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

Percentage Accurate: 99.9% → 99.9%

Time: 5.3s

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: 69.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y - x\right) \cdot 0.5\\ \mathbf{if}\;x \leq -4.4 \cdot 10^{-253}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-158} \lor \neg \left(x \leq 7.5 \cdot 10^{+144}\right):\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 \cdot t_0 - x \cdot x}{y \cdot -0.5 + x \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- y x) 0.5)))
   (if (<= x -4.4e-253)
     (* 0.5 (+ x y))
     (if (or (<= x 3.1e-158) (not (<= x 7.5e+144)))
       (* (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 <= -4.4e-253) {
		tmp = 0.5 * (x + y);
	} else if ((x <= 3.1e-158) || !(x <= 7.5e+144)) {
		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 <= (-4.4d-253)) then
        tmp = 0.5d0 * (x + y)
    else if ((x <= 3.1d-158) .or. (.not. (x <= 7.5d+144))) 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 <= -4.4e-253) {
		tmp = 0.5 * (x + y);
	} else if ((x <= 3.1e-158) || !(x <= 7.5e+144)) {
		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 <= -4.4e-253:
		tmp = 0.5 * (x + y)
	elif (x <= 3.1e-158) or not (x <= 7.5e+144):
		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 <= -4.4e-253)
		tmp = Float64(0.5 * Float64(x + y));
	elseif ((x <= 3.1e-158) || !(x <= 7.5e+144))
		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 <= -4.4e-253)
		tmp = 0.5 * (x + y);
	elseif ((x <= 3.1e-158) || ~((x <= 7.5e+144)))
		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, -4.4e-253], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, 3.1e-158], N[Not[LessEqual[x, 7.5e+144]], $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 < -4.39999999999999992e-253

   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 80.4%

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

      5. fabs-sqr 80.4%

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

      6. add-sqr-sqrt 81.2%

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

      7. metadata-eval 81.2%

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

   3. Applied egg-rr 81.2%

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

   4. Taylor expanded in y around 0 81.2%

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

      1. +-commutative 81.2%

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

      2. associate-+r+ 81.2%

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

      3. distribute-lft1-in 81.2%

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

      4. metadata-eval 81.2%

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

      5. distribute-lft-out 81.2%

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

   6. Simplified 81.2%

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

   if -4.39999999999999992e-253 < x < 3.10000000000000018e-158 or 7.5000000000000006e144 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 67.3%

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

   if 3.10000000000000018e-158 < x < 7.5000000000000006e144

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. div-inv 99.8%

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

      3. fma-def 99.8%

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

      4. add-sqr-sqrt 12.6%

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

      5. fabs-sqr 12.6%

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

      6. add-sqr-sqrt 24.0%

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

      7. metadata-eval 24.0%

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

   3. Applied egg-rr 24.0%

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

      1. fma-udef 24.0%

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

      2. flip-+ 16.6%

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

   5. Applied egg-rr 16.6%

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

      1. add-sqr-sqrt 5.3%

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

      2. sqrt-prod 84.6%

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

      3. associate-*r* 84.5%

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

      4. sqrt-prod 84.4%

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

      5. fma-neg 84.4%

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

      6. *-commutative 84.4%

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

      7. associate-*l* 84.4%

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

      8. pow2 84.4%

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

   7. Applied egg-rr 84.4%

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

   8. Taylor expanded in y around -inf 79.4%

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

      1. sub-neg 79.4%

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

      2. +-commutative 79.4%

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

      3. associate-+l+ 79.4%

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

      4. *-commutative 79.4%

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

      5. *-commutative 79.4%

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

      6. unpow2 79.4%

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

      7. rem-square-sqrt 79.7%

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

      8. associate-*l* 79.7%

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

      9. metadata-eval 79.7%

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

      10. unpow2 79.7%

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

      11. rem-square-sqrt 80.3%

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

      12. neg-mul-1 80.3%

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

      13. distribute-rgt-out 80.3%

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

      14. metadata-eval 80.3%

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

   10. Simplified 80.3%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-253}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-158} \lor \neg \left(x \leq 7.5 \cdot 10^{+144}\right):\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(\left(y - x\right) \cdot 0.5\right) - x \cdot x}{y \cdot -0.5 + x \cdot -0.5}\\ \end{array} \]

Alternative 3: 71.5% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y - x\right) \cdot 0.5\\ \mathbf{if}\;y \leq -1 \cdot 10^{+42}:\\ \;\;\;\;\frac{y \cdot -0.25}{0.5}\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-239}:\\ \;\;\;\;\frac{t_0 \cdot t_0 - x \cdot x}{y \cdot -0.5 + x \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- y x) 0.5)))
   (if (<= y -1e+42)
     (/ (* y -0.25) 0.5)
     (if (<= y -1.6e-239)
       (/ (- (* 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 <= -1e+42) {
		tmp = (y * -0.25) / 0.5;
	} else if (y <= -1.6e-239) {
		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 <= (-1d+42)) then
        tmp = (y * (-0.25d0)) / 0.5d0
    else if (y <= (-1.6d-239)) 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 <= -1e+42) {
		tmp = (y * -0.25) / 0.5;
	} else if (y <= -1.6e-239) {
		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 <= -1e+42:
		tmp = (y * -0.25) / 0.5
	elif y <= -1.6e-239:
		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 <= -1e+42)
		tmp = Float64(Float64(y * -0.25) / 0.5);
	elseif (y <= -1.6e-239)
		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 <= -1e+42)
		tmp = (y * -0.25) / 0.5;
	elseif (y <= -1.6e-239)
		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, -1e+42], N[(N[(y * -0.25), $MachinePrecision] / 0.5), $MachinePrecision], If[LessEqual[y, -1.6e-239], 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 < -1.00000000000000004e42

   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 15.8%

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

      5. fabs-sqr 15.8%

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

      6. add-sqr-sqrt 17.1%

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

      7. metadata-eval 17.1%

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

   3. Applied egg-rr 17.1%

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

      1. fma-udef 17.1%

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

      2. flip-+ 0.3%

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

   5. Applied egg-rr 0.3%

      \[\leadsto \color{blue}{\frac{\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(\left(y - x\right) \cdot 0.5\right) - x \cdot x}{\left(y - x\right) \cdot 0.5 - x}} \]
   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 32.0%

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

      3. associate-*r* 32.0%

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

      4. sqrt-prod 31.7%

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

      5. fma-neg 31.7%

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

      6. *-commutative 31.7%

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

      7. associate-*l* 31.7%

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

      8. pow2 31.7%

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

   7. Applied egg-rr 31.7%

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

   8. Taylor expanded in y around -inf 78.7%

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

      1. associate-*r/ 78.7%

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

      2. unpow2 78.7%

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

      3. rem-square-sqrt 80.4%

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

   10. Simplified 80.4%

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

   if -1.00000000000000004e42 < y < -1.6e-239

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. div-inv 99.8%

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

      3. fma-def 99.8%

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

      4. add-sqr-sqrt 22.6%

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

      5. fabs-sqr 22.6%

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

      6. add-sqr-sqrt 30.9%

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

      7. metadata-eval 30.9%

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

   3. Applied egg-rr 30.9%

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

      1. fma-udef 30.9%

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

      2. flip-+ 16.6%

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

   5. Applied egg-rr 16.6%

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

      1. add-sqr-sqrt 10.4%

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

      2. sqrt-prod 66.3%

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

      3. associate-*r* 66.2%

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

      4. sqrt-prod 66.1%

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

      5. fma-neg 66.1%

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

      6. *-commutative 66.1%

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

      7. associate-*l* 66.1%

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

      8. pow2 66.1%

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

   7. Applied egg-rr 66.1%

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

   8. Taylor expanded in y around -inf 56.8%

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

      1. sub-neg 56.8%

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

      2. +-commutative 56.8%

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

      3. associate-+l+ 56.8%

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

      4. *-commutative 56.8%

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

      5. *-commutative 56.8%

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

      6. unpow2 56.8%

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

      7. rem-square-sqrt 57.4%

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

      8. associate-*l* 57.4%

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

      9. metadata-eval 57.4%

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

      10. unpow2 57.4%

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

      11. rem-square-sqrt 57.7%

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

      12. neg-mul-1 57.7%

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

      13. distribute-rgt-out 57.7%

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

      14. metadata-eval 57.7%

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

   10. Simplified 57.7%

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

   if -1.6e-239 < 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 71.9%

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

      5. fabs-sqr 71.9%

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

      6. add-sqr-sqrt 77.2%

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

      7. metadata-eval 77.2%

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

   3. Applied egg-rr 77.2%

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

   4. Taylor expanded in y around 0 77.2%

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

      1. +-commutative 77.2%

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

      2. associate-+r+ 77.2%

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

      3. distribute-lft1-in 77.2%

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

      4. metadata-eval 77.2%

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

      5. distribute-lft-out 77.2%

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

   6. Simplified 77.2%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+42}:\\ \;\;\;\;\frac{y \cdot -0.25}{0.5}\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-239}:\\ \;\;\;\;\frac{\left(\left(y - x\right) \cdot 0.5\right) \cdot \left(\left(y - x\right) \cdot 0.5\right) - x \cdot x}{y \cdot -0.5 + x \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \]

Alternative 4: 70.1% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{-122}:\\ \;\;\;\;\frac{y \cdot -0.25}{0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.02e-122) (/ (* y -0.25) 0.5) (* 0.5 (+ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.02e-122) {
		tmp = (y * -0.25) / 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) :: tmp
    if (y <= (-1.02d-122)) then
        tmp = (y * (-0.25d0)) / 0.5d0
    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 <= -1.02e-122) {
		tmp = (y * -0.25) / 0.5;
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.02e-122:
		tmp = (y * -0.25) / 0.5
	else:
		tmp = 0.5 * (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.02e-122)
		tmp = Float64(Float64(y * -0.25) / 0.5);
	else
		tmp = Float64(0.5 * Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.02e-122)
		tmp = (y * -0.25) / 0.5;
	else
		tmp = 0.5 * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.02e-122], N[(N[(y * -0.25), $MachinePrecision] / 0.5), $MachinePrecision], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.02000000000000002e-122

   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 18.9%

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

      5. fabs-sqr 18.9%

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

      6. add-sqr-sqrt 22.4%

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

      7. metadata-eval 22.4%

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

   3. Applied egg-rr 22.4%

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

      1. fma-udef 22.4%

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

      2. flip-+ 6.5%

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

   5. Applied egg-rr 6.5%

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

      1. add-sqr-sqrt 4.7%

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

      2. sqrt-prod 48.6%

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

      3. associate-*r* 48.6%

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

      4. sqrt-prod 48.4%

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

      5. fma-neg 48.4%

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

      6. *-commutative 48.4%

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

      7. associate-*l* 48.4%

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

      8. pow2 48.4%

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

   7. Applied egg-rr 48.4%

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

   8. Taylor expanded in y around -inf 65.0%

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

      1. associate-*r/ 65.0%

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

      2. unpow2 65.0%

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

      3. rem-square-sqrt 66.3%

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

   10. Simplified 66.3%

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

   if -1.02000000000000002e-122 < 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 65.2%

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

      5. fabs-sqr 65.2%

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

      6. add-sqr-sqrt 71.1%

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

      7. metadata-eval 71.1%

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

   3. Applied egg-rr 71.1%

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

   4. Taylor expanded in y around 0 71.1%

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

      1. +-commutative 71.1%

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

      2. associate-+r+ 71.1%

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

      3. distribute-lft1-in 71.1%

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

      4. metadata-eval 71.1%

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

      5. distribute-lft-out 71.1%

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

   6. Simplified 71.1%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{-122}:\\ \;\;\;\;\frac{y \cdot -0.25}{0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \]

Alternative 5: 31.8% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{-185}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 3.4e-185) x (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.4e-185) {
		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 <= 3.4d-185) 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 <= 3.4e-185) {
		tmp = x;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.4e-185:
		tmp = x
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.4e-185)
		tmp = x;
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.4e-185)
		tmp = x;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.4e-185], x, N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.3999999999999998e-185

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 13.3%

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

   if 3.3999999999999998e-185 < 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 83.9%

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

      5. fabs-sqr 83.9%

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

      6. add-sqr-sqrt 87.3%

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

      7. metadata-eval 87.3%

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

   3. Applied egg-rr 87.3%

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

   4. Taylor expanded in y around inf 62.7%

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

4. Final simplification 30.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{-185}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 6: 45.7% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{-101}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.6e-101) (* x 0.5) (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.6e-101) {
		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 <= 2.6d-101) 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 <= 2.6e-101) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.6e-101:
		tmp = x * 0.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.6e-101)
		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 <= 2.6e-101)
		tmp = x * 0.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.6e-101], N[(x * 0.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.6000000000000001e-101

   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 35.0%

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

      5. fabs-sqr 35.0%

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

      6. add-sqr-sqrt 41.2%

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

      7. metadata-eval 41.2%

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

   3. Applied egg-rr 41.2%

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

   4. Taylor expanded in y around 0 38.1%

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

      1. distribute-lft1-in 38.1%

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

      2. metadata-eval 38.1%

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

      3. *-commutative 38.1%

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

   6. Simplified 38.1%

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

   if 2.6000000000000001e-101 < 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 88.2%

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

      5. fabs-sqr 88.2%

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

      6. add-sqr-sqrt 90.9%

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

      7. metadata-eval 90.9%

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

   3. Applied egg-rr 90.9%

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

   4. Taylor expanded in y around inf 72.0%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{-101}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 7: 54.7% 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 50.0%

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

   5. fabs-sqr 50.0%

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

   6. add-sqr-sqrt 55.1%

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

   7. metadata-eval 55.1%

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

3. Applied egg-rr 55.1%

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

4. Taylor expanded in y around 0 55.1%

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

   1. +-commutative 55.1%

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

   2. associate-+r+ 55.1%

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

   3. distribute-lft1-in 55.1%

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

   4. metadata-eval 55.1%

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

   5. distribute-lft-out 55.1%

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

6. Simplified 55.1%

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

7. Final simplification 55.1%

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

Alternative 8: 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 11.7%

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

3. Final simplification 11.7%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023238 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))