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

Percentage Accurate: 99.9% → 99.9%

Time: 8.5s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

Alternative 2: 79.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-164}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{+113}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -9.5e-164)
   (* 0.5 (+ x y))
   (if (<= x 2.55e+113) (+ x (/ (fabs y) 2.0)) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -9.5e-164) {
		tmp = 0.5 * (x + y);
	} else if (x <= 2.55e+113) {
		tmp = x + (fabs(y) / 2.0);
	} 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 <= (-9.5d-164)) then
        tmp = 0.5d0 * (x + y)
    else if (x <= 2.55d+113) then
        tmp = x + (abs(y) / 2.0d0)
    else
        tmp = x * 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -9.5e-164) {
		tmp = 0.5 * (x + y);
	} else if (x <= 2.55e+113) {
		tmp = x + (Math.abs(y) / 2.0);
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -9.5e-164:
		tmp = 0.5 * (x + y)
	elif x <= 2.55e+113:
		tmp = x + (math.fabs(y) / 2.0)
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -9.5e-164)
		tmp = Float64(0.5 * Float64(x + y));
	elseif (x <= 2.55e+113)
		tmp = Float64(x + Float64(abs(y) / 2.0));
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9.5e-164)
		tmp = 0.5 * (x + y);
	elseif (x <= 2.55e+113)
		tmp = x + (abs(y) / 2.0);
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -9.5e-164], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.55e+113], N[(x + N[(N[Abs[y], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.5000000000000001e-164

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   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 85.4%

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

      8. fabs-sqr 85.4%

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

      9. rem-square-sqrt 86.0%

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

   5. Simplified 86.0%

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

   6. Taylor expanded in x around 0 87.1%

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

      1. distribute-lft-out 87.1%

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

      2. +-commutative 87.1%

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

   8. Simplified 87.1%

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

   if -9.5000000000000001e-164 < x < 2.54999999999999997e113

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 79.5%

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

   if 2.54999999999999997e113 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 88.7%

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

      1. neg-mul-1 88.7%

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

   5. Simplified 88.7%

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

   6. Taylor expanded in x around 0 88.7%

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

      1. *-commutative 88.7%

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

      2. fabs-neg 88.7%

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

      3. rem-square-sqrt 88.7%

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

      4. fabs-sqr 88.7%

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

      5. rem-square-sqrt 88.7%

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

      6. *-rgt-identity 88.7%

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

      7. distribute-lft-out 88.7%

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

      8. metadata-eval 88.7%

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

   8. Simplified 88.7%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-164}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{+113}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 58.9% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{-163}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-83}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.12e-163) (* x 0.5) (if (<= x 7e-83) (* y 0.5) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.12e-163) {
		tmp = x * 0.5;
	} else if (x <= 7e-83) {
		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 <= (-1.12d-163)) then
        tmp = x * 0.5d0
    else if (x <= 7d-83) 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 <= -1.12e-163) {
		tmp = x * 0.5;
	} else if (x <= 7e-83) {
		tmp = y * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.12e-163:
		tmp = x * 0.5
	elif x <= 7e-83:
		tmp = y * 0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.12e-163)
		tmp = Float64(x * 0.5);
	elseif (x <= 7e-83)
		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 <= -1.12e-163)
		tmp = x * 0.5;
	elseif (x <= 7e-83)
		tmp = y * 0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.12e-163], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, 7e-83], N[(y * 0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.12e-163

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   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 85.4%

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

      8. fabs-sqr 85.4%

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

      9. rem-square-sqrt 86.0%

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

   5. Simplified 86.0%

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

   6. Taylor expanded in y around 0 70.6%

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

   if -1.12e-163 < x < 7.00000000000000061e-83

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 89.4%

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

      1. add-sqr-sqrt 88.7%

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

      2. associate-/l* 88.7%

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

      3. add-sqr-sqrt 40.0%

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

      4. fabs-sqr 40.0%

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

      5. add-sqr-sqrt 40.0%

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

      6. add-sqr-sqrt 40.0%

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

      7. fabs-sqr 40.0%

         \[\leadsto x + \sqrt{y} \cdot \frac{\sqrt{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}}{2} \]

      8. add-sqr-sqrt 40.0%

         \[\leadsto x + \sqrt{y} \cdot \frac{\sqrt{\color{blue}{y}}}{2} \]

   5. Applied egg-rr 40.0%

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

      1. associate-*r/ 40.0%

         \[\leadsto x + \color{blue}{\frac{\sqrt{y} \cdot \sqrt{y}}{2}} \]

      2. rem-square-sqrt 42.5%

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

   7. Simplified 42.5%

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

   8. Taylor expanded in y around inf 42.5%

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

   9. Taylor expanded in x around 0 41.1%

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

   if 7.00000000000000061e-83 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 69.3%

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

      1. neg-mul-1 69.3%

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

   5. Simplified 69.3%

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

   6. Taylor expanded in x around 0 69.3%

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

      1. *-commutative 69.3%

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

      2. fabs-neg 69.3%

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

      3. rem-square-sqrt 69.3%

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

      4. fabs-sqr 69.3%

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

      5. rem-square-sqrt 69.3%

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

      6. *-rgt-identity 69.3%

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

      7. distribute-lft-out 69.3%

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

      8. metadata-eval 69.3%

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

   8. Simplified 69.3%

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

Alternative 4: 68.2% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.3 \cdot 10^{-83}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 3.3e-83) (* 0.5 (+ x y)) (* x 1.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 3.3e-83) {
		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 <= 3.3d-83) 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 <= 3.3e-83) {
		tmp = 0.5 * (x + y);
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 3.3e-83:
		tmp = 0.5 * (x + y)
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 3.3e-83)
		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 <= 3.3e-83)
		tmp = 0.5 * (x + y);
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 3.3e-83], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.2999999999999999e-83

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 85.0%

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

      1. +-commutative 85.0%

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

      2. sub-neg 85.0%

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

      3. neg-mul-1 85.0%

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

      4. fma-define 85.0%

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

      5. neg-mul-1 85.0%

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

      6. sub-neg 85.0%

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

      7. rem-square-sqrt 61.0%

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

      8. fabs-sqr 61.0%

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

      9. rem-square-sqrt 62.3%

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

   5. Simplified 62.3%

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

   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 3.2999999999999999e-83 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 69.3%

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

      1. neg-mul-1 69.3%

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

   5. Simplified 69.3%

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

   6. Taylor expanded in x around 0 69.3%

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

      1. *-commutative 69.3%

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

      2. fabs-neg 69.3%

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

      3. rem-square-sqrt 69.3%

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

      4. fabs-sqr 69.3%

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

      5. rem-square-sqrt 69.3%

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

      6. *-rgt-identity 69.3%

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

      7. distribute-lft-out 69.3%

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

      8. metadata-eval 69.3%

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

   8. Simplified 69.3%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.3 \cdot 10^{-83}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.6% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -5e-310) (* x 0.5) (* x 1.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5e-310) {
		tmp = x * 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 <= (-5d-310)) then
        tmp = x * 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 <= -5e-310) {
		tmp = x * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5e-310:
		tmp = x * 0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(x * 0.5);
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5e-310)
		tmp = x * 0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5e-310], N[(x * 0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 88.1%

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

      1. +-commutative 88.1%

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

      2. sub-neg 88.1%

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

      3. neg-mul-1 88.1%

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

      4. fma-define 88.1%

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

      5. neg-mul-1 88.1%

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

      6. sub-neg 88.1%

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

      7. rem-square-sqrt 72.3%

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

      8. fabs-sqr 72.3%

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

      9. rem-square-sqrt 73.0%

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

   5. Simplified 73.0%

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

   6. Taylor expanded in y around 0 53.8%

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

   if -4.999999999999985e-310 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 48.3%

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

      1. neg-mul-1 48.3%

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

   5. Simplified 48.3%

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

   6. Taylor expanded in x around 0 48.3%

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

      1. *-commutative 48.3%

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

      2. fabs-neg 48.3%

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

      3. rem-square-sqrt 48.3%

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

      4. fabs-sqr 48.3%

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

      5. rem-square-sqrt 48.3%

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

      6. *-rgt-identity 48.3%

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

      7. distribute-lft-out 48.3%

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

      8. metadata-eval 48.3%

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

   8. Simplified 48.3%

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

Alternative 6: 30.8% 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 89.6%

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

   1. +-commutative 89.6%

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

   2. sub-neg 89.6%

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

   3. neg-mul-1 89.6%

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

   4. fma-define 89.6%

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

   5. neg-mul-1 89.6%

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

   6. sub-neg 89.6%

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

   7. rem-square-sqrt 45.6%

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

   8. fabs-sqr 45.6%

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

   9. rem-square-sqrt 50.4%

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

5. Simplified 50.4%

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

6. Taylor expanded in y around 0 32.2%

   \[\leadsto x \cdot \color{blue}{0.5} \]
7. Add Preprocessing

Alternative 7: 11.3% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 x)

C

double code(double x, double y) {
	return x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x
end function

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

function tmp = code(x, y)
	tmp = x;
end

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around inf 11.6%

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

Reproduce

herbie shell --seed 2024186 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))