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

Percentage Accurate: 99.9% → 99.9%

Time: 9.3s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 84.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-198}:\\ \;\;\;\;y \cdot -0.5 + x \cdot 1.5\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-99}:\\ \;\;\;\;x + \frac{\left|x\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.1e-198)
   (+ (* y -0.5) (* x 1.5))
   (if (<= y 1.25e-99) (+ x (/ (fabs x) 2.0)) (* 0.5 (+ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.1e-198) {
		tmp = (y * -0.5) + (x * 1.5);
	} else if (y <= 1.25e-99) {
		tmp = x + (fabs(x) / 2.0);
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.1d-198)) then
        tmp = (y * (-0.5d0)) + (x * 1.5d0)
    else if (y <= 1.25d-99) then
        tmp = x + (abs(x) / 2.0d0)
    else
        tmp = 0.5d0 * (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.1e-198) {
		tmp = (y * -0.5) + (x * 1.5);
	} else if (y <= 1.25e-99) {
		tmp = x + (Math.abs(x) / 2.0);
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.1e-198:
		tmp = (y * -0.5) + (x * 1.5)
	elif y <= 1.25e-99:
		tmp = x + (math.fabs(x) / 2.0)
	else:
		tmp = 0.5 * (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.1e-198)
		tmp = Float64(Float64(y * -0.5) + Float64(x * 1.5));
	elseif (y <= 1.25e-99)
		tmp = Float64(x + Float64(abs(x) / 2.0));
	else
		tmp = Float64(0.5 * Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.1e-198)
		tmp = (y * -0.5) + (x * 1.5);
	elseif (y <= 1.25e-99)
		tmp = x + (abs(x) / 2.0);
	else
		tmp = 0.5 * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.1e-198], N[(N[(y * -0.5), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.25e-99], N[(x + N[(N[Abs[x], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.1e-198

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 90.8%

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

      1. mul-1-neg 90.8%

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

      2. unsub-neg 90.8%

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

   5. Simplified 90.8%

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

      1. add-sqr-sqrt 14.2%

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

      2. fabs-sqr 14.2%

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

      3. add-sqr-sqrt 18.0%

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

      4. div-inv 17.9%

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

      5. cancel-sign-sub-inv 17.9%

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

      6. add-sqr-sqrt 14.4%

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

      7. sqrt-unprod 18.0%

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

      8. sqr-neg 18.0%

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

      9. sqrt-unprod 17.9%

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

      10. add-sqr-sqrt 21.0%

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

      11. div-inv 21.0%

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

      12. flip-+ 14.2%

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

      13. associate-*r/ 11.4%

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

   7. Applied egg-rr 11.5%

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

      1. distribute-lft-in 11.5%

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

      2. *-rgt-identity 11.5%

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

      3. unpow2 11.5%

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

      4. associate-*r* 13.7%

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

      5. *-commutative 13.7%

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

      6. associate-*l/ 11.7%

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

      7. associate-/l* 13.7%

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

      8. *-inverses 13.7%

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

      9. *-rgt-identity 13.7%

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

   9. Simplified 13.7%

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

   10. Taylor expanded in y around 0 85.1%

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

       1. +-commutative 85.1%

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

       2. *-commutative 85.1%

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

       3. associate-+l+ 85.1%

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

       4. *-commutative 85.1%

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

       5. *-commutative 85.1%

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

       6. distribute-lft1-in 85.1%

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

       7. metadata-eval 85.1%

          \[\leadsto y \cdot -0.5 + \color{blue}{1.5} \cdot x \]

       8. *-commutative 85.1%

          \[\leadsto y \cdot -0.5 + \color{blue}{x \cdot 1.5} \]

   12. Simplified 85.1%

       \[\leadsto \color{blue}{y \cdot -0.5 + x \cdot 1.5} \]

   if -1.1e-198 < y < 1.24999999999999992e-99

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 89.5%

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

      1. neg-mul-1 89.5%

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

   5. Simplified 89.5%

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

   if 1.24999999999999992e-99 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around -inf 100.0%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-sub 100.0%

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

      5. rem-square-sqrt 78.9%

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

      6. fabs-sqr 78.9%

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

      7. rem-square-sqrt 83.1%

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

   5. Simplified 83.1%

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

   6. Taylor expanded in x around 0 83.1%

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

      1. +-commutative 83.1%

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

      2. distribute-lft-in 83.1%

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

   8. Simplified 83.1%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-198}:\\ \;\;\;\;y \cdot -0.5 + x \cdot 1.5\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-99}:\\ \;\;\;\;x + \frac{\left|x\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+59}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-87}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5 + x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.4e+59)
   (* 0.5 (+ x y))
   (if (<= x 5.8e-87) (+ x (/ (fabs y) 2.0)) (+ (* y -0.5) (* x 1.5)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.4e+59) {
		tmp = 0.5 * (x + y);
	} else if (x <= 5.8e-87) {
		tmp = x + (fabs(y) / 2.0);
	} else {
		tmp = (y * -0.5) + (x * 1.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.4d+59)) then
        tmp = 0.5d0 * (x + y)
    else if (x <= 5.8d-87) then
        tmp = x + (abs(y) / 2.0d0)
    else
        tmp = (y * (-0.5d0)) + (x * 1.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.4e+59) {
		tmp = 0.5 * (x + y);
	} else if (x <= 5.8e-87) {
		tmp = x + (Math.abs(y) / 2.0);
	} else {
		tmp = (y * -0.5) + (x * 1.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.4e+59:
		tmp = 0.5 * (x + y)
	elif x <= 5.8e-87:
		tmp = x + (math.fabs(y) / 2.0)
	else:
		tmp = (y * -0.5) + (x * 1.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.4e+59)
		tmp = Float64(0.5 * Float64(x + y));
	elseif (x <= 5.8e-87)
		tmp = Float64(x + Float64(abs(y) / 2.0));
	else
		tmp = Float64(Float64(y * -0.5) + Float64(x * 1.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.4e+59)
		tmp = 0.5 * (x + y);
	elseif (x <= 5.8e-87)
		tmp = x + (abs(y) / 2.0);
	else
		tmp = (y * -0.5) + (x * 1.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.4e+59], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.8e-87], N[(x + N[(N[Abs[y], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * -0.5), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.4000000000000002e59

   1. Initial program 100.0%

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

   3. Taylor expanded in y around -inf 100.0%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-sub 100.0%

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

      5. rem-square-sqrt 91.8%

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

      6. fabs-sqr 91.8%

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

      7. rem-square-sqrt 92.6%

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

   5. Simplified 92.6%

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

   6. Taylor expanded in x around 0 92.6%

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

      1. +-commutative 92.6%

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

      2. distribute-lft-in 92.6%

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

   8. Simplified 92.6%

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

   if -2.4000000000000002e59 < x < 5.7999999999999998e-87

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 79.5%

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

   if 5.7999999999999998e-87 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 78.5%

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

      1. mul-1-neg 78.5%

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

      2. unsub-neg 78.5%

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

   5. Simplified 78.5%

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

      1. add-sqr-sqrt 16.0%

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

      2. fabs-sqr 16.0%

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

      3. add-sqr-sqrt 24.6%

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

      4. div-inv 24.6%

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

      5. cancel-sign-sub-inv 24.6%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 39.5%

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

      8. sqr-neg 39.5%

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

      9. sqrt-unprod 64.0%

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

      10. add-sqr-sqrt 64.1%

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

      11. div-inv 64.1%

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

      12. flip-+ 41.8%

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

      13. associate-*r/ 32.2%

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

   7. Applied egg-rr 32.5%

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

      1. distribute-lft-in 32.5%

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

      2. *-rgt-identity 32.5%

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

      3. unpow2 32.5%

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

      4. associate-*r* 39.2%

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

      5. *-commutative 39.2%

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

      6. associate-*l/ 27.7%

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

      7. associate-/l* 39.2%

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

      8. *-inverses 39.2%

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

      9. *-rgt-identity 39.2%

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

   9. Simplified 39.2%

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

   10. Taylor expanded in y around 0 83.9%

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

       1. +-commutative 83.9%

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

       2. *-commutative 83.9%

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

       3. associate-+l+ 83.9%

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

       4. *-commutative 83.9%

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

       5. *-commutative 83.9%

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

       6. distribute-lft1-in 83.9%

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

       7. metadata-eval 83.9%

          \[\leadsto y \cdot -0.5 + \color{blue}{1.5} \cdot x \]

       8. *-commutative 83.9%

          \[\leadsto y \cdot -0.5 + \color{blue}{x \cdot 1.5} \]

   12. Simplified 83.9%

       \[\leadsto \color{blue}{y \cdot -0.5 + x \cdot 1.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+59}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-87}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5 + x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.4% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-201}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-84}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.8e-201) (* x 0.5) (if (<= x 1.75e-84) (* y 0.5) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.8e-201) {
		tmp = x * 0.5;
	} else if (x <= 1.75e-84) {
		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 <= (-4.8d-201)) then
        tmp = x * 0.5d0
    else if (x <= 1.75d-84) 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 <= -4.8e-201) {
		tmp = x * 0.5;
	} else if (x <= 1.75e-84) {
		tmp = y * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.8e-201:
		tmp = x * 0.5
	elif x <= 1.75e-84:
		tmp = y * 0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.8e-201)
		tmp = Float64(x * 0.5);
	elseif (x <= 1.75e-84)
		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 <= -4.8e-201)
		tmp = x * 0.5;
	elseif (x <= 1.75e-84)
		tmp = y * 0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.8e-201], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, 1.75e-84], N[(y * 0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.80000000000000018e-201

   1. Initial program 100.0%

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

   3. Taylor expanded in y around -inf 100.0%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-sub 100.0%

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

      5. rem-square-sqrt 76.8%

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

      6. fabs-sqr 76.8%

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

      7. rem-square-sqrt 77.6%

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

   5. Simplified 77.6%

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

   6. Taylor expanded in x around inf 63.6%

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

   if -4.80000000000000018e-201 < x < 1.7500000000000001e-84

   1. Initial program 99.9%

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

   3. Taylor expanded in y around -inf 99.9%

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

      1. fabs-neg 99.9%

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

      2. mul-1-neg 99.9%

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

      3. sub-neg 99.9%

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

      4. fabs-sub 99.9%

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

      5. rem-square-sqrt 47.1%

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

      6. fabs-sqr 47.1%

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

      7. rem-square-sqrt 50.4%

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

   5. Simplified 50.4%

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

   6. Taylor expanded in x around 0 46.0%

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

   if 1.7500000000000001e-84 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 71.1%

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

      1. neg-mul-1 71.1%

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

   5. Simplified 71.1%

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

   6. Taylor expanded in x around 0 71.1%

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

      1. +-commutative 71.1%

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

      2. fma-define 71.1%

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

      3. fabs-neg 71.1%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\left|x\right|}, x\right) \]

      4. rem-square-sqrt 71.0%

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

      5. fabs-sqr 71.0%

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

      6. rem-square-sqrt 71.1%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{x}, x\right) \]

      7. fma-define 71.1%

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

      8. *-lft-identity 71.1%

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

      9. distribute-rgt-out 71.1%

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

      10. metadata-eval 71.1%

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

   8. Simplified 71.1%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-201}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-84}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.8% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{-227}:\\ \;\;\;\;y \cdot -0.5 + x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.7e-227) (+ (* y -0.5) (* x 1.5)) (* 0.5 (+ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.7e-227) {
		tmp = (y * -0.5) + (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 <= (-3.7d-227)) then
        tmp = (y * (-0.5d0)) + (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 <= -3.7e-227) {
		tmp = (y * -0.5) + (x * 1.5);
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.7e-227:
		tmp = (y * -0.5) + (x * 1.5)
	else:
		tmp = 0.5 * (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.7e-227)
		tmp = Float64(Float64(y * -0.5) + 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 <= -3.7e-227)
		tmp = (y * -0.5) + (x * 1.5);
	else
		tmp = 0.5 * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.7e-227], N[(N[(y * -0.5), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.69999999999999978e-227

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 86.9%

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

      1. mul-1-neg 86.9%

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

      2. unsub-neg 86.9%

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

   5. Simplified 86.9%

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

      1. add-sqr-sqrt 14.3%

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

      2. fabs-sqr 14.3%

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

      3. add-sqr-sqrt 18.4%

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

      4. div-inv 18.4%

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

      5. cancel-sign-sub-inv 18.4%

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

      6. add-sqr-sqrt 14.5%

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

      7. sqrt-unprod 20.2%

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

      8. sqr-neg 20.2%

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

      9. sqrt-unprod 20.3%

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

      10. add-sqr-sqrt 23.5%

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

      11. div-inv 23.5%

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

      12. flip-+ 14.7%

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

      13. associate-*r/ 12.3%

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

   7. Applied egg-rr 12.3%

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

      1. distribute-lft-in 12.3%

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

      2. *-rgt-identity 12.3%

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

      3. unpow2 12.3%

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

      4. associate-*r* 15.9%

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

      5. *-commutative 15.9%

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

      6. associate-*l/ 12.7%

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

      7. associate-/l* 15.9%

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

      8. *-inverses 15.9%

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

      9. *-rgt-identity 15.9%

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

   9. Simplified 15.9%

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

   10. Taylor expanded in y around 0 83.9%

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

       1. +-commutative 83.9%

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

       2. *-commutative 83.9%

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

       3. associate-+l+ 83.9%

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

       4. *-commutative 83.9%

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

       5. *-commutative 83.9%

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

       6. distribute-lft1-in 83.9%

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

       7. metadata-eval 83.9%

          \[\leadsto y \cdot -0.5 + \color{blue}{1.5} \cdot x \]

       8. *-commutative 83.9%

          \[\leadsto y \cdot -0.5 + \color{blue}{x \cdot 1.5} \]

   12. Simplified 83.9%

       \[\leadsto \color{blue}{y \cdot -0.5 + x \cdot 1.5} \]

   if -3.69999999999999978e-227 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around -inf 100.0%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-sub 100.0%

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

      5. rem-square-sqrt 71.2%

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

      6. fabs-sqr 71.2%

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

      7. rem-square-sqrt 76.7%

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

   5. Simplified 76.7%

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

   6. Taylor expanded in x around 0 76.7%

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

      1. +-commutative 76.7%

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

      2. distribute-lft-in 76.7%

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

   8. Simplified 76.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(y + x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{-227}:\\ \;\;\;\;y \cdot -0.5 + x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.8% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{-227}:\\ \;\;\;\;x + \frac{x - y}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.7e-227) (+ x (/ (- x y) 2.0)) (* 0.5 (+ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.7e-227) {
		tmp = x + ((x - y) / 2.0);
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3.7d-227)) then
        tmp = x + ((x - y) / 2.0d0)
    else
        tmp = 0.5d0 * (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.7e-227) {
		tmp = x + ((x - y) / 2.0);
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.7e-227:
		tmp = x + ((x - y) / 2.0)
	else:
		tmp = 0.5 * (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.7e-227)
		tmp = Float64(x + Float64(Float64(x - y) / 2.0));
	else
		tmp = Float64(0.5 * Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.7e-227)
		tmp = x + ((x - y) / 2.0);
	else
		tmp = 0.5 * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.7e-227], N[(x + N[(N[(x - y), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.69999999999999978e-227

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 86.9%

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

      1. mul-1-neg 86.9%

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

      2. unsub-neg 86.9%

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

   5. Simplified 86.9%

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

      1. add-sqr-sqrt 14.3%

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

      2. fabs-sqr 14.3%

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

      3. add-sqr-sqrt 18.4%

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

      4. div-inv 18.4%

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

      5. cancel-sign-sub-inv 18.4%

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

      6. add-sqr-sqrt 14.5%

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

      7. sqrt-unprod 20.2%

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

      8. sqr-neg 20.2%

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

      9. sqrt-unprod 20.3%

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

      10. add-sqr-sqrt 23.5%

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

      11. div-inv 23.5%

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

      12. flip-+ 14.7%

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

      13. associate-*r/ 12.3%

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

   7. Applied egg-rr 12.3%

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

      1. distribute-lft-in 12.3%

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

      2. *-rgt-identity 12.3%

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

      3. unpow2 12.3%

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

      4. associate-*r* 15.9%

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

      5. *-commutative 15.9%

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

      6. associate-*l/ 12.7%

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

      7. associate-/l* 15.9%

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

      8. *-inverses 15.9%

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

      9. *-rgt-identity 15.9%

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

   9. Simplified 15.9%

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

   10. Taylor expanded in y around 0 83.9%

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

       1. neg-mul-1 83.9%

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

       2. unsub-neg 83.9%

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

   12. Simplified 83.9%

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

   if -3.69999999999999978e-227 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around -inf 100.0%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-sub 100.0%

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

      5. rem-square-sqrt 71.2%

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

      6. fabs-sqr 71.2%

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

      7. rem-square-sqrt 76.7%

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

   5. Simplified 76.7%

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

   6. Taylor expanded in x around 0 76.7%

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

      1. +-commutative 76.7%

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

      2. distribute-lft-in 76.7%

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

   8. Simplified 76.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(y + x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{-227}:\\ \;\;\;\;x + \frac{x - y}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.0% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{-81}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 7.2e-81) (* 0.5 (+ x y)) (* x 1.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 7.2e-81) {
		tmp = 0.5 * (x + y);
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 7.2d-81) then
        tmp = 0.5d0 * (x + y)
    else
        tmp = x * 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 7.2e-81) {
		tmp = 0.5 * (x + y);
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 7.2e-81:
		tmp = 0.5 * (x + y)
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 7.2e-81)
		tmp = Float64(0.5 * Float64(x + y));
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 7.2e-81)
		tmp = 0.5 * (x + y);
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 7.2e-81], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.1999999999999997e-81

   1. Initial program 100.0%

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

   3. Taylor expanded in y around -inf 100.0%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-sub 100.0%

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

      5. rem-square-sqrt 63.4%

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

      6. fabs-sqr 63.4%

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

      7. rem-square-sqrt 65.4%

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

   5. Simplified 65.4%

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

   6. Taylor expanded in x around 0 65.4%

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

      1. +-commutative 65.4%

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

      2. distribute-lft-in 65.4%

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

   8. Simplified 65.4%

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

   if 7.1999999999999997e-81 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 71.1%

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

      1. neg-mul-1 71.1%

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

   5. Simplified 71.1%

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

   6. Taylor expanded in x around 0 71.1%

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

      1. +-commutative 71.1%

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

      2. fma-define 71.1%

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

      3. fabs-neg 71.1%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\left|x\right|}, x\right) \]

      4. rem-square-sqrt 71.0%

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

      5. fabs-sqr 71.0%

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

      6. rem-square-sqrt 71.1%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{x}, x\right) \]

      7. fma-define 71.1%

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

      8. *-lft-identity 71.1%

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

      9. distribute-rgt-out 71.1%

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

      10. metadata-eval 71.1%

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

   8. Simplified 71.1%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{-81}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.7% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-98}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 8.5e-98) (* x 0.5) (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8.5e-98) {
		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 <= 8.5d-98) 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 <= 8.5e-98) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8.5e-98:
		tmp = x * 0.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8.5e-98)
		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 <= 8.5e-98)
		tmp = x * 0.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8.5e-98], N[(x * 0.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.4999999999999997e-98

   1. Initial program 99.9%

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

   3. Taylor expanded in y around -inf 99.9%

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

      1. fabs-neg 99.9%

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

      2. mul-1-neg 99.9%

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

      3. sub-neg 99.9%

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

      4. fabs-sub 99.9%

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

      5. rem-square-sqrt 33.8%

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

      6. fabs-sqr 33.8%

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

      7. rem-square-sqrt 40.0%

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

   5. Simplified 40.0%

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

   6. Taylor expanded in x around inf 36.4%

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

   if 8.4999999999999997e-98 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around -inf 100.0%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-sub 100.0%

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

      5. rem-square-sqrt 78.6%

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

      6. fabs-sqr 78.6%

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

      7. rem-square-sqrt 82.9%

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

   5. Simplified 82.9%

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

   6. Taylor expanded in x around 0 68.7%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-98}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 30.7% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around -inf 100.0%

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

   1. fabs-neg 100.0%

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

   2. mul-1-neg 100.0%

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

   3. sub-neg 100.0%

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

   4. fabs-sub 100.0%

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

   5. rem-square-sqrt 47.3%

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

   6. fabs-sqr 47.3%

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

   7. rem-square-sqrt 52.9%

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

5. Simplified 52.9%

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

6. Taylor expanded in x around inf 30.4%

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

7. Final simplification 30.4%

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

Alternative 10: 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 100.0%

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

3. Taylor expanded in x around inf 11.9%

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

Reproduce

herbie shell --seed 2024145 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))