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

Percentage Accurate: 99.9% → 99.9%

Time: 4.6s

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: 70.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{-63}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.95e-63) (* (fabs (- y x)) 0.5) (* 0.5 (+ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.95e-63) {
		tmp = fabs((y - x)) * 0.5;
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.95d-63)) then
        tmp = abs((y - x)) * 0.5d0
    else
        tmp = 0.5d0 * (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.95e-63) {
		tmp = Math.abs((y - x)) * 0.5;
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.95e-63:
		tmp = math.fabs((y - x)) * 0.5
	else:
		tmp = 0.5 * (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.95e-63)
		tmp = Float64(abs(Float64(y - x)) * 0.5);
	else
		tmp = Float64(0.5 * Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.95e-63)
		tmp = abs((y - x)) * 0.5;
	else
		tmp = 0.5 * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.95e-63], N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.95000000000000011e-63

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 72.6%

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

   if -1.95000000000000011e-63 < y

   1. Initial program 99.9%

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

      1. flip-+ 56.3%

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

      2. div-sub 56.3%

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

      3. pow2 56.3%

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

      4. div-inv 56.3%

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

      5. add-sqr-sqrt 34.4%

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

      6. fabs-sqr 34.4%

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

      7. add-sqr-sqrt 39.6%

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

      8. metadata-eval 39.6%

         \[\leadsto \frac{{x}^{2}}{x - \left(y - x\right) \cdot \color{blue}{0.5}} - \frac{\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2}}{x - \frac{\left|y - x\right|}{2}} \]

   4. Applied egg-rr 38.1%

      \[\leadsto \color{blue}{\frac{{x}^{2}}{x - \left(y - x\right) \cdot 0.5} - \frac{{\left(y - x\right)}^{2} \cdot 0.25}{x - \left(y - x\right) \cdot 0.5}} \]
   5. Step-by-step derivation

      1. div-sub 38.1%

         \[\leadsto \color{blue}{\frac{{x}^{2} - {\left(y - x\right)}^{2} \cdot 0.25}{x - \left(y - x\right) \cdot 0.5}} \]

      2. unpow2 38.1%

         \[\leadsto \frac{\color{blue}{x \cdot x} - {\left(y - x\right)}^{2} \cdot 0.25}{x - \left(y - x\right) \cdot 0.5} \]

      3. fma-neg 38.2%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, -{\left(y - x\right)}^{2} \cdot 0.25\right)}}{x - \left(y - x\right) \cdot 0.5} \]

      4. distribute-rgt-neg-in 38.2%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, \color{blue}{{\left(y - x\right)}^{2} \cdot \left(-0.25\right)}\right)}{x - \left(y - x\right) \cdot 0.5} \]

      5. metadata-eval 38.2%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot \color{blue}{-0.25}\right)}{x - \left(y - x\right) \cdot 0.5} \]

      6. sub-neg 38.2%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{\color{blue}{x + \left(-\left(y - x\right) \cdot 0.5\right)}} \]

      7. *-commutative 38.2%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + \left(-\color{blue}{0.5 \cdot \left(y - x\right)}\right)} \]

      8. distribute-rgt-neg-in 38.2%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + \color{blue}{0.5 \cdot \left(-\left(y - x\right)\right)}} \]

      9. sub-neg 38.2%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + 0.5 \cdot \left(-\color{blue}{\left(y + \left(-x\right)\right)}\right)} \]

      10. +-commutative 38.2%

          \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + 0.5 \cdot \left(-\color{blue}{\left(\left(-x\right) + y\right)}\right)} \]

      11. distribute-neg-in 38.2%

          \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + 0.5 \cdot \color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)}} \]

      12. unsub-neg 38.2%

          \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + 0.5 \cdot \color{blue}{\left(\left(-\left(-x\right)\right) - y\right)}} \]

      13. remove-double-neg 38.2%

          \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + 0.5 \cdot \left(\color{blue}{x} - y\right)} \]

   6. Simplified 38.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + 0.5 \cdot \left(x - y\right)}} \]

   7. Taylor expanded in x around 0 67.0%

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

      1. distribute-lft-out 67.0%

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

      2. +-commutative 67.0%

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

   9. Simplified 67.0%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{-63}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 42.9% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{-33}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+106}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.85e-33) (* x 0.5) (if (<= x 1.5e+106) (* y 0.5) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.85e-33) {
		tmp = x * 0.5;
	} else if (x <= 1.5e+106) {
		tmp = y * 0.5;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.85d-33)) then
        tmp = x * 0.5d0
    else if (x <= 1.5d+106) then
        tmp = y * 0.5d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.85e-33) {
		tmp = x * 0.5;
	} else if (x <= 1.5e+106) {
		tmp = y * 0.5;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.85e-33:
		tmp = x * 0.5
	elif x <= 1.5e+106:
		tmp = y * 0.5
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.85e-33)
		tmp = Float64(x * 0.5);
	elseif (x <= 1.5e+106)
		tmp = Float64(y * 0.5);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.85e-33)
		tmp = x * 0.5;
	elseif (x <= 1.5e+106)
		tmp = y * 0.5;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.85e-33], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, 1.5e+106], N[(y * 0.5), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if x < -1.85000000000000007e-33

   1. Initial program 100.0%

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

      1. flip-+ 38.0%

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

      2. div-sub 37.9%

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

      3. pow2 37.9%

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

      4. div-inv 37.9%

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

      5. add-sqr-sqrt 32.3%

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

      6. fabs-sqr 32.3%

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

      7. add-sqr-sqrt 36.9%

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

      8. metadata-eval 36.9%

         \[\leadsto \frac{{x}^{2}}{x - \left(y - x\right) \cdot \color{blue}{0.5}} - \frac{\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2}}{x - \frac{\left|y - x\right|}{2}} \]

   4. Applied egg-rr 32.4%

      \[\leadsto \color{blue}{\frac{{x}^{2}}{x - \left(y - x\right) \cdot 0.5} - \frac{{\left(y - x\right)}^{2} \cdot 0.25}{x - \left(y - x\right) \cdot 0.5}} \]
   5. Step-by-step derivation

      1. div-sub 32.4%

         \[\leadsto \color{blue}{\frac{{x}^{2} - {\left(y - x\right)}^{2} \cdot 0.25}{x - \left(y - x\right) \cdot 0.5}} \]

      2. unpow2 32.4%

         \[\leadsto \frac{\color{blue}{x \cdot x} - {\left(y - x\right)}^{2} \cdot 0.25}{x - \left(y - x\right) \cdot 0.5} \]

      3. fma-neg 32.5%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, -{\left(y - x\right)}^{2} \cdot 0.25\right)}}{x - \left(y - x\right) \cdot 0.5} \]

      4. distribute-rgt-neg-in 32.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, \color{blue}{{\left(y - x\right)}^{2} \cdot \left(-0.25\right)}\right)}{x - \left(y - x\right) \cdot 0.5} \]

      5. metadata-eval 32.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot \color{blue}{-0.25}\right)}{x - \left(y - x\right) \cdot 0.5} \]

      6. sub-neg 32.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{\color{blue}{x + \left(-\left(y - x\right) \cdot 0.5\right)}} \]

      7. *-commutative 32.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + \left(-\color{blue}{0.5 \cdot \left(y - x\right)}\right)} \]

      8. distribute-rgt-neg-in 32.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + \color{blue}{0.5 \cdot \left(-\left(y - x\right)\right)}} \]

      9. sub-neg 32.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + 0.5 \cdot \left(-\color{blue}{\left(y + \left(-x\right)\right)}\right)} \]

      10. +-commutative 32.5%

          \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + 0.5 \cdot \left(-\color{blue}{\left(\left(-x\right) + y\right)}\right)} \]

      11. distribute-neg-in 32.5%

          \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + 0.5 \cdot \color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)}} \]

      12. unsub-neg 32.5%

          \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + 0.5 \cdot \color{blue}{\left(\left(-\left(-x\right)\right) - y\right)}} \]

      13. remove-double-neg 32.5%

          \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + 0.5 \cdot \left(\color{blue}{x} - y\right)} \]

   6. Simplified 32.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + 0.5 \cdot \left(x - y\right)}} \]

   7. Taylor expanded in x around inf 74.8%

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

   if -1.85000000000000007e-33 < x < 1.5e106

   1. Initial program 99.9%

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

      1. flip-+ 70.6%

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

      2. div-sub 70.5%

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

      3. pow2 70.5%

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

      4. div-inv 70.5%

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

      5. add-sqr-sqrt 24.9%

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

      6. fabs-sqr 24.9%

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

      7. add-sqr-sqrt 52.8%

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

      8. metadata-eval 52.8%

         \[\leadsto \frac{{x}^{2}}{x - \left(y - x\right) \cdot \color{blue}{0.5}} - \frac{\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2}}{x - \frac{\left|y - x\right|}{2}} \]

   4. Applied egg-rr 29.2%

      \[\leadsto \color{blue}{\frac{{x}^{2}}{x - \left(y - x\right) \cdot 0.5} - \frac{{\left(y - x\right)}^{2} \cdot 0.25}{x - \left(y - x\right) \cdot 0.5}} \]
   5. Step-by-step derivation

      1. div-sub 29.2%

         \[\leadsto \color{blue}{\frac{{x}^{2} - {\left(y - x\right)}^{2} \cdot 0.25}{x - \left(y - x\right) \cdot 0.5}} \]

      2. unpow2 29.2%

         \[\leadsto \frac{\color{blue}{x \cdot x} - {\left(y - x\right)}^{2} \cdot 0.25}{x - \left(y - x\right) \cdot 0.5} \]

      3. fma-neg 29.2%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, -{\left(y - x\right)}^{2} \cdot 0.25\right)}}{x - \left(y - x\right) \cdot 0.5} \]

      4. distribute-rgt-neg-in 29.2%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, \color{blue}{{\left(y - x\right)}^{2} \cdot \left(-0.25\right)}\right)}{x - \left(y - x\right) \cdot 0.5} \]

      5. metadata-eval 29.2%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot \color{blue}{-0.25}\right)}{x - \left(y - x\right) \cdot 0.5} \]

      6. sub-neg 29.2%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{\color{blue}{x + \left(-\left(y - x\right) \cdot 0.5\right)}} \]

      7. *-commutative 29.2%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + \left(-\color{blue}{0.5 \cdot \left(y - x\right)}\right)} \]

      8. distribute-rgt-neg-in 29.2%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + \color{blue}{0.5 \cdot \left(-\left(y - x\right)\right)}} \]

      9. sub-neg 29.2%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + 0.5 \cdot \left(-\color{blue}{\left(y + \left(-x\right)\right)}\right)} \]

      10. +-commutative 29.2%

          \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + 0.5 \cdot \left(-\color{blue}{\left(\left(-x\right) + y\right)}\right)} \]

      11. distribute-neg-in 29.2%

          \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + 0.5 \cdot \color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)}} \]

      12. unsub-neg 29.2%

          \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + 0.5 \cdot \color{blue}{\left(\left(-\left(-x\right)\right) - y\right)}} \]

      13. remove-double-neg 29.2%

          \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + 0.5 \cdot \left(\color{blue}{x} - y\right)} \]

   6. Simplified 29.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + 0.5 \cdot \left(x - y\right)}} \]

   7. Taylor expanded in x around 0 36.5%

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

   if 1.5e106 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 19.8%

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

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{-33}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+106}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.5% 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. Step-by-step derivation

   1. flip-+ 55.0%

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

   2. div-sub 55.0%

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

   3. pow2 55.0%

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

   4. div-inv 55.0%

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

   5. add-sqr-sqrt 22.5%

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

   6. fabs-sqr 22.5%

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

   7. add-sqr-sqrt 41.0%

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

   8. metadata-eval 41.0%

      \[\leadsto \frac{{x}^{2}}{x - \left(y - x\right) \cdot \color{blue}{0.5}} - \frac{\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2}}{x - \frac{\left|y - x\right|}{2}} \]

4. Applied egg-rr 25.8%

   \[\leadsto \color{blue}{\frac{{x}^{2}}{x - \left(y - x\right) \cdot 0.5} - \frac{{\left(y - x\right)}^{2} \cdot 0.25}{x - \left(y - x\right) \cdot 0.5}} \]
5. Step-by-step derivation

   1. div-sub 25.8%

      \[\leadsto \color{blue}{\frac{{x}^{2} - {\left(y - x\right)}^{2} \cdot 0.25}{x - \left(y - x\right) \cdot 0.5}} \]

   2. unpow2 25.8%

      \[\leadsto \frac{\color{blue}{x \cdot x} - {\left(y - x\right)}^{2} \cdot 0.25}{x - \left(y - x\right) \cdot 0.5} \]

   3. fma-neg 25.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, -{\left(y - x\right)}^{2} \cdot 0.25\right)}}{x - \left(y - x\right) \cdot 0.5} \]

   4. distribute-rgt-neg-in 25.8%

      \[\leadsto \frac{\mathsf{fma}\left(x, x, \color{blue}{{\left(y - x\right)}^{2} \cdot \left(-0.25\right)}\right)}{x - \left(y - x\right) \cdot 0.5} \]

   5. metadata-eval 25.8%

      \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot \color{blue}{-0.25}\right)}{x - \left(y - x\right) \cdot 0.5} \]

   6. sub-neg 25.8%

      \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{\color{blue}{x + \left(-\left(y - x\right) \cdot 0.5\right)}} \]

   7. *-commutative 25.8%

      \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + \left(-\color{blue}{0.5 \cdot \left(y - x\right)}\right)} \]

   8. distribute-rgt-neg-in 25.8%

      \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + \color{blue}{0.5 \cdot \left(-\left(y - x\right)\right)}} \]

   9. sub-neg 25.8%

      \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + 0.5 \cdot \left(-\color{blue}{\left(y + \left(-x\right)\right)}\right)} \]

   10. +-commutative 25.8%

       \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + 0.5 \cdot \left(-\color{blue}{\left(\left(-x\right) + y\right)}\right)} \]

   11. distribute-neg-in 25.8%

       \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + 0.5 \cdot \color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)}} \]

   12. unsub-neg 25.8%

       \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + 0.5 \cdot \color{blue}{\left(\left(-\left(-x\right)\right) - y\right)}} \]

   13. remove-double-neg 25.8%

       \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + 0.5 \cdot \left(\color{blue}{x} - y\right)} \]

6. Simplified 25.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + 0.5 \cdot \left(x - y\right)}} \]

7. Taylor expanded in x around 0 48.5%

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

   1. distribute-lft-out 48.5%

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

   2. +-commutative 48.5%

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

9. Simplified 48.5%

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

10. Final simplification 48.5%

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

Alternative 5: 30.7% 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. flip-+ 55.0%

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

   2. div-sub 55.0%

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

   3. pow2 55.0%

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

   4. div-inv 55.0%

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

   5. add-sqr-sqrt 22.5%

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

   6. fabs-sqr 22.5%

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

   7. add-sqr-sqrt 41.0%

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

   8. metadata-eval 41.0%

      \[\leadsto \frac{{x}^{2}}{x - \left(y - x\right) \cdot \color{blue}{0.5}} - \frac{\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2}}{x - \frac{\left|y - x\right|}{2}} \]

4. Applied egg-rr 25.8%

   \[\leadsto \color{blue}{\frac{{x}^{2}}{x - \left(y - x\right) \cdot 0.5} - \frac{{\left(y - x\right)}^{2} \cdot 0.25}{x - \left(y - x\right) \cdot 0.5}} \]
5. Step-by-step derivation

   1. div-sub 25.8%

      \[\leadsto \color{blue}{\frac{{x}^{2} - {\left(y - x\right)}^{2} \cdot 0.25}{x - \left(y - x\right) \cdot 0.5}} \]

   2. unpow2 25.8%

      \[\leadsto \frac{\color{blue}{x \cdot x} - {\left(y - x\right)}^{2} \cdot 0.25}{x - \left(y - x\right) \cdot 0.5} \]

   3. fma-neg 25.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, -{\left(y - x\right)}^{2} \cdot 0.25\right)}}{x - \left(y - x\right) \cdot 0.5} \]

   4. distribute-rgt-neg-in 25.8%

      \[\leadsto \frac{\mathsf{fma}\left(x, x, \color{blue}{{\left(y - x\right)}^{2} \cdot \left(-0.25\right)}\right)}{x - \left(y - x\right) \cdot 0.5} \]

   5. metadata-eval 25.8%

      \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot \color{blue}{-0.25}\right)}{x - \left(y - x\right) \cdot 0.5} \]

   6. sub-neg 25.8%

      \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{\color{blue}{x + \left(-\left(y - x\right) \cdot 0.5\right)}} \]

   7. *-commutative 25.8%

      \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + \left(-\color{blue}{0.5 \cdot \left(y - x\right)}\right)} \]

   8. distribute-rgt-neg-in 25.8%

      \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + \color{blue}{0.5 \cdot \left(-\left(y - x\right)\right)}} \]

   9. sub-neg 25.8%

      \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + 0.5 \cdot \left(-\color{blue}{\left(y + \left(-x\right)\right)}\right)} \]

   10. +-commutative 25.8%

       \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + 0.5 \cdot \left(-\color{blue}{\left(\left(-x\right) + y\right)}\right)} \]

   11. distribute-neg-in 25.8%

       \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + 0.5 \cdot \color{blue}{\left(\left(-\left(-x\right)\right) + \left(-y\right)\right)}} \]

   12. unsub-neg 25.8%

       \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + 0.5 \cdot \color{blue}{\left(\left(-\left(-x\right)\right) - y\right)}} \]

   13. remove-double-neg 25.8%

       \[\leadsto \frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + 0.5 \cdot \left(\color{blue}{x} - y\right)} \]

6. Simplified 25.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, {\left(y - x\right)}^{2} \cdot -0.25\right)}{x + 0.5 \cdot \left(x - y\right)}} \]

7. Taylor expanded in x around inf 25.6%

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

8. Final simplification 25.6%

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

Alternative 6: 11.5% 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 11.0%

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

4. Final simplification 11.0%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024021 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))