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

Percentage Accurate: 99.9% → 99.9%

Time: 9.1s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 82.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + y\right) \cdot 0.5\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{+34}:\\ \;\;\;\;x - t\_0\\ \mathbf{elif}\;x \leq -4.9 \cdot 10^{-114}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+18}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 1.5, y \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (+ x y) 0.5)))
   (if (<= x -8.5e+34)
     (- x t_0)
     (if (<= x -4.9e-114)
       t_0
       (if (<= x 2.05e+18) (+ x (/ (fabs y) 2.0)) (fma x 1.5 (* y -0.5)))))))

C

double code(double x, double y) {
	double t_0 = (x + y) * 0.5;
	double tmp;
	if (x <= -8.5e+34) {
		tmp = x - t_0;
	} else if (x <= -4.9e-114) {
		tmp = t_0;
	} else if (x <= 2.05e+18) {
		tmp = x + (fabs(y) / 2.0);
	} else {
		tmp = fma(x, 1.5, (y * -0.5));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x + y) * 0.5)
	tmp = 0.0
	if (x <= -8.5e+34)
		tmp = Float64(x - t_0);
	elseif (x <= -4.9e-114)
		tmp = t_0;
	elseif (x <= 2.05e+18)
		tmp = Float64(x + Float64(abs(y) / 2.0));
	else
		tmp = fma(x, 1.5, Float64(y * -0.5));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[x, -8.5e+34], N[(x - t$95$0), $MachinePrecision], If[LessEqual[x, -4.9e-114], t$95$0, If[LessEqual[x, 2.05e+18], N[(x + N[(N[Abs[y], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(x * 1.5 + N[(y * -0.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.5000000000000003e34

   1. Initial program 100.0%

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

      1. add-cube-cbrt 97.7%

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

      2. fabs-mul 97.7%

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

      3. pow2 97.7%

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

   4. Applied egg-rr 97.7%

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

   6. Applied egg-rr 96.1%

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

   if -8.5000000000000003e34 < x < -4.8999999999999997e-114

   1. Initial program 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. fma-define 100.0%

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

      3. add-sqr-sqrt 88.7%

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

      4. fabs-sqr 88.7%

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

      5. add-sqr-sqrt 89.4%

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

      6. frac-2neg 89.4%

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

      7. distribute-frac-neg 89.4%

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

      8. fmm-undef 89.4%

         \[\leadsto \color{blue}{1 \cdot x - \frac{y - x}{-2}} \]

      9. *-un-lft-identity 89.4%

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

      10. metadata-eval 89.4%

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

   4. Applied egg-rr 89.4%

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

   5. Taylor expanded in x around 0 89.4%

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

      1. cancel-sign-sub-inv 89.4%

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

      2. metadata-eval 89.4%

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

      3. distribute-lft-in 89.4%

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

      4. +-commutative 89.4%

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

   7. Simplified 89.4%

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

   if -4.8999999999999997e-114 < x < 2.05e18

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 83.6%

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

   if 2.05e18 < x

   1. Initial program 99.8%

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

      1. add-cube-cbrt 98.9%

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

      2. fabs-mul 98.9%

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

      3. pow2 98.9%

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

   4. Applied egg-rr 98.9%

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

      1. add-sqr-sqrt 98.8%

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

      2. sqrt-unprod 44.2%

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

      3. frac-2neg 44.2%

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

      4. metadata-eval 44.2%

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

      5. frac-2neg 44.2%

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

      6. metadata-eval 44.2%

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

      7. frac-times 44.2%

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

   6. Applied egg-rr 96.4%

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

   7. Taylor expanded in x around 0 96.4%

      \[\leadsto \color{blue}{-0.5 \cdot y + 1.5 \cdot x} \]
   8. Step-by-step derivation

      1. +-commutative 96.4%

         \[\leadsto \color{blue}{1.5 \cdot x + -0.5 \cdot y} \]

      2. *-commutative 96.4%

         \[\leadsto \color{blue}{x \cdot 1.5} + -0.5 \cdot y \]

      3. fma-define 96.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1.5, -0.5 \cdot y\right)} \]

      4. *-commutative 96.7%

         \[\leadsto \mathsf{fma}\left(x, 1.5, \color{blue}{y \cdot -0.5}\right) \]

   9. Simplified 96.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1.5, y \cdot -0.5\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+34}:\\ \;\;\;\;x - \left(x + y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq -4.9 \cdot 10^{-114}:\\ \;\;\;\;\left(x + y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+18}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 1.5, y \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + y\right) \cdot 0.5\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+35}:\\ \;\;\;\;x - t\_0\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-114}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{+17}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5 + x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (+ x y) 0.5)))
   (if (<= x -1.35e+35)
     (- x t_0)
     (if (<= x -9e-114)
       t_0
       (if (<= x 5.1e+17) (+ x (/ (fabs y) 2.0)) (+ (* y -0.5) (* x 1.5)))))))

C

double code(double x, double y) {
	double t_0 = (x + y) * 0.5;
	double tmp;
	if (x <= -1.35e+35) {
		tmp = x - t_0;
	} else if (x <= -9e-114) {
		tmp = t_0;
	} else if (x <= 5.1e+17) {
		tmp = x + (fabs(y) / 2.0);
	} else {
		tmp = (y * -0.5) + (x * 1.5);
	}
	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 = (x + y) * 0.5d0
    if (x <= (-1.35d+35)) then
        tmp = x - t_0
    else if (x <= (-9d-114)) then
        tmp = t_0
    else if (x <= 5.1d+17) then
        tmp = x + (abs(y) / 2.0d0)
    else
        tmp = (y * (-0.5d0)) + (x * 1.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x + y) * 0.5;
	double tmp;
	if (x <= -1.35e+35) {
		tmp = x - t_0;
	} else if (x <= -9e-114) {
		tmp = t_0;
	} else if (x <= 5.1e+17) {
		tmp = x + (Math.abs(y) / 2.0);
	} else {
		tmp = (y * -0.5) + (x * 1.5);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x + y) * 0.5
	tmp = 0
	if x <= -1.35e+35:
		tmp = x - t_0
	elif x <= -9e-114:
		tmp = t_0
	elif x <= 5.1e+17:
		tmp = x + (math.fabs(y) / 2.0)
	else:
		tmp = (y * -0.5) + (x * 1.5)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x + y) * 0.5)
	tmp = 0.0
	if (x <= -1.35e+35)
		tmp = Float64(x - t_0);
	elseif (x <= -9e-114)
		tmp = t_0;
	elseif (x <= 5.1e+17)
		tmp = Float64(x + Float64(abs(y) / 2.0));
	else
		tmp = Float64(Float64(y * -0.5) + Float64(x * 1.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x + y) * 0.5;
	tmp = 0.0;
	if (x <= -1.35e+35)
		tmp = x - t_0;
	elseif (x <= -9e-114)
		tmp = t_0;
	elseif (x <= 5.1e+17)
		tmp = x + (abs(y) / 2.0);
	else
		tmp = (y * -0.5) + (x * 1.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[x, -1.35e+35], N[(x - t$95$0), $MachinePrecision], If[LessEqual[x, -9e-114], t$95$0, If[LessEqual[x, 5.1e+17], N[(x + N[(N[Abs[y], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * -0.5), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.35000000000000001e35

   1. Initial program 100.0%

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

      1. add-cube-cbrt 97.7%

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

      2. fabs-mul 97.7%

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

      3. pow2 97.7%

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

   4. Applied egg-rr 97.7%

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

   6. Applied egg-rr 96.1%

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

   if -1.35000000000000001e35 < x < -8.99999999999999937e-114

   1. Initial program 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. fma-define 100.0%

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

      3. add-sqr-sqrt 88.7%

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

      4. fabs-sqr 88.7%

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

      5. add-sqr-sqrt 89.4%

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

      6. frac-2neg 89.4%

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

      7. distribute-frac-neg 89.4%

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

      8. fmm-undef 89.4%

         \[\leadsto \color{blue}{1 \cdot x - \frac{y - x}{-2}} \]

      9. *-un-lft-identity 89.4%

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

      10. metadata-eval 89.4%

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

   4. Applied egg-rr 89.4%

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

   5. Taylor expanded in x around 0 89.4%

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

      1. cancel-sign-sub-inv 89.4%

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

      2. metadata-eval 89.4%

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

      3. distribute-lft-in 89.4%

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

      4. +-commutative 89.4%

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

   7. Simplified 89.4%

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

   if -8.99999999999999937e-114 < x < 5.1e17

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 83.6%

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

   if 5.1e17 < x

   1. Initial program 99.8%

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

      1. add-cube-cbrt 98.9%

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

      2. fabs-mul 98.9%

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

      3. pow2 98.9%

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

   4. Applied egg-rr 98.9%

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

      1. add-sqr-sqrt 98.8%

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

      2. sqrt-unprod 44.2%

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

      3. frac-2neg 44.2%

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

      4. metadata-eval 44.2%

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

      5. frac-2neg 44.2%

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

      6. metadata-eval 44.2%

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

      7. frac-times 44.2%

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

   6. Applied egg-rr 96.4%

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

   7. Taylor expanded in x around 0 96.4%

      \[\leadsto \color{blue}{-0.5 \cdot y + 1.5 \cdot x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+35}:\\ \;\;\;\;x - \left(x + y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-114}:\\ \;\;\;\;\left(x + y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{+17}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5 + x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + y\right) \cdot 0.5\\ \mathbf{if}\;y \leq -2.05 \cdot 10^{-49}:\\ \;\;\;\;x - t\_0\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-7}:\\ \;\;\;\;x + \frac{\left|x\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (+ x y) 0.5)))
   (if (<= y -2.05e-49)
     (- x t_0)
     (if (<= y 3.6e-7) (+ x (/ (fabs x) 2.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = (x + y) * 0.5;
	double tmp;
	if (y <= -2.05e-49) {
		tmp = x - t_0;
	} else if (y <= 3.6e-7) {
		tmp = x + (fabs(x) / 2.0);
	} else {
		tmp = t_0;
	}
	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 = (x + y) * 0.5d0
    if (y <= (-2.05d-49)) then
        tmp = x - t_0
    else if (y <= 3.6d-7) then
        tmp = x + (abs(x) / 2.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x + y) * 0.5;
	double tmp;
	if (y <= -2.05e-49) {
		tmp = x - t_0;
	} else if (y <= 3.6e-7) {
		tmp = x + (Math.abs(x) / 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x + y) * 0.5
	tmp = 0
	if y <= -2.05e-49:
		tmp = x - t_0
	elif y <= 3.6e-7:
		tmp = x + (math.fabs(x) / 2.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x + y) * 0.5)
	tmp = 0.0
	if (y <= -2.05e-49)
		tmp = Float64(x - t_0);
	elseif (y <= 3.6e-7)
		tmp = Float64(x + Float64(abs(x) / 2.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x + y) * 0.5;
	tmp = 0.0;
	if (y <= -2.05e-49)
		tmp = x - t_0;
	elseif (y <= 3.6e-7)
		tmp = x + (abs(x) / 2.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[y, -2.05e-49], N[(x - t$95$0), $MachinePrecision], If[LessEqual[y, 3.6e-7], N[(x + N[(N[Abs[x], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.0500000000000001e-49

   1. Initial program 100.0%

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

      1. add-cube-cbrt 98.0%

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

      2. fabs-mul 98.0%

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

      3. pow2 98.0%

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

   4. Applied egg-rr 98.0%

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

   6. Applied egg-rr 90.5%

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

   if -2.0500000000000001e-49 < y < 3.59999999999999994e-7

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 84.8%

      \[\leadsto x + \frac{\left|\color{blue}{-1 \cdot x}\right|}{2} \]
   4. Step-by-step derivation

      1. neg-mul-1 84.8%

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

   5. Simplified 84.8%

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

   if 3.59999999999999994e-7 < y

   1. Initial program 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. fma-define 100.0%

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

      3. add-sqr-sqrt 92.0%

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

      4. fabs-sqr 92.0%

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

      5. add-sqr-sqrt 94.0%

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

      6. frac-2neg 94.0%

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

      7. distribute-frac-neg 94.0%

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

      8. fmm-undef 94.0%

         \[\leadsto \color{blue}{1 \cdot x - \frac{y - x}{-2}} \]

      9. *-un-lft-identity 94.0%

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

      10. metadata-eval 94.0%

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

   4. Applied egg-rr 94.0%

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

   5. Taylor expanded in x around 0 94.0%

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

      1. cancel-sign-sub-inv 94.0%

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

      2. metadata-eval 94.0%

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

      3. distribute-lft-in 94.0%

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

      4. +-commutative 94.0%

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

   7. Simplified 94.0%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{-49}:\\ \;\;\;\;x - \left(x + y\right) \cdot 0.5\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-7}:\\ \;\;\;\;x + \frac{\left|x\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.9% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + y\right) \cdot 0.5\\ \mathbf{if}\;y \leq -9.6 \cdot 10^{-178}:\\ \;\;\;\;x - t\_0\\ \mathbf{elif}\;y \leq -4.1 \cdot 10^{-271}:\\ \;\;\;\;y \cdot -0.5 + x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (+ x y) 0.5)))
   (if (<= y -9.6e-178)
     (- x t_0)
     (if (<= y -4.1e-271) (+ (* y -0.5) (* x 1.5)) t_0))))

C

double code(double x, double y) {
	double t_0 = (x + y) * 0.5;
	double tmp;
	if (y <= -9.6e-178) {
		tmp = x - t_0;
	} else if (y <= -4.1e-271) {
		tmp = (y * -0.5) + (x * 1.5);
	} else {
		tmp = t_0;
	}
	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 = (x + y) * 0.5d0
    if (y <= (-9.6d-178)) then
        tmp = x - t_0
    else if (y <= (-4.1d-271)) then
        tmp = (y * (-0.5d0)) + (x * 1.5d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x + y) * 0.5;
	double tmp;
	if (y <= -9.6e-178) {
		tmp = x - t_0;
	} else if (y <= -4.1e-271) {
		tmp = (y * -0.5) + (x * 1.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x + y) * 0.5
	tmp = 0
	if y <= -9.6e-178:
		tmp = x - t_0
	elif y <= -4.1e-271:
		tmp = (y * -0.5) + (x * 1.5)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x + y) * 0.5)
	tmp = 0.0
	if (y <= -9.6e-178)
		tmp = Float64(x - t_0);
	elseif (y <= -4.1e-271)
		tmp = Float64(Float64(y * -0.5) + Float64(x * 1.5));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x + y) * 0.5;
	tmp = 0.0;
	if (y <= -9.6e-178)
		tmp = x - t_0;
	elseif (y <= -4.1e-271)
		tmp = (y * -0.5) + (x * 1.5);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[y, -9.6e-178], N[(x - t$95$0), $MachinePrecision], If[LessEqual[y, -4.1e-271], N[(N[(y * -0.5), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.6000000000000002e-178

   1. Initial program 100.0%

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

      1. add-cube-cbrt 98.2%

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

      2. fabs-mul 98.2%

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

      3. pow2 98.2%

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

   4. Applied egg-rr 98.2%

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

   6. Applied egg-rr 85.3%

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

   if -9.6000000000000002e-178 < y < -4.1000000000000003e-271

   1. Initial program 99.9%

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

      1. add-cube-cbrt 99.0%

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

      2. fabs-mul 99.0%

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

      3. pow2 99.0%

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

   4. Applied egg-rr 99.0%

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

      1. add-sqr-sqrt 98.9%

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

      2. sqrt-unprod 53.0%

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

      3. frac-2neg 53.0%

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

      4. metadata-eval 53.0%

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

      5. frac-2neg 53.0%

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

      6. metadata-eval 53.0%

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

      7. frac-times 53.0%

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

   6. Applied egg-rr 80.3%

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

   7. Taylor expanded in x around 0 80.3%

      \[\leadsto \color{blue}{-0.5 \cdot y + 1.5 \cdot x} \]

   if -4.1000000000000003e-271 < y

   1. Initial program 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. fma-define 99.9%

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

      3. add-sqr-sqrt 77.8%

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

      4. fabs-sqr 77.8%

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

      5. add-sqr-sqrt 82.3%

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

      6. frac-2neg 82.3%

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

      7. distribute-frac-neg 82.3%

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

      8. fmm-undef 82.3%

         \[\leadsto \color{blue}{1 \cdot x - \frac{y - x}{-2}} \]

      9. *-un-lft-identity 82.3%

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

      10. metadata-eval 82.3%

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

   4. Applied egg-rr 82.3%

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

   5. Taylor expanded in x around 0 82.3%

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

      1. cancel-sign-sub-inv 82.3%

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

      2. metadata-eval 82.3%

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

      3. distribute-lft-in 82.3%

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

      4. +-commutative 82.3%

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

   7. Simplified 82.3%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.6 \cdot 10^{-178}:\\ \;\;\;\;x - \left(x + y\right) \cdot 0.5\\ \mathbf{elif}\;y \leq -4.1 \cdot 10^{-271}:\\ \;\;\;\;y \cdot -0.5 + x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.8% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-139}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{+18}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1e-139) (* x 0.5) (if (<= x 1.12e+18) (* y 0.5) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1e-139) {
		tmp = x * 0.5;
	} else if (x <= 1.12e+18) {
		tmp = y * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1d-139)) then
        tmp = x * 0.5d0
    else if (x <= 1.12d+18) then
        tmp = y * 0.5d0
    else
        tmp = x * 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1e-139) {
		tmp = x * 0.5;
	} else if (x <= 1.12e+18) {
		tmp = y * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1e-139:
		tmp = x * 0.5
	elif x <= 1.12e+18:
		tmp = y * 0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1e-139)
		tmp = Float64(x * 0.5);
	elseif (x <= 1.12e+18)
		tmp = Float64(y * 0.5);
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1e-139)
		tmp = x * 0.5;
	elseif (x <= 1.12e+18)
		tmp = y * 0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1e-139], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, 1.12e+18], N[(y * 0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.00000000000000003e-139

   1. Initial program 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. fma-define 100.0%

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

      3. add-sqr-sqrt 81.2%

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

      4. fabs-sqr 81.2%

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

      5. add-sqr-sqrt 81.9%

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

      6. frac-2neg 81.9%

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

      7. distribute-frac-neg 81.9%

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

      8. fmm-undef 81.9%

         \[\leadsto \color{blue}{1 \cdot x - \frac{y - x}{-2}} \]

      9. *-un-lft-identity 81.9%

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

      10. metadata-eval 81.9%

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

   4. Applied egg-rr 81.9%

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

   5. Taylor expanded in x around inf 66.1%

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

   if -1.00000000000000003e-139 < x < 1.12e18

   1. Initial program 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. fma-define 100.0%

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

      3. add-sqr-sqrt 50.1%

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

      4. fabs-sqr 50.1%

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

      5. add-sqr-sqrt 53.0%

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

      6. frac-2neg 53.0%

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

      7. distribute-frac-neg 53.0%

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

      8. fmm-undef 53.0%

         \[\leadsto \color{blue}{1 \cdot x - \frac{y - x}{-2}} \]

      9. *-un-lft-identity 53.0%

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

      10. metadata-eval 53.0%

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

   4. Applied egg-rr 53.0%

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

   5. Taylor expanded in x around 0 44.9%

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

   if 1.12e18 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 82.6%

      \[\leadsto x + \frac{\left|\color{blue}{-1 \cdot x}\right|}{2} \]
   4. Step-by-step derivation

      1. neg-mul-1 82.6%

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

   5. Simplified 82.6%

      \[\leadsto x + \frac{\left|\color{blue}{-x}\right|}{2} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 82.4%

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

      2. associate-/l* 82.4%

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

      3. fabs-neg 82.4%

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

      4. add-sqr-sqrt 82.4%

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

      5. fabs-sqr 82.4%

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

      6. add-sqr-sqrt 82.4%

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

      7. fabs-neg 82.4%

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

      8. add-sqr-sqrt 82.4%

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

      9. fabs-sqr 82.4%

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

      10. add-sqr-sqrt 82.4%

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

   7. Applied egg-rr 82.4%

      \[\leadsto x + \color{blue}{\sqrt{x} \cdot \frac{\sqrt{x}}{2}} \]
   8. Step-by-step derivation

      1. associate-*r/ 82.4%

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

      2. rem-square-sqrt 82.6%

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

   9. Simplified 82.6%

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

   10. Taylor expanded in x around 0 82.6%

       \[\leadsto \color{blue}{1.5 \cdot x} \]
   11. Step-by-step derivation

       1. *-commutative 82.6%

          \[\leadsto \color{blue}{x \cdot 1.5} \]

   12. Simplified 82.6%

       \[\leadsto \color{blue}{x \cdot 1.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-139}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{+18}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.9% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{-272}:\\ \;\;\;\;y \cdot -0.5 + x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.8e-272) (+ (* y -0.5) (* x 1.5)) (* (+ x y) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.8e-272) {
		tmp = (y * -0.5) + (x * 1.5);
	} else {
		tmp = (x + 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 <= (-3.8d-272)) then
        tmp = (y * (-0.5d0)) + (x * 1.5d0)
    else
        tmp = (x + y) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.8e-272) {
		tmp = (y * -0.5) + (x * 1.5);
	} else {
		tmp = (x + y) * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.8e-272:
		tmp = (y * -0.5) + (x * 1.5)
	else:
		tmp = (x + y) * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.8e-272)
		tmp = Float64(Float64(y * -0.5) + Float64(x * 1.5));
	else
		tmp = Float64(Float64(x + y) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.8e-272)
		tmp = (y * -0.5) + (x * 1.5);
	else
		tmp = (x + y) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.8e-272], N[(N[(y * -0.5), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision], N[(N[(x + y), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.7999999999999997e-272

   1. Initial program 100.0%

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

      1. add-cube-cbrt 98.3%

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

      2. fabs-mul 98.3%

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

      3. pow2 98.3%

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

   4. Applied egg-rr 98.3%

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

      1. add-sqr-sqrt 98.2%

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

      2. sqrt-unprod 46.5%

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

      3. frac-2neg 46.5%

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

      4. metadata-eval 46.5%

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

      5. frac-2neg 46.5%

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

      6. metadata-eval 46.5%

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

      7. frac-times 46.5%

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

   6. Applied egg-rr 80.4%

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

   7. Taylor expanded in x around 0 80.4%

      \[\leadsto \color{blue}{-0.5 \cdot y + 1.5 \cdot x} \]

   if -3.7999999999999997e-272 < y

   1. Initial program 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. fma-define 99.9%

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

      3. add-sqr-sqrt 77.8%

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

      4. fabs-sqr 77.8%

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

      5. add-sqr-sqrt 82.3%

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

      6. frac-2neg 82.3%

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

      7. distribute-frac-neg 82.3%

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

      8. fmm-undef 82.3%

         \[\leadsto \color{blue}{1 \cdot x - \frac{y - x}{-2}} \]

      9. *-un-lft-identity 82.3%

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

      10. metadata-eval 82.3%

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

   4. Applied egg-rr 82.3%

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

   5. Taylor expanded in x around 0 82.3%

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

      1. cancel-sign-sub-inv 82.3%

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

      2. metadata-eval 82.3%

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

      3. distribute-lft-in 82.3%

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

      4. +-commutative 82.3%

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

   7. Simplified 82.3%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{-272}:\\ \;\;\;\;y \cdot -0.5 + x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.5% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-126}:\\ \;\;\;\;x + y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6e-126) (+ x (* y -0.5)) (* (+ x y) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6e-126) {
		tmp = x + (y * -0.5);
	} else {
		tmp = (x + 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 <= (-6d-126)) then
        tmp = x + (y * (-0.5d0))
    else
        tmp = (x + y) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -6e-126) {
		tmp = x + (y * -0.5);
	} else {
		tmp = (x + y) * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -6e-126:
		tmp = x + (y * -0.5)
	else:
		tmp = (x + y) * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6e-126)
		tmp = Float64(x + Float64(y * -0.5));
	else
		tmp = Float64(Float64(x + y) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -6e-126)
		tmp = x + (y * -0.5);
	else
		tmp = (x + y) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6e-126], N[(x + N[(y * -0.5), $MachinePrecision]), $MachinePrecision], N[(N[(x + y), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.0000000000000003e-126

   1. Initial program 99.9%

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

      1. add-cube-cbrt 98.1%

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

      2. fabs-mul 98.1%

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

      3. pow2 98.1%

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

   4. Applied egg-rr 98.1%

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

      1. add-sqr-sqrt 98.1%

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

      2. sqrt-unprod 45.1%

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

      3. frac-2neg 45.1%

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

      4. metadata-eval 45.1%

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

      5. frac-2neg 45.1%

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

      6. metadata-eval 45.1%

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

      7. frac-times 45.1%

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

   6. Applied egg-rr 86.5%

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

   7. Taylor expanded in y around inf 73.6%

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

      1. *-commutative 73.6%

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

   9. Simplified 73.6%

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

   if -6.0000000000000003e-126 < y

   1. Initial program 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. fma-define 99.9%

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

      3. add-sqr-sqrt 70.7%

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

      4. fabs-sqr 70.7%

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

      5. add-sqr-sqrt 76.3%

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

      6. frac-2neg 76.3%

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

      7. distribute-frac-neg 76.3%

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

      8. fmm-undef 76.3%

         \[\leadsto \color{blue}{1 \cdot x - \frac{y - x}{-2}} \]

      9. *-un-lft-identity 76.3%

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

      10. metadata-eval 76.3%

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

   4. Applied egg-rr 76.3%

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

   5. Taylor expanded in x around 0 76.3%

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

      1. cancel-sign-sub-inv 76.3%

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

      2. metadata-eval 76.3%

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

      3. distribute-lft-in 76.3%

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

      4. +-commutative 76.3%

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

   7. Simplified 76.3%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-126}:\\ \;\;\;\;x + y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.6% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.12 \cdot 10^{+18}:\\ \;\;\;\;\left(x + y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.12e+18) (* (+ x y) 0.5) (* x 1.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.12e+18) {
		tmp = (x + y) * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.12d+18) then
        tmp = (x + y) * 0.5d0
    else
        tmp = x * 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.12e+18) {
		tmp = (x + y) * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.12e+18:
		tmp = (x + y) * 0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.12e+18)
		tmp = Float64(Float64(x + y) * 0.5);
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.12e+18)
		tmp = (x + y) * 0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.12e+18], N[(N[(x + y), $MachinePrecision] * 0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.12e18

   1. Initial program 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. fma-define 100.0%

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

      3. add-sqr-sqrt 65.8%

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

      4. fabs-sqr 65.8%

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

      5. add-sqr-sqrt 67.6%

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

      6. frac-2neg 67.6%

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

      7. distribute-frac-neg 67.6%

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

      8. fmm-undef 67.6%

         \[\leadsto \color{blue}{1 \cdot x - \frac{y - x}{-2}} \]

      9. *-un-lft-identity 67.6%

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

      10. metadata-eval 67.6%

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

   4. Applied egg-rr 67.6%

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

   5. Taylor expanded in x around 0 67.6%

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

      1. cancel-sign-sub-inv 67.6%

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

      2. metadata-eval 67.6%

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

      3. distribute-lft-in 67.6%

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

      4. +-commutative 67.6%

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

   7. Simplified 67.6%

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

   if 1.12e18 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 82.6%

      \[\leadsto x + \frac{\left|\color{blue}{-1 \cdot x}\right|}{2} \]
   4. Step-by-step derivation

      1. neg-mul-1 82.6%

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

   5. Simplified 82.6%

      \[\leadsto x + \frac{\left|\color{blue}{-x}\right|}{2} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 82.4%

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

      2. associate-/l* 82.4%

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

      3. fabs-neg 82.4%

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

      4. add-sqr-sqrt 82.4%

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

      5. fabs-sqr 82.4%

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

      6. add-sqr-sqrt 82.4%

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

      7. fabs-neg 82.4%

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

      8. add-sqr-sqrt 82.4%

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

      9. fabs-sqr 82.4%

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

      10. add-sqr-sqrt 82.4%

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

   7. Applied egg-rr 82.4%

      \[\leadsto x + \color{blue}{\sqrt{x} \cdot \frac{\sqrt{x}}{2}} \]
   8. Step-by-step derivation

      1. associate-*r/ 82.4%

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

      2. rem-square-sqrt 82.6%

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

   9. Simplified 82.6%

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

   10. Taylor expanded in x around 0 82.6%

       \[\leadsto \color{blue}{1.5 \cdot x} \]
   11. Step-by-step derivation

       1. *-commutative 82.6%

          \[\leadsto \color{blue}{x \cdot 1.5} \]

   12. Simplified 82.6%

       \[\leadsto \color{blue}{x \cdot 1.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.12 \cdot 10^{+18}:\\ \;\;\;\;\left(x + y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 45.7% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 36:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 36.0) (* x 0.5) (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 36.0) {
		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 <= 36.0d0) 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 <= 36.0) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 36.0:
		tmp = x * 0.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 36.0)
		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 <= 36.0)
		tmp = x * 0.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 36.0], N[(x * 0.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 36

   1. Initial program 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. fma-define 99.9%

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

      3. add-sqr-sqrt 36.4%

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

      4. fabs-sqr 36.4%

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

      5. add-sqr-sqrt 42.2%

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

      6. frac-2neg 42.2%

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

      7. distribute-frac-neg 42.2%

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

      8. fmm-undef 42.2%

         \[\leadsto \color{blue}{1 \cdot x - \frac{y - x}{-2}} \]

      9. *-un-lft-identity 42.2%

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

      10. metadata-eval 42.2%

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

   4. Applied egg-rr 42.2%

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

   5. Taylor expanded in x around inf 37.4%

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

   if 36 < y

   1. Initial program 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. fma-define 100.0%

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

      3. add-sqr-sqrt 91.9%

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

      4. fabs-sqr 91.9%

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

      5. add-sqr-sqrt 93.9%

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

      6. frac-2neg 93.9%

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

      7. distribute-frac-neg 93.9%

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

      8. fmm-undef 93.9%

         \[\leadsto \color{blue}{1 \cdot x - \frac{y - x}{-2}} \]

      9. *-un-lft-identity 93.9%

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

      10. metadata-eval 93.9%

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

   4. Applied egg-rr 93.9%

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

   5. Taylor expanded in x around 0 72.8%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 36:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 31.1% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. fma-define 99.9%

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

   3. add-sqr-sqrt 51.2%

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

   4. fabs-sqr 51.2%

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

   5. add-sqr-sqrt 55.9%

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

   6. frac-2neg 55.9%

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

   7. distribute-frac-neg 55.9%

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

   8. fmm-undef 55.9%

      \[\leadsto \color{blue}{1 \cdot x - \frac{y - x}{-2}} \]

   9. *-un-lft-identity 55.9%

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

   10. metadata-eval 55.9%

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

4. Applied egg-rr 55.9%

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

5. Taylor expanded in x around inf 33.4%

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

6. Final simplification 33.4%

   \[\leadsto x \cdot 0.5 \]
7. Add Preprocessing

Alternative 12: 11.4% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around inf 11.4%

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

Reproduce

herbie shell --seed 2024152 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))