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

Percentage Accurate: 99.9% → 99.9%

Time: 6.2s

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. Add Preprocessing

3. Final simplification 99.9%

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

Alternative 2: 70.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.16 \cdot 10^{-184}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.16e-184) (* 0.5 (+ x y)) (* (fabs (- y x)) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.16e-184) {
		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 <= (-1.16d-184)) 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 <= -1.16e-184) {
		tmp = 0.5 * (x + y);
	} else {
		tmp = Math.abs((y - x)) * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.16e-184:
		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 <= -1.16e-184)
		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 <= -1.16e-184)
		tmp = 0.5 * (x + y);
	else
		tmp = abs((y - x)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.16e-184], 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 < -1.15999999999999998e-184

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 97.9%

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

   4. 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 87.1%

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

      4. fabs-sqr 87.1%

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

      5. rem-square-sqrt 88.0%

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

      6. fma-undefine 88.0%

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

      7. +-commutative 88.0%

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

      8. sub-neg 88.0%

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

      9. distribute-lft-in 88.0%

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

      10. distribute-rgt-neg-in 88.0%

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

      11. distribute-lft-neg-in 88.0%

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

      12. metadata-eval 88.0%

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

      13. +-commutative 88.0%

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

      14. associate-+r+ 88.0%

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

      15. distribute-rgt1-in 88.0%

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

      16. metadata-eval 88.0%

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

      17. distribute-lft-out 88.0%

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

      18. +-commutative 88.0%

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

   6. Simplified 88.0%

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

   if -1.15999999999999998e-184 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 64.4%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.16 \cdot 10^{-184}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 45.4% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-111} \lor \neg \left(y \leq 3.6 \cdot 10^{-41}\right) \land y \leq 5.4 \cdot 10^{-17}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y 1.15e-111) (and (not (<= y 3.6e-41)) (<= y 5.4e-17)))
   (* x 0.5)
   (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= 1.15e-111) || (!(y <= 3.6e-41) && (y <= 5.4e-17))) {
		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 <= 1.15d-111) .or. (.not. (y <= 3.6d-41)) .and. (y <= 5.4d-17)) 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 <= 1.15e-111) || (!(y <= 3.6e-41) && (y <= 5.4e-17))) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= 1.15e-111) or (not (y <= 3.6e-41) and (y <= 5.4e-17)):
		tmp = x * 0.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= 1.15e-111) || (!(y <= 3.6e-41) && (y <= 5.4e-17)))
		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 <= 1.15e-111) || (~((y <= 3.6e-41)) && (y <= 5.4e-17)))
		tmp = x * 0.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, 1.15e-111], And[N[Not[LessEqual[y, 3.6e-41]], $MachinePrecision], LessEqual[y, 5.4e-17]]], N[(x * 0.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.15e-111 or 3.6e-41 < y < 5.4000000000000002e-17

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing
   3. 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-define 99.9%

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

      4. add-sqr-sqrt 32.3%

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

      5. fabs-sqr 32.3%

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

      6. add-sqr-sqrt 38.8%

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

      7. fma-define 38.8%

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

      8. div-inv 38.8%

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

      9. add-sqr-sqrt 32.3%

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

      10. fabs-sqr 32.3%

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

      11. add-sqr-sqrt 99.9%

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

      12. add-cube-cbrt 98.3%

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

      13. associate-/l* 98.3%

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

      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 38.1%

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

      1. fma-undefine 38.1%

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

      2. +-commutative 38.1%

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

      3. associate-*r/ 38.1%

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

      4. unpow2 38.1%

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

      5. rem-3cbrt-lft 38.8%

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

   6. Simplified 38.8%

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

   7. Taylor expanded in x around inf 36.2%

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

   if 1.15e-111 < y < 3.6e-41 or 5.4000000000000002e-17 < y

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Add Preprocessing
   3. 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-define 99.9%

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

      4. add-sqr-sqrt 86.9%

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

      5. fabs-sqr 86.9%

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

      6. add-sqr-sqrt 89.8%

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

      7. fma-define 89.8%

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

      8. div-inv 89.8%

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

      9. add-sqr-sqrt 86.9%

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

      10. fabs-sqr 86.9%

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

      11. add-sqr-sqrt 99.9%

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

      12. add-cube-cbrt 97.9%

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

      13. associate-/l* 97.9%

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

      14. fma-define 97.9%

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

   4. Applied egg-rr 87.9%

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

      1. fma-undefine 87.9%

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

      2. +-commutative 87.9%

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

      3. associate-*r/ 87.9%

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

      4. unpow2 87.9%

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

      5. rem-3cbrt-lft 89.8%

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

   6. Simplified 89.8%

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

   7. Taylor expanded in x around 0 73.8%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-111} \lor \neg \left(y \leq 3.6 \cdot 10^{-41}\right) \land y \leq 5.4 \cdot 10^{-17}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.3% 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. Add Preprocessing

3. Taylor expanded in x around inf 88.5%

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

4. Taylor expanded in x around 0 99.9%

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

   1. +-commutative 99.9%

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

   2. fma-define 99.9%

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

   3. rem-square-sqrt 49.6%

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

   4. fabs-sqr 49.6%

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

   5. rem-square-sqrt 54.9%

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

   6. fma-undefine 54.9%

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

   7. +-commutative 54.9%

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

   8. sub-neg 54.9%

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

   9. distribute-lft-in 54.9%

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

   10. distribute-rgt-neg-in 54.9%

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

   11. distribute-lft-neg-in 54.9%

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

   12. metadata-eval 54.9%

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

   13. +-commutative 54.9%

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

   14. associate-+r+ 54.9%

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

   15. distribute-rgt1-in 54.9%

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

   16. metadata-eval 54.9%

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

   17. distribute-lft-out 54.9%

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

   18. +-commutative 54.9%

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

6. Simplified 54.9%

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

7. Final simplification 54.9%

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

Alternative 5: 29.8% accurate, 35.7× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.5 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[x + \frac{\left|y - x\right|}{2} \]
2. Add Preprocessing
3. 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-define 99.9%

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

   4. add-sqr-sqrt 49.6%

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

   5. fabs-sqr 49.6%

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

   6. add-sqr-sqrt 54.9%

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

   7. fma-define 54.9%

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

   8. div-inv 54.9%

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

   9. add-sqr-sqrt 49.6%

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

   10. fabs-sqr 49.6%

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

   11. add-sqr-sqrt 99.9%

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

   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 53.9%

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

   1. fma-undefine 53.8%

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

   2. +-commutative 53.8%

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

   3. associate-*r/ 53.8%

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

   4. unpow2 53.8%

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

   5. rem-3cbrt-lft 54.9%

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

6. Simplified 54.9%

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

7. Taylor expanded in x around inf 30.5%

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

8. Final simplification 30.5%

   \[\leadsto x \cdot 0.5 \]
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 99.9%

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

3. Taylor expanded in x around inf 11.6%

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

4. Final simplification 11.6%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024067 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))