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

Percentage Accurate: 99.9% → 99.9%

Time: 10.0s

Alternatives: 10

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: 79.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-97}:\\ \;\;\;\;x + \frac{x - y}{-2}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-90}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left|x\right|}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.5e-97)
   (+ x (/ (- x y) -2.0))
   (if (<= x 5.5e-90) (+ x (/ (fabs y) 2.0)) (+ x (/ (fabs x) 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.5e-97) {
		tmp = x + ((x - y) / -2.0);
	} else if (x <= 5.5e-90) {
		tmp = x + (fabs(y) / 2.0);
	} else {
		tmp = x + (fabs(x) / 2.0);
	}
	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-97)) then
        tmp = x + ((x - y) / (-2.0d0))
    else if (x <= 5.5d-90) then
        tmp = x + (abs(y) / 2.0d0)
    else
        tmp = x + (abs(x) / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.5e-97) {
		tmp = x + ((x - y) / -2.0);
	} else if (x <= 5.5e-90) {
		tmp = x + (Math.abs(y) / 2.0);
	} else {
		tmp = x + (Math.abs(x) / 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.5e-97:
		tmp = x + ((x - y) / -2.0)
	elif x <= 5.5e-90:
		tmp = x + (math.fabs(y) / 2.0)
	else:
		tmp = x + (math.fabs(x) / 2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.5e-97)
		tmp = Float64(x + Float64(Float64(x - y) / -2.0));
	elseif (x <= 5.5e-90)
		tmp = Float64(x + Float64(abs(y) / 2.0));
	else
		tmp = Float64(x + Float64(abs(x) / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.5e-97)
		tmp = x + ((x - y) / -2.0);
	elseif (x <= 5.5e-90)
		tmp = x + (abs(y) / 2.0);
	else
		tmp = x + (abs(x) / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.5e-97], N[(x + N[(N[(x - y), $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.5e-90], N[(x + N[(N[Abs[y], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Abs[x], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.50000000000000019e-97

   1. Initial program 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. fma-define 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(1, x, \frac{\left|y - x\right|}{2}\right)} \]

      3. add-sqr-sqrt 92.0%

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

      4. fabs-sqr 92.0%

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

      5. add-sqr-sqrt 92.7%

         \[\leadsto \mathsf{fma}\left(1, x, \frac{\color{blue}{y - x}}{2}\right) \]

      6. frac-2neg 92.7%

         \[\leadsto \mathsf{fma}\left(1, x, \color{blue}{\frac{-\left(y - x\right)}{-2}}\right) \]

      7. distribute-frac-neg 92.7%

         \[\leadsto \mathsf{fma}\left(1, x, \color{blue}{-\frac{y - x}{-2}}\right) \]

      8. fmm-undef 92.7%

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

      9. *-un-lft-identity 92.7%

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

      10. metadata-eval 92.7%

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

   4. Applied egg-rr 92.7%

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

   if -3.50000000000000019e-97 < x < 5.5000000000000003e-90

   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 5.5000000000000003e-90 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 82.4%

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

      1. neg-mul-1 82.4%

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

   5. Simplified 82.4%

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

   6. Taylor expanded in x around -inf 82.4%

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

      1. neg-mul-1 82.4%

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

      2. fabs-neg 82.4%

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

   8. Simplified 82.4%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-97}:\\ \;\;\;\;x + \frac{x - y}{-2}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-90}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left|x\right|}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{-90}:\\ \;\;\;\;x + \frac{x - y}{-2}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left|x\right|}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 9.5e-90) (+ x (/ (- x y) -2.0)) (+ x (/ (fabs x) 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 9.5e-90) {
		tmp = x + ((x - y) / -2.0);
	} else {
		tmp = x + (fabs(x) / 2.0);
	}
	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-90) then
        tmp = x + ((x - y) / (-2.0d0))
    else
        tmp = x + (abs(x) / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 9.5e-90) {
		tmp = x + ((x - y) / -2.0);
	} else {
		tmp = x + (Math.abs(x) / 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 9.5e-90:
		tmp = x + ((x - y) / -2.0)
	else:
		tmp = x + (math.fabs(x) / 2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 9.5e-90)
		tmp = Float64(x + Float64(Float64(x - y) / -2.0));
	else
		tmp = Float64(x + Float64(abs(x) / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 9.5e-90)
		tmp = x + ((x - y) / -2.0);
	else
		tmp = x + (abs(x) / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 9.5e-90], N[(x + N[(N[(x - y), $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Abs[x], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.5000000000000003e-90

   1. Initial program 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. fma-define 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(1, x, \frac{\left|y - x\right|}{2}\right)} \]

      3. add-sqr-sqrt 71.8%

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

      4. fabs-sqr 71.8%

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

      5. add-sqr-sqrt 73.5%

         \[\leadsto \mathsf{fma}\left(1, x, \frac{\color{blue}{y - x}}{2}\right) \]

      6. frac-2neg 73.5%

         \[\leadsto \mathsf{fma}\left(1, x, \color{blue}{\frac{-\left(y - x\right)}{-2}}\right) \]

      7. distribute-frac-neg 73.5%

         \[\leadsto \mathsf{fma}\left(1, x, \color{blue}{-\frac{y - x}{-2}}\right) \]

      8. fmm-undef 73.5%

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

      9. *-un-lft-identity 73.5%

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

      10. metadata-eval 73.5%

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

   4. Applied egg-rr 73.5%

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

   if 9.5000000000000003e-90 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 82.4%

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

      1. neg-mul-1 82.4%

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

   5. Simplified 82.4%

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

   6. Taylor expanded in x around -inf 82.4%

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

      1. neg-mul-1 82.4%

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

      2. fabs-neg 82.4%

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

   8. Simplified 82.4%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{-90}:\\ \;\;\;\;x + \frac{x - y}{-2}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left|x\right|}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.7% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-47}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-93}:\\ \;\;\;\;y \cdot \left(0.5 + \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.7e-47)
   (* x 0.5)
   (if (<= x 1.3e-93) (* y (+ 0.5 (/ x y))) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.7e-47) {
		tmp = x * 0.5;
	} else if (x <= 1.3e-93) {
		tmp = y * (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 <= (-3.7d-47)) then
        tmp = x * 0.5d0
    else if (x <= 1.3d-93) then
        tmp = y * (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 <= -3.7e-47) {
		tmp = x * 0.5;
	} else if (x <= 1.3e-93) {
		tmp = y * (0.5 + (x / y));
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.7e-47:
		tmp = x * 0.5
	elif x <= 1.3e-93:
		tmp = y * (0.5 + (x / y))
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.7e-47)
		tmp = Float64(x * 0.5);
	elseif (x <= 1.3e-93)
		tmp = Float64(y * 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 <= -3.7e-47)
		tmp = x * 0.5;
	elseif (x <= 1.3e-93)
		tmp = y * (0.5 + (x / y));
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.7e-47], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, 1.3e-93], N[(y * N[(0.5 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.7e-47

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 77.8%

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

      1. neg-mul-1 77.8%

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

   5. Simplified 77.8%

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

   6. Taylor expanded in x around -inf 77.8%

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

      1. neg-mul-1 77.8%

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

      2. rem-square-sqrt 77.2%

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

      3. fabs-sqr 77.2%

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

      4. rem-square-sqrt 77.8%

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

   8. Simplified 77.8%

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

   9. Taylor expanded in x around 0 77.8%

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

   if -3.7e-47 < x < 1.2999999999999999e-93

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 86.2%

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

      1. div-inv 86.2%

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

      2. add-sqr-sqrt 50.1%

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

      3. fabs-sqr 50.1%

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

      4. add-sqr-sqrt 52.5%

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

      5. metadata-eval 52.5%

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

   5. Applied egg-rr 52.5%

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

   6. Taylor expanded in y around inf 52.5%

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

   if 1.2999999999999999e-93 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 82.4%

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

      1. neg-mul-1 82.4%

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

   5. Simplified 82.4%

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

      1. clear-num 82.4%

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

      2. inv-pow 82.4%

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

      3. fabs-neg 82.4%

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

   7. Applied egg-rr 82.4%

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

      1. unpow-1 82.4%

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

   9. Simplified 82.4%

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

   10. Taylor expanded in x around 0 82.4%

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

       1. *-commutative 82.4%

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

       2. rem-square-sqrt 82.3%

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

       3. fabs-sqr 82.3%

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

       4. rem-square-sqrt 82.4%

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

       5. *-rgt-identity 82.4%

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

       6. distribute-lft-out 82.4%

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

       7. metadata-eval 82.4%

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

   12. Simplified 82.4%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-47}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-93}:\\ \;\;\;\;y \cdot \left(0.5 + \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.6% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.26 \cdot 10^{-45}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-91}:\\ \;\;\;\;x + y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.26e-45)
   (* x 0.5)
   (if (<= x 1.45e-91) (+ x (* y 0.5)) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.26e-45) {
		tmp = x * 0.5;
	} else if (x <= 1.45e-91) {
		tmp = x + (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 <= (-1.26d-45)) then
        tmp = x * 0.5d0
    else if (x <= 1.45d-91) then
        tmp = x + (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 <= -1.26e-45) {
		tmp = x * 0.5;
	} else if (x <= 1.45e-91) {
		tmp = x + (y * 0.5);
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.26e-45:
		tmp = x * 0.5
	elif x <= 1.45e-91:
		tmp = x + (y * 0.5)
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.26e-45)
		tmp = Float64(x * 0.5);
	elseif (x <= 1.45e-91)
		tmp = Float64(x + 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 <= -1.26e-45)
		tmp = x * 0.5;
	elseif (x <= 1.45e-91)
		tmp = x + (y * 0.5);
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.26e-45], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, 1.45e-91], N[(x + N[(y * 0.5), $MachinePrecision]), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.26e-45

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 77.8%

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

      1. neg-mul-1 77.8%

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

   5. Simplified 77.8%

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

   6. Taylor expanded in x around -inf 77.8%

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

      1. neg-mul-1 77.8%

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

      2. rem-square-sqrt 77.2%

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

      3. fabs-sqr 77.2%

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

      4. rem-square-sqrt 77.8%

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

   8. Simplified 77.8%

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

   9. Taylor expanded in x around 0 77.8%

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

   if -1.26e-45 < x < 1.45e-91

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 86.2%

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

      1. div-inv 86.2%

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

      2. add-sqr-sqrt 50.1%

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

      3. fabs-sqr 50.1%

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

      4. add-sqr-sqrt 52.5%

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

      5. metadata-eval 52.5%

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

   5. Applied egg-rr 52.5%

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

   if 1.45e-91 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 82.4%

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

      1. neg-mul-1 82.4%

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

   5. Simplified 82.4%

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

      1. clear-num 82.4%

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

      2. inv-pow 82.4%

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

      3. fabs-neg 82.4%

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

   7. Applied egg-rr 82.4%

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

      1. unpow-1 82.4%

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

   9. Simplified 82.4%

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

   10. Taylor expanded in x around 0 82.4%

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

       1. *-commutative 82.4%

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

       2. rem-square-sqrt 82.3%

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

       3. fabs-sqr 82.3%

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

       4. rem-square-sqrt 82.4%

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

       5. *-rgt-identity 82.4%

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

       6. distribute-lft-out 82.4%

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

       7. metadata-eval 82.4%

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

   12. Simplified 82.4%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.26 \cdot 10^{-45}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-91}:\\ \;\;\;\;x + y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.6% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-46}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-90}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.2e-46) (* x 0.5) (if (<= x 5.8e-90) (* y 0.5) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.2e-46) {
		tmp = x * 0.5;
	} else if (x <= 5.8e-90) {
		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 <= (-2.2d-46)) then
        tmp = x * 0.5d0
    else if (x <= 5.8d-90) 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 <= -2.2e-46) {
		tmp = x * 0.5;
	} else if (x <= 5.8e-90) {
		tmp = y * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.2e-46:
		tmp = x * 0.5
	elif x <= 5.8e-90:
		tmp = y * 0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.2e-46)
		tmp = Float64(x * 0.5);
	elseif (x <= 5.8e-90)
		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 <= -2.2e-46)
		tmp = x * 0.5;
	elseif (x <= 5.8e-90)
		tmp = y * 0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.2e-46], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, 5.8e-90], N[(y * 0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.2000000000000001e-46

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 77.8%

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

      1. neg-mul-1 77.8%

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

   5. Simplified 77.8%

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

   6. Taylor expanded in x around -inf 77.8%

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

      1. neg-mul-1 77.8%

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

      2. rem-square-sqrt 77.2%

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

      3. fabs-sqr 77.2%

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

      4. rem-square-sqrt 77.8%

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

   8. Simplified 77.8%

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

   9. Taylor expanded in x around 0 77.8%

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

   if -2.2000000000000001e-46 < x < 5.79999999999999967e-90

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 86.2%

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

      1. div-inv 86.2%

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

      2. add-sqr-sqrt 50.1%

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

      3. fabs-sqr 50.1%

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

      4. add-sqr-sqrt 52.5%

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

      5. metadata-eval 52.5%

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

   5. Applied egg-rr 52.5%

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

   6. Taylor expanded in x around inf 32.5%

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

      1. associate-*r/ 32.5%

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

   8. Simplified 32.5%

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

   9. Taylor expanded in x around 0 50.3%

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

   if 5.79999999999999967e-90 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 82.4%

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

      1. neg-mul-1 82.4%

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

   5. Simplified 82.4%

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

      1. clear-num 82.4%

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

      2. inv-pow 82.4%

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

      3. fabs-neg 82.4%

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

   7. Applied egg-rr 82.4%

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

      1. unpow-1 82.4%

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

   9. Simplified 82.4%

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

   10. Taylor expanded in x around 0 82.4%

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

       1. *-commutative 82.4%

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

       2. rem-square-sqrt 82.3%

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

       3. fabs-sqr 82.3%

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

       4. rem-square-sqrt 82.4%

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

       5. *-rgt-identity 82.4%

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

       6. distribute-lft-out 82.4%

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

       7. metadata-eval 82.4%

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

   12. Simplified 82.4%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-46}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-90}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.8% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.4 \cdot 10^{-94}:\\ \;\;\;\;x + \frac{x - y}{-2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 6.4e-94) (+ x (/ (- x y) -2.0)) (* x 1.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 6.4e-94) {
		tmp = x + ((x - 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 <= 6.4d-94) then
        tmp = x + ((x - 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 <= 6.4e-94) {
		tmp = x + ((x - y) / -2.0);
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 6.4e-94:
		tmp = x + ((x - y) / -2.0)
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 6.4e-94)
		tmp = Float64(x + Float64(Float64(x - 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 <= 6.4e-94)
		tmp = x + ((x - y) / -2.0);
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 6.4e-94], N[(x + N[(N[(x - y), $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.39999999999999993e-94

   1. Initial program 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. fma-define 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(1, x, \frac{\left|y - x\right|}{2}\right)} \]

      3. add-sqr-sqrt 71.8%

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

      4. fabs-sqr 71.8%

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

      5. add-sqr-sqrt 73.5%

         \[\leadsto \mathsf{fma}\left(1, x, \frac{\color{blue}{y - x}}{2}\right) \]

      6. frac-2neg 73.5%

         \[\leadsto \mathsf{fma}\left(1, x, \color{blue}{\frac{-\left(y - x\right)}{-2}}\right) \]

      7. distribute-frac-neg 73.5%

         \[\leadsto \mathsf{fma}\left(1, x, \color{blue}{-\frac{y - x}{-2}}\right) \]

      8. fmm-undef 73.5%

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

      9. *-un-lft-identity 73.5%

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

      10. metadata-eval 73.5%

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

   4. Applied egg-rr 73.5%

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

   if 6.39999999999999993e-94 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 82.4%

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

      1. neg-mul-1 82.4%

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

   5. Simplified 82.4%

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

      1. clear-num 82.4%

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

      2. inv-pow 82.4%

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

      3. fabs-neg 82.4%

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

   7. Applied egg-rr 82.4%

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

      1. unpow-1 82.4%

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

   9. Simplified 82.4%

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

   10. Taylor expanded in x around 0 82.4%

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

       1. *-commutative 82.4%

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

       2. rem-square-sqrt 82.3%

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

       3. fabs-sqr 82.3%

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

       4. rem-square-sqrt 82.4%

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

       5. *-rgt-identity 82.4%

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

       6. distribute-lft-out 82.4%

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

       7. metadata-eval 82.4%

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

   12. Simplified 82.4%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.4 \cdot 10^{-94}:\\ \;\;\;\;x + \frac{x - y}{-2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.8% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-115}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 8e-115) (* x 0.5) (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8e-115) {
		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 <= 8d-115) 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 <= 8e-115) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8e-115:
		tmp = x * 0.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8e-115)
		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 <= 8e-115)
		tmp = x * 0.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8e-115], N[(x * 0.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.0000000000000004e-115

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 67.1%

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

      1. neg-mul-1 67.1%

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

   5. Simplified 67.1%

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

   6. Taylor expanded in x around -inf 67.1%

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

      1. neg-mul-1 67.1%

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

      2. rem-square-sqrt 32.5%

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

      3. fabs-sqr 32.5%

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

      4. rem-square-sqrt 39.6%

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

   8. Simplified 39.6%

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

   9. Taylor expanded in x around 0 39.6%

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

   if 8.0000000000000004e-115 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 70.8%

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

      1. div-inv 70.8%

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

      2. add-sqr-sqrt 70.4%

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

      3. fabs-sqr 70.4%

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

      4. add-sqr-sqrt 70.8%

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

      5. metadata-eval 70.8%

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

   5. Applied egg-rr 70.8%

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

   6. Taylor expanded in x around inf 48.7%

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

      1. associate-*r/ 48.7%

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

   8. Simplified 48.7%

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

   9. Taylor expanded in x around 0 65.5%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-115}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 30.0% 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 y around 0 55.1%

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

   1. neg-mul-1 55.1%

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

5. Simplified 55.1%

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

6. Taylor expanded in x around -inf 55.1%

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

   1. neg-mul-1 55.1%

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

   2. rem-square-sqrt 25.8%

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

   3. fabs-sqr 25.8%

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

   4. rem-square-sqrt 32.1%

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

8. Simplified 32.1%

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

9. Taylor expanded in x around 0 32.1%

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

10. Final simplification 32.1%

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

Alternative 10: 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 12.0%

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

Reproduce

herbie shell --seed 2024185 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))