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

Percentage Accurate: 99.9% → 99.9%

Time: 5.1s

Alternatives: 8

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: 72.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y - x\right) \cdot 0.5\\ \mathbf{if}\;x \leq -1 \cdot 10^{-32}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-90} \lor \neg \left(x \leq 5.9 \cdot 10^{+147}\right):\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 \cdot t_0 - x \cdot x}{y \cdot -0.5 + x \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- y x) 0.5)))
   (if (<= x -1e-32)
     (* 0.5 (+ x y))
     (if (or (<= x 9.5e-90) (not (<= x 5.9e+147)))
       (* (fabs (- y x)) 0.5)
       (/ (- (* t_0 t_0) (* x x)) (+ (* y -0.5) (* x -0.5)))))))

C

double code(double x, double y) {
	double t_0 = (y - x) * 0.5;
	double tmp;
	if (x <= -1e-32) {
		tmp = 0.5 * (x + y);
	} else if ((x <= 9.5e-90) || !(x <= 5.9e+147)) {
		tmp = fabs((y - x)) * 0.5;
	} else {
		tmp = ((t_0 * t_0) - (x * x)) / ((y * -0.5) + (x * -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y - x) * 0.5d0
    if (x <= (-1d-32)) then
        tmp = 0.5d0 * (x + y)
    else if ((x <= 9.5d-90) .or. (.not. (x <= 5.9d+147))) then
        tmp = abs((y - x)) * 0.5d0
    else
        tmp = ((t_0 * t_0) - (x * x)) / ((y * (-0.5d0)) + (x * (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y - x) * 0.5;
	double tmp;
	if (x <= -1e-32) {
		tmp = 0.5 * (x + y);
	} else if ((x <= 9.5e-90) || !(x <= 5.9e+147)) {
		tmp = Math.abs((y - x)) * 0.5;
	} else {
		tmp = ((t_0 * t_0) - (x * x)) / ((y * -0.5) + (x * -0.5));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y - x) * 0.5
	tmp = 0
	if x <= -1e-32:
		tmp = 0.5 * (x + y)
	elif (x <= 9.5e-90) or not (x <= 5.9e+147):
		tmp = math.fabs((y - x)) * 0.5
	else:
		tmp = ((t_0 * t_0) - (x * x)) / ((y * -0.5) + (x * -0.5))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y - x) * 0.5)
	tmp = 0.0
	if (x <= -1e-32)
		tmp = Float64(0.5 * Float64(x + y));
	elseif ((x <= 9.5e-90) || !(x <= 5.9e+147))
		tmp = Float64(abs(Float64(y - x)) * 0.5);
	else
		tmp = Float64(Float64(Float64(t_0 * t_0) - Float64(x * x)) / Float64(Float64(y * -0.5) + Float64(x * -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y - x) * 0.5;
	tmp = 0.0;
	if (x <= -1e-32)
		tmp = 0.5 * (x + y);
	elseif ((x <= 9.5e-90) || ~((x <= 5.9e+147)))
		tmp = abs((y - x)) * 0.5;
	else
		tmp = ((t_0 * t_0) - (x * x)) / ((y * -0.5) + (x * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y - x), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[x, -1e-32], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, 9.5e-90], N[Not[LessEqual[x, 5.9e+147]], $MachinePrecision]], N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(N[(y * -0.5), $MachinePrecision] + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.00000000000000006e-32

   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 88.4%

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

      5. fabs-sqr 88.4%

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

      6. add-sqr-sqrt 89.1%

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

      7. metadata-eval 89.1%

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

   3. Applied egg-rr 89.1%

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

   4. Taylor expanded in y around 0 89.1%

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

      1. +-commutative 89.1%

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

      2. associate-+r+ 89.1%

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

      3. distribute-lft1-in 89.1%

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

      4. metadata-eval 89.1%

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

      5. distribute-lft-out 89.1%

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

   6. Simplified 89.1%

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

   if -1.00000000000000006e-32 < x < 9.5000000000000003e-90 or 5.9000000000000001e147 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 66.8%

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

   if 9.5000000000000003e-90 < x < 5.9000000000000001e147

   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 15.1%

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

      5. fabs-sqr 15.1%

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

      6. add-sqr-sqrt 26.9%

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

      7. metadata-eval 26.9%

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

   3. Applied egg-rr 26.9%

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

      1. fma-udef 26.9%

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

      2. flip-+ 17.6%

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

   5. Applied egg-rr 17.6%

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

      1. add-sqr-sqrt 6.0%

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

      2. sqrt-prod 76.6%

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

      3. associate-*r* 76.6%

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

      4. sqrt-prod 76.2%

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

      5. fma-neg 76.2%

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

      6. *-commutative 76.2%

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

      7. associate-*l* 76.2%

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

      8. pow2 76.2%

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

   7. Applied egg-rr 76.2%

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

   8. Taylor expanded in y around -inf 71.2%

      \[\leadsto \frac{\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(\left(y - x\right) \cdot 0.5\right) - x \cdot x}{\color{blue}{\left({\left(\sqrt{0.5}\right)}^{2} \cdot x + -1 \cdot \left({\left(\sqrt{0.5}\right)}^{2} \cdot y\right)\right) - x}} \]
   9. Step-by-step derivation

      1. sub-neg 71.2%

         \[\leadsto \frac{\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(\left(y - x\right) \cdot 0.5\right) - x \cdot x}{\color{blue}{\left({\left(\sqrt{0.5}\right)}^{2} \cdot x + -1 \cdot \left({\left(\sqrt{0.5}\right)}^{2} \cdot y\right)\right) + \left(-x\right)}} \]

      2. +-commutative 71.2%

         \[\leadsto \frac{\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(\left(y - x\right) \cdot 0.5\right) - x \cdot x}{\color{blue}{\left(-1 \cdot \left({\left(\sqrt{0.5}\right)}^{2} \cdot y\right) + {\left(\sqrt{0.5}\right)}^{2} \cdot x\right)} + \left(-x\right)} \]

      3. associate-+l+ 71.2%

         \[\leadsto \frac{\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(\left(y - x\right) \cdot 0.5\right) - x \cdot x}{\color{blue}{-1 \cdot \left({\left(\sqrt{0.5}\right)}^{2} \cdot y\right) + \left({\left(\sqrt{0.5}\right)}^{2} \cdot x + \left(-x\right)\right)}} \]

      4. *-commutative 71.2%

         \[\leadsto \frac{\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(\left(y - x\right) \cdot 0.5\right) - x \cdot x}{\color{blue}{\left({\left(\sqrt{0.5}\right)}^{2} \cdot y\right) \cdot -1} + \left({\left(\sqrt{0.5}\right)}^{2} \cdot x + \left(-x\right)\right)} \]

      5. *-commutative 71.2%

         \[\leadsto \frac{\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(\left(y - x\right) \cdot 0.5\right) - x \cdot x}{\color{blue}{\left(y \cdot {\left(\sqrt{0.5}\right)}^{2}\right)} \cdot -1 + \left({\left(\sqrt{0.5}\right)}^{2} \cdot x + \left(-x\right)\right)} \]

      6. unpow2 71.2%

         \[\leadsto \frac{\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(\left(y - x\right) \cdot 0.5\right) - x \cdot x}{\left(y \cdot \color{blue}{\left(\sqrt{0.5} \cdot \sqrt{0.5}\right)}\right) \cdot -1 + \left({\left(\sqrt{0.5}\right)}^{2} \cdot x + \left(-x\right)\right)} \]

      7. rem-square-sqrt 71.4%

         \[\leadsto \frac{\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(\left(y - x\right) \cdot 0.5\right) - x \cdot x}{\left(y \cdot \color{blue}{0.5}\right) \cdot -1 + \left({\left(\sqrt{0.5}\right)}^{2} \cdot x + \left(-x\right)\right)} \]

      8. associate-*l* 71.4%

         \[\leadsto \frac{\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(\left(y - x\right) \cdot 0.5\right) - x \cdot x}{\color{blue}{y \cdot \left(0.5 \cdot -1\right)} + \left({\left(\sqrt{0.5}\right)}^{2} \cdot x + \left(-x\right)\right)} \]

      9. metadata-eval 71.4%

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

      10. unpow2 71.4%

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

      11. rem-square-sqrt 72.3%

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

      12. neg-mul-1 72.3%

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

      13. distribute-rgt-out 72.3%

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

      14. metadata-eval 72.3%

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

   10. Simplified 72.3%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-32}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-90} \lor \neg \left(x \leq 5.9 \cdot 10^{+147}\right):\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(\left(y - x\right) \cdot 0.5\right) - x \cdot x}{y \cdot -0.5 + x \cdot -0.5}\\ \end{array} \]

Alternative 3: 72.4% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y - x\right) \cdot 0.5\\ \mathbf{if}\;y \leq -3.3 \cdot 10^{+127}:\\ \;\;\;\;\frac{y}{0.5} \cdot -0.25\\ \mathbf{elif}\;y \leq -1 \cdot 10^{-124}:\\ \;\;\;\;\frac{t_0 \cdot t_0 - x \cdot x}{y \cdot -0.5 + x \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- y x) 0.5)))
   (if (<= y -3.3e+127)
     (* (/ y 0.5) -0.25)
     (if (<= y -1e-124)
       (/ (- (* t_0 t_0) (* x x)) (+ (* y -0.5) (* x -0.5)))
       (* 0.5 (+ x y))))))

C

double code(double x, double y) {
	double t_0 = (y - x) * 0.5;
	double tmp;
	if (y <= -3.3e+127) {
		tmp = (y / 0.5) * -0.25;
	} else if (y <= -1e-124) {
		tmp = ((t_0 * t_0) - (x * x)) / ((y * -0.5) + (x * -0.5));
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y - x) * 0.5d0
    if (y <= (-3.3d+127)) then
        tmp = (y / 0.5d0) * (-0.25d0)
    else if (y <= (-1d-124)) then
        tmp = ((t_0 * t_0) - (x * x)) / ((y * (-0.5d0)) + (x * (-0.5d0)))
    else
        tmp = 0.5d0 * (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y - x) * 0.5;
	double tmp;
	if (y <= -3.3e+127) {
		tmp = (y / 0.5) * -0.25;
	} else if (y <= -1e-124) {
		tmp = ((t_0 * t_0) - (x * x)) / ((y * -0.5) + (x * -0.5));
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y - x) * 0.5
	tmp = 0
	if y <= -3.3e+127:
		tmp = (y / 0.5) * -0.25
	elif y <= -1e-124:
		tmp = ((t_0 * t_0) - (x * x)) / ((y * -0.5) + (x * -0.5))
	else:
		tmp = 0.5 * (x + y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y - x) * 0.5)
	tmp = 0.0
	if (y <= -3.3e+127)
		tmp = Float64(Float64(y / 0.5) * -0.25);
	elseif (y <= -1e-124)
		tmp = Float64(Float64(Float64(t_0 * t_0) - Float64(x * x)) / Float64(Float64(y * -0.5) + Float64(x * -0.5)));
	else
		tmp = Float64(0.5 * Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y - x) * 0.5;
	tmp = 0.0;
	if (y <= -3.3e+127)
		tmp = (y / 0.5) * -0.25;
	elseif (y <= -1e-124)
		tmp = ((t_0 * t_0) - (x * x)) / ((y * -0.5) + (x * -0.5));
	else
		tmp = 0.5 * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y - x), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[y, -3.3e+127], N[(N[(y / 0.5), $MachinePrecision] * -0.25), $MachinePrecision], If[LessEqual[y, -1e-124], N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(N[(y * -0.5), $MachinePrecision] + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.29999999999999977e127

   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 5.2%

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

      5. fabs-sqr 5.2%

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

      6. add-sqr-sqrt 7.0%

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

      7. metadata-eval 7.0%

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

   3. Applied egg-rr 7.0%

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

      1. fma-udef 7.0%

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

      2. flip-+ 0.2%

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

   5. Applied egg-rr 0.2%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-prod 10.5%

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

      3. associate-*r* 10.5%

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

      4. sqrt-prod 10.4%

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

      5. fma-neg 10.4%

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

      6. *-commutative 10.4%

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

      7. associate-*l* 10.4%

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

      8. pow2 10.4%

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

   7. Applied egg-rr 10.4%

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

   8. Taylor expanded in y around -inf 85.0%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{y}{{\left(\sqrt{0.5}\right)}^{2}}} \]
   9. Step-by-step derivation

      1. *-commutative 85.0%

         \[\leadsto \color{blue}{\frac{y}{{\left(\sqrt{0.5}\right)}^{2}} \cdot -0.25} \]

      2. unpow2 85.0%

         \[\leadsto \frac{y}{\color{blue}{\sqrt{0.5} \cdot \sqrt{0.5}}} \cdot -0.25 \]

      3. rem-square-sqrt 86.7%

         \[\leadsto \frac{y}{\color{blue}{0.5}} \cdot -0.25 \]

   10. Simplified 86.7%

       \[\leadsto \color{blue}{\frac{y}{0.5} \cdot -0.25} \]

   if -3.29999999999999977e127 < y < -9.99999999999999933e-125

   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 20.2%

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

      5. fabs-sqr 20.2%

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

      6. add-sqr-sqrt 26.1%

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

      7. metadata-eval 26.1%

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

   3. Applied egg-rr 26.1%

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

      1. fma-udef 26.1%

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

      2. flip-+ 11.9%

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

   5. Applied egg-rr 11.9%

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

      1. add-sqr-sqrt 9.2%

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

      2. sqrt-prod 72.0%

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

      3. associate-*r* 72.0%

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

      4. sqrt-prod 71.7%

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

      5. fma-neg 71.7%

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

      6. *-commutative 71.7%

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

      7. associate-*l* 71.7%

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

      8. pow2 71.7%

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

   7. Applied egg-rr 71.7%

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

   8. Taylor expanded in y around -inf 63.5%

      \[\leadsto \frac{\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(\left(y - x\right) \cdot 0.5\right) - x \cdot x}{\color{blue}{\left({\left(\sqrt{0.5}\right)}^{2} \cdot x + -1 \cdot \left({\left(\sqrt{0.5}\right)}^{2} \cdot y\right)\right) - x}} \]
   9. Step-by-step derivation

      1. sub-neg 63.5%

         \[\leadsto \frac{\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(\left(y - x\right) \cdot 0.5\right) - x \cdot x}{\color{blue}{\left({\left(\sqrt{0.5}\right)}^{2} \cdot x + -1 \cdot \left({\left(\sqrt{0.5}\right)}^{2} \cdot y\right)\right) + \left(-x\right)}} \]

      2. +-commutative 63.5%

         \[\leadsto \frac{\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(\left(y - x\right) \cdot 0.5\right) - x \cdot x}{\color{blue}{\left(-1 \cdot \left({\left(\sqrt{0.5}\right)}^{2} \cdot y\right) + {\left(\sqrt{0.5}\right)}^{2} \cdot x\right)} + \left(-x\right)} \]

      3. associate-+l+ 63.5%

         \[\leadsto \frac{\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(\left(y - x\right) \cdot 0.5\right) - x \cdot x}{\color{blue}{-1 \cdot \left({\left(\sqrt{0.5}\right)}^{2} \cdot y\right) + \left({\left(\sqrt{0.5}\right)}^{2} \cdot x + \left(-x\right)\right)}} \]

      4. *-commutative 63.5%

         \[\leadsto \frac{\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(\left(y - x\right) \cdot 0.5\right) - x \cdot x}{\color{blue}{\left({\left(\sqrt{0.5}\right)}^{2} \cdot y\right) \cdot -1} + \left({\left(\sqrt{0.5}\right)}^{2} \cdot x + \left(-x\right)\right)} \]

      5. *-commutative 63.5%

         \[\leadsto \frac{\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(\left(y - x\right) \cdot 0.5\right) - x \cdot x}{\color{blue}{\left(y \cdot {\left(\sqrt{0.5}\right)}^{2}\right)} \cdot -1 + \left({\left(\sqrt{0.5}\right)}^{2} \cdot x + \left(-x\right)\right)} \]

      6. unpow2 63.5%

         \[\leadsto \frac{\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(\left(y - x\right) \cdot 0.5\right) - x \cdot x}{\left(y \cdot \color{blue}{\left(\sqrt{0.5} \cdot \sqrt{0.5}\right)}\right) \cdot -1 + \left({\left(\sqrt{0.5}\right)}^{2} \cdot x + \left(-x\right)\right)} \]

      7. rem-square-sqrt 64.3%

         \[\leadsto \frac{\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(\left(y - x\right) \cdot 0.5\right) - x \cdot x}{\left(y \cdot \color{blue}{0.5}\right) \cdot -1 + \left({\left(\sqrt{0.5}\right)}^{2} \cdot x + \left(-x\right)\right)} \]

      8. associate-*l* 64.3%

         \[\leadsto \frac{\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(\left(y - x\right) \cdot 0.5\right) - x \cdot x}{\color{blue}{y \cdot \left(0.5 \cdot -1\right)} + \left({\left(\sqrt{0.5}\right)}^{2} \cdot x + \left(-x\right)\right)} \]

      9. metadata-eval 64.3%

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

      10. unpow2 64.3%

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

      11. rem-square-sqrt 64.5%

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

      12. neg-mul-1 64.5%

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

      13. distribute-rgt-out 64.5%

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

      14. metadata-eval 64.5%

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

   10. Simplified 64.5%

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

   if -9.99999999999999933e-125 < 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 63.5%

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

      5. fabs-sqr 63.5%

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

      6. add-sqr-sqrt 70.1%

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

      7. metadata-eval 70.1%

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

   3. Applied egg-rr 70.1%

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

   4. Taylor expanded in y around 0 70.0%

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

      1. +-commutative 70.0%

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

      2. associate-+r+ 70.1%

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

      3. distribute-lft1-in 70.1%

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

      4. metadata-eval 70.1%

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

      5. distribute-lft-out 70.1%

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

   6. Simplified 70.1%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+127}:\\ \;\;\;\;\frac{y}{0.5} \cdot -0.25\\ \mathbf{elif}\;y \leq -1 \cdot 10^{-124}:\\ \;\;\;\;\frac{\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(\left(y - x\right) \cdot 0.5\right) - x \cdot x}{y \cdot -0.5 + x \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \]

Alternative 4: 69.7% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{-109}:\\ \;\;\;\;\frac{y}{0.5} \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.4e-109) (* (/ y 0.5) -0.25) (* 0.5 (+ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.4e-109) {
		tmp = (y / 0.5) * -0.25;
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.4d-109)) then
        tmp = (y / 0.5d0) * (-0.25d0)
    else
        tmp = 0.5d0 * (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.4e-109) {
		tmp = (y / 0.5) * -0.25;
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.4e-109:
		tmp = (y / 0.5) * -0.25
	else:
		tmp = 0.5 * (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.4e-109)
		tmp = Float64(Float64(y / 0.5) * -0.25);
	else
		tmp = Float64(0.5 * Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.4e-109)
		tmp = (y / 0.5) * -0.25;
	else
		tmp = 0.5 * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.4e-109], N[(N[(y / 0.5), $MachinePrecision] * -0.25), $MachinePrecision], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.39999999999999989e-109

   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 14.2%

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

      5. fabs-sqr 14.2%

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

      6. add-sqr-sqrt 18.2%

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

      7. metadata-eval 18.2%

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

   3. Applied egg-rr 18.2%

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

      1. fma-udef 18.2%

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

      2. flip-+ 7.0%

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

   5. Applied egg-rr 7.0%

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

      1. add-sqr-sqrt 5.4%

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

      2. sqrt-prod 46.0%

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

      3. associate-*r* 46.0%

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

      4. sqrt-prod 45.8%

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

      5. fma-neg 45.8%

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

      6. *-commutative 45.8%

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

      7. associate-*l* 45.8%

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

      8. pow2 45.8%

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

   7. Applied egg-rr 45.8%

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

   8. Taylor expanded in y around -inf 66.6%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{y}{{\left(\sqrt{0.5}\right)}^{2}}} \]
   9. Step-by-step derivation

      1. *-commutative 66.6%

         \[\leadsto \color{blue}{\frac{y}{{\left(\sqrt{0.5}\right)}^{2}} \cdot -0.25} \]

      2. unpow2 66.6%

         \[\leadsto \frac{y}{\color{blue}{\sqrt{0.5} \cdot \sqrt{0.5}}} \cdot -0.25 \]

      3. rem-square-sqrt 67.8%

         \[\leadsto \frac{y}{\color{blue}{0.5}} \cdot -0.25 \]

   10. Simplified 67.8%

       \[\leadsto \color{blue}{\frac{y}{0.5} \cdot -0.25} \]

   if -1.39999999999999989e-109 < 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 63.1%

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

      5. fabs-sqr 63.1%

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

      6. add-sqr-sqrt 69.8%

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

      7. metadata-eval 69.8%

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

   3. Applied egg-rr 69.8%

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

   4. Taylor expanded in y around 0 69.7%

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

      1. +-commutative 69.7%

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

      2. associate-+r+ 69.8%

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

      3. distribute-lft1-in 69.8%

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

      4. metadata-eval 69.8%

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

      5. distribute-lft-out 69.8%

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

   6. Simplified 69.8%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{-109}:\\ \;\;\;\;\frac{y}{0.5} \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \]

Alternative 5: 32.1% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-166}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2e-166) x (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2e-166) {
		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 <= 2d-166) 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 <= 2e-166) {
		tmp = x;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2e-166:
		tmp = x
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2e-166)
		tmp = x;
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2e-166)
		tmp = x;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2e-166], x, N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.00000000000000008e-166

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 12.8%

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

   if 2.00000000000000008e-166 < 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 76.6%

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

      5. fabs-sqr 76.6%

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

      6. add-sqr-sqrt 81.3%

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

      7. metadata-eval 81.3%

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

   3. Applied egg-rr 81.3%

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

   4. Taylor expanded in y around inf 58.4%

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

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-166}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 6: 46.2% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{-71}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.45e-71) (* x 0.5) (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.45e-71) {
		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 <= 1.45d-71) 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 <= 1.45e-71) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.45e-71:
		tmp = x * 0.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.45e-71)
		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 <= 1.45e-71)
		tmp = x * 0.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.45e-71], N[(x * 0.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.4499999999999999e-71

   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 29.1%

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

      5. fabs-sqr 29.1%

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

      6. add-sqr-sqrt 35.8%

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

      7. metadata-eval 35.8%

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

   3. Applied egg-rr 35.8%

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

   4. Taylor expanded in y around 0 34.0%

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

      1. distribute-lft1-in 34.0%

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

      2. metadata-eval 34.0%

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

      3. *-commutative 34.0%

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

   6. Simplified 34.0%

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

   if 1.4499999999999999e-71 < 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 82.8%

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

      5. fabs-sqr 82.8%

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

      6. add-sqr-sqrt 86.5%

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

      7. metadata-eval 86.5%

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

   3. Applied egg-rr 86.5%

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

   4. Taylor expanded in y around inf 66.1%

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

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{-71}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 7: 54.8% 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. 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 45.7%

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

   5. fabs-sqr 45.7%

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

   6. add-sqr-sqrt 51.4%

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

   7. metadata-eval 51.4%

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

3. Applied egg-rr 51.4%

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

4. Taylor expanded in y around 0 51.4%

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

   1. +-commutative 51.4%

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

   2. associate-+r+ 51.4%

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

   3. distribute-lft1-in 51.4%

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

   4. metadata-eval 51.4%

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

   5. distribute-lft-out 51.4%

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

6. Simplified 51.4%

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

7. Final simplification 51.4%

   \[\leadsto 0.5 \cdot \left(x + y\right) \]

Alternative 8: 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. Taylor expanded in x around inf 11.7%

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

3. Final simplification 11.7%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023230 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))