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

Percentage Accurate: 99.9% → 99.9%

Time: 2.4s

Alternatives: 6

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 70.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{-50}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.9e-50) (* 0.5 (+ x y)) (* (fabs (- y x)) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.9e-50) {
		tmp = 0.5 * (x + y);
	} else {
		tmp = fabs((y - x)) * 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 (x <= (-4.9d-50)) then
        tmp = 0.5d0 * (x + y)
    else
        tmp = abs((y - x)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4.9e-50) {
		tmp = 0.5 * (x + y);
	} else {
		tmp = Math.abs((y - x)) * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.9e-50:
		tmp = 0.5 * (x + y)
	else:
		tmp = math.fabs((y - x)) * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.9e-50)
		tmp = Float64(0.5 * Float64(x + y));
	else
		tmp = Float64(abs(Float64(y - x)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.9e-50)
		tmp = 0.5 * (x + y);
	else
		tmp = abs((y - x)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.9e-50], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.8999999999999999e-50

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 23.2%

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

      2. pow2 23.2%

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

      3. +-commutative 23.2%

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

      4. div-inv 23.2%

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

      5. fma-def 23.2%

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

      6. add-sqr-sqrt 13.4%

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

      7. fabs-sqr 13.4%

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

      8. add-sqr-sqrt 13.4%

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

      9. metadata-eval 13.4%

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(y - x, \color{blue}{0.5}, x\right)}\right)}^{2} \]

   4. Applied egg-rr 13.4%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(y - x, 0.5, x\right)}\right)}^{2}} \]

   5. Taylor expanded in y around 0 90.2%

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

      1. associate-+r+ 90.2%

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

      2. distribute-rgt1-in 90.2%

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

      3. metadata-eval 90.2%

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

      4. distribute-lft-out 90.2%

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

   7. Simplified 90.2%

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

   if -4.8999999999999999e-50 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 67.2%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{-50}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 31.7% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{-198}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.2e-198) x (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.2e-198) {
		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 <= 1.2d-198) 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 <= 1.2e-198) {
		tmp = x;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.2e-198:
		tmp = x
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.2e-198)
		tmp = x;
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.2e-198)
		tmp = x;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.2e-198], x, N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.19999999999999993e-198

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 12.4%

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

   if 1.19999999999999993e-198 < y

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 72.6%

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

      2. pow2 72.6%

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

      3. +-commutative 72.6%

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

      4. div-inv 72.6%

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

      5. fma-def 72.6%

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

      6. add-sqr-sqrt 55.9%

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

      7. fabs-sqr 55.9%

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

      8. add-sqr-sqrt 59.0%

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

      9. metadata-eval 59.0%

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(y - x, \color{blue}{0.5}, x\right)}\right)}^{2} \]

   4. Applied egg-rr 59.0%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(y - x, 0.5, x\right)}\right)}^{2}} \]

   5. Taylor expanded in x around 0 55.8%

      \[\leadsto \color{blue}{y \cdot {\left(\sqrt{0.5}\right)}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative 55.8%

         \[\leadsto \color{blue}{{\left(\sqrt{0.5}\right)}^{2} \cdot y} \]

      2. unpow2 55.8%

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

      3. rem-square-sqrt 56.8%

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

   7. Simplified 56.8%

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

4. Final simplification 31.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{-198}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 46.0% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-63}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.3e-63) (* x 0.5) (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.3e-63) {
		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 <= 1.3d-63) 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 <= 1.3e-63) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.3e-63:
		tmp = x * 0.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.3e-63)
		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 <= 1.3e-63)
		tmp = x * 0.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.3e-63], N[(x * 0.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.3000000000000001e-63

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 67.6%

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

      2. pow2 67.6%

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

      3. +-commutative 67.6%

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

      4. div-inv 67.6%

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

      5. fma-def 67.6%

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

      6. add-sqr-sqrt 2.9%

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

      7. fabs-sqr 2.9%

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

      8. add-sqr-sqrt 7.7%

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

      9. metadata-eval 7.7%

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(y - x, \color{blue}{0.5}, x\right)}\right)}^{2} \]

   4. Applied egg-rr 7.7%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(y - x, 0.5, x\right)}\right)}^{2}} \]

   5. Taylor expanded in y around 0 38.0%

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

      1. distribute-rgt1-in 38.0%

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

      2. metadata-eval 38.0%

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

      3. *-commutative 38.0%

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

   7. Simplified 38.0%

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

   if 1.3000000000000001e-63 < y

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 81.9%

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

      2. pow2 81.9%

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

      3. +-commutative 81.9%

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

      4. div-inv 81.9%

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

      5. fma-def 81.9%

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

      6. add-sqr-sqrt 64.5%

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

      7. fabs-sqr 64.5%

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

      8. add-sqr-sqrt 67.7%

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

      9. metadata-eval 67.7%

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(y - x, \color{blue}{0.5}, x\right)}\right)}^{2} \]

   4. Applied egg-rr 67.7%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(y - x, 0.5, x\right)}\right)}^{2}} \]

   5. Taylor expanded in x around 0 63.9%

      \[\leadsto \color{blue}{y \cdot {\left(\sqrt{0.5}\right)}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative 63.9%

         \[\leadsto \color{blue}{{\left(\sqrt{0.5}\right)}^{2} \cdot y} \]

      2. unpow2 63.9%

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

      3. rem-square-sqrt 65.1%

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

   7. Simplified 65.1%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-63}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.9% accurate, 21.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. add-sqr-sqrt 72.4%

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

   2. pow2 72.4%

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

   3. +-commutative 72.4%

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

   4. div-inv 72.4%

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

   5. fma-def 72.4%

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

   6. add-sqr-sqrt 23.6%

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

   7. fabs-sqr 23.6%

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

   8. add-sqr-sqrt 27.9%

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

   9. metadata-eval 27.9%

      \[\leadsto {\left(\sqrt{\mathsf{fma}\left(y - x, \color{blue}{0.5}, x\right)}\right)}^{2} \]

4. Applied egg-rr 27.9%

   \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(y - x, 0.5, x\right)}\right)}^{2}} \]

5. Taylor expanded in y around 0 55.3%

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

   1. associate-+r+ 55.3%

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

   2. distribute-rgt1-in 55.3%

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

   3. metadata-eval 55.3%

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

   4. distribute-lft-out 55.3%

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

7. Simplified 55.3%

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

8. Final simplification 55.3%

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

Alternative 6: 11.3% accurate, 107.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around inf 11.4%

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

4. Final simplification 11.4%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))