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

Percentage Accurate: 99.9% → 99.9%

Time: 11.1s

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. Add Preprocessing

Alternative 2: 55.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-250}:\\ \;\;\;\;\left|y\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.75 + y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.6e-250) (* (fabs y) 0.5) (+ (* x 0.75) (* y 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.6e-250) {
		tmp = fabs(y) * 0.5;
	} else {
		tmp = (x * 0.75) + (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 <= (-1.6d-250)) then
        tmp = abs(y) * 0.5d0
    else
        tmp = (x * 0.75d0) + (y * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.6e-250) {
		tmp = Math.abs(y) * 0.5;
	} else {
		tmp = (x * 0.75) + (y * 0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.6e-250:
		tmp = math.fabs(y) * 0.5
	else:
		tmp = (x * 0.75) + (y * 0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.6e-250)
		tmp = Float64(abs(y) * 0.5);
	else
		tmp = Float64(Float64(x * 0.75) + Float64(y * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.6e-250)
		tmp = abs(y) * 0.5;
	else
		tmp = (x * 0.75) + (y * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.6e-250], N[(N[Abs[y], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(x * 0.75), $MachinePrecision] + N[(y * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.60000000000000002e-250

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\color{blue}{y}\right), 2\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 57.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left|y\right|} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\left|y\right|\right)}\right) \]

      2. fabs-lowering-fabs.f64 48.8%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{fabs.f64}\left(y\right)\right) \]

   8. Simplified 48.8%

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

   if -1.60000000000000002e-250 < y

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{\left|y - x\right|}{2} - x}{\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot x}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\left|y - x\right|}{2} - x\right), \color{blue}{\left(\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot x\right)}\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{\left|y - x\right|}{2}\right), x\right), \left(\color{blue}{\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2}} - x \cdot x\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\left|y - x\right|\right), 2\right), x\right), \left(\color{blue}{\frac{\left|y - x\right|}{2}} \cdot \frac{\left|y - x\right|}{2} - x \cdot x\right)\right)\right) \]

      8. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\left(y - x\right)\right), 2\right), x\right), \left(\frac{\color{blue}{\left|y - x\right|}}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot x\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(y, x\right)\right), 2\right), x\right), \left(\frac{\left|\color{blue}{y - x}\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot x\right)\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(y, x\right)\right), 2\right), x\right), \mathsf{\_.f64}\left(\left(\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2}\right), \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

   4. Applied egg-rr 53.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\left|y - x\right|}{2} - x}{\frac{\left(y - x\right) \cdot \left(y - x\right)}{4} - x \cdot x}}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot x\right)}, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{\_.f64}\left(y, x\right)\right), 4\right), \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(x\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{\_.f64}\left(y, x\right)\right), 4\right)}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(0 - x\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{\_.f64}\left(y, x\right)\right), 4\right)}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

      3. --lowering--.f64 7.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{\_.f64}\left(y, x\right)\right), 4\right)}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

   7. Simplified 7.0%

      \[\leadsto \frac{1}{\frac{\color{blue}{0 - x}}{\frac{\left(y - x\right) \cdot \left(y - x\right)}{4} - x \cdot x}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{\frac{1}{4} \cdot {x}^{2} - {x}^{2}}{x} + \frac{1}{2} \cdot y} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\frac{1}{4} \cdot {x}^{2} - {x}^{2}}{x}\right)\right) + \color{blue}{\frac{1}{2}} \cdot y \]

      2. div-sub N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\frac{\frac{1}{4} \cdot {x}^{2}}{x} - \frac{{x}^{2}}{x}\right)\right)\right) + \frac{1}{2} \cdot y \]

      3. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\frac{1}{4} \cdot \frac{{x}^{2}}{x} - \frac{{x}^{2}}{x}\right)\right)\right) + \frac{1}{2} \cdot y \]

      4. unpow2 N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\frac{1}{4} \cdot \frac{x \cdot x}{x} - \frac{{x}^{2}}{x}\right)\right)\right) + \frac{1}{2} \cdot y \]

      5. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\frac{1}{4} \cdot \left(x \cdot \frac{x}{x}\right) - \frac{{x}^{2}}{x}\right)\right)\right) + \frac{1}{2} \cdot y \]

      6. *-inverses N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\frac{1}{4} \cdot \left(x \cdot 1\right) - \frac{{x}^{2}}{x}\right)\right)\right) + \frac{1}{2} \cdot y \]

      7. *-rgt-identity N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\frac{1}{4} \cdot x - \frac{{x}^{2}}{x}\right)\right)\right) + \frac{1}{2} \cdot y \]

      8. unpow2 N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\frac{1}{4} \cdot x - \frac{x \cdot x}{x}\right)\right)\right) + \frac{1}{2} \cdot y \]

      9. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\frac{1}{4} \cdot x - x \cdot \frac{x}{x}\right)\right)\right) + \frac{1}{2} \cdot y \]

      10. *-inverses N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(\frac{1}{4} \cdot x - x \cdot 1\right)\right)\right) + \frac{1}{2} \cdot y \]

      11. *-rgt-identity N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(\frac{1}{4} \cdot x - x\right)\right)\right) + \frac{1}{2} \cdot y \]

      12. sub-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(\frac{1}{4} \cdot x + \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right) + \frac{1}{2} \cdot y \]

      13. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(\frac{1}{4} \cdot x + -1 \cdot x\right)\right)\right) + \frac{1}{2} \cdot y \]

      14. distribute-rgt-out N/A

          \[\leadsto \left(\mathsf{neg}\left(x \cdot \left(\frac{1}{4} + -1\right)\right)\right) + \frac{1}{2} \cdot y \]

      15. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{-3}{4}\right)\right) + \frac{1}{2} \cdot y \]

      16. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{-3}{4} \cdot x\right)\right) + \frac{1}{2} \cdot y \]

      17. distribute-lft-neg-in N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{-3}{4}\right)\right) \cdot x + \color{blue}{\frac{1}{2}} \cdot y \]

      18. metadata-eval N/A

          \[\leadsto \frac{3}{4} \cdot x + \frac{1}{2} \cdot y \]

   10. Simplified 53.1%

       \[\leadsto \color{blue}{x \cdot 0.75 + 0.5 \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-250}:\\ \;\;\;\;\left|y\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.75 + y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around inf

   \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\color{blue}{y}\right), 2\right)\right) \]
4. Step-by-step derivation

5. Simplified 55.2%

   \[\leadsto x + \frac{\left|\color{blue}{y}\right|}{2} \]
6. Add Preprocessing

Alternative 4: 35.0% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.75 + y \cdot 0.5 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (* x 0.75) (* y 0.5)))

C

double code(double x, double y) {
	return (x * 0.75) + (y * 0.5);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * 0.75d0) + (y * 0.5d0)
end function

Java

public static double code(double x, double y) {
	return (x * 0.75) + (y * 0.5);
}

Python

def code(x, y):
	return (x * 0.75) + (y * 0.5)

Julia

function code(x, y)
	return Float64(Float64(x * 0.75) + Float64(y * 0.5))
end

MATLAB

function tmp = code(x, y)
	tmp = (x * 0.75) + (y * 0.5);
end

Wolfram

code[x_, y_] := N[(N[(x * 0.75), $MachinePrecision] + N[(y * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. +-commutative N/A

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

   2. flip-+ N/A

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

   3. clear-num N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{\left|y - x\right|}{2} - x}{\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot x}\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\left|y - x\right|}{2} - x\right), \color{blue}{\left(\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot x\right)}\right)\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{\left|y - x\right|}{2}\right), x\right), \left(\color{blue}{\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2}} - x \cdot x\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\left|y - x\right|\right), 2\right), x\right), \left(\color{blue}{\frac{\left|y - x\right|}{2}} \cdot \frac{\left|y - x\right|}{2} - x \cdot x\right)\right)\right) \]

   8. fabs-lowering-fabs.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\left(y - x\right)\right), 2\right), x\right), \left(\frac{\color{blue}{\left|y - x\right|}}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot x\right)\right)\right) \]

   9. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(y, x\right)\right), 2\right), x\right), \left(\frac{\left|\color{blue}{y - x}\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot x\right)\right)\right) \]

   10. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(y, x\right)\right), 2\right), x\right), \mathsf{\_.f64}\left(\left(\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2}\right), \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

4. Applied egg-rr 51.8%

   \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\left|y - x\right|}{2} - x}{\frac{\left(y - x\right) \cdot \left(y - x\right)}{4} - x \cdot x}}} \]

5. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot x\right)}, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{\_.f64}\left(y, x\right)\right), 4\right), \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(x\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{\_.f64}\left(y, x\right)\right), 4\right)}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(0 - x\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{\_.f64}\left(y, x\right)\right), 4\right)}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

   3. --lowering--.f64 7.0%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{\_.f64}\left(y, x\right)\right), 4\right)}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

7. Simplified 7.0%

   \[\leadsto \frac{1}{\frac{\color{blue}{0 - x}}{\frac{\left(y - x\right) \cdot \left(y - x\right)}{4} - x \cdot x}} \]

8. Taylor expanded in y around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{\frac{1}{4} \cdot {x}^{2} - {x}^{2}}{x} + \frac{1}{2} \cdot y} \]
9. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{\frac{1}{4} \cdot {x}^{2} - {x}^{2}}{x}\right)\right) + \color{blue}{\frac{1}{2}} \cdot y \]

   2. div-sub N/A

      \[\leadsto \left(\mathsf{neg}\left(\left(\frac{\frac{1}{4} \cdot {x}^{2}}{x} - \frac{{x}^{2}}{x}\right)\right)\right) + \frac{1}{2} \cdot y \]

   3. associate-/l* N/A

      \[\leadsto \left(\mathsf{neg}\left(\left(\frac{1}{4} \cdot \frac{{x}^{2}}{x} - \frac{{x}^{2}}{x}\right)\right)\right) + \frac{1}{2} \cdot y \]

   4. unpow2 N/A

      \[\leadsto \left(\mathsf{neg}\left(\left(\frac{1}{4} \cdot \frac{x \cdot x}{x} - \frac{{x}^{2}}{x}\right)\right)\right) + \frac{1}{2} \cdot y \]

   5. associate-/l* N/A

      \[\leadsto \left(\mathsf{neg}\left(\left(\frac{1}{4} \cdot \left(x \cdot \frac{x}{x}\right) - \frac{{x}^{2}}{x}\right)\right)\right) + \frac{1}{2} \cdot y \]

   6. *-inverses N/A

      \[\leadsto \left(\mathsf{neg}\left(\left(\frac{1}{4} \cdot \left(x \cdot 1\right) - \frac{{x}^{2}}{x}\right)\right)\right) + \frac{1}{2} \cdot y \]

   7. *-rgt-identity N/A

      \[\leadsto \left(\mathsf{neg}\left(\left(\frac{1}{4} \cdot x - \frac{{x}^{2}}{x}\right)\right)\right) + \frac{1}{2} \cdot y \]

   8. unpow2 N/A

      \[\leadsto \left(\mathsf{neg}\left(\left(\frac{1}{4} \cdot x - \frac{x \cdot x}{x}\right)\right)\right) + \frac{1}{2} \cdot y \]

   9. associate-/l* N/A

      \[\leadsto \left(\mathsf{neg}\left(\left(\frac{1}{4} \cdot x - x \cdot \frac{x}{x}\right)\right)\right) + \frac{1}{2} \cdot y \]

   10. *-inverses N/A

       \[\leadsto \left(\mathsf{neg}\left(\left(\frac{1}{4} \cdot x - x \cdot 1\right)\right)\right) + \frac{1}{2} \cdot y \]

   11. *-rgt-identity N/A

       \[\leadsto \left(\mathsf{neg}\left(\left(\frac{1}{4} \cdot x - x\right)\right)\right) + \frac{1}{2} \cdot y \]

   12. sub-neg N/A

       \[\leadsto \left(\mathsf{neg}\left(\left(\frac{1}{4} \cdot x + \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right) + \frac{1}{2} \cdot y \]

   13. mul-1-neg N/A

       \[\leadsto \left(\mathsf{neg}\left(\left(\frac{1}{4} \cdot x + -1 \cdot x\right)\right)\right) + \frac{1}{2} \cdot y \]

   14. distribute-rgt-out N/A

       \[\leadsto \left(\mathsf{neg}\left(x \cdot \left(\frac{1}{4} + -1\right)\right)\right) + \frac{1}{2} \cdot y \]

   15. metadata-eval N/A

       \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{-3}{4}\right)\right) + \frac{1}{2} \cdot y \]

   16. *-commutative N/A

       \[\leadsto \left(\mathsf{neg}\left(\frac{-3}{4} \cdot x\right)\right) + \frac{1}{2} \cdot y \]

   17. distribute-lft-neg-in N/A

       \[\leadsto \left(\mathsf{neg}\left(\frac{-3}{4}\right)\right) \cdot x + \color{blue}{\frac{1}{2}} \cdot y \]

   18. metadata-eval N/A

       \[\leadsto \frac{3}{4} \cdot x + \frac{1}{2} \cdot y \]

10. Simplified 29.6%

    \[\leadsto \color{blue}{x \cdot 0.75 + 0.5 \cdot y} \]

11. Final simplification 29.6%

    \[\leadsto x \cdot 0.75 + y \cdot 0.5 \]
12. Add Preprocessing

Alternative 5: 11.3% accurate, 35.7× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.75 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x 0.75))

C

double code(double x, double y) {
	return x * 0.75;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * 0.75d0
end function

Java

public static double code(double x, double y) {
	return x * 0.75;
}

Python

def code(x, y):
	return x * 0.75

Julia

function code(x, y)
	return Float64(x * 0.75)
end

MATLAB

function tmp = code(x, y)
	tmp = x * 0.75;
end

Wolfram

code[x_, y_] := N[(x * 0.75), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. +-commutative N/A

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

   2. flip-+ N/A

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

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot x\right), \color{blue}{\left(\frac{\left|y - x\right|}{2} - x\right)}\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2}\right), \left(x \cdot x\right)\right), \left(\color{blue}{\frac{\left|y - x\right|}{2}} - x\right)\right) \]

   5. frac-times N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{\left|y - x\right| \cdot \left|y - x\right|}{2 \cdot 2}\right), \left(x \cdot x\right)\right), \left(\frac{\color{blue}{\left|y - x\right|}}{2} - x\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\left|y - x\right| \cdot \left|y - x\right|\right), \left(2 \cdot 2\right)\right), \left(x \cdot x\right)\right), \left(\frac{\color{blue}{\left|y - x\right|}}{2} - x\right)\right) \]

   7. sqr-abs N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\left(y - x\right) \cdot \left(y - x\right)\right), \left(2 \cdot 2\right)\right), \left(x \cdot x\right)\right), \left(\frac{\left|\color{blue}{y - x}\right|}{2} - x\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(y - x\right), \left(y - x\right)\right), \left(2 \cdot 2\right)\right), \left(x \cdot x\right)\right), \left(\frac{\left|\color{blue}{y - x}\right|}{2} - x\right)\right) \]

   9. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(y - x\right)\right), \left(2 \cdot 2\right)\right), \left(x \cdot x\right)\right), \left(\frac{\left|\color{blue}{y} - x\right|}{2} - x\right)\right) \]

   10. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{\_.f64}\left(y, x\right)\right), \left(2 \cdot 2\right)\right), \left(x \cdot x\right)\right), \left(\frac{\left|y - \color{blue}{x}\right|}{2} - x\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{\_.f64}\left(y, x\right)\right), 4\right), \left(x \cdot x\right)\right), \left(\frac{\left|y - x\right|}{2} - x\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{\_.f64}\left(y, x\right)\right), 4\right), \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{\left|y - x\right|}{\color{blue}{2}} - x\right)\right) \]

   13. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{\_.f64}\left(y, x\right)\right), 4\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\left(\frac{\left|y - x\right|}{2}\right), \color{blue}{x}\right)\right) \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{\_.f64}\left(y, x\right)\right), 4\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\left|y - x\right|\right), 2\right), x\right)\right) \]

   15. fabs-lowering-fabs.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{\_.f64}\left(y, x\right)\right), 4\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\left(y - x\right)\right), 2\right), x\right)\right) \]

   16. --lowering--.f64 51.9%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{\_.f64}\left(y, x\right)\right), 4\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(y, x\right)\right), 2\right), x\right)\right) \]

4. Applied egg-rr 51.9%

   \[\leadsto \color{blue}{\frac{\frac{\left(y - x\right) \cdot \left(y - x\right)}{4} - x \cdot x}{\frac{\left|y - x\right|}{2} - x}} \]

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{3}{4} \cdot x} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\frac{3}{4}} \]

   2. *-lowering-*.f64 12.0%

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\frac{3}{4}}\right) \]

7. Simplified 12.0%

   \[\leadsto \color{blue}{x \cdot 0.75} \]
8. 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

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Simplified 12.0%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024161 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))