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

Percentage Accurate: 99.9% → 99.9%

Time: 10.4s

Alternatives: 5

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: 51.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{+148}:\\ \;\;\;\;\left|y\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.0833333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 3.5e+148) (* (fabs y) 0.5) (* x 1.0833333333333333)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 3.5e+148) {
		tmp = fabs(y) * 0.5;
	} else {
		tmp = x * 1.0833333333333333;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 3.5d+148) then
        tmp = abs(y) * 0.5d0
    else
        tmp = x * 1.0833333333333333d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 3.5e+148) {
		tmp = Math.abs(y) * 0.5;
	} else {
		tmp = x * 1.0833333333333333;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 3.5e+148:
		tmp = math.fabs(y) * 0.5
	else:
		tmp = x * 1.0833333333333333
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 3.5e+148)
		tmp = Float64(abs(y) * 0.5);
	else
		tmp = Float64(x * 1.0833333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 3.5e+148)
		tmp = abs(y) * 0.5;
	else
		tmp = x * 1.0833333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 3.5e+148], N[(N[Abs[y], $MachinePrecision] * 0.5), $MachinePrecision], N[(x * 1.0833333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.4999999999999999e148

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   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 67.5%

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

   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 61.1%

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

   8. Simplified 61.1%

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

   if 3.4999999999999999e148 < x

   1. Initial program 99.7%

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

      1. flip-+ N/A

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

      2. div-sub N/A

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

      3. flip-- N/A

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

      4. associate-/r/ N/A

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

      5. fmm-def N/A

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

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

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

   4. Applied egg-rr 3.0%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{13}{12} \cdot x} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\frac{13}{12}} \]

      2. *-lowering-*.f64 19.8%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\frac{13}{12}}\right) \]

   7. Simplified 19.8%

      \[\leadsto \color{blue}{x \cdot 1.0833333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{+148}:\\ \;\;\;\;\left|y\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.0833333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.4% 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 61.7%

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

Alternative 4: 11.3% accurate, 35.7× speedup

Math

\[\begin{array}{l} \\ x \cdot 1.0833333333333333 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x 1.0833333333333333))

C

double code(double x, double y) {
	return x * 1.0833333333333333;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * 1.0833333333333333d0
end function

Java

public static double code(double x, double y) {
	return x * 1.0833333333333333;
}

Python

def code(x, y):
	return x * 1.0833333333333333

Julia

function code(x, y)
	return Float64(x * 1.0833333333333333)
end

MATLAB

function tmp = code(x, y)
	tmp = x * 1.0833333333333333;
end

Wolfram

code[x_, y_] := N[(x * 1.0833333333333333), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. flip-+ N/A

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

   2. div-sub N/A

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

   3. flip-- N/A

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

   4. associate-/r/ N/A

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

   5. fmm-def N/A

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

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

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

4. Applied egg-rr 54.2%

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

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{13}{12} \cdot x} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\frac{13}{12}} \]

   2. *-lowering-*.f64 10.9%

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\frac{13}{12}}\right) \]

7. Simplified 10.9%

   \[\leadsto \color{blue}{x \cdot 1.0833333333333333} \]
8. Add Preprocessing

Alternative 5: 11.2% 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 10.9%

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

Reproduce

herbie shell --seed 2024158 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))