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

Percentage Accurate: 99.9% → 99.9%

Time: 3.1s

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. Final simplification 99.9%

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

Alternative 2: 70.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{-182}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5.5e-182) (* (fabs (- y x)) 0.5) (/ (+ x y) 2.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5.5e-182) {
		tmp = fabs((y - x)) * 0.5;
	} else {
		tmp = (x + y) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-5.5d-182)) then
        tmp = abs((y - x)) * 0.5d0
    else
        tmp = (x + y) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -5.5e-182) {
		tmp = Math.abs((y - x)) * 0.5;
	} else {
		tmp = (x + y) / 2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5.5e-182:
		tmp = math.fabs((y - x)) * 0.5
	else:
		tmp = (x + y) / 2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5.5e-182)
		tmp = Float64(abs(Float64(y - x)) * 0.5);
	else
		tmp = Float64(Float64(x + y) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -5.5e-182)
		tmp = abs((y - x)) * 0.5;
	else
		tmp = (x + y) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5.5e-182], N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(x + y), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.49999999999999993e-182

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 63.0%

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

   if -5.49999999999999993e-182 < y

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. 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-def 99.9%

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

      4. add-sqr-sqrt 75.2%

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

      5. fabs-sqr 75.2%

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

      6. add-sqr-sqrt 79.8%

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

      7. metadata-eval 79.8%

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

   3. Applied egg-rr 79.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 0.5, x\right)} \]
   4. Step-by-step derivation

      1. fma-udef 79.8%

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

      2. flip-+ 45.3%

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

      3. pow2 45.3%

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

      4. pow2 45.3%

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

   5. Applied egg-rr 45.3%

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

      1. unpow2 45.3%

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

      2. unpow2 45.3%

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

      3. difference-of-squares 46.9%

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

      4. fma-def 46.9%

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

      5. associate-/l* 79.8%

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

      6. fma-def 79.8%

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

      7. *-commutative 79.8%

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

      8. sub-neg 79.8%

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

      9. +-commutative 79.8%

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

      10. distribute-lft-in 79.8%

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

      11. distribute-rgt-neg-in 79.8%

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

      12. distribute-lft-neg-in 79.8%

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

      13. metadata-eval 79.8%

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

      14. +-commutative 79.8%

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

      15. associate-+r+ 79.8%

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

      16. distribute-rgt1-in 79.8%

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

      17. metadata-eval 79.8%

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

      18. distribute-lft-out 79.8%

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

      19. *-inverses 79.8%

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

   7. Simplified 79.8%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{-182}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{2}\\ \end{array} \]

Alternative 3: 31.4% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.25 \cdot 10^{-140}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.25e-140) x (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.25e-140) {
		tmp = x;
	} else {
		tmp = 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 <= 2.25d-140) then
        tmp = x
    else
        tmp = y * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.25e-140) {
		tmp = x;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.25e-140:
		tmp = x
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.25e-140)
		tmp = x;
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.25e-140)
		tmp = x;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.25e-140], x, N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.25000000000000002e-140

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 13.1%

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

   if 2.25000000000000002e-140 < y

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. 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-def 99.9%

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

      4. add-sqr-sqrt 89.6%

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

      5. fabs-sqr 89.6%

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

      6. add-sqr-sqrt 92.0%

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

      7. metadata-eval 92.0%

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

   3. Applied egg-rr 92.0%

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

   4. Taylor expanded in y around inf 68.0%

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

4. Final simplification 32.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.25 \cdot 10^{-140}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 4: 45.7% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.85 \cdot 10^{-105}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.85e-105) (* x 0.5) (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.85e-105) {
		tmp = x * 0.5;
	} else {
		tmp = 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 <= 2.85d-105) then
        tmp = x * 0.5d0
    else
        tmp = y * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.85e-105) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.85e-105:
		tmp = x * 0.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.85e-105)
		tmp = Float64(x * 0.5);
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.85e-105)
		tmp = x * 0.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.85e-105], N[(x * 0.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.84999999999999981e-105

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. 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-def 99.9%

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

      4. add-sqr-sqrt 30.0%

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

      5. fabs-sqr 30.0%

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

      6. add-sqr-sqrt 36.3%

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

      7. metadata-eval 36.3%

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

   3. Applied egg-rr 36.3%

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

   4. Taylor expanded in y around 0 34.3%

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

      1. distribute-rgt1-in 34.3%

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

      2. metadata-eval 34.3%

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

      3. *-commutative 34.3%

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

   6. Simplified 34.3%

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

   if 2.84999999999999981e-105 < y

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. 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-def 100.0%

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

      4. add-sqr-sqrt 92.2%

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

      5. fabs-sqr 92.2%

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

      6. add-sqr-sqrt 94.1%

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

      7. metadata-eval 94.1%

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

   3. Applied egg-rr 94.1%

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

   4. Taylor expanded in y around inf 71.9%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.85 \cdot 10^{-105}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 5: 55.5% accurate, 21.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[x + \frac{\left|y - x\right|}{2} \]
2. 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-def 99.9%

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

   4. add-sqr-sqrt 50.4%

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

   5. fabs-sqr 50.4%

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

   6. add-sqr-sqrt 55.3%

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

   7. metadata-eval 55.3%

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

3. Applied egg-rr 55.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 0.5, x\right)} \]
4. Step-by-step derivation

   1. fma-udef 55.3%

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

   2. flip-+ 29.3%

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

   3. pow2 29.3%

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

   4. pow2 29.3%

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

5. Applied egg-rr 29.3%

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

   1. unpow2 29.3%

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

   2. unpow2 29.3%

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

   3. difference-of-squares 31.0%

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

   4. fma-def 31.0%

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

   5. associate-/l* 55.3%

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

   6. fma-def 55.3%

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

   7. *-commutative 55.3%

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

   8. sub-neg 55.3%

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

   9. +-commutative 55.3%

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

   10. distribute-lft-in 55.3%

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

   11. distribute-rgt-neg-in 55.3%

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

   12. distribute-lft-neg-in 55.3%

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

   13. metadata-eval 55.3%

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

   14. +-commutative 55.3%

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

   15. associate-+r+ 55.3%

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

   16. distribute-rgt1-in 55.3%

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

   17. metadata-eval 55.3%

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

   18. distribute-lft-out 55.3%

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

   19. *-inverses 55.3%

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

7. Simplified 55.3%

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

8. Final simplification 55.3%

   \[\leadsto \frac{x + y}{2} \]

Alternative 6: 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. Taylor expanded in x around inf 11.3%

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

3. Final simplification 11.3%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023322 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))