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

Percentage Accurate: 99.9% → 99.9%

Time: 9.9s

Alternatives: 11

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: 80.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-81}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-147}:\\ \;\;\;\;x + \frac{\left|x\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7e-81)
   (+ x (/ (fabs y) 2.0))
   (if (<= y 3.3e-147) (+ x (/ (fabs x) 2.0)) (* 0.5 (+ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7e-81) {
		tmp = x + (fabs(y) / 2.0);
	} else if (y <= 3.3e-147) {
		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 <= (-7d-81)) then
        tmp = x + (abs(y) / 2.0d0)
    else if (y <= 3.3d-147) 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 <= -7e-81) {
		tmp = x + (Math.abs(y) / 2.0);
	} else if (y <= 3.3e-147) {
		tmp = x + (Math.abs(x) / 2.0);
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7e-81:
		tmp = x + (math.fabs(y) / 2.0)
	elif y <= 3.3e-147:
		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 <= -7e-81)
		tmp = Float64(x + Float64(abs(y) / 2.0));
	elseif (y <= 3.3e-147)
		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 <= -7e-81)
		tmp = x + (abs(y) / 2.0);
	elseif (y <= 3.3e-147)
		tmp = x + (abs(x) / 2.0);
	else
		tmp = 0.5 * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7e-81], N[(x + N[(N[Abs[y], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.3e-147], 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 < -6.99999999999999973e-81

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 78.2%

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

   if -6.99999999999999973e-81 < y < 3.29999999999999987e-147

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 85.6%

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

      1. neg-mul-1 85.6%

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

   5. Simplified 85.6%

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

   if 3.29999999999999987e-147 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 88.9%

      \[\leadsto \color{blue}{x \cdot \left(1 + 0.5 \cdot \frac{\left|y - x\right|}{x}\right)} \]
   4. Step-by-step derivation

      1. +-commutative 88.9%

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

      2. sub-neg 88.9%

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

      3. neg-mul-1 88.9%

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

      4. fma-define 88.9%

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{\left|y + -1 \cdot x\right|}{x}, 1\right)} \]

      5. neg-mul-1 88.9%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\left|y + \color{blue}{\left(-x\right)}\right|}{x}, 1\right) \]

      6. sub-neg 88.9%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\left|\color{blue}{y - x}\right|}{x}, 1\right) \]

      7. rem-square-sqrt 68.6%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right|}{x}, 1\right) \]

      8. fabs-sqr 68.6%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}}{x}, 1\right) \]

      9. rem-square-sqrt 72.5%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\color{blue}{y - x}}{x}, 1\right) \]

   5. Simplified 72.5%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(0.5, \frac{y - x}{x}, 1\right)} \]

   6. Taylor expanded in x around 0 83.5%

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

      1. distribute-lft-out 83.5%

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

      2. +-commutative 83.5%

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

   8. Simplified 83.5%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-81}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-147}:\\ \;\;\;\;x + \frac{\left|x\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-100}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+103}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{\frac{2}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.4e-100)
   (* 0.5 (+ x y))
   (if (<= x 2.3e+103) (+ x (/ (fabs y) 2.0)) (+ x (/ 1.0 (/ 2.0 x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.4e-100) {
		tmp = 0.5 * (x + y);
	} else if (x <= 2.3e+103) {
		tmp = x + (fabs(y) / 2.0);
	} else {
		tmp = x + (1.0 / (2.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.4d-100)) then
        tmp = 0.5d0 * (x + y)
    else if (x <= 2.3d+103) then
        tmp = x + (abs(y) / 2.0d0)
    else
        tmp = x + (1.0d0 / (2.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.4e-100) {
		tmp = 0.5 * (x + y);
	} else if (x <= 2.3e+103) {
		tmp = x + (Math.abs(y) / 2.0);
	} else {
		tmp = x + (1.0 / (2.0 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.4e-100:
		tmp = 0.5 * (x + y)
	elif x <= 2.3e+103:
		tmp = x + (math.fabs(y) / 2.0)
	else:
		tmp = x + (1.0 / (2.0 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.4e-100)
		tmp = Float64(0.5 * Float64(x + y));
	elseif (x <= 2.3e+103)
		tmp = Float64(x + Float64(abs(y) / 2.0));
	else
		tmp = Float64(x + Float64(1.0 / Float64(2.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.4e-100)
		tmp = 0.5 * (x + y);
	elseif (x <= 2.3e+103)
		tmp = x + (abs(y) / 2.0);
	else
		tmp = x + (1.0 / (2.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.4e-100], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.3e+103], N[(x + N[(N[Abs[y], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.4000000000000003e-100

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 98.7%

      \[\leadsto \color{blue}{x \cdot \left(1 + 0.5 \cdot \frac{\left|y - x\right|}{x}\right)} \]
   4. Step-by-step derivation

      1. +-commutative 98.7%

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

      2. sub-neg 98.7%

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

      3. neg-mul-1 98.7%

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

      4. fma-define 98.7%

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{\left|y + -1 \cdot x\right|}{x}, 1\right)} \]

      5. neg-mul-1 98.7%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\left|y + \color{blue}{\left(-x\right)}\right|}{x}, 1\right) \]

      6. sub-neg 98.7%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\left|\color{blue}{y - x}\right|}{x}, 1\right) \]

      7. rem-square-sqrt 82.9%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right|}{x}, 1\right) \]

      8. fabs-sqr 82.9%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}}{x}, 1\right) \]

      9. rem-square-sqrt 83.4%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\color{blue}{y - x}}{x}, 1\right) \]

   5. Simplified 83.4%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(0.5, \frac{y - x}{x}, 1\right)} \]

   6. Taylor expanded in x around 0 83.4%

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

      1. distribute-lft-out 83.4%

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

      2. +-commutative 83.4%

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

   8. Simplified 83.4%

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

   if -2.4000000000000003e-100 < x < 2.30000000000000008e103

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 77.4%

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

   if 2.30000000000000008e103 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 73.8%

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

      1. neg-mul-1 73.8%

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

   5. Simplified 73.8%

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

      1. clear-num 73.9%

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

      2. inv-pow 73.9%

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

      3. fabs-neg 73.9%

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

      4. add-sqr-sqrt 73.8%

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

      5. fabs-sqr 73.8%

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

      6. add-sqr-sqrt 73.9%

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

   7. Applied egg-rr 73.9%

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

      1. unpow-1 73.9%

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

   9. Simplified 73.9%

      \[\leadsto x + \color{blue}{\frac{1}{\frac{2}{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-100}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+103}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{\frac{2}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.45 \cdot 10^{-99}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-67}:\\ \;\;\;\;\left|y\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{\frac{2}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.45e-99)
   (* 0.5 (+ x y))
   (if (<= x 5.5e-67) (* (fabs y) 0.5) (+ x (/ 1.0 (/ 2.0 x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.45e-99) {
		tmp = 0.5 * (x + y);
	} else if (x <= 5.5e-67) {
		tmp = fabs(y) * 0.5;
	} else {
		tmp = x + (1.0 / (2.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.45d-99)) then
        tmp = 0.5d0 * (x + y)
    else if (x <= 5.5d-67) then
        tmp = abs(y) * 0.5d0
    else
        tmp = x + (1.0d0 / (2.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.45e-99) {
		tmp = 0.5 * (x + y);
	} else if (x <= 5.5e-67) {
		tmp = Math.abs(y) * 0.5;
	} else {
		tmp = x + (1.0 / (2.0 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.45e-99:
		tmp = 0.5 * (x + y)
	elif x <= 5.5e-67:
		tmp = math.fabs(y) * 0.5
	else:
		tmp = x + (1.0 / (2.0 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.45e-99)
		tmp = Float64(0.5 * Float64(x + y));
	elseif (x <= 5.5e-67)
		tmp = Float64(abs(y) * 0.5);
	else
		tmp = Float64(x + Float64(1.0 / Float64(2.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.45e-99)
		tmp = 0.5 * (x + y);
	elseif (x <= 5.5e-67)
		tmp = abs(y) * 0.5;
	else
		tmp = x + (1.0 / (2.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.45e-99], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.5e-67], N[(N[Abs[y], $MachinePrecision] * 0.5), $MachinePrecision], N[(x + N[(1.0 / N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.4500000000000002e-99

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 98.7%

      \[\leadsto \color{blue}{x \cdot \left(1 + 0.5 \cdot \frac{\left|y - x\right|}{x}\right)} \]
   4. Step-by-step derivation

      1. +-commutative 98.7%

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

      2. sub-neg 98.7%

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

      3. neg-mul-1 98.7%

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

      4. fma-define 98.7%

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{\left|y + -1 \cdot x\right|}{x}, 1\right)} \]

      5. neg-mul-1 98.7%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\left|y + \color{blue}{\left(-x\right)}\right|}{x}, 1\right) \]

      6. sub-neg 98.7%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\left|\color{blue}{y - x}\right|}{x}, 1\right) \]

      7. rem-square-sqrt 82.9%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right|}{x}, 1\right) \]

      8. fabs-sqr 82.9%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}}{x}, 1\right) \]

      9. rem-square-sqrt 83.4%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\color{blue}{y - x}}{x}, 1\right) \]

   5. Simplified 83.4%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(0.5, \frac{y - x}{x}, 1\right)} \]

   6. Taylor expanded in x around 0 83.4%

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

      1. distribute-lft-out 83.4%

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

      2. +-commutative 83.4%

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

   8. Simplified 83.4%

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

   if -3.4500000000000002e-99 < x < 5.5000000000000003e-67

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 85.1%

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

   4. Taylor expanded in x around 0 82.8%

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

   if 5.5000000000000003e-67 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 65.0%

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

      1. neg-mul-1 65.0%

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

   5. Simplified 65.0%

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

      1. clear-num 65.0%

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

      2. inv-pow 65.0%

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

      3. fabs-neg 65.0%

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

      4. add-sqr-sqrt 64.9%

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

      5. fabs-sqr 64.9%

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

      6. add-sqr-sqrt 65.0%

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

   7. Applied egg-rr 65.0%

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

      1. unpow-1 65.0%

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

   9. Simplified 65.0%

      \[\leadsto x + \color{blue}{\frac{1}{\frac{2}{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.45 \cdot 10^{-99}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-67}:\\ \;\;\;\;\left|y\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{\frac{2}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.4% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-82}:\\ \;\;\;\;x \cdot \left(1 - 0.5 \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.2e-82) (* x (- 1.0 (* 0.5 (/ y x)))) (* 0.5 (+ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.2e-82) {
		tmp = x * (1.0 - (0.5 * (y / x)));
	} 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 <= (-2.2d-82)) then
        tmp = x * (1.0d0 - (0.5d0 * (y / x)))
    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 <= -2.2e-82) {
		tmp = x * (1.0 - (0.5 * (y / x)));
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.2e-82:
		tmp = x * (1.0 - (0.5 * (y / x)))
	else:
		tmp = 0.5 * (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.2e-82)
		tmp = Float64(x * Float64(1.0 - Float64(0.5 * Float64(y / x))));
	else
		tmp = Float64(0.5 * Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.2e-82)
		tmp = x * (1.0 - (0.5 * (y / x)));
	else
		tmp = 0.5 * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.2e-82], N[(x * N[(1.0 - N[(0.5 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.19999999999999986e-82

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 78.2%

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

   4. Taylor expanded in x around inf 63.2%

      \[\leadsto \color{blue}{x \cdot \left(1 + 0.5 \cdot \frac{\left|y\right|}{x}\right)} \]
   5. Step-by-step derivation

      1. associate-*r/ 63.2%

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

      2. rem-square-sqrt 0.0%

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

      3. fabs-sqr 0.0%

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

      4. rem-square-sqrt 5.2%

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

   6. Simplified 5.2%

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{0.5 \cdot y}{x}\right)} \]
   7. Step-by-step derivation

      1. frac-2neg 5.2%

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

      2. add-sqr-sqrt 2.8%

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

      3. sqrt-unprod 26.9%

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

      4. sqr-neg 26.9%

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

      5. sqrt-unprod 31.9%

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

      6. add-sqr-sqrt 63.2%

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

      7. distribute-frac-neg 63.2%

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

      8. sub-neg 63.2%

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

      9. associate-/l* 63.2%

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

   8. Applied egg-rr 63.2%

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

   if -2.19999999999999986e-82 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 94.1%

      \[\leadsto \color{blue}{x \cdot \left(1 + 0.5 \cdot \frac{\left|y - x\right|}{x}\right)} \]
   4. Step-by-step derivation

      1. +-commutative 94.1%

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

      2. sub-neg 94.1%

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

      3. neg-mul-1 94.1%

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

      4. fma-define 94.1%

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{\left|y + -1 \cdot x\right|}{x}, 1\right)} \]

      5. neg-mul-1 94.1%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\left|y + \color{blue}{\left(-x\right)}\right|}{x}, 1\right) \]

      6. sub-neg 94.1%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\left|\color{blue}{y - x}\right|}{x}, 1\right) \]

      7. rem-square-sqrt 59.1%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right|}{x}, 1\right) \]

      8. fabs-sqr 59.1%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}}{x}, 1\right) \]

      9. rem-square-sqrt 64.9%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\color{blue}{y - x}}{x}, 1\right) \]

   5. Simplified 64.9%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(0.5, \frac{y - x}{x}, 1\right)} \]

   6. Taylor expanded in x around 0 70.7%

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

      1. distribute-lft-out 70.7%

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

      2. +-commutative 70.7%

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

   8. Simplified 70.7%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-82}:\\ \;\;\;\;x \cdot \left(1 - 0.5 \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.2% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-150}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-68}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.5e-150) (* x 0.5) (if (<= x 5e-68) (* y 0.5) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.5e-150) {
		tmp = x * 0.5;
	} else if (x <= 5e-68) {
		tmp = y * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.5d-150)) then
        tmp = x * 0.5d0
    else if (x <= 5d-68) then
        tmp = y * 0.5d0
    else
        tmp = x * 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.5e-150) {
		tmp = x * 0.5;
	} else if (x <= 5e-68) {
		tmp = y * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.5e-150:
		tmp = x * 0.5
	elif x <= 5e-68:
		tmp = y * 0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.5e-150)
		tmp = Float64(x * 0.5);
	elseif (x <= 5e-68)
		tmp = Float64(y * 0.5);
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.5e-150)
		tmp = x * 0.5;
	elseif (x <= 5e-68)
		tmp = y * 0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.5e-150], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, 5e-68], N[(y * 0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.4999999999999998e-150

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 97.9%

      \[\leadsto \color{blue}{x \cdot \left(1 + 0.5 \cdot \frac{\left|y - x\right|}{x}\right)} \]
   4. Step-by-step derivation

      1. +-commutative 97.9%

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

      2. sub-neg 97.9%

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

      3. neg-mul-1 97.9%

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

      4. fma-define 97.9%

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{\left|y + -1 \cdot x\right|}{x}, 1\right)} \]

      5. neg-mul-1 97.9%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\left|y + \color{blue}{\left(-x\right)}\right|}{x}, 1\right) \]

      6. sub-neg 97.9%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\left|\color{blue}{y - x}\right|}{x}, 1\right) \]

      7. rem-square-sqrt 79.3%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right|}{x}, 1\right) \]

      8. fabs-sqr 79.3%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}}{x}, 1\right) \]

      9. rem-square-sqrt 79.9%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\color{blue}{y - x}}{x}, 1\right) \]

   5. Simplified 79.9%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(0.5, \frac{y - x}{x}, 1\right)} \]

   6. Taylor expanded in y around 0 63.9%

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

   if -3.4999999999999998e-150 < x < 4.99999999999999971e-68

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 87.1%

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

   4. Taylor expanded in x around 0 85.2%

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

      1. rem-square-sqrt 45.9%

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

      2. fabs-sqr 45.9%

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

      3. rem-square-sqrt 47.4%

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

   6. Simplified 47.4%

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

   if 4.99999999999999971e-68 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 65.0%

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

      1. neg-mul-1 65.0%

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

   5. Simplified 65.0%

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

   6. Taylor expanded in x around 0 65.0%

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

      1. fabs-neg 65.0%

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

      2. rem-square-sqrt 64.9%

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

      3. fabs-sqr 64.9%

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

      4. rem-square-sqrt 65.0%

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

      5. *-commutative 65.0%

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

      6. *-rgt-identity 65.0%

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

      7. distribute-lft-out 65.0%

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

      8. metadata-eval 65.0%

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

   8. Simplified 65.0%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-150}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-68}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.4% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.9 \cdot 10^{-67}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{\frac{2}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.9e-67) (* 0.5 (+ x y)) (+ x (/ 1.0 (/ 2.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.9e-67) {
		tmp = 0.5 * (x + y);
	} else {
		tmp = x + (1.0 / (2.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 2.9d-67) then
        tmp = 0.5d0 * (x + y)
    else
        tmp = x + (1.0d0 / (2.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2.9e-67) {
		tmp = 0.5 * (x + y);
	} else {
		tmp = x + (1.0 / (2.0 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.9e-67:
		tmp = 0.5 * (x + y)
	else:
		tmp = x + (1.0 / (2.0 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.9e-67)
		tmp = Float64(0.5 * Float64(x + y));
	else
		tmp = Float64(x + Float64(1.0 / Float64(2.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.9e-67)
		tmp = 0.5 * (x + y);
	else
		tmp = x + (1.0 / (2.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.9e-67], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.90000000000000005e-67

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 87.2%

      \[\leadsto \color{blue}{x \cdot \left(1 + 0.5 \cdot \frac{\left|y - x\right|}{x}\right)} \]
   4. Step-by-step derivation

      1. +-commutative 87.2%

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

      2. sub-neg 87.2%

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

      3. neg-mul-1 87.2%

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

      4. fma-define 87.2%

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{\left|y + -1 \cdot x\right|}{x}, 1\right)} \]

      5. neg-mul-1 87.2%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\left|y + \color{blue}{\left(-x\right)}\right|}{x}, 1\right) \]

      6. sub-neg 87.2%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\left|\color{blue}{y - x}\right|}{x}, 1\right) \]

      7. rem-square-sqrt 60.0%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right|}{x}, 1\right) \]

      8. fabs-sqr 60.0%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}}{x}, 1\right) \]

      9. rem-square-sqrt 61.5%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\color{blue}{y - x}}{x}, 1\right) \]

   5. Simplified 61.5%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(0.5, \frac{y - x}{x}, 1\right)} \]

   6. Taylor expanded in x around 0 67.7%

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

      1. distribute-lft-out 67.7%

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

      2. +-commutative 67.7%

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

   8. Simplified 67.7%

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

   if 2.90000000000000005e-67 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 65.0%

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

      1. neg-mul-1 65.0%

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

   5. Simplified 65.0%

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

      1. clear-num 65.0%

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

      2. inv-pow 65.0%

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

      3. fabs-neg 65.0%

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

      4. add-sqr-sqrt 64.9%

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

      5. fabs-sqr 64.9%

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

      6. add-sqr-sqrt 65.0%

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

   7. Applied egg-rr 65.0%

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

      1. unpow-1 65.0%

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

   9. Simplified 65.0%

      \[\leadsto x + \color{blue}{\frac{1}{\frac{2}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.9 \cdot 10^{-67}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{\frac{2}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.4% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{-68}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 9e-68) (* 0.5 (+ x y)) (* x 1.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 9e-68) {
		tmp = 0.5 * (x + y);
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 9d-68) then
        tmp = 0.5d0 * (x + y)
    else
        tmp = x * 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 9e-68) {
		tmp = 0.5 * (x + y);
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 9e-68:
		tmp = 0.5 * (x + y)
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 9e-68)
		tmp = Float64(0.5 * Float64(x + y));
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 9e-68)
		tmp = 0.5 * (x + y);
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 9e-68], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.99999999999999998e-68

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 87.2%

      \[\leadsto \color{blue}{x \cdot \left(1 + 0.5 \cdot \frac{\left|y - x\right|}{x}\right)} \]
   4. Step-by-step derivation

      1. +-commutative 87.2%

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

      2. sub-neg 87.2%

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

      3. neg-mul-1 87.2%

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

      4. fma-define 87.2%

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{\left|y + -1 \cdot x\right|}{x}, 1\right)} \]

      5. neg-mul-1 87.2%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\left|y + \color{blue}{\left(-x\right)}\right|}{x}, 1\right) \]

      6. sub-neg 87.2%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\left|\color{blue}{y - x}\right|}{x}, 1\right) \]

      7. rem-square-sqrt 60.0%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right|}{x}, 1\right) \]

      8. fabs-sqr 60.0%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}}{x}, 1\right) \]

      9. rem-square-sqrt 61.5%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\color{blue}{y - x}}{x}, 1\right) \]

   5. Simplified 61.5%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(0.5, \frac{y - x}{x}, 1\right)} \]

   6. Taylor expanded in x around 0 67.7%

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

      1. distribute-lft-out 67.7%

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

      2. +-commutative 67.7%

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

   8. Simplified 67.7%

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

   if 8.99999999999999998e-68 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 65.0%

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

      1. neg-mul-1 65.0%

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

   5. Simplified 65.0%

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

   6. Taylor expanded in x around 0 65.0%

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

      1. fabs-neg 65.0%

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

      2. rem-square-sqrt 64.9%

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

      3. fabs-sqr 64.9%

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

      4. rem-square-sqrt 65.0%

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

      5. *-commutative 65.0%

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

      6. *-rgt-identity 65.0%

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

      7. distribute-lft-out 65.0%

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

      8. metadata-eval 65.0%

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

   8. Simplified 65.0%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{-68}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 45.7% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-60}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 6.5e-60) (* x 0.5) (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6.5e-60) {
		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 <= 6.5d-60) 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 <= 6.5e-60) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 6.5e-60:
		tmp = x * 0.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6.5e-60)
		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 <= 6.5e-60)
		tmp = x * 0.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6.5e-60], N[(x * 0.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.49999999999999995e-60

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 93.7%

      \[\leadsto \color{blue}{x \cdot \left(1 + 0.5 \cdot \frac{\left|y - x\right|}{x}\right)} \]
   4. Step-by-step derivation

      1. +-commutative 93.7%

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

      2. sub-neg 93.7%

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

      3. neg-mul-1 93.7%

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

      4. fma-define 93.7%

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{\left|y + -1 \cdot x\right|}{x}, 1\right)} \]

      5. neg-mul-1 93.7%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\left|y + \color{blue}{\left(-x\right)}\right|}{x}, 1\right) \]

      6. sub-neg 93.7%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\left|\color{blue}{y - x}\right|}{x}, 1\right) \]

      7. rem-square-sqrt 37.0%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right|}{x}, 1\right) \]

      8. fabs-sqr 37.0%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}}{x}, 1\right) \]

      9. rem-square-sqrt 42.4%

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, \frac{\color{blue}{y - x}}{x}, 1\right) \]

   5. Simplified 42.4%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(0.5, \frac{y - x}{x}, 1\right)} \]

   6. Taylor expanded in y around 0 37.6%

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

   if 6.49999999999999995e-60 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 78.6%

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

   4. Taylor expanded in x around 0 75.3%

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

      1. rem-square-sqrt 74.8%

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

      2. fabs-sqr 74.8%

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

      3. rem-square-sqrt 75.3%

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

   6. Simplified 75.3%

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

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-60}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 31.5% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-182}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 8.5e-182) x (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8.5e-182) {
		tmp = x;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 8.5d-182) then
        tmp = x
    else
        tmp = y * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 8.5e-182) {
		tmp = x;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8.5e-182:
		tmp = x
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8.5e-182)
		tmp = x;
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8.5e-182)
		tmp = x;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8.5e-182], x, N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.5000000000000001e-182

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 12.7%

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

   if 8.5000000000000001e-182 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 68.4%

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

   4. Taylor expanded in x around 0 62.7%

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

      1. rem-square-sqrt 62.3%

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

      2. fabs-sqr 62.3%

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

      3. rem-square-sqrt 62.7%

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

   6. Simplified 62.7%

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

4. Final simplification 32.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-182}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 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 11.4%

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

Reproduce

herbie shell --seed 2024157 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))