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

Percentage Accurate: 99.9% → 99.9%

Time: 8.9s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

Alternative 2: 71.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-34}:\\ \;\;\;\;\left(y - x\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.5e-34) (* (- y x) -0.5) (+ x (/ (fabs y) 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.5e-34) {
		tmp = (y - x) * -0.5;
	} else {
		tmp = x + (fabs(y) / 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.5d-34)) then
        tmp = (y - x) * (-0.5d0)
    else
        tmp = x + (abs(y) / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.5e-34) {
		tmp = (y - x) * -0.5;
	} else {
		tmp = x + (Math.abs(y) / 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.5e-34:
		tmp = (y - x) * -0.5
	else:
		tmp = x + (math.fabs(y) / 2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.5e-34)
		tmp = Float64(Float64(y - x) * -0.5);
	else
		tmp = Float64(x + Float64(abs(y) / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.5e-34)
		tmp = (y - x) * -0.5;
	else
		tmp = x + (abs(y) / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.5e-34], N[(N[(y - x), $MachinePrecision] * -0.5), $MachinePrecision], N[(x + N[(N[Abs[y], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.5e-34

   1. Initial program 100.0%

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

      1. flip3-+ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{3} + {\left(\frac{\left|y - x\right|}{2}\right)}^{3}\right), \color{blue}{\left(x \cdot x + \left(\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot \frac{\left|y - x\right|}{2}\right)\right)}\right) \]

   4. Applied egg-rr 29.0%

      \[\leadsto \color{blue}{\frac{x \cdot \left(x \cdot x\right) + {\left(\frac{\left(y - x\right) \cdot \left(y - x\right)}{4}\right)}^{1.5}}{\frac{\left(y - x\right) \cdot \left(y - x\right)}{4} + x \cdot \left(x + \frac{\left|y - x\right|}{-2}\right)}} \]

   5. Taylor expanded in y around -inf

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot y\right), \color{blue}{\left(\frac{1}{2} + -1 \cdot \frac{\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x}{y}\right)}\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left(y\right)\right), \left(\color{blue}{\frac{1}{2}} + -1 \cdot \frac{\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x}{y}\right)\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(0 - y\right), \left(\color{blue}{\frac{1}{2}} + -1 \cdot \frac{\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x}{y}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \left(\color{blue}{\frac{1}{2}} + -1 \cdot \frac{\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x}{y}\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x}{y}\right)\right)\right)\right) \]

      7. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \left(\frac{1}{2} - \color{blue}{\frac{\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x}{y}}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \mathsf{\_.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x}{y}\right)}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x\right), \color{blue}{y}\right)\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \left(x + 2 \cdot x\right)\right), x\right), y\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(x + 2 \cdot x\right)\right), x\right), y\right)\right)\right) \]

      12. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\left(2 + 1\right) \cdot x\right)\right), x\right), y\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(3 \cdot x\right)\right), x\right), y\right)\right)\right) \]

      14. *-lowering-*.f64 59.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(3, x\right)\right), x\right), y\right)\right)\right) \]

   7. Simplified 59.9%

      \[\leadsto \color{blue}{\left(0 - y\right) \cdot \left(0.5 - \frac{0.5 \cdot \left(3 \cdot x\right) - x}{y}\right)} \]

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{-1}{2} \cdot y + \left(\mathsf{neg}\left(\left(x - \frac{3}{2} \cdot x\right)\right)\right) \]

      3. unsub-neg N/A

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

      4. cancel-sign-sub-inv N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-out-- N/A

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

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left(y - x\right)}\right) \]

      10. --lowering--.f64 80.5%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{\_.f64}\left(y, \color{blue}{x}\right)\right) \]

   10. Simplified 80.5%

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

   if -3.5e-34 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\color{blue}{y}\right), 2\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 69.8%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-34}:\\ \;\;\;\;\left(y - x\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.9% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-151}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.2e-151) (* y -0.5) (if (<= y 2.5e-5) (* x 0.5) (* y 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.2e-151) {
		tmp = y * -0.5;
	} else if (y <= 2.5e-5) {
		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 <= (-1.2d-151)) then
        tmp = y * (-0.5d0)
    else if (y <= 2.5d-5) 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 <= -1.2e-151) {
		tmp = y * -0.5;
	} else if (y <= 2.5e-5) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.2e-151:
		tmp = y * -0.5
	elif y <= 2.5e-5:
		tmp = x * 0.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.2e-151)
		tmp = Float64(y * -0.5);
	elseif (y <= 2.5e-5)
		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 <= -1.2e-151)
		tmp = y * -0.5;
	elseif (y <= 2.5e-5)
		tmp = x * 0.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.2e-151], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 2.5e-5], N[(x * 0.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.2e-151

   1. Initial program 100.0%

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

      1. flip3-+ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{3} + {\left(\frac{\left|y - x\right|}{2}\right)}^{3}\right), \color{blue}{\left(x \cdot x + \left(\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot \frac{\left|y - x\right|}{2}\right)\right)}\right) \]

   4. Applied egg-rr 35.9%

      \[\leadsto \color{blue}{\frac{x \cdot \left(x \cdot x\right) + {\left(\frac{\left(y - x\right) \cdot \left(y - x\right)}{4}\right)}^{1.5}}{\frac{\left(y - x\right) \cdot \left(y - x\right)}{4} + x \cdot \left(x + \frac{\left|y - x\right|}{-2}\right)}} \]

   5. Taylor expanded in y around -inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 63.2%

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\frac{-1}{2}}\right) \]

   7. Simplified 63.2%

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

   if -1.2e-151 < y < 2.50000000000000012e-5

   1. Initial program 99.9%

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

      1. flip3-+ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{3} + {\left(\frac{\left|y - x\right|}{2}\right)}^{3}\right), \color{blue}{\left(x \cdot x + \left(\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot \frac{\left|y - x\right|}{2}\right)\right)}\right) \]

   4. Applied egg-rr 37.8%

      \[\leadsto \color{blue}{\frac{x \cdot \left(x \cdot x\right) + {\left(\frac{\left(y - x\right) \cdot \left(y - x\right)}{4}\right)}^{1.5}}{\frac{\left(y - x\right) \cdot \left(y - x\right)}{4} + x \cdot \left(x + \frac{\left|y - x\right|}{-2}\right)}} \]

   5. Taylor expanded in y around -inf

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot y\right), \color{blue}{\left(\frac{1}{2} + -1 \cdot \frac{\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x}{y}\right)}\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left(y\right)\right), \left(\color{blue}{\frac{1}{2}} + -1 \cdot \frac{\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x}{y}\right)\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(0 - y\right), \left(\color{blue}{\frac{1}{2}} + -1 \cdot \frac{\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x}{y}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \left(\color{blue}{\frac{1}{2}} + -1 \cdot \frac{\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x}{y}\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x}{y}\right)\right)\right)\right) \]

      7. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \left(\frac{1}{2} - \color{blue}{\frac{\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x}{y}}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \mathsf{\_.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x}{y}\right)}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x\right), \color{blue}{y}\right)\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \left(x + 2 \cdot x\right)\right), x\right), y\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(x + 2 \cdot x\right)\right), x\right), y\right)\right)\right) \]

      12. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\left(2 + 1\right) \cdot x\right)\right), x\right), y\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(3 \cdot x\right)\right), x\right), y\right)\right)\right) \]

      14. *-lowering-*.f64 36.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(3, x\right)\right), x\right), y\right)\right)\right) \]

   7. Simplified 36.5%

      \[\leadsto \color{blue}{\left(0 - y\right) \cdot \left(0.5 - \frac{0.5 \cdot \left(3 \cdot x\right) - x}{y}\right)} \]

   8. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. distribute-lft-in N/A

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

      3. neg-mul-1 N/A

         \[\leadsto -1 \cdot x + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{3}{2} \cdot x\right)\right)\right)\right) \]

      4. remove-double-neg N/A

         \[\leadsto -1 \cdot x + \frac{3}{2} \cdot \color{blue}{x} \]

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

         \[\leadsto x \cdot \frac{1}{2} \]

      7. *-commutative N/A

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

      8. *-lowering-*.f64 48.3%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{x}\right) \]

   10. Simplified 48.3%

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

   if 2.50000000000000012e-5 < y

   1. Initial program 100.0%

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

      1. flip3-+ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{3} + {\left(\frac{\left|y - x\right|}{2}\right)}^{3}\right), \color{blue}{\left(x \cdot x + \left(\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot \frac{\left|y - x\right|}{2}\right)\right)}\right) \]

   4. Applied egg-rr 24.0%

      \[\leadsto \color{blue}{\frac{x \cdot \left(x \cdot x\right) + {\left(\frac{\left(y - x\right) \cdot \left(y - x\right)}{4}\right)}^{1.5}}{\frac{\left(y - x\right) \cdot \left(y - x\right)}{4} + x \cdot \left(x + \frac{\left|y - x\right|}{-2}\right)}} \]

   5. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 79.8%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{y}\right) \]

   7. Simplified 79.8%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-151}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.8% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-5}:\\ \;\;\;\;\left(y - x\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.8e-5) (* (- y x) -0.5) (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.8e-5) {
		tmp = (y - 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 <= 3.8d-5) then
        tmp = (y - 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 <= 3.8e-5) {
		tmp = (y - x) * -0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.8e-5:
		tmp = (y - x) * -0.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.8e-5)
		tmp = Float64(Float64(y - 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 <= 3.8e-5)
		tmp = (y - x) * -0.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.8e-5], N[(N[(y - x), $MachinePrecision] * -0.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.8000000000000002e-5

   1. Initial program 99.9%

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

      1. flip3-+ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{3} + {\left(\frac{\left|y - x\right|}{2}\right)}^{3}\right), \color{blue}{\left(x \cdot x + \left(\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot \frac{\left|y - x\right|}{2}\right)\right)}\right) \]

   4. Applied egg-rr 36.9%

      \[\leadsto \color{blue}{\frac{x \cdot \left(x \cdot x\right) + {\left(\frac{\left(y - x\right) \cdot \left(y - x\right)}{4}\right)}^{1.5}}{\frac{\left(y - x\right) \cdot \left(y - x\right)}{4} + x \cdot \left(x + \frac{\left|y - x\right|}{-2}\right)}} \]

   5. Taylor expanded in y around -inf

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot y\right), \color{blue}{\left(\frac{1}{2} + -1 \cdot \frac{\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x}{y}\right)}\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left(y\right)\right), \left(\color{blue}{\frac{1}{2}} + -1 \cdot \frac{\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x}{y}\right)\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(0 - y\right), \left(\color{blue}{\frac{1}{2}} + -1 \cdot \frac{\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x}{y}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \left(\color{blue}{\frac{1}{2}} + -1 \cdot \frac{\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x}{y}\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x}{y}\right)\right)\right)\right) \]

      7. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \left(\frac{1}{2} - \color{blue}{\frac{\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x}{y}}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \mathsf{\_.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x}{y}\right)}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x\right), \color{blue}{y}\right)\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \left(x + 2 \cdot x\right)\right), x\right), y\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(x + 2 \cdot x\right)\right), x\right), y\right)\right)\right) \]

      12. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\left(2 + 1\right) \cdot x\right)\right), x\right), y\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(3 \cdot x\right)\right), x\right), y\right)\right)\right) \]

      14. *-lowering-*.f64 57.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(3, x\right)\right), x\right), y\right)\right)\right) \]

   7. Simplified 57.4%

      \[\leadsto \color{blue}{\left(0 - y\right) \cdot \left(0.5 - \frac{0.5 \cdot \left(3 \cdot x\right) - x}{y}\right)} \]

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{-1}{2} \cdot y + \left(\mathsf{neg}\left(\left(x - \frac{3}{2} \cdot x\right)\right)\right) \]

      3. unsub-neg N/A

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

      4. cancel-sign-sub-inv N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-out-- N/A

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

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left(y - x\right)}\right) \]

      10. --lowering--.f64 66.8%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{\_.f64}\left(y, \color{blue}{x}\right)\right) \]

   10. Simplified 66.8%

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

   if 3.8000000000000002e-5 < y

   1. Initial program 100.0%

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

      1. flip3-+ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{3} + {\left(\frac{\left|y - x\right|}{2}\right)}^{3}\right), \color{blue}{\left(x \cdot x + \left(\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot \frac{\left|y - x\right|}{2}\right)\right)}\right) \]

   4. Applied egg-rr 24.0%

      \[\leadsto \color{blue}{\frac{x \cdot \left(x \cdot x\right) + {\left(\frac{\left(y - x\right) \cdot \left(y - x\right)}{4}\right)}^{1.5}}{\frac{\left(y - x\right) \cdot \left(y - x\right)}{4} + x \cdot \left(x + \frac{\left|y - x\right|}{-2}\right)}} \]

   5. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 79.8%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{y}\right) \]

   7. Simplified 79.8%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-5}:\\ \;\;\;\;\left(y - x\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 45.5% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{-5}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 5.6e-5) (* x 0.5) (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.6e-5) {
		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 <= 5.6d-5) 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 <= 5.6e-5) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.6e-5:
		tmp = x * 0.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.6e-5)
		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 <= 5.6e-5)
		tmp = x * 0.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.6e-5], N[(x * 0.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.59999999999999992e-5

   1. Initial program 99.9%

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

      1. flip3-+ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{3} + {\left(\frac{\left|y - x\right|}{2}\right)}^{3}\right), \color{blue}{\left(x \cdot x + \left(\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot \frac{\left|y - x\right|}{2}\right)\right)}\right) \]

   4. Applied egg-rr 36.9%

      \[\leadsto \color{blue}{\frac{x \cdot \left(x \cdot x\right) + {\left(\frac{\left(y - x\right) \cdot \left(y - x\right)}{4}\right)}^{1.5}}{\frac{\left(y - x\right) \cdot \left(y - x\right)}{4} + x \cdot \left(x + \frac{\left|y - x\right|}{-2}\right)}} \]

   5. Taylor expanded in y around -inf

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot y\right), \color{blue}{\left(\frac{1}{2} + -1 \cdot \frac{\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x}{y}\right)}\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left(y\right)\right), \left(\color{blue}{\frac{1}{2}} + -1 \cdot \frac{\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x}{y}\right)\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(0 - y\right), \left(\color{blue}{\frac{1}{2}} + -1 \cdot \frac{\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x}{y}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \left(\color{blue}{\frac{1}{2}} + -1 \cdot \frac{\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x}{y}\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x}{y}\right)\right)\right)\right) \]

      7. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \left(\frac{1}{2} - \color{blue}{\frac{\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x}{y}}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \mathsf{\_.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x}{y}\right)}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x\right), \color{blue}{y}\right)\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \left(x + 2 \cdot x\right)\right), x\right), y\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(x + 2 \cdot x\right)\right), x\right), y\right)\right)\right) \]

      12. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\left(2 + 1\right) \cdot x\right)\right), x\right), y\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(3 \cdot x\right)\right), x\right), y\right)\right)\right) \]

      14. *-lowering-*.f64 57.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(3, x\right)\right), x\right), y\right)\right)\right) \]

   7. Simplified 57.4%

      \[\leadsto \color{blue}{\left(0 - y\right) \cdot \left(0.5 - \frac{0.5 \cdot \left(3 \cdot x\right) - x}{y}\right)} \]

   8. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. distribute-lft-in N/A

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

      3. neg-mul-1 N/A

         \[\leadsto -1 \cdot x + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{3}{2} \cdot x\right)\right)\right)\right) \]

      4. remove-double-neg N/A

         \[\leadsto -1 \cdot x + \frac{3}{2} \cdot \color{blue}{x} \]

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

         \[\leadsto x \cdot \frac{1}{2} \]

      7. *-commutative N/A

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

      8. *-lowering-*.f64 37.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{x}\right) \]

   10. Simplified 37.0%

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

   if 5.59999999999999992e-5 < y

   1. Initial program 100.0%

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

      1. flip3-+ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{3} + {\left(\frac{\left|y - x\right|}{2}\right)}^{3}\right), \color{blue}{\left(x \cdot x + \left(\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot \frac{\left|y - x\right|}{2}\right)\right)}\right) \]

   4. Applied egg-rr 24.0%

      \[\leadsto \color{blue}{\frac{x \cdot \left(x \cdot x\right) + {\left(\frac{\left(y - x\right) \cdot \left(y - x\right)}{4}\right)}^{1.5}}{\frac{\left(y - x\right) \cdot \left(y - x\right)}{4} + x \cdot \left(x + \frac{\left|y - x\right|}{-2}\right)}} \]

   5. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 79.8%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{y}\right) \]

   7. Simplified 79.8%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{-5}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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 99.9%

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

   1. flip3-+ N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({x}^{3} + {\left(\frac{\left|y - x\right|}{2}\right)}^{3}\right), \color{blue}{\left(x \cdot x + \left(\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot \frac{\left|y - x\right|}{2}\right)\right)}\right) \]

4. Applied egg-rr 33.3%

   \[\leadsto \color{blue}{\frac{x \cdot \left(x \cdot x\right) + {\left(\frac{\left(y - x\right) \cdot \left(y - x\right)}{4}\right)}^{1.5}}{\frac{\left(y - x\right) \cdot \left(y - x\right)}{4} + x \cdot \left(x + \frac{\left|y - x\right|}{-2}\right)}} \]

5. Taylor expanded in y around -inf

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

   1. associate-*r* N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot y\right), \color{blue}{\left(\frac{1}{2} + -1 \cdot \frac{\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x}{y}\right)}\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left(y\right)\right), \left(\color{blue}{\frac{1}{2}} + -1 \cdot \frac{\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x}{y}\right)\right) \]

   4. neg-sub0 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(0 - y\right), \left(\color{blue}{\frac{1}{2}} + -1 \cdot \frac{\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x}{y}\right)\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \left(\color{blue}{\frac{1}{2}} + -1 \cdot \frac{\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x}{y}\right)\right) \]

   6. mul-1-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x}{y}\right)\right)\right)\right) \]

   7. unsub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \left(\frac{1}{2} - \color{blue}{\frac{\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x}{y}}\right)\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \mathsf{\_.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x}{y}\right)}\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(x + 2 \cdot x\right) - x\right), \color{blue}{y}\right)\right)\right) \]

   10. --lowering--.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \left(x + 2 \cdot x\right)\right), x\right), y\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(x + 2 \cdot x\right)\right), x\right), y\right)\right)\right) \]

   12. distribute-rgt1-in N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\left(2 + 1\right) \cdot x\right)\right), x\right), y\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(3 \cdot x\right)\right), x\right), y\right)\right)\right) \]

   14. *-lowering-*.f64 45.3%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, y\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(3, x\right)\right), x\right), y\right)\right)\right) \]

7. Simplified 45.3%

   \[\leadsto \color{blue}{\left(0 - y\right) \cdot \left(0.5 - \frac{0.5 \cdot \left(3 \cdot x\right) - x}{y}\right)} \]

8. Taylor expanded in y around 0

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

   1. sub-neg N/A

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

   2. distribute-lft-in N/A

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

   3. neg-mul-1 N/A

      \[\leadsto -1 \cdot x + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{3}{2} \cdot x\right)\right)\right)\right) \]

   4. remove-double-neg N/A

      \[\leadsto -1 \cdot x + \frac{3}{2} \cdot \color{blue}{x} \]

   5. distribute-rgt-out N/A

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

   6. metadata-eval N/A

      \[\leadsto x \cdot \frac{1}{2} \]

   7. *-commutative N/A

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

   8. *-lowering-*.f64 31.1%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{x}\right) \]

10. Simplified 31.1%

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

11. Final simplification 31.1%

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

Alternative 7: 11.3% 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

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Simplified 11.1%

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

Reproduce

herbie shell --seed 2024149 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))