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

Percentage Accurate: 99.9% → 99.9%

Time: 2.7s

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(x + N[(N[Abs[N[(x - y), $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%

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

Alternative 2: 53.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left|x - y\right| \cdot 0.5 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return Math.abs((x - y)) * 0.5;
}

Python

def code(x, y):
	return math.fabs((x - y)) * 0.5

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0 55.0%

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

4. Final simplification 55.0%

   \[\leadsto \left|x - y\right| \cdot 0.5 \]
5. Add Preprocessing

Alternative 3: 53.8% 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. add-sqr-sqrt 77.9%

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

   2. pow2 77.9%

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

   3. +-commutative 77.9%

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

   4. div-inv 77.9%

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

   5. fma-def 77.9%

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

   6. add-sqr-sqrt 25.1%

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

   7. fabs-sqr 25.1%

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

   8. add-sqr-sqrt 30.1%

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

   9. metadata-eval 30.1%

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

4. Applied egg-rr 30.1%

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

5. Taylor expanded in y around 0 52.1%

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

   1. associate-+r+ 52.1%

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

   2. distribute-rgt1-in 52.1%

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

   3. metadata-eval 52.1%

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

   4. distribute-lft-out 52.1%

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

7. Simplified 52.1%

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

8. Final simplification 52.1%

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

Alternative 4: 26.3% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(y * 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. add-sqr-sqrt 77.9%

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

   2. pow2 77.9%

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

   3. +-commutative 77.9%

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

   4. div-inv 77.9%

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

   5. fma-def 77.9%

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

   6. add-sqr-sqrt 25.1%

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

   7. fabs-sqr 25.1%

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

   8. add-sqr-sqrt 30.1%

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

   9. metadata-eval 30.1%

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

4. Applied egg-rr 30.1%

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

5. Taylor expanded in x around 0 26.6%

   \[\leadsto \color{blue}{y \cdot {\left(\sqrt{0.5}\right)}^{2}} \]
6. Step-by-step derivation

   1. *-commutative 26.6%

      \[\leadsto \color{blue}{{\left(\sqrt{0.5}\right)}^{2} \cdot y} \]

   2. unpow2 26.6%

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

   3. rem-square-sqrt 27.1%

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

7. Simplified 27.1%

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

8. Final simplification 27.1%

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

Alternative 5: 11.4% 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 11.3%

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

4. Final simplification 11.3%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024033 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))