub-neg 44.2%

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

   10. Applied egg-rr 48.3%

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

       1. associate-/r/ 48.4%

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

       2. *-commutative 48.4%

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

       3. +-commutative 48.4%

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

       4. +-commutative 48.4%

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

   12. Simplified 48.4%

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

       1. associate-/l* 48.4%

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

       2. unpow2 48.4%

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

       3. associate-*l* 77.2%

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

       4. +-commutative 77.2%

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

       5. +-commutative 77.2%

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

       6. associate-/l/ 77.2%

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

       7. +-commutative 77.2%

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

   14. Applied egg-rr 77.2%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-208}:\\ \;\;\;\;x - 0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \frac{1}{2 \cdot \left(x + y\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.3% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(x + y\right)\\ \mathbf{if}\;y \leq -4 \cdot 10^{-208}:\\ \;\;\;\;x - t\_0\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{-245}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ x y))))
   (if (<= y -4e-208) (- x t_0) (if (<= y 7.4e-245) (* x 1.5) t_0))))

C

double code(double x, double y) {
	double t_0 = 0.5 * (x + y);
	double tmp;
	if (y <= -4e-208) {
		tmp = x - t_0;
	} else if (y <= 7.4e-245) {
		tmp = x * 1.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (x + y)
    if (y <= (-4d-208)) then
        tmp = x - t_0
    else if (y <= 7.4d-245) then
        tmp = x * 1.5d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.5 * (x + y);
	double tmp;
	if (y <= -4e-208) {
		tmp = x - t_0;
	} else if (y <= 7.4e-245) {
		tmp = x * 1.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.5 * (x + y)
	tmp = 0
	if y <= -4e-208:
		tmp = x - t_0
	elif y <= 7.4e-245:
		tmp = x * 1.5
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.5 * Float64(x + y))
	tmp = 0.0
	if (y <= -4e-208)
		tmp = Float64(x - t_0);
	elseif (y <= 7.4e-245)
		tmp = Float64(x * 1.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.5 * (x + y);
	tmp = 0.0;
	if (y <= -4e-208)
		tmp = x - t_0;
	elseif (y <= 7.4e-245)
		tmp = x * 1.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4e-208], N[(x - t$95$0), $MachinePrecision], If[LessEqual[y, 7.4e-245], N[(x * 1.5), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.0000000000000004e-208

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 85.5%

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

      1. *-inverses 85.5%

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

      2. div-sub 85.5%

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

      3. add-sqr-sqrt 41.5%

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

      4. associate-/r* 41.6%

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

   5. Applied egg-rr 41.6%

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

      1. div-inv 41.6%

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

      2. metadata-eval 41.6%

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

   7. Applied egg-rr 71.1%

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

      1. associate-*r* 71.1%

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

      2. associate-*r/ 56.8%

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

      3. *-commutative 56.8%

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

      4. associate-/l* 85.6%

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

      5. *-inverses 85.6%

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

      6. *-rgt-identity 85.6%

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

      7. *-commutative 85.6%

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

      8. +-commutative 85.6%

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

   9. Simplified 85.6%

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

   if -4.0000000000000004e-208 < y < 7.4000000000000005e-245

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 91.6%

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

      1. neg-mul-1 91.6%

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

   5. Simplified 91.6%

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

   6. Taylor expanded in x around 0 91.6%

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

      1. *-rgt-identity 91.6%

         \[\leadsto \color{blue}{x \cdot 1} + 0.5 \cdot \left|-x\right| \]

      2. *-commutative 91.6%

         \[\leadsto x \cdot 1 + \color{blue}{\left|-x\right| \cdot 0.5} \]

      3. fabs-neg 91.6%

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

      4. rem-square-sqrt 66.2%

         \[\leadsto x \cdot 1 + \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot 0.5 \]

      5. fabs-sqr 66.2%

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

      6. rem-square-sqrt 71.0%

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

      7. metadata-eval 71.0%

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

      8. *-inverses 71.0%

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

      9. rem-square-sqrt 66.2%

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

      10. fabs-sqr 66.2%

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

      11. rem-square-sqrt 91.6%

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

      12. fabs-neg 91.6%

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

      13. distribute-lft-in 91.6%

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

      14. fabs-neg 91.6%

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

      15. rem-square-sqrt 66.3%

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

      16. fabs-sqr 66.3%

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

      17. rem-square-sqrt 71.0%

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

      18. *-inverses 71.0%

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

      19. metadata-eval 71.0%

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

      20. metadata-eval 71.0%

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

   8. Simplified 71.0%

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

   if 7.4000000000000005e-245 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 84.7%

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

      1. +-commutative 84.7%

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

      2. sub-neg 84.7%

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

      3. neg-mul-1 84.7%

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

      4. fma-define 84.7%

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

      5. neg-mul-1 84.7%

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

      6. sub-neg 84.7%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\left|\color{blue}{y - x}\right|}{x}, 1\right) \]

      7. rem-square-sqrt 60.9%

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

      8. fabs-sqr 60.9%

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

      9. rem-square-sqrt 65.6%

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

   5. Simplified 65.6%

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

   6. Taylor expanded in x around 0 80.8%

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

      1. distribute-lft-out 80.8%

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

      2. +-commutative 80.8%

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

   8. Simplified 80.8%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-208}:\\ \;\;\;\;x - 0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{-245}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.5% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-207}:\\ \;\;\;\;x - y \cdot 0.5\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-256}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.5e-207)
   (- x (* y 0.5))
   (if (<= y 1.2e-256) (* x 1.5) (* 0.5 (+ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.5e-207) {
		tmp = x - (y * 0.5);
	} else if (y <= 1.2e-256) {
		tmp = x * 1.5;
	} 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 <= (-2.5d-207)) then
        tmp = x - (y * 0.5d0)
    else if (y <= 1.2d-256) then
        tmp = x * 1.5d0
    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 <= -2.5e-207) {
		tmp = x - (y * 0.5);
	} else if (y <= 1.2e-256) {
		tmp = x * 1.5;
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.5e-207:
		tmp = x - (y * 0.5)
	elif y <= 1.2e-256:
		tmp = x * 1.5
	else:
		tmp = 0.5 * (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.5e-207)
		tmp = Float64(x - Float64(y * 0.5));
	elseif (y <= 1.2e-256)
		tmp = Float64(x * 1.5);
	else
		tmp = Float64(0.5 * Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.5e-207)
		tmp = x - (y * 0.5);
	elseif (y <= 1.2e-256)
		tmp = x * 1.5;
	else
		tmp = 0.5 * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.5e-207], N[(x - N[(y * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.2e-256], N[(x * 1.5), $MachinePrecision], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.50000000000000007e-207

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 85.5%

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

      1. *-inverses 85.5%

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

      2. div-sub 85.5%

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

      3. add-sqr-sqrt 41.5%

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

      4. associate-/r* 41.6%

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

   5. Applied egg-rr 41.6%

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

      1. div-inv 41.6%

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

      2. metadata-eval 41.6%

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

   7. Applied egg-rr 71.1%

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

      1. associate-*r* 71.1%

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

      2. associate-*r/ 56.8%

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

      3. *-commutative 56.8%

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

      4. associate-/l* 85.6%

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

      5. *-inverses 85.6%

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

      6. *-rgt-identity 85.6%

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

      7. *-commutative 85.6%

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

      8. +-commutative 85.6%

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

   9. Simplified 85.6%

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

   10. Taylor expanded in x around 0 70.2%

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

   if -2.50000000000000007e-207 < y < 1.2e-256

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 91.6%

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

      1. neg-mul-1 91.6%

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

   5. Simplified 91.6%

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

   6. Taylor expanded in x around 0 91.6%

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

      1. *-rgt-identity 91.6%

         \[\leadsto \color{blue}{x \cdot 1} + 0.5 \cdot \left|-x\right| \]

      2. *-commutative 91.6%

         \[\leadsto x \cdot 1 + \color{blue}{\left|-x\right| \cdot 0.5} \]

      3. fabs-neg 91.6%

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

      4. rem-square-sqrt 66.2%

         \[\leadsto x \cdot 1 + \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot 0.5 \]

      5. fabs-sqr 66.2%

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

      6. rem-square-sqrt 71.0%

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

      7. metadata-eval 71.0%

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

      8. *-inverses 71.0%

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

      9. rem-square-sqrt 66.2%

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

      10. fabs-sqr 66.2%

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

      11. rem-square-sqrt 91.6%

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

      12. fabs-neg 91.6%

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

      13. distribute-lft-in 91.6%

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

      14. fabs-neg 91.6%

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

      15. rem-square-sqrt 66.3%

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

      16. fabs-sqr 66.3%

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

      17. rem-square-sqrt 71.0%

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

      18. *-inverses 71.0%

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

      19. metadata-eval 71.0%

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

      20. metadata-eval 71.0%

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

   8. Simplified 71.0%

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

   if 1.2e-256 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 84.7%

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

      1. +-commutative 84.7%

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

      2. sub-neg 84.7%

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

      3. neg-mul-1 84.7%

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

      4. fma-define 84.7%

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

      5. neg-mul-1 84.7%

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

      6. sub-neg 84.7%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\left|\color{blue}{y - x}\right|}{x}, 1\right) \]

      7. rem-square-sqrt 60.9%

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

      8. fabs-sqr 60.9%

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

      9. rem-square-sqrt 65.6%

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

   5. Simplified 65.6%

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

   6. Taylor expanded in x around 0 80.8%

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

      1. distribute-lft-out 80.8%

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

      2. +-commutative 80.8%

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

   8. Simplified 80.8%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-207}:\\ \;\;\;\;x - y \cdot 0.5\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-256}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.9% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-195}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 10^{-249}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.3e-195) (* y -0.5) (if (<= y 1e-249) (* x 1.5) (* 0.5 (+ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.3e-195) {
		tmp = y * -0.5;
	} else if (y <= 1e-249) {
		tmp = x * 1.5;
	} 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 <= (-3.3d-195)) then
        tmp = y * (-0.5d0)
    else if (y <= 1d-249) then
        tmp = x * 1.5d0
    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 <= -3.3e-195) {
		tmp = y * -0.5;
	} else if (y <= 1e-249) {
		tmp = x * 1.5;
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.3e-195:
		tmp = y * -0.5
	elif y <= 1e-249:
		tmp = x * 1.5
	else:
		tmp = 0.5 * (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.3e-195)
		tmp = Float64(y * -0.5);
	elseif (y <= 1e-249)
		tmp = Float64(x * 1.5);
	else
		tmp = Float64(0.5 * Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.3e-195)
		tmp = y * -0.5;
	elseif (y <= 1e-249)
		tmp = x * 1.5;
	else
		tmp = 0.5 * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.3e-195], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 1e-249], N[(x * 1.5), $MachinePrecision], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.3e-195

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 85.2%

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

      1. *-inverses 85.2%

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

      2. div-sub 85.2%

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

      3. add-sqr-sqrt 42.3%

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

      4. associate-/r* 42.4%

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

   5. Applied egg-rr 42.4%

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

      1. div-inv 42.4%

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

      2. metadata-eval 42.4%

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

   7. Applied egg-rr 70.6%

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

      1. associate-*r* 70.6%

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

      2. associate-*r/ 55.9%

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

      3. *-commutative 55.9%

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

      4. associate-/l* 85.3%

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

      5. *-inverses 85.3%

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

      6. *-rgt-identity 85.3%

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

      7. *-commutative 85.3%

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

      8. +-commutative 85.3%

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

   9. Simplified 85.3%

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

   10. Taylor expanded in x around 0 65.6%

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

       1. *-commutative 65.6%

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

   12. Simplified 65.6%

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

   if -3.3e-195 < y < 1.00000000000000005e-249

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 92.1%

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

      1. neg-mul-1 92.1%

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

   5. Simplified 92.1%

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

   6. Taylor expanded in x around 0 92.1%

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

      1. *-rgt-identity 92.1%

         \[\leadsto \color{blue}{x \cdot 1} + 0.5 \cdot \left|-x\right| \]

      2. *-commutative 92.1%

         \[\leadsto x \cdot 1 + \color{blue}{\left|-x\right| \cdot 0.5} \]

      3. fabs-neg 92.1%

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

      4. rem-square-sqrt 61.8%

         \[\leadsto x \cdot 1 + \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot 0.5 \]

      5. fabs-sqr 61.8%

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

      6. rem-square-sqrt 67.4%

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

      7. metadata-eval 67.4%

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

      8. *-inverses 67.4%

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

      9. rem-square-sqrt 61.8%

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

      10. fabs-sqr 61.8%

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

      11. rem-square-sqrt 92.1%

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

      12. fabs-neg 92.1%

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

      13. distribute-lft-in 92.1%

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

      14. fabs-neg 92.1%

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

      15. rem-square-sqrt 61.9%

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

      16. fabs-sqr 61.9%

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

      17. rem-square-sqrt 67.4%

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

      18. *-inverses 67.4%

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

      19. metadata-eval 67.4%

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

      20. metadata-eval 67.4%

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

   8. Simplified 67.4%

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

   if 1.00000000000000005e-249 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 84.7%

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

      1. +-commutative 84.7%

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

      2. sub-neg 84.7%

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

      3. neg-mul-1 84.7%

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

      4. fma-define 84.7%

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

      5. neg-mul-1 84.7%

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

      6. sub-neg 84.7%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\left|\color{blue}{y - x}\right|}{x}, 1\right) \]

      7. rem-square-sqrt 60.9%

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

      8. fabs-sqr 60.9%

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

      9. rem-square-sqrt 65.6%

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

   5. Simplified 65.6%

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

   6. Taylor expanded in x around 0 80.8%

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

      1. distribute-lft-out 80.8%

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

      2. +-commutative 80.8%

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

   8. Simplified 80.8%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-195}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 10^{-249}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.1% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-195}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 6.9 \cdot 10^{-77}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.3e-195) (* y -0.5) (if (<= y 6.9e-77) (* x 1.5) (* y 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.3e-195) {
		tmp = y * -0.5;
	} else if (y <= 6.9e-77) {
		tmp = x * 1.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3.3d-195)) then
        tmp = y * (-0.5d0)
    else if (y <= 6.9d-77) then
        tmp = x * 1.5d0
    else
        tmp = y * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.3e-195) {
		tmp = y * -0.5;
	} else if (y <= 6.9e-77) {
		tmp = x * 1.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.3e-195:
		tmp = y * -0.5
	elif y <= 6.9e-77:
		tmp = x * 1.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.3e-195)
		tmp = Float64(y * -0.5);
	elseif (y <= 6.9e-77)
		tmp = Float64(x * 1.5);
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.3e-195)
		tmp = y * -0.5;
	elseif (y <= 6.9e-77)
		tmp = x * 1.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.3e-195], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 6.9e-77], N[(x * 1.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.3e-195

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 85.2%

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

      1. *-inverses 85.2%

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

      2. div-sub 85.2%

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

      3. add-sqr-sqrt 42.3%

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

      4. associate-/r* 42.4%

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

   5. Applied egg-rr 42.4%

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

      1. div-inv 42.4%

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

      2. metadata-eval 42.4%

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

   7. Applied egg-rr 70.6%

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

      1. associate-*r* 70.6%

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

      2. associate-*r/ 55.9%

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

      3. *-commutative 55.9%

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

      4. associate-/l* 85.3%

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

      5. *-inverses 85.3%

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

      6. *-rgt-identity 85.3%

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

      7. *-commutative 85.3%

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

      8. +-commutative 85.3%

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

   9. Simplified 85.3%

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

   10. Taylor expanded in x around 0 65.6%

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

       1. *-commutative 65.6%

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

   12. Simplified 65.6%

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

   if -3.3e-195 < y < 6.90000000000000034e-77

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 92.4%

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

      1. neg-mul-1 92.4%

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

   5. Simplified 92.4%

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

   6. Taylor expanded in x around 0 92.4%

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

      1. *-rgt-identity 92.4%

         \[\leadsto \color{blue}{x \cdot 1} + 0.5 \cdot \left|-x\right| \]

      2. *-commutative 92.4%

         \[\leadsto x \cdot 1 + \color{blue}{\left|-x\right| \cdot 0.5} \]

      3. fabs-neg 92.4%

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

      4. rem-square-sqrt 50.8%

         \[\leadsto x \cdot 1 + \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot 0.5 \]

      5. fabs-sqr 50.8%

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

      6. rem-square-sqrt 58.4%

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

      7. metadata-eval 58.4%

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

      8. *-inverses 58.4%

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

      9. rem-square-sqrt 50.8%

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

      10. fabs-sqr 50.8%

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

      11. rem-square-sqrt 92.4%

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

      12. fabs-neg 92.4%

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

      13. distribute-lft-in 92.4%

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

      14. fabs-neg 92.4%

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

      15. rem-square-sqrt 50.9%

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

      16. fabs-sqr 50.9%

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

      17. rem-square-sqrt 58.4%

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

      18. *-inverses 58.4%

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

      19. metadata-eval 58.4%

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

      20. metadata-eval 58.4%

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

   8. Simplified 58.4%

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

   if 6.90000000000000034e-77 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 77.3%

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

   4. Taylor expanded in x around 0 73.3%

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

      1. rem-square-sqrt 72.7%

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

      2. fabs-sqr 72.7%

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

      3. rem-square-sqrt 73.3%

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

   6. Simplified 73.3%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-195}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 6.9 \cdot 10^{-77}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.1% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-70}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-71}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.5e-70) (* x 0.5) (if (<= x 7e-71) (* y 0.5) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.5e-70) {
		tmp = x * 0.5;
	} else if (x <= 7e-71) {
		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 <= (-4.5d-70)) then
        tmp = x * 0.5d0
    else if (x <= 7d-71) 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 <= -4.5e-70) {
		tmp = x * 0.5;
	} else if (x <= 7e-71) {
		tmp = y * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.5e-70:
		tmp = x * 0.5
	elif x <= 7e-71:
		tmp = y * 0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.5e-70)
		tmp = Float64(x * 0.5);
	elseif (x <= 7e-71)
		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 <= -4.5e-70)
		tmp = x * 0.5;
	elseif (x <= 7e-71)
		tmp = y * 0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.5e-70], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, 7e-71], N[(y * 0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.50000000000000022e-70

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 98.7%

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

      1. +-commutative 98.7%

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

      2. sub-neg 98.7%

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

      3. neg-mul-1 98.7%

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

      4. fma-define 98.7%

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

      5. neg-mul-1 98.7%

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

      6. sub-neg 98.7%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\left|\color{blue}{y - x}\right|}{x}, 1\right) \]

      7. rem-square-sqrt 79.3%

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

      8. fabs-sqr 79.3%

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

      9. rem-square-sqrt 79.9%

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

   5. Simplified 79.9%

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

   6. Taylor expanded in y around 0 68.5%

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

   if -4.50000000000000022e-70 < x < 6.9999999999999998e-71

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 87.8%

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

   4. Taylor expanded in x around 0 85.9%

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

      1. rem-square-sqrt 46.4%

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

      2. fabs-sqr 46.4%

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

      3. rem-square-sqrt 47.8%

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

   6. Simplified 47.8%

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

   if 6.9999999999999998e-71 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 66.6%

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

      1. neg-mul-1 66.6%

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

   5. Simplified 66.6%

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

   6. Taylor expanded in x around 0 66.6%

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

      1. *-rgt-identity 66.6%

         \[\leadsto \color{blue}{x \cdot 1} + 0.5 \cdot \left|-x\right| \]

      2. *-commutative 66.6%

         \[\leadsto x \cdot 1 + \color{blue}{\left|-x\right| \cdot 0.5} \]

      3. fabs-neg 66.6%

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

      4. rem-square-sqrt 66.6%

         \[\leadsto x \cdot 1 + \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot 0.5 \]

      5. fabs-sqr 66.6%

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

      6. rem-square-sqrt 66.6%

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

      7. metadata-eval 66.6%

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

      8. *-inverses 66.6%

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

      9. rem-square-sqrt 66.6%

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

      10. fabs-sqr 66.6%

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

      11. rem-square-sqrt 66.6%

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

      12. fabs-neg 66.6%

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

      13. distribute-lft-in 66.6%

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

      14. fabs-neg 66.6%

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

      15. rem-square-sqrt 66.6%

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

      16. fabs-sqr 66.6%

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

      17. rem-square-sqrt 66.6%

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

      18. *-inverses 66.6%

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

      19. metadata-eval 66.6%

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

      20. metadata-eval 66.6%

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

   8. Simplified 66.6%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-70}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-71}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.0% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.88 \cdot 10^{-84}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.88e-84) (* x 0.5) (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.88e-84) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.88d-84) then
        tmp = x * 0.5d0
    else
        tmp = y * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.88e-84) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.88e-84:
		tmp = x * 0.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.88e-84)
		tmp = Float64(x * 0.5);
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.88e-84)
		tmp = x * 0.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.88e-84], N[(x * 0.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.88000000000000004e-84

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 91.0%

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

      1. +-commutative 91.0%

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

      2. sub-neg 91.0%

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

      3. neg-mul-1 91.0%

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

      4. fma-define 91.0%

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

      5. neg-mul-1 91.0%

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

      6. sub-neg 91.0%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\left|\color{blue}{y - x}\right|}{x}, 1\right) \]

      7. rem-square-sqrt 29.2%

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

      8. fabs-sqr 29.2%

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

      9. rem-square-sqrt 35.2%

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

   5. Simplified 35.2%

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

   6. Taylor expanded in y around 0 34.7%

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

   if 1.88000000000000004e-84 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 76.8%

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

   4. Taylor expanded in x around 0 72.8%

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

      1. rem-square-sqrt 72.2%

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

      2. fabs-sqr 72.2%

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

      3. rem-square-sqrt 72.8%

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

   6. Simplified 72.8%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.88 \cdot 10^{-84}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 30.3% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.8 \cdot 10^{-86}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 8.8e-86) x (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8.8e-86) {
		tmp = x;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 8.8d-86) then
        tmp = x
    else
        tmp = y * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 8.8e-86) {
		tmp = x;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8.8e-86:
		tmp = x
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8.8e-86)
		tmp = x;
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8.8e-86)
		tmp = x;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8.8e-86], x, N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.8000000000000006e-86

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 12.8%

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

   if 8.8000000000000006e-86 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 76.8%

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

   4. Taylor expanded in x around 0 72.8%

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

      1. rem-square-sqrt 72.2%

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

      2. fabs-sqr 72.2%

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

      3. rem-square-sqrt 72.8%

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

   6. Simplified 72.8%

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

4. Final simplification 33.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.8 \cdot 10^{-86}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 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 11.0%

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

Reproduce

herbie shell --seed 2024170 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))