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

Percentage Accurate: 99.9% → 99.9%

Time: 4.8s

Alternatives: 8

Speedup: 1.7×

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, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left|y - x\right|, 0.5, x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma (fabs (- y x)) 0.5 x))

C

double code(double x, double y) {
	return fma(fabs((y - x)), 0.5, x);
}

Julia

function code(x, y)
	return fma(abs(Float64(y - x)), 0.5, x)
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 99.9%

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

6. Final simplification 99.9%

   \[\leadsto \mathsf{fma}\left(\left|y - x\right|, 0.5, x\right) \]
7. Add Preprocessing

Alternative 2: 62.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{-145}:\\ \;\;\;\;x + -0.5 \cdot y\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-70}:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left|-y\right| \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.35e-145)
   (+ x (* -0.5 y))
   (if (<= y 2.3e-70) (* 1.5 x) (* (fabs (- y)) 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.35e-145) {
		tmp = x + (-0.5 * y);
	} else if (y <= 2.3e-70) {
		tmp = 1.5 * x;
	} else {
		tmp = fabs(-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 <= (-1.35d-145)) then
        tmp = x + ((-0.5d0) * y)
    else if (y <= 2.3d-70) then
        tmp = 1.5d0 * x
    else
        tmp = abs(-y) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.35e-145) {
		tmp = x + (-0.5 * y);
	} else if (y <= 2.3e-70) {
		tmp = 1.5 * x;
	} else {
		tmp = Math.abs(-y) * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.35e-145:
		tmp = x + (-0.5 * y)
	elif y <= 2.3e-70:
		tmp = 1.5 * x
	else:
		tmp = math.fabs(-y) * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.35e-145)
		tmp = Float64(x + Float64(-0.5 * y));
	elseif (y <= 2.3e-70)
		tmp = Float64(1.5 * x);
	else
		tmp = Float64(abs(Float64(-y)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.35e-145)
		tmp = x + (-0.5 * y);
	elseif (y <= 2.3e-70)
		tmp = 1.5 * x;
	else
		tmp = abs(-y) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.35e-145], N[(x + N[(-0.5 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.3e-70], N[(1.5 * x), $MachinePrecision], N[(N[Abs[(-y)], $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.35e-145

   1. Initial program 100.0%

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

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 71.6

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

   7. Applied rewrites 71.6%

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

   if -1.35e-145 < y < 2.30000000000000001e-70

   1. Initial program 99.8%

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

   4. Applied rewrites 43.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{y - x}{-2}}, \sqrt{\frac{y - x}{-2}}, x\right)} \]

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 45.5

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

   7. Applied rewrites 45.5%

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

   if 2.30000000000000001e-70 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 69.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left|-1 \cdot y\right| \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 68.2%

      \[\leadsto \left|-y\right| \cdot 0.5 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 70.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-220}:\\ \;\;\;\;\mathsf{fma}\left(y - x, -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left|-y\right|, 0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.6e-220) (fma (- y x) -0.5 x) (fma (fabs (- y)) 0.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.6e-220) {
		tmp = fma((y - x), -0.5, x);
	} else {
		tmp = fma(fabs(-y), 0.5, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.6e-220)
		tmp = fma(Float64(y - x), -0.5, x);
	else
		tmp = fma(abs(Float64(-y)), 0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.6e-220], N[(N[(y - x), $MachinePrecision] * -0.5 + x), $MachinePrecision], N[(N[Abs[(-y)], $MachinePrecision] * 0.5 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.60000000000000003e-220

   1. Initial program 99.9%

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

   4. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{y - x}{-2}}, \sqrt{\frac{y - x}{-2}}, x\right)} \]

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 48.3

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

   7. Applied rewrites 48.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 74.2%

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

   if 1.60000000000000003e-220 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\left|-1 \cdot y\right|, \frac{1}{2}, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 63.1%

      \[\leadsto \mathsf{fma}\left(\left|-y\right|, 0.5, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 70.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-70}:\\ \;\;\;\;\mathsf{fma}\left(y - x, -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2e-70) (fma (- y x) -0.5 x) (* (fabs (- y x)) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2e-70) {
		tmp = fma((y - x), -0.5, x);
	} else {
		tmp = fabs((y - x)) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2e-70)
		tmp = fma(Float64(y - x), -0.5, x);
	else
		tmp = Float64(abs(Float64(y - x)) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2e-70], N[(N[(y - x), $MachinePrecision] * -0.5 + x), $MachinePrecision], N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.99999999999999999e-70

   1. Initial program 99.9%

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

   4. Applied rewrites 61.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{y - x}{-2}}, \sqrt{\frac{y - x}{-2}}, x\right)} \]

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 41.4

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

   7. Applied rewrites 41.4%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 68.2%

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

   if 1.99999999999999999e-70 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 69.4%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-70}:\\ \;\;\;\;\mathsf{fma}\left(y - x, -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{-70}:\\ \;\;\;\;\mathsf{fma}\left(y - x, -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left|-y\right| \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.3e-70) (fma (- y x) -0.5 x) (* (fabs (- y)) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.3e-70) {
		tmp = fma((y - x), -0.5, x);
	} else {
		tmp = fabs(-y) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.3e-70)
		tmp = fma(Float64(y - x), -0.5, x);
	else
		tmp = Float64(abs(Float64(-y)) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.3e-70], N[(N[(y - x), $MachinePrecision] * -0.5 + x), $MachinePrecision], N[(N[Abs[(-y)], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.30000000000000001e-70

   1. Initial program 99.9%

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

   4. Applied rewrites 61.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{y - x}{-2}}, \sqrt{\frac{y - x}{-2}}, x\right)} \]

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 41.4

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

   7. Applied rewrites 41.4%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 68.2%

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

   if 2.30000000000000001e-70 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 69.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left|-1 \cdot y\right| \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 68.2%

      \[\leadsto \left|-y\right| \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 48.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{-145}:\\ \;\;\;\;x + -0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.35e-145) (+ x (* -0.5 y)) (* 1.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.35e-145) {
		tmp = x + (-0.5 * y);
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.35d-145)) then
        tmp = x + ((-0.5d0) * y)
    else
        tmp = 1.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.35e-145) {
		tmp = x + (-0.5 * y);
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.35e-145:
		tmp = x + (-0.5 * y)
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.35e-145)
		tmp = Float64(x + Float64(-0.5 * y));
	else
		tmp = Float64(1.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.35e-145)
		tmp = x + (-0.5 * y);
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.35e-145], N[(x + N[(-0.5 * y), $MachinePrecision]), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.35e-145

   1. Initial program 100.0%

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

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 71.6

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

   7. Applied rewrites 71.6%

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

   if -1.35e-145 < y

   1. Initial program 99.9%

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

   4. Applied rewrites 28.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{y - x}{-2}}, \sqrt{\frac{y - x}{-2}}, x\right)} \]

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 32.0

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

   7. Applied rewrites 32.0%

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

Alternative 7: 45.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-139}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -7.2e-139) (* -0.5 y) (* 1.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.2e-139) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-7.2d-139)) then
        tmp = (-0.5d0) * y
    else
        tmp = 1.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7.2e-139) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.2e-139:
		tmp = -0.5 * y
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.2e-139)
		tmp = Float64(-0.5 * y);
	else
		tmp = Float64(1.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7.2e-139)
		tmp = -0.5 * y;
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.2e-139], N[(-0.5 * y), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.20000000000000007e-139

   1. Initial program 100.0%

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

   4. Applied rewrites 78.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{y - x}{-2}}, \sqrt{\frac{y - x}{-2}}, x\right)} \]

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 67.4

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

   7. Applied rewrites 67.4%

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

   if -7.20000000000000007e-139 < y

   1. Initial program 99.9%

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

   4. Applied rewrites 27.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{y - x}{-2}}, \sqrt{\frac{y - x}{-2}}, x\right)} \]

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 31.8

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

   7. Applied rewrites 31.8%

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

Alternative 8: 26.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 47.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{y - x}{-2}}, \sqrt{\frac{y - x}{-2}}, x\right)} \]

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. rem-square-sqrt N/A

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

   4. lower-*.f64 30.1

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

7. Applied rewrites 30.1%

   \[\leadsto \color{blue}{-0.5 \cdot y} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024343 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))