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

Percentage Accurate: 99.9% → 99.9%

Time: 2.6s

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: 71.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-98}:\\ \;\;\;\;x + \frac{y - x}{2}\\ \mathbf{else}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1e-98) (+ x (/ (- y x) 2.0)) (* (fabs (- y x)) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1e-98) {
		tmp = x + ((y - x) / 2.0);
	} 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 <= (-1d-98)) then
        tmp = x + ((y - x) / 2.0d0)
    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 <= -1e-98) {
		tmp = x + ((y - x) / 2.0);
	} else {
		tmp = Math.abs((y - x)) * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1e-98:
		tmp = x + ((y - x) / 2.0)
	else:
		tmp = math.fabs((y - x)) * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1e-98)
		tmp = Float64(x + Float64(Float64(y - x) / 2.0));
	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 <= -1e-98)
		tmp = x + ((y - x) / 2.0);
	else
		tmp = abs((y - x)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1e-98], N[(x + N[(N[(y - x), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.99999999999999939e-99

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. div-inv 100.0%

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

      3. fma-define 100.0%

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

      4. add-sqr-sqrt 90.3%

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

      5. fabs-sqr 90.3%

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

      6. add-sqr-sqrt 91.0%

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

      7. fma-define 91.0%

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

      8. div-inv 91.0%

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

      9. add-sqr-sqrt 90.3%

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

      10. fabs-sqr 90.3%

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

      11. add-sqr-sqrt 100.0%

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

      12. add-cube-cbrt 98.0%

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

      13. associate-/l* 98.0%

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

      14. fma-define 98.0%

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

   4. Applied egg-rr 89.2%

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

      1. fma-undefine 89.2%

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

      2. +-commutative 89.2%

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

      3. associate-*r/ 89.2%

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

      4. unpow2 89.2%

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

      5. rem-3cbrt-lft 91.0%

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

   6. Simplified 91.0%

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

   if -9.99999999999999939e-99 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 63.5%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-98}:\\ \;\;\;\;x + \frac{y - x}{2}\\ \mathbf{else}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 30.3% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.35 \cdot 10^{-119}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.35e-119) x (* (- y x) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.35e-119) {
		tmp = x;
	} else {
		tmp = (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 (y <= 2.35d-119) then
        tmp = x
    else
        tmp = (y - x) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.35e-119) {
		tmp = x;
	} else {
		tmp = (y - x) * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.35e-119:
		tmp = x
	else:
		tmp = (y - x) * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.35e-119)
		tmp = x;
	else
		tmp = Float64(Float64(y - x) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.35e-119)
		tmp = x;
	else
		tmp = (y - x) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.35e-119], x, N[(N[(y - x), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.35000000000000001e-119

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 13.1%

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

   if 2.35000000000000001e-119 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 64.4%

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

      1. rem-square-sqrt 61.1%

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

      2. fabs-sqr 61.1%

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

      3. rem-square-sqrt 61.6%

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

   5. Simplified 61.6%

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

4. Final simplification 32.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.35 \cdot 10^{-119}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.8% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ x + \frac{y - x}{2} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x + ((y - x) / 2.0)

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(x + N[(N[(y - x), $MachinePrecision] / 2.0), $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 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 50.8%

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

   5. fabs-sqr 50.8%

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

   6. add-sqr-sqrt 56.0%

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

   7. fma-define 56.0%

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

   8. div-inv 56.0%

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

   9. add-sqr-sqrt 50.8%

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

   10. fabs-sqr 50.8%

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

   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 98.3%

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

4. Applied egg-rr 54.9%

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

   1. fma-undefine 54.9%

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

   2. +-commutative 54.9%

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

   3. associate-*r/ 54.9%

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

   4. unpow2 54.9%

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

   5. rem-3cbrt-lft 56.0%

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

6. Simplified 56.0%

   \[\leadsto \color{blue}{x + \frac{y - x}{2}} \]
7. Add Preprocessing

Alternative 5: 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 11.4%

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

Reproduce

herbie shell --seed 2024103 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))