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

Percentage Accurate: 99.9% → 99.9%

Time: 6.2s

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 100.0%

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

2. Final simplification 100.0%

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

Alternative 2: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{+75}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 20:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + \left(y - x\right) \cdot 0.5} \cdot \left(x \cdot 0.75\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.3e+75)
   (* 0.5 (+ x y))
   (if (<= x 20.0)
     (* (fabs (- y x)) 0.5)
     (* (/ x (+ x (* (- y x) 0.5))) (* x 0.75)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.3e+75) {
		tmp = 0.5 * (x + y);
	} else if (x <= 20.0) {
		tmp = fabs((y - x)) * 0.5;
	} else {
		tmp = (x / (x + ((y - x) * 0.5))) * (x * 0.75);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.3d+75)) then
        tmp = 0.5d0 * (x + y)
    else if (x <= 20.0d0) then
        tmp = abs((y - x)) * 0.5d0
    else
        tmp = (x / (x + ((y - x) * 0.5d0))) * (x * 0.75d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.3e+75) {
		tmp = 0.5 * (x + y);
	} else if (x <= 20.0) {
		tmp = Math.abs((y - x)) * 0.5;
	} else {
		tmp = (x / (x + ((y - x) * 0.5))) * (x * 0.75);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.3e+75:
		tmp = 0.5 * (x + y)
	elif x <= 20.0:
		tmp = math.fabs((y - x)) * 0.5
	else:
		tmp = (x / (x + ((y - x) * 0.5))) * (x * 0.75)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.3e+75)
		tmp = Float64(0.5 * Float64(x + y));
	elseif (x <= 20.0)
		tmp = Float64(abs(Float64(y - x)) * 0.5);
	else
		tmp = Float64(Float64(x / Float64(x + Float64(Float64(y - x) * 0.5))) * Float64(x * 0.75));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.3e+75)
		tmp = 0.5 * (x + y);
	elseif (x <= 20.0)
		tmp = abs((y - x)) * 0.5;
	else
		tmp = (x / (x + ((y - x) * 0.5))) * (x * 0.75);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.3e+75], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 20.0], N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(x / N[(x + N[(N[(y - x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * 0.75), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.29999999999999998e75

   1. Initial program 100.0%

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

      1. flip-+ 30.3%

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

   3. Applied egg-rr 28.1%

      \[\leadsto \color{blue}{\frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{x - \left(y - x\right) \cdot 0.5}} \]

   4. Taylor expanded in x around 0 94.0%

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

      1. +-commutative 94.0%

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

      2. distribute-lft-out 94.0%

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

   6. Simplified 94.0%

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

   if -3.29999999999999998e75 < x < 20

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 77.7%

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

   if 20 < x

   1. Initial program 99.9%

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

      1. flip-+ 30.7%

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

   3. Applied egg-rr 6.0%

      \[\leadsto \color{blue}{\frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{x - \left(y - x\right) \cdot 0.5}} \]

   4. Taylor expanded in x around inf 9.8%

      \[\leadsto \frac{\color{blue}{0.75 \cdot {x}^{2}}}{x - \left(y - x\right) \cdot 0.5} \]
   5. Step-by-step derivation

      1. unpow2 9.8%

         \[\leadsto \frac{0.75 \cdot \color{blue}{\left(x \cdot x\right)}}{x - \left(y - x\right) \cdot 0.5} \]

      2. *-commutative 9.8%

         \[\leadsto \frac{\color{blue}{\left(x \cdot x\right) \cdot 0.75}}{x - \left(y - x\right) \cdot 0.5} \]

      3. associate-*l* 9.8%

         \[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot 0.75\right)}}{x - \left(y - x\right) \cdot 0.5} \]

   6. Simplified 9.8%

      \[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot 0.75\right)}}{x - \left(y - x\right) \cdot 0.5} \]
   7. Step-by-step derivation

      1. associate-/l* 14.8%

         \[\leadsto \color{blue}{\frac{x}{\frac{x - \left(y - x\right) \cdot 0.5}{x \cdot 0.75}}} \]

      2. associate-/r/ 14.8%

         \[\leadsto \color{blue}{\frac{x}{x - \left(y - x\right) \cdot 0.5} \cdot \left(x \cdot 0.75\right)} \]

      3. sub-neg 14.8%

         \[\leadsto \frac{x}{\color{blue}{x + \left(-\left(y - x\right) \cdot 0.5\right)}} \cdot \left(x \cdot 0.75\right) \]

      4. distribute-rgt-neg-in 14.8%

         \[\leadsto \frac{x}{x + \color{blue}{\left(y - x\right) \cdot \left(-0.5\right)}} \cdot \left(x \cdot 0.75\right) \]

      5. metadata-eval 14.8%

         \[\leadsto \frac{x}{x + \left(y - x\right) \cdot \color{blue}{-0.5}} \cdot \left(x \cdot 0.75\right) \]

   8. Applied egg-rr 14.8%

      \[\leadsto \color{blue}{\frac{x}{x + \left(y - x\right) \cdot -0.5} \cdot \left(x \cdot 0.75\right)} \]
   9. Step-by-step derivation

      1. +-commutative 14.8%

         \[\leadsto \frac{x}{\color{blue}{\left(y - x\right) \cdot -0.5 + x}} \cdot \left(x \cdot 0.75\right) \]

      2. add-cube-cbrt 14.8%

         \[\leadsto \frac{x}{\color{blue}{\left(\sqrt[3]{\left(y - x\right) \cdot -0.5} \cdot \sqrt[3]{\left(y - x\right) \cdot -0.5}\right) \cdot \sqrt[3]{\left(y - x\right) \cdot -0.5}} + x} \cdot \left(x \cdot 0.75\right) \]

      3. fma-def 14.8%

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\left(y - x\right) \cdot -0.5} \cdot \sqrt[3]{\left(y - x\right) \cdot -0.5}, \sqrt[3]{\left(y - x\right) \cdot -0.5}, x\right)}} \cdot \left(x \cdot 0.75\right) \]

   10. Applied egg-rr 77.5%

       \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\left(y - x\right) \cdot 0.5}\right)}^{2}, \sqrt[3]{\left(y - x\right) \cdot 0.5}, x\right)}} \cdot \left(x \cdot 0.75\right) \]
   11. Step-by-step derivation

       1. fma-udef 77.5%

          \[\leadsto \frac{x}{\color{blue}{{\left(\sqrt[3]{\left(y - x\right) \cdot 0.5}\right)}^{2} \cdot \sqrt[3]{\left(y - x\right) \cdot 0.5} + x}} \cdot \left(x \cdot 0.75\right) \]

       2. unpow2 77.5%

          \[\leadsto \frac{x}{\color{blue}{\left(\sqrt[3]{\left(y - x\right) \cdot 0.5} \cdot \sqrt[3]{\left(y - x\right) \cdot 0.5}\right)} \cdot \sqrt[3]{\left(y - x\right) \cdot 0.5} + x} \cdot \left(x \cdot 0.75\right) \]

       3. rem-3cbrt-lft 79.2%

          \[\leadsto \frac{x}{\color{blue}{\left(y - x\right) \cdot 0.5} + x} \cdot \left(x \cdot 0.75\right) \]

       4. *-commutative 79.2%

          \[\leadsto \frac{x}{\color{blue}{0.5 \cdot \left(y - x\right)} + x} \cdot \left(x \cdot 0.75\right) \]

   12. Simplified 79.2%

       \[\leadsto \frac{x}{\color{blue}{0.5 \cdot \left(y - x\right) + x}} \cdot \left(x \cdot 0.75\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{+75}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 20:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + \left(y - x\right) \cdot 0.5} \cdot \left(x \cdot 0.75\right)\\ \end{array} \]

Alternative 3: 67.7% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{-12}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + \left(y - x\right) \cdot 0.5} \cdot \left(x \cdot 0.75\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 6.5e-12)
   (* 0.5 (+ x y))
   (* (/ x (+ x (* (- y x) 0.5))) (* x 0.75))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 6.5e-12) {
		tmp = 0.5 * (x + y);
	} else {
		tmp = (x / (x + ((y - x) * 0.5))) * (x * 0.75);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 6.5d-12) then
        tmp = 0.5d0 * (x + y)
    else
        tmp = (x / (x + ((y - x) * 0.5d0))) * (x * 0.75d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 6.5e-12) {
		tmp = 0.5 * (x + y);
	} else {
		tmp = (x / (x + ((y - x) * 0.5))) * (x * 0.75);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 6.5e-12:
		tmp = 0.5 * (x + y)
	else:
		tmp = (x / (x + ((y - x) * 0.5))) * (x * 0.75)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 6.5e-12)
		tmp = Float64(0.5 * Float64(x + y));
	else
		tmp = Float64(Float64(x / Float64(x + Float64(Float64(y - x) * 0.5))) * Float64(x * 0.75));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 6.5e-12)
		tmp = 0.5 * (x + y);
	else
		tmp = (x / (x + ((y - x) * 0.5))) * (x * 0.75);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 6.5e-12], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + N[(N[(y - x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * 0.75), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.5000000000000002e-12

   1. Initial program 100.0%

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

      1. flip-+ 53.6%

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

   3. Applied egg-rr 32.8%

      \[\leadsto \color{blue}{\frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{x - \left(y - x\right) \cdot 0.5}} \]

   4. Taylor expanded in x around 0 63.8%

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

      1. +-commutative 63.8%

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

      2. distribute-lft-out 63.8%

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

   6. Simplified 63.8%

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

   if 6.5000000000000002e-12 < x

   1. Initial program 99.9%

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

      1. flip-+ 30.3%

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

   3. Applied egg-rr 5.9%

      \[\leadsto \color{blue}{\frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{x - \left(y - x\right) \cdot 0.5}} \]

   4. Taylor expanded in x around inf 9.6%

      \[\leadsto \frac{\color{blue}{0.75 \cdot {x}^{2}}}{x - \left(y - x\right) \cdot 0.5} \]
   5. Step-by-step derivation

      1. unpow2 9.6%

         \[\leadsto \frac{0.75 \cdot \color{blue}{\left(x \cdot x\right)}}{x - \left(y - x\right) \cdot 0.5} \]

      2. *-commutative 9.6%

         \[\leadsto \frac{\color{blue}{\left(x \cdot x\right) \cdot 0.75}}{x - \left(y - x\right) \cdot 0.5} \]

      3. associate-*l* 9.6%

         \[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot 0.75\right)}}{x - \left(y - x\right) \cdot 0.5} \]

   6. Simplified 9.6%

      \[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot 0.75\right)}}{x - \left(y - x\right) \cdot 0.5} \]
   7. Step-by-step derivation

      1. associate-/l* 14.6%

         \[\leadsto \color{blue}{\frac{x}{\frac{x - \left(y - x\right) \cdot 0.5}{x \cdot 0.75}}} \]

      2. associate-/r/ 14.6%

         \[\leadsto \color{blue}{\frac{x}{x - \left(y - x\right) \cdot 0.5} \cdot \left(x \cdot 0.75\right)} \]

      3. sub-neg 14.6%

         \[\leadsto \frac{x}{\color{blue}{x + \left(-\left(y - x\right) \cdot 0.5\right)}} \cdot \left(x \cdot 0.75\right) \]

      4. distribute-rgt-neg-in 14.6%

         \[\leadsto \frac{x}{x + \color{blue}{\left(y - x\right) \cdot \left(-0.5\right)}} \cdot \left(x \cdot 0.75\right) \]

      5. metadata-eval 14.6%

         \[\leadsto \frac{x}{x + \left(y - x\right) \cdot \color{blue}{-0.5}} \cdot \left(x \cdot 0.75\right) \]

   8. Applied egg-rr 14.6%

      \[\leadsto \color{blue}{\frac{x}{x + \left(y - x\right) \cdot -0.5} \cdot \left(x \cdot 0.75\right)} \]
   9. Step-by-step derivation

      1. +-commutative 14.6%

         \[\leadsto \frac{x}{\color{blue}{\left(y - x\right) \cdot -0.5 + x}} \cdot \left(x \cdot 0.75\right) \]

      2. add-cube-cbrt 14.6%

         \[\leadsto \frac{x}{\color{blue}{\left(\sqrt[3]{\left(y - x\right) \cdot -0.5} \cdot \sqrt[3]{\left(y - x\right) \cdot -0.5}\right) \cdot \sqrt[3]{\left(y - x\right) \cdot -0.5}} + x} \cdot \left(x \cdot 0.75\right) \]

      3. fma-def 14.6%

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\left(y - x\right) \cdot -0.5} \cdot \sqrt[3]{\left(y - x\right) \cdot -0.5}, \sqrt[3]{\left(y - x\right) \cdot -0.5}, x\right)}} \cdot \left(x \cdot 0.75\right) \]

   10. Applied egg-rr 76.2%

       \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\left(y - x\right) \cdot 0.5}\right)}^{2}, \sqrt[3]{\left(y - x\right) \cdot 0.5}, x\right)}} \cdot \left(x \cdot 0.75\right) \]
   11. Step-by-step derivation

       1. fma-udef 76.2%

          \[\leadsto \frac{x}{\color{blue}{{\left(\sqrt[3]{\left(y - x\right) \cdot 0.5}\right)}^{2} \cdot \sqrt[3]{\left(y - x\right) \cdot 0.5} + x}} \cdot \left(x \cdot 0.75\right) \]

       2. unpow2 76.2%

          \[\leadsto \frac{x}{\color{blue}{\left(\sqrt[3]{\left(y - x\right) \cdot 0.5} \cdot \sqrt[3]{\left(y - x\right) \cdot 0.5}\right)} \cdot \sqrt[3]{\left(y - x\right) \cdot 0.5} + x} \cdot \left(x \cdot 0.75\right) \]

       3. rem-3cbrt-lft 77.8%

          \[\leadsto \frac{x}{\color{blue}{\left(y - x\right) \cdot 0.5} + x} \cdot \left(x \cdot 0.75\right) \]

       4. *-commutative 77.8%

          \[\leadsto \frac{x}{\color{blue}{0.5 \cdot \left(y - x\right)} + x} \cdot \left(x \cdot 0.75\right) \]

   12. Simplified 77.8%

       \[\leadsto \frac{x}{\color{blue}{0.5 \cdot \left(y - x\right) + x}} \cdot \left(x \cdot 0.75\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{-12}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + \left(y - x\right) \cdot 0.5} \cdot \left(x \cdot 0.75\right)\\ \end{array} \]

Alternative 4: 46.7% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-56}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 3.2e-56) (* x 0.5) (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.2e-56) {
		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 <= 3.2d-56) 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 <= 3.2e-56) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.2e-56:
		tmp = x * 0.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.2e-56)
		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 <= 3.2e-56)
		tmp = x * 0.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.2e-56], N[(x * 0.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.19999999999999986e-56

   1. Initial program 99.9%

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

      1. flip-+ 49.0%

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

   3. Applied egg-rr 20.0%

      \[\leadsto \color{blue}{\frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{x - \left(y - x\right) \cdot 0.5}} \]

   4. Taylor expanded in x around inf 36.8%

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

   if 3.19999999999999986e-56 < y

   1. Initial program 100.0%

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

      1. flip-+ 47.1%

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

   3. Applied egg-rr 40.2%

      \[\leadsto \color{blue}{\frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{x - \left(y - x\right) \cdot 0.5}} \]

   4. Taylor expanded in x around 0 72.1%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-56}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 5: 55.4% accurate, 21.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(x + y\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. flip-+ 48.4%

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

3. Applied egg-rr 26.7%

   \[\leadsto \color{blue}{\frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{x - \left(y - x\right) \cdot 0.5}} \]

4. Taylor expanded in x around 0 54.4%

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

   1. +-commutative 54.4%

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

   2. distribute-lft-out 54.4%

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

6. Simplified 54.4%

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

7. Final simplification 54.4%

   \[\leadsto 0.5 \cdot \left(x + y\right) \]

Alternative 6: 30.6% 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. Step-by-step derivation

   1. flip-+ 48.4%

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

3. Applied egg-rr 26.7%

   \[\leadsto \color{blue}{\frac{x \cdot x - {\left(y - x\right)}^{2} \cdot 0.25}{x - \left(y - x\right) \cdot 0.5}} \]

4. Taylor expanded in x around inf 29.2%

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

5. Final simplification 29.2%

   \[\leadsto x \cdot 0.5 \]

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 100.0%

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

2. Taylor expanded in x around inf 10.8%

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

3. Final simplification 10.8%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023290 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))