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

Percentage Accurate: 99.9% → 99.9%

Time: 10.1s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-14}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-109}:\\ \;\;\;\;x + \frac{\left|x\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.7e-14)
   (+ x (/ (fabs y) 2.0))
   (if (<= y 3e-109) (+ x (/ (fabs x) 2.0)) (* 0.5 (+ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.7e-14) {
		tmp = x + (fabs(y) / 2.0);
	} else if (y <= 3e-109) {
		tmp = x + (fabs(x) / 2.0);
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.7d-14)) then
        tmp = x + (abs(y) / 2.0d0)
    else if (y <= 3d-109) then
        tmp = x + (abs(x) / 2.0d0)
    else
        tmp = 0.5d0 * (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.7e-14) {
		tmp = x + (Math.abs(y) / 2.0);
	} else if (y <= 3e-109) {
		tmp = x + (Math.abs(x) / 2.0);
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.7e-14:
		tmp = x + (math.fabs(y) / 2.0)
	elif y <= 3e-109:
		tmp = x + (math.fabs(x) / 2.0)
	else:
		tmp = 0.5 * (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.7e-14)
		tmp = Float64(x + Float64(abs(y) / 2.0));
	elseif (y <= 3e-109)
		tmp = Float64(x + Float64(abs(x) / 2.0));
	else
		tmp = Float64(0.5 * Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.7e-14)
		tmp = x + (abs(y) / 2.0);
	elseif (y <= 3e-109)
		tmp = x + (abs(x) / 2.0);
	else
		tmp = 0.5 * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.7e-14], N[(x + N[(N[Abs[y], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3e-109], N[(x + N[(N[Abs[x], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.6999999999999999e-14

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 88.0%

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

   if -2.6999999999999999e-14 < y < 3.00000000000000021e-109

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 85.4%

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

      1. neg-mul-1 85.4%

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

   5. Simplified 85.4%

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

   if 3.00000000000000021e-109 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 83.7%

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

      1. add-sqr-sqrt 72.4%

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

      2. fabs-sqr 72.4%

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

      3. add-sqr-sqrt 74.8%

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

      4. *-commutative 74.8%

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

      5. sub-neg 74.8%

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

      6. metadata-eval 74.8%

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

   5. Applied egg-rr 74.8%

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

   6. Taylor expanded in x around 0 91.2%

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

      1. distribute-lft-out 91.2%

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

      2. +-commutative 91.2%

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

   8. Simplified 91.2%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-14}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-109}:\\ \;\;\;\;x + \frac{\left|x\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{-111}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.8e-111) (+ x (/ (fabs y) 2.0)) (* 0.5 (+ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.8e-111) {
		tmp = x + (fabs(y) / 2.0);
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-7.8d-111)) then
        tmp = x + (abs(y) / 2.0d0)
    else
        tmp = 0.5d0 * (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7.8e-111) {
		tmp = x + (Math.abs(y) / 2.0);
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.8e-111:
		tmp = x + (math.fabs(y) / 2.0)
	else:
		tmp = 0.5 * (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.8e-111)
		tmp = Float64(x + Float64(abs(y) / 2.0));
	else
		tmp = Float64(0.5 * Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7.8e-111)
		tmp = x + (abs(y) / 2.0);
	else
		tmp = 0.5 * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.8e-111], N[(x + N[(N[Abs[y], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.8000000000000006e-111

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 80.2%

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

   if -7.8000000000000006e-111 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 91.7%

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

      1. add-sqr-sqrt 58.5%

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

      2. fabs-sqr 58.5%

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

      3. add-sqr-sqrt 64.2%

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

      4. *-commutative 64.2%

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

      5. sub-neg 64.2%

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

      6. metadata-eval 64.2%

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

   5. Applied egg-rr 64.2%

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

   6. Taylor expanded in x around 0 72.5%

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

      1. distribute-lft-out 72.5%

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

      2. +-commutative 72.5%

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

   8. Simplified 72.5%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{-111}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-70}:\\ \;\;\;\;\left|y\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7e-70) (* (fabs y) 0.5) (* 0.5 (+ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7e-70) {
		tmp = fabs(y) * 0.5;
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-7d-70)) then
        tmp = abs(y) * 0.5d0
    else
        tmp = 0.5d0 * (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7e-70) {
		tmp = Math.abs(y) * 0.5;
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7e-70:
		tmp = math.fabs(y) * 0.5
	else:
		tmp = 0.5 * (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7e-70)
		tmp = Float64(abs(y) * 0.5);
	else
		tmp = Float64(0.5 * Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7e-70)
		tmp = abs(y) * 0.5;
	else
		tmp = 0.5 * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -6.99999999999999949e-70

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 84.2%

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

   4. Taylor expanded in x around 0 81.4%

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

   if -6.99999999999999949e-70 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 92.0%

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

      1. add-sqr-sqrt 57.4%

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

      2. fabs-sqr 57.4%

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

      3. add-sqr-sqrt 63.2%

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

      4. *-commutative 63.2%

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

      5. sub-neg 63.2%

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

      6. metadata-eval 63.2%

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

   5. Applied egg-rr 63.2%

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

   6. Taylor expanded in x around 0 71.2%

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

      1. distribute-lft-out 71.2%

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

      2. +-commutative 71.2%

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

   8. Simplified 71.2%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-70}:\\ \;\;\;\;\left|y\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 45.9% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{-37}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.6e-37) (* x 0.5) (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.6e-37) {
		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 <= 2.6d-37) 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 <= 2.6e-37) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.6e-37:
		tmp = x * 0.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.6e-37)
		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 <= 2.6e-37)
		tmp = x * 0.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.6e-37], N[(x * 0.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.5999999999999998e-37

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 92.5%

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

      1. add-sqr-sqrt 32.1%

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

      2. fabs-sqr 32.1%

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

      3. add-sqr-sqrt 38.4%

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

      4. *-commutative 38.4%

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

      5. sub-neg 38.4%

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

      6. metadata-eval 38.4%

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

   5. Applied egg-rr 38.4%

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

   6. Taylor expanded in x around inf 34.2%

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

   if 2.5999999999999998e-37 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 79.7%

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

   4. Taylor expanded in x around 0 76.3%

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

      1. rem-square-sqrt 75.8%

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

      2. fabs-sqr 75.8%

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

      3. rem-square-sqrt 76.3%

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

   6. Simplified 76.3%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{-37}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.2% 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 99.9%

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

3. Taylor expanded in x around inf 88.9%

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

   1. add-sqr-sqrt 44.1%

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

   2. fabs-sqr 44.1%

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

   3. add-sqr-sqrt 49.2%

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

   4. *-commutative 49.2%

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

   5. sub-neg 49.2%

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

   6. metadata-eval 49.2%

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

5. Applied egg-rr 49.2%

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

6. Taylor expanded in x around 0 55.1%

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

   1. distribute-lft-out 55.1%

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

   2. +-commutative 55.1%

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

8. Simplified 55.1%

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

9. Final simplification 55.1%

   \[\leadsto 0.5 \cdot \left(x + y\right) \]
10. Add Preprocessing

Alternative 7: 31.0% 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. Taylor expanded in x around inf 88.9%

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

   1. add-sqr-sqrt 44.1%

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

   2. fabs-sqr 44.1%

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

   3. add-sqr-sqrt 49.2%

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

   4. *-commutative 49.2%

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

   5. sub-neg 49.2%

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

   6. metadata-eval 49.2%

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

5. Applied egg-rr 49.2%

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

6. Taylor expanded in x around inf 28.1%

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

7. Final simplification 28.1%

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

Alternative 8: 11.5% 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 10.9%

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

Reproduce

herbie shell --seed 2024132 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))