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

Percentage Accurate: 99.9% → 99.9%

Time: 2.3s

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 99.9%

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

2. Final simplification 99.9%

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

Alternative 2: 70.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-170}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.1e-170) (* 0.5 (+ x y)) (* (fabs (- y x)) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.1e-170) {
		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 <= (-3.1d-170)) 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 <= -3.1e-170) {
		tmp = 0.5 * (x + y);
	} else {
		tmp = Math.abs((y - x)) * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.1e-170:
		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 <= -3.1e-170)
		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 <= -3.1e-170)
		tmp = 0.5 * (x + y);
	else
		tmp = abs((y - x)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.1e-170], 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 < -3.09999999999999986e-170

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. div-inv 100.0%

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

      3. fma-def 100.0%

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

      4. add-sqr-sqrt 89.2%

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

      5. fabs-sqr 89.2%

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

      6. add-sqr-sqrt 90.0%

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

      7. fma-def 90.0%

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

      8. add-sqr-sqrt 89.2%

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

      9. fabs-sqr 89.2%

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

      10. add-sqr-sqrt 100.0%

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

      11. add-sqr-sqrt 99.2%

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

      12. associate-*l* 99.2%

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

      13. fma-def 99.2%

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

   3. Applied egg-rr 89.2%

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

   4. Taylor expanded in y around 0 0.0%

      \[\leadsto \color{blue}{x + \left(0.5 \cdot y + 0.5 \cdot \left(x \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative 0.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot y + 0.5 \cdot \left(x \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) + x} \]

      2. associate-+l+ 0.0%

         \[\leadsto \color{blue}{0.5 \cdot y + \left(0.5 \cdot \left(x \cdot {\left(\sqrt{-1}\right)}^{2}\right) + x\right)} \]

      3. *-commutative 0.0%

         \[\leadsto 0.5 \cdot y + \left(0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot x\right)} + x\right) \]

      4. associate-*r* 0.0%

         \[\leadsto 0.5 \cdot y + \left(\color{blue}{\left(0.5 \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot x} + x\right) \]

      5. unpow2 0.0%

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

      6. rem-square-sqrt 90.0%

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

      7. metadata-eval 90.0%

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

      8. distribute-lft1-in 90.0%

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

      9. metadata-eval 90.0%

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

      10. distribute-lft-in 90.0%

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

      11. +-commutative 90.0%

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

   6. Simplified 90.0%

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

   if -3.09999999999999986e-170 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 67.6%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-170}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \end{array} \]

Alternative 3: 31.9% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{-210}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 6.2e-210) x (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6.2e-210) {
		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 <= 6.2d-210) 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 <= 6.2e-210) {
		tmp = x;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 6.2e-210:
		tmp = x
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6.2e-210)
		tmp = x;
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.2e-210)
		tmp = x;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6.2e-210], x, N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.19999999999999973e-210

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 12.8%

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

   if 6.19999999999999973e-210 < y

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. div-inv 100.0%

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

      3. fma-def 100.0%

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

      4. add-sqr-sqrt 84.8%

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

      5. fabs-sqr 84.8%

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

      6. add-sqr-sqrt 88.0%

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

      7. fma-def 88.0%

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

      8. add-sqr-sqrt 84.8%

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

      9. fabs-sqr 84.8%

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

      10. add-sqr-sqrt 100.0%

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

      11. add-sqr-sqrt 99.4%

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

      12. associate-*l* 99.4%

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

      13. fma-def 99.4%

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

   3. Applied egg-rr 84.8%

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

   4. Taylor expanded in x around 0 58.7%

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

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{-210}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 4: 46.4% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{-112}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.65e-112) (* x 0.5) (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.65e-112) {
		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 <= 1.65d-112) 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 <= 1.65e-112) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.65e-112:
		tmp = x * 0.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.65e-112)
		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 <= 1.65e-112)
		tmp = x * 0.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.65e-112], N[(x * 0.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.65e-112

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. div-inv 99.9%

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

      3. fma-def 99.9%

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

      4. add-sqr-sqrt 36.2%

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

      5. fabs-sqr 36.2%

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

      6. add-sqr-sqrt 41.9%

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

      7. fma-def 41.9%

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

      8. add-sqr-sqrt 36.2%

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

      9. fabs-sqr 36.2%

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

      10. add-sqr-sqrt 99.9%

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

      11. add-sqr-sqrt 99.3%

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

      12. associate-*l* 99.3%

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

      13. fma-def 99.4%

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

   3. Applied egg-rr 36.2%

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

   4. Taylor expanded in y around 0 0.0%

      \[\leadsto \color{blue}{x + 0.5 \cdot \left(x \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto x + 0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot x\right)} \]

      2. associate-*r* 0.0%

         \[\leadsto x + \color{blue}{\left(0.5 \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot x} \]

      3. unpow2 0.0%

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

      4. rem-square-sqrt 38.9%

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

      5. metadata-eval 38.9%

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

      6. distribute-rgt1-in 38.9%

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

      7. metadata-eval 38.9%

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

      8. *-commutative 38.9%

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

   6. Simplified 38.9%

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

   if 1.65e-112 < y

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. div-inv 100.0%

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

      3. fma-def 100.0%

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

      4. add-sqr-sqrt 88.1%

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

      5. fabs-sqr 88.1%

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

      6. add-sqr-sqrt 90.7%

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

      7. fma-def 90.7%

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

      8. add-sqr-sqrt 88.1%

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

      9. fabs-sqr 88.1%

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

      10. add-sqr-sqrt 100.0%

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

      11. add-sqr-sqrt 99.3%

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

      12. associate-*l* 99.3%

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

      13. fma-def 99.3%

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

   3. Applied egg-rr 88.1%

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

   4. Taylor expanded in x around 0 67.5%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{-112}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 5: 55.9% 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 99.9%

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

   1. +-commutative 99.9%

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

   2. div-inv 99.9%

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

   3. fma-def 99.9%

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

   4. add-sqr-sqrt 54.3%

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

   5. fabs-sqr 54.3%

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

   6. add-sqr-sqrt 58.9%

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

   7. fma-def 58.9%

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

   8. add-sqr-sqrt 54.3%

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

   9. fabs-sqr 54.3%

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

   10. add-sqr-sqrt 99.9%

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

   11. add-sqr-sqrt 99.3%

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

   12. associate-*l* 99.3%

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

   13. fma-def 99.4%

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

3. Applied egg-rr 54.3%

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

4. Taylor expanded in y around 0 0.0%

   \[\leadsto \color{blue}{x + \left(0.5 \cdot y + 0.5 \cdot \left(x \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]
5. Step-by-step derivation

   1. +-commutative 0.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot y + 0.5 \cdot \left(x \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) + x} \]

   2. associate-+l+ 0.0%

      \[\leadsto \color{blue}{0.5 \cdot y + \left(0.5 \cdot \left(x \cdot {\left(\sqrt{-1}\right)}^{2}\right) + x\right)} \]

   3. *-commutative 0.0%

      \[\leadsto 0.5 \cdot y + \left(0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot x\right)} + x\right) \]

   4. associate-*r* 0.0%

      \[\leadsto 0.5 \cdot y + \left(\color{blue}{\left(0.5 \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot x} + x\right) \]

   5. unpow2 0.0%

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

   6. rem-square-sqrt 58.9%

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

   7. metadata-eval 58.9%

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

   8. distribute-lft1-in 58.9%

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

   9. metadata-eval 58.9%

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

   10. distribute-lft-in 58.9%

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

   11. +-commutative 58.9%

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

6. Simplified 58.9%

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

7. Final simplification 58.9%

   \[\leadsto 0.5 \cdot \left(x + y\right) \]

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 99.9%

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

2. Taylor expanded in x around inf 11.6%

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

3. Final simplification 11.6%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023271 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))