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

Percentage Accurate: 100.0% → 100.0%

Time: 4.0s

Alternatives: 6

Speedup: 1.0×

• could not determine a ground truth

  1. x = 4.37522298520882e-45
  2. y = 8.610061295766484e-155

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: 100.0% 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: 100.0% 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: 71.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-138}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6.8e-138) (* 0.5 (+ x y)) (* (fabs (- y x)) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6.8e-138) {
		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 <= (-6.8d-138)) 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 <= -6.8e-138) {
		tmp = 0.5 * (x + y);
	} else {
		tmp = Math.abs((y - x)) * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6.8e-138:
		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 <= -6.8e-138)
		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 <= -6.8e-138)
		tmp = 0.5 * (x + y);
	else
		tmp = abs((y - x)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6.8e-138], 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 < -6.8000000000000003e-138

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 99.0%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. rem-square-sqrt 85.0%

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

      4. fabs-sqr 85.0%

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

      5. rem-square-sqrt 85.6%

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

      6. fma-undefine 85.6%

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

      7. +-commutative 85.6%

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

      8. sub-neg 85.6%

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

      9. distribute-lft-in 85.6%

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

      10. distribute-rgt-neg-in 85.6%

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

      11. distribute-lft-neg-in 85.6%

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

      12. metadata-eval 85.6%

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

      13. +-commutative 85.6%

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

      14. associate-+r+ 85.6%

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

      15. distribute-rgt1-in 85.6%

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

      16. metadata-eval 85.6%

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

      17. distribute-lft-out 85.6%

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

      18. +-commutative 85.6%

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

   6. Simplified 85.6%

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

   if -6.8000000000000003e-138 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 65.8%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-138}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 45.7% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.7 \cdot 10^{-171}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 5.7e-171) (* x 0.5) (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.7e-171) {
		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 <= 5.7d-171) 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 <= 5.7e-171) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.7e-171:
		tmp = x * 0.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.7e-171)
		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 <= 5.7e-171)
		tmp = x * 0.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.7e-171], N[(x * 0.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.6999999999999996e-171

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing
   3. 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-define 100.0%

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

      4. add-sqr-sqrt 37.1%

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

      5. fabs-sqr 37.1%

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

      6. add-sqr-sqrt 42.1%

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

      7. fma-define 42.1%

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

      8. div-inv 42.1%

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

      9. add-sqr-sqrt 37.1%

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

      10. fabs-sqr 37.1%

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

      11. add-sqr-sqrt 100.0%

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

      12. add-cube-cbrt 98.2%

          \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\left|y - x\right|} \cdot \sqrt[3]{\left|y - x\right|}\right) \cdot \sqrt[3]{\left|y - x\right|}}}{2} + x \]

      13. associate-/l* 98.2%

          \[\leadsto \color{blue}{\left(\sqrt[3]{\left|y - x\right|} \cdot \sqrt[3]{\left|y - x\right|}\right) \cdot \frac{\sqrt[3]{\left|y - x\right|}}{2}} + x \]

      14. fma-define 98.2%

          \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\left|y - x\right|} \cdot \sqrt[3]{\left|y - x\right|}, \frac{\sqrt[3]{\left|y - x\right|}}{2}, x\right)} \]

   4. Applied egg-rr 41.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{y - x}\right)}^{2}, \frac{\sqrt[3]{y - x}}{2}, x\right)} \]
   5. Step-by-step derivation

      1. fma-undefine 41.4%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{y - x}\right)}^{2} \cdot \frac{\sqrt[3]{y - x}}{2} + x} \]

      2. +-commutative 41.4%

         \[\leadsto \color{blue}{x + {\left(\sqrt[3]{y - x}\right)}^{2} \cdot \frac{\sqrt[3]{y - x}}{2}} \]

      3. associate-*r/ 41.4%

         \[\leadsto x + \color{blue}{\frac{{\left(\sqrt[3]{y - x}\right)}^{2} \cdot \sqrt[3]{y - x}}{2}} \]

      4. unpow2 41.4%

         \[\leadsto x + \frac{\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right)} \cdot \sqrt[3]{y - x}}{2} \]

      5. rem-3cbrt-lft 42.1%

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

   6. Simplified 42.1%

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

   7. Taylor expanded in x around inf 39.7%

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

   if 5.6999999999999996e-171 < y

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing
   3. 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-define 100.0%

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

      4. add-sqr-sqrt 84.1%

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

      5. fabs-sqr 84.1%

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

      6. add-sqr-sqrt 87.4%

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

      7. fma-define 87.4%

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

      8. div-inv 87.4%

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

      9. add-sqr-sqrt 84.1%

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

      10. fabs-sqr 84.1%

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

      11. add-sqr-sqrt 100.0%

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

      12. add-cube-cbrt 98.2%

          \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\left|y - x\right|} \cdot \sqrt[3]{\left|y - x\right|}\right) \cdot \sqrt[3]{\left|y - x\right|}}}{2} + x \]

      13. associate-/l* 98.2%

          \[\leadsto \color{blue}{\left(\sqrt[3]{\left|y - x\right|} \cdot \sqrt[3]{\left|y - x\right|}\right) \cdot \frac{\sqrt[3]{\left|y - x\right|}}{2}} + x \]

      14. fma-define 98.3%

          \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\left|y - x\right|} \cdot \sqrt[3]{\left|y - x\right|}, \frac{\sqrt[3]{\left|y - x\right|}}{2}, x\right)} \]

   4. Applied egg-rr 85.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{y - x}\right)}^{2}, \frac{\sqrt[3]{y - x}}{2}, x\right)} \]
   5. Step-by-step derivation

      1. fma-undefine 85.8%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{y - x}\right)}^{2} \cdot \frac{\sqrt[3]{y - x}}{2} + x} \]

      2. +-commutative 85.8%

         \[\leadsto \color{blue}{x + {\left(\sqrt[3]{y - x}\right)}^{2} \cdot \frac{\sqrt[3]{y - x}}{2}} \]

      3. associate-*r/ 85.8%

         \[\leadsto x + \color{blue}{\frac{{\left(\sqrt[3]{y - x}\right)}^{2} \cdot \sqrt[3]{y - x}}{2}} \]

      4. unpow2 85.8%

         \[\leadsto x + \frac{\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right)} \cdot \sqrt[3]{y - x}}{2} \]

      5. rem-3cbrt-lft 87.4%

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

   6. Simplified 87.4%

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

   7. 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 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.7 \cdot 10^{-171}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.7% 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. Taylor expanded in x around inf 87.4%

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

4. Taylor expanded in x around 0 100.0%

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

   1. +-commutative 100.0%

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

   2. fma-define 100.0%

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

   3. rem-square-sqrt 55.1%

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

   4. fabs-sqr 55.1%

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

   5. rem-square-sqrt 59.4%

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

   6. fma-undefine 59.4%

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

   7. +-commutative 59.4%

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

   8. sub-neg 59.4%

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

   9. distribute-lft-in 59.4%

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

   10. distribute-rgt-neg-in 59.4%

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

   11. distribute-lft-neg-in 59.4%

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

   12. metadata-eval 59.4%

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

   13. +-commutative 59.4%

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

   14. associate-+r+ 59.4%

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

   15. distribute-rgt1-in 59.4%

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

   16. metadata-eval 59.4%

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

   17. distribute-lft-out 59.4%

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

   18. +-commutative 59.4%

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

6. Simplified 59.4%

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

7. Final simplification 59.4%

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

Alternative 5: 31.6% 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 100.0%

   \[x + \frac{\left|y - x\right|}{2} \]
2. Add Preprocessing
3. 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-define 100.0%

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

   4. add-sqr-sqrt 55.1%

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

   5. fabs-sqr 55.1%

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

   6. add-sqr-sqrt 59.4%

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

   7. fma-define 59.4%

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

   8. div-inv 59.4%

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

   9. add-sqr-sqrt 55.1%

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

   10. fabs-sqr 55.1%

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

   11. add-sqr-sqrt 100.0%

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

   12. add-cube-cbrt 98.2%

       \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\left|y - x\right|} \cdot \sqrt[3]{\left|y - x\right|}\right) \cdot \sqrt[3]{\left|y - x\right|}}}{2} + x \]

   13. associate-/l* 98.2%

       \[\leadsto \color{blue}{\left(\sqrt[3]{\left|y - x\right|} \cdot \sqrt[3]{\left|y - x\right|}\right) \cdot \frac{\sqrt[3]{\left|y - x\right|}}{2}} + x \]

   14. fma-define 98.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\left|y - x\right|} \cdot \sqrt[3]{\left|y - x\right|}, \frac{\sqrt[3]{\left|y - x\right|}}{2}, x\right)} \]

4. Applied egg-rr 58.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{y - x}\right)}^{2}, \frac{\sqrt[3]{y - x}}{2}, x\right)} \]
5. Step-by-step derivation

   1. fma-undefine 58.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{y - x}\right)}^{2} \cdot \frac{\sqrt[3]{y - x}}{2} + x} \]

   2. +-commutative 58.4%

      \[\leadsto \color{blue}{x + {\left(\sqrt[3]{y - x}\right)}^{2} \cdot \frac{\sqrt[3]{y - x}}{2}} \]

   3. associate-*r/ 58.4%

      \[\leadsto x + \color{blue}{\frac{{\left(\sqrt[3]{y - x}\right)}^{2} \cdot \sqrt[3]{y - x}}{2}} \]

   4. unpow2 58.4%

      \[\leadsto x + \frac{\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right)} \cdot \sqrt[3]{y - x}}{2} \]

   5. rem-3cbrt-lft 59.4%

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

6. Simplified 59.4%

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

7. Taylor expanded in x around inf 33.3%

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

8. Final simplification 33.3%

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

Alternative 6: 11.1% 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 11.3%

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

4. Final simplification 11.3%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024073 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))