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

Percentage Accurate: 99.9% → 99.9%

Time: 13.1s

Alternatives: 11

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: 84.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-180}:\\ \;\;\;\;y \cdot -0.5 + x \cdot 1.5\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-189}:\\ \;\;\;\;x + \frac{\left|x\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.1e-180)
   (+ (* y -0.5) (* x 1.5))
   (if (<= y 6e-189) (+ x (/ (fabs x) 2.0)) (* 0.5 (+ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.1e-180) {
		tmp = (y * -0.5) + (x * 1.5);
	} else if (y <= 6e-189) {
		tmp = x + (fabs(x) / 2.0);
	} 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 <= (-2.1d-180)) then
        tmp = (y * (-0.5d0)) + (x * 1.5d0)
    else if (y <= 6d-189) then
        tmp = x + (abs(x) / 2.0d0)
    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 <= -2.1e-180) {
		tmp = (y * -0.5) + (x * 1.5);
	} else if (y <= 6e-189) {
		tmp = x + (Math.abs(x) / 2.0);
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.1e-180:
		tmp = (y * -0.5) + (x * 1.5)
	elif y <= 6e-189:
		tmp = x + (math.fabs(x) / 2.0)
	else:
		tmp = 0.5 * (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.1e-180)
		tmp = Float64(Float64(y * -0.5) + Float64(x * 1.5));
	elseif (y <= 6e-189)
		tmp = Float64(x + Float64(abs(x) / 2.0));
	else
		tmp = Float64(0.5 * Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.1e-180)
		tmp = (y * -0.5) + (x * 1.5);
	elseif (y <= 6e-189)
		tmp = x + (abs(x) / 2.0);
	else
		tmp = 0.5 * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.1e-180], N[(N[(y * -0.5), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6e-189], N[(x + N[(N[Abs[x], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.0999999999999999e-180

   1. Initial program 99.9%

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

      1. add-cube-cbrt 98.2%

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

      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%

      \[\leadsto x + \frac{\color{blue}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{2} \]
   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 53.1%

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

      3. frac-2neg 53.1%

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

      4. metadata-eval 53.1%

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

      5. frac-2neg 53.1%

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

      6. metadata-eval 53.1%

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

      7. frac-times 53.1%

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

   6. Applied egg-rr 86.2%

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

   7. Taylor expanded in x around 0 86.2%

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

   if -2.0999999999999999e-180 < y < 6e-189

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 92.9%

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

      1. neg-mul-1 92.9%

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

   5. Simplified 92.9%

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

   if 6e-189 < 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 82.5%

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

      4. fabs-sqr 82.5%

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

      5. add-sqr-sqrt 86.0%

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

      6. frac-2neg 86.0%

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

      7. distribute-frac-neg 86.0%

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

      8. fmm-undef 86.0%

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

      9. *-un-lft-identity 86.0%

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

      10. metadata-eval 86.0%

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

   4. Applied egg-rr 86.0%

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

   5. Taylor expanded in x around 0 86.0%

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

      1. cancel-sign-sub-inv 86.0%

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

      2. metadata-eval 86.0%

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

      3. distribute-lft-in 86.0%

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

      4. +-commutative 86.0%

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

   7. Simplified 86.0%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-180}:\\ \;\;\;\;y \cdot -0.5 + x \cdot 1.5\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-189}:\\ \;\;\;\;x + \frac{\left|x\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -75000000000:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 8000000:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1.5 + -0.5 \cdot \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -75000000000.0)
   (* 0.5 (+ x y))
   (if (<= x 8000000.0)
     (+ x (/ (fabs y) 2.0))
     (* x (+ 1.5 (* -0.5 (/ y x)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -75000000000.0) {
		tmp = 0.5 * (x + y);
	} else if (x <= 8000000.0) {
		tmp = x + (fabs(y) / 2.0);
	} else {
		tmp = x * (1.5 + (-0.5 * (y / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-75000000000.0d0)) then
        tmp = 0.5d0 * (x + y)
    else if (x <= 8000000.0d0) then
        tmp = x + (abs(y) / 2.0d0)
    else
        tmp = x * (1.5d0 + ((-0.5d0) * (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -75000000000.0) {
		tmp = 0.5 * (x + y);
	} else if (x <= 8000000.0) {
		tmp = x + (Math.abs(y) / 2.0);
	} else {
		tmp = x * (1.5 + (-0.5 * (y / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -75000000000.0:
		tmp = 0.5 * (x + y)
	elif x <= 8000000.0:
		tmp = x + (math.fabs(y) / 2.0)
	else:
		tmp = x * (1.5 + (-0.5 * (y / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -75000000000.0)
		tmp = Float64(0.5 * Float64(x + y));
	elseif (x <= 8000000.0)
		tmp = Float64(x + Float64(abs(y) / 2.0));
	else
		tmp = Float64(x * Float64(1.5 + Float64(-0.5 * Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -75000000000.0)
		tmp = 0.5 * (x + y);
	elseif (x <= 8000000.0)
		tmp = x + (abs(y) / 2.0);
	else
		tmp = x * (1.5 + (-0.5 * (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -75000000000.0], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8000000.0], N[(x + N[(N[Abs[y], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.5 + N[(-0.5 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.5e10

   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 93.3%

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

      4. fabs-sqr 93.3%

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

      5. add-sqr-sqrt 94.1%

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

      6. frac-2neg 94.1%

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

      7. distribute-frac-neg 94.1%

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

      8. fmm-undef 94.1%

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

      9. *-un-lft-identity 94.1%

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

      10. metadata-eval 94.1%

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

   4. Applied egg-rr 94.1%

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

   5. Taylor expanded in x around 0 94.1%

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

      1. cancel-sign-sub-inv 94.1%

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

      2. metadata-eval 94.1%

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

      3. distribute-lft-in 94.1%

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

      4. +-commutative 94.1%

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

   7. Simplified 94.1%

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

   if -7.5e10 < x < 8e6

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 79.2%

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

   if 8e6 < x

   1. Initial program 99.7%

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

      1. add-cube-cbrt 98.8%

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

      2. fabs-mul 98.8%

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

      3. pow2 98.8%

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

   4. Applied egg-rr 98.8%

      \[\leadsto x + \frac{\color{blue}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{2} \]
   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 29.1%

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

      3. frac-2neg 29.1%

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

      4. metadata-eval 29.1%

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

      5. frac-2neg 29.1%

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

      6. metadata-eval 29.1%

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

      7. frac-times 29.1%

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

   6. Applied egg-rr 92.0%

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

   7. Taylor expanded in x around inf 92.0%

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

      1. *-commutative 92.0%

         \[\leadsto x \cdot \left(1.5 + \color{blue}{\frac{y}{x} \cdot -0.5}\right) \]

   9. Simplified 92.0%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -75000000000:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 8000000:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1.5 + -0.5 \cdot \frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.6% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-90}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-245}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-117}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5e-90)
   (* y -0.5)
   (if (<= y 8.5e-245) (* x 1.5) (if (<= y 5.2e-117) (* x 0.5) (* y 0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5e-90) {
		tmp = y * -0.5;
	} else if (y <= 8.5e-245) {
		tmp = x * 1.5;
	} else if (y <= 5.2e-117) {
		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 <= (-5d-90)) then
        tmp = y * (-0.5d0)
    else if (y <= 8.5d-245) then
        tmp = x * 1.5d0
    else if (y <= 5.2d-117) 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 <= -5e-90) {
		tmp = y * -0.5;
	} else if (y <= 8.5e-245) {
		tmp = x * 1.5;
	} else if (y <= 5.2e-117) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5e-90:
		tmp = y * -0.5
	elif y <= 8.5e-245:
		tmp = x * 1.5
	elif y <= 5.2e-117:
		tmp = x * 0.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5e-90)
		tmp = Float64(y * -0.5);
	elseif (y <= 8.5e-245)
		tmp = Float64(x * 1.5);
	elseif (y <= 5.2e-117)
		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 <= -5e-90)
		tmp = y * -0.5;
	elseif (y <= 8.5e-245)
		tmp = x * 1.5;
	elseif (y <= 5.2e-117)
		tmp = x * 0.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5e-90], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 8.5e-245], N[(x * 1.5), $MachinePrecision], If[LessEqual[y, 5.2e-117], N[(x * 0.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -5.00000000000000019e-90

   1. Initial program 99.9%

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

      1. add-cube-cbrt 98.2%

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

      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%

      \[\leadsto x + \frac{\color{blue}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{2} \]
   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 52.0%

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

      3. frac-2neg 52.0%

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

      4. metadata-eval 52.0%

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

      5. frac-2neg 52.0%

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

      6. metadata-eval 52.0%

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

      7. frac-times 52.0%

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

   6. Applied egg-rr 87.6%

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

   7. Taylor expanded in x around inf 76.0%

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

      1. *-commutative 76.0%

         \[\leadsto x \cdot \left(1.5 + \color{blue}{\frac{y}{x} \cdot -0.5}\right) \]

   9. Simplified 76.0%

      \[\leadsto \color{blue}{x \cdot \left(1.5 + \frac{y}{x} \cdot -0.5\right)} \]

   10. Taylor expanded in x around 0 68.2%

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

       1. *-commutative 68.2%

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

   12. Simplified 68.2%

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

   if -5.00000000000000019e-90 < y < 8.50000000000000022e-245

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 83.8%

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

      1. neg-mul-1 83.8%

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

   5. Simplified 83.8%

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

      1. add-sqr-sqrt 83.4%

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

      2. associate-/l* 83.4%

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

      3. fabs-neg 83.4%

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

      4. add-sqr-sqrt 49.2%

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

      5. fabs-sqr 49.2%

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

      6. add-sqr-sqrt 49.2%

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

      7. fabs-neg 49.2%

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

      8. add-sqr-sqrt 49.2%

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

      9. fabs-sqr 49.2%

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

      10. add-sqr-sqrt 49.2%

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

   7. Applied egg-rr 49.2%

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

      1. associate-*r/ 49.2%

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

      2. rem-square-sqrt 55.9%

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

   9. Simplified 55.9%

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

   10. Taylor expanded in x around 0 55.9%

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

       1. *-commutative 55.9%

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

   12. Simplified 55.9%

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

   if 8.50000000000000022e-245 < y < 5.19999999999999966e-117

   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 67.4%

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

      4. fabs-sqr 67.4%

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

      5. add-sqr-sqrt 73.7%

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

      6. frac-2neg 73.7%

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

      7. distribute-frac-neg 73.7%

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

      8. fmm-undef 73.7%

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

      9. *-un-lft-identity 73.7%

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

      10. metadata-eval 73.7%

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

   4. Applied egg-rr 73.7%

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

   5. Taylor expanded in x around inf 58.4%

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

   if 5.19999999999999966e-117 < 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 83.1%

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

      4. fabs-sqr 83.1%

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

      5. add-sqr-sqrt 86.5%

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

      6. frac-2neg 86.5%

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

      7. distribute-frac-neg 86.5%

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

      8. fmm-undef 86.5%

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

      9. *-un-lft-identity 86.5%

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

      10. metadata-eval 86.5%

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

   4. Applied egg-rr 86.5%

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

   5. Taylor expanded in x around 0 69.5%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-90}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-245}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-117}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.8% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{-182}:\\ \;\;\;\;x + y \cdot -0.5\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-284}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.6e-182)
   (+ x (* y -0.5))
   (if (<= y 8e-284) (* x 1.5) (* 0.5 (+ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.6e-182) {
		tmp = x + (y * -0.5);
	} else if (y <= 8e-284) {
		tmp = x * 1.5;
	} 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 <= (-3.6d-182)) then
        tmp = x + (y * (-0.5d0))
    else if (y <= 8d-284) then
        tmp = x * 1.5d0
    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 <= -3.6e-182) {
		tmp = x + (y * -0.5);
	} else if (y <= 8e-284) {
		tmp = x * 1.5;
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.6e-182:
		tmp = x + (y * -0.5)
	elif y <= 8e-284:
		tmp = x * 1.5
	else:
		tmp = 0.5 * (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.6e-182)
		tmp = Float64(x + Float64(y * -0.5));
	elseif (y <= 8e-284)
		tmp = Float64(x * 1.5);
	else
		tmp = Float64(0.5 * Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.6e-182)
		tmp = x + (y * -0.5);
	elseif (y <= 8e-284)
		tmp = x * 1.5;
	else
		tmp = 0.5 * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.6e-182], N[(x + N[(y * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8e-284], N[(x * 1.5), $MachinePrecision], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.59999999999999977e-182

   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 53.6%

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

      3. frac-2neg 53.6%

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

      4. metadata-eval 53.6%

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

      5. frac-2neg 53.6%

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

      6. metadata-eval 53.6%

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

      7. frac-times 53.6%

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

   6. Applied egg-rr 85.5%

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

   7. Taylor expanded in y around inf 68.8%

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

      1. *-commutative 68.8%

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

   9. Simplified 68.8%

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

   if -3.59999999999999977e-182 < y < 8.00000000000000029e-284

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 92.6%

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

      1. neg-mul-1 92.6%

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

   5. Simplified 92.6%

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

      1. add-sqr-sqrt 92.2%

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

      2. associate-/l* 92.2%

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

      3. fabs-neg 92.2%

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

      4. add-sqr-sqrt 57.5%

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

      5. fabs-sqr 57.5%

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

      6. add-sqr-sqrt 57.5%

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

      7. fabs-neg 57.5%

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

      8. add-sqr-sqrt 57.5%

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

      9. fabs-sqr 57.5%

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

      10. add-sqr-sqrt 57.5%

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

   7. Applied egg-rr 57.5%

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

      1. associate-*r/ 57.5%

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

      2. rem-square-sqrt 64.1%

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

   9. Simplified 64.1%

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

   10. Taylor expanded in x around 0 64.1%

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

       1. *-commutative 64.1%

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

   12. Simplified 64.1%

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

   if 8.00000000000000029e-284 < 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 76.1%

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

      4. fabs-sqr 76.1%

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

      5. add-sqr-sqrt 80.8%

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

      6. frac-2neg 80.8%

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

      7. distribute-frac-neg 80.8%

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

      8. fmm-undef 80.8%

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

      9. *-un-lft-identity 80.8%

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

      10. metadata-eval 80.8%

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

   4. Applied egg-rr 80.8%

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

   5. Taylor expanded in x around 0 80.8%

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

      1. cancel-sign-sub-inv 80.8%

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

      2. metadata-eval 80.8%

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

      3. distribute-lft-in 80.8%

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

      4. +-commutative 80.8%

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

   7. Simplified 80.8%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{-182}:\\ \;\;\;\;x + y \cdot -0.5\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-284}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.2% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-164}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 8000000:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.5e-164) (* x 0.5) (if (<= x 8000000.0) (* y 0.5) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.5e-164) {
		tmp = x * 0.5;
	} else if (x <= 8000000.0) {
		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 <= (-5.5d-164)) then
        tmp = x * 0.5d0
    else if (x <= 8000000.0d0) 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 <= -5.5e-164) {
		tmp = x * 0.5;
	} else if (x <= 8000000.0) {
		tmp = y * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.5e-164:
		tmp = x * 0.5
	elif x <= 8000000.0:
		tmp = y * 0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.5e-164)
		tmp = Float64(x * 0.5);
	elseif (x <= 8000000.0)
		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 <= -5.5e-164)
		tmp = x * 0.5;
	elseif (x <= 8000000.0)
		tmp = y * 0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.5e-164], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, 8000000.0], N[(y * 0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.50000000000000027e-164

   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 81.3%

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

      4. fabs-sqr 81.3%

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

      5. add-sqr-sqrt 82.1%

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

      6. frac-2neg 82.1%

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

      7. distribute-frac-neg 82.1%

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

      8. fmm-undef 82.1%

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

      9. *-un-lft-identity 82.1%

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

      10. metadata-eval 82.1%

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

   4. Applied egg-rr 82.1%

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

   5. Taylor expanded in x around inf 63.0%

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

   if -5.50000000000000027e-164 < x < 8e6

   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 48.3%

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

      4. fabs-sqr 48.3%

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

      5. add-sqr-sqrt 52.0%

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

      6. frac-2neg 52.0%

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

      7. distribute-frac-neg 52.0%

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

      8. fmm-undef 52.0%

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

      9. *-un-lft-identity 52.0%

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

      10. metadata-eval 52.0%

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

   4. Applied egg-rr 52.0%

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

   5. Taylor expanded in x around 0 47.1%

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

   if 8e6 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 76.5%

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

      1. neg-mul-1 76.5%

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

   5. Simplified 76.5%

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

      1. add-sqr-sqrt 76.4%

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

      2. associate-/l* 76.4%

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

      3. fabs-neg 76.4%

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

      4. add-sqr-sqrt 76.4%

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

      5. fabs-sqr 76.4%

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

      6. add-sqr-sqrt 76.4%

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

      7. fabs-neg 76.4%

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

      8. add-sqr-sqrt 76.4%

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

      9. fabs-sqr 76.4%

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

      10. add-sqr-sqrt 76.4%

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

   7. Applied egg-rr 76.4%

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

      1. associate-*r/ 76.4%

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

      2. rem-square-sqrt 76.5%

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

   9. Simplified 76.5%

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

   10. Taylor expanded in x around 0 76.5%

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

       1. *-commutative 76.5%

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

   12. Simplified 76.5%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-164}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 8000000:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.4% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.12 \cdot 10^{-282}:\\ \;\;\;\;y \cdot -0.5 + x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.12e-282) (+ (* y -0.5) (* x 1.5)) (* 0.5 (+ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.12e-282) {
		tmp = (y * -0.5) + (x * 1.5);
	} 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 <= 1.12d-282) then
        tmp = (y * (-0.5d0)) + (x * 1.5d0)
    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 <= 1.12e-282) {
		tmp = (y * -0.5) + (x * 1.5);
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.12e-282:
		tmp = (y * -0.5) + (x * 1.5)
	else:
		tmp = 0.5 * (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.12e-282)
		tmp = Float64(Float64(y * -0.5) + Float64(x * 1.5));
	else
		tmp = Float64(0.5 * Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.12e-282)
		tmp = (y * -0.5) + (x * 1.5);
	else
		tmp = 0.5 * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.12e-282], N[(N[(y * -0.5), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.11999999999999992e-282

   1. Initial program 99.9%

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

      1. add-cube-cbrt 98.2%

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

      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%

      \[\leadsto x + \frac{\color{blue}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{2} \]
   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 52.1%

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

      3. frac-2neg 52.1%

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

      4. metadata-eval 52.1%

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

      5. frac-2neg 52.1%

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

      6. metadata-eval 52.1%

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

      7. frac-times 52.1%

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

   6. Applied egg-rr 82.5%

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

   7. Taylor expanded in x around 0 82.5%

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

   if 1.11999999999999992e-282 < 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 76.1%

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

      4. fabs-sqr 76.1%

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

      5. add-sqr-sqrt 80.8%

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

      6. frac-2neg 80.8%

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

      7. distribute-frac-neg 80.8%

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

      8. fmm-undef 80.8%

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

      9. *-un-lft-identity 80.8%

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

      10. metadata-eval 80.8%

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

   4. Applied egg-rr 80.8%

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

   5. Taylor expanded in x around 0 80.8%

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

      1. cancel-sign-sub-inv 80.8%

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

      2. metadata-eval 80.8%

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

      3. distribute-lft-in 80.8%

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

      4. +-commutative 80.8%

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

   7. Simplified 80.8%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.12 \cdot 10^{-282}:\\ \;\;\;\;y \cdot -0.5 + x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.1% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{-75}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8.8e-75) (* y -0.5) (* 0.5 (+ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8.8e-75) {
		tmp = y * -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 <= (-8.8d-75)) then
        tmp = y * (-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 <= -8.8e-75) {
		tmp = y * -0.5;
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8.8e-75:
		tmp = y * -0.5
	else:
		tmp = 0.5 * (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8.8e-75)
		tmp = Float64(y * -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 <= -8.8e-75)
		tmp = y * -0.5;
	else
		tmp = 0.5 * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8.8e-75], N[(y * -0.5), $MachinePrecision], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.80000000000000022e-75

   1. Initial program 99.9%

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

      1. add-cube-cbrt 98.2%

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

      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%

      \[\leadsto x + \frac{\color{blue}{\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right| \cdot \left|\sqrt[3]{y - x}\right|}}{2} \]
   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 50.8%

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

      3. frac-2neg 50.8%

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

      4. metadata-eval 50.8%

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

      5. frac-2neg 50.8%

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

      6. metadata-eval 50.8%

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

      7. frac-times 50.8%

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

   6. Applied egg-rr 90.2%

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

   7. Taylor expanded in x around inf 77.9%

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

      1. *-commutative 77.9%

         \[\leadsto x \cdot \left(1.5 + \color{blue}{\frac{y}{x} \cdot -0.5}\right) \]

   9. Simplified 77.9%

      \[\leadsto \color{blue}{x \cdot \left(1.5 + \frac{y}{x} \cdot -0.5\right)} \]

   10. Taylor expanded in x around 0 70.8%

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

       1. *-commutative 70.8%

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

   12. Simplified 70.8%

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

   if -8.80000000000000022e-75 < 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 65.6%

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

      4. fabs-sqr 65.6%

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

      5. add-sqr-sqrt 71.4%

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

      6. frac-2neg 71.4%

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

      7. distribute-frac-neg 71.4%

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

      8. fmm-undef 71.4%

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

      9. *-un-lft-identity 71.4%

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

      10. metadata-eval 71.4%

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

   4. Applied egg-rr 71.4%

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

   5. Taylor expanded in x around 0 71.4%

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

      1. cancel-sign-sub-inv 71.4%

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

      2. metadata-eval 71.4%

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

      3. distribute-lft-in 71.4%

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

      4. +-commutative 71.4%

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

   7. Simplified 71.4%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{-75}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 45.3% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-117}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 8e-117) (* x 0.5) (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8e-117) {
		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 <= 8d-117) 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 <= 8e-117) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8e-117:
		tmp = x * 0.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8e-117)
		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 <= 8e-117)
		tmp = x * 0.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8e-117], N[(x * 0.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.00000000000000024e-117

   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 30.8%

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

      4. fabs-sqr 30.8%

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

      5. add-sqr-sqrt 37.0%

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

      6. frac-2neg 37.0%

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

      7. distribute-frac-neg 37.0%

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

      8. fmm-undef 37.0%

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

      9. *-un-lft-identity 37.0%

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

      10. metadata-eval 37.0%

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

   4. Applied egg-rr 37.0%

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

   5. Taylor expanded in x around inf 34.3%

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

   if 8.00000000000000024e-117 < 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 83.1%

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

      4. fabs-sqr 83.1%

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

      5. add-sqr-sqrt 86.5%

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

      6. frac-2neg 86.5%

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

      7. distribute-frac-neg 86.5%

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

      8. fmm-undef 86.5%

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

      9. *-un-lft-identity 86.5%

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

      10. metadata-eval 86.5%

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

   4. Applied egg-rr 86.5%

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

   5. Taylor expanded in x around 0 69.5%

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

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-117}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 30.3% 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 49.6%

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

   4. fabs-sqr 49.6%

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

   5. add-sqr-sqrt 54.8%

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

   6. frac-2neg 54.8%

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

   7. distribute-frac-neg 54.8%

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

   8. fmm-undef 54.8%

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

   9. *-un-lft-identity 54.8%

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

   10. metadata-eval 54.8%

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

4. Applied egg-rr 54.8%

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

5. Taylor expanded in x around inf 29.0%

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

6. Final simplification 29.0%

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

Alternative 11: 11.4% accurate, 107.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around inf 11.3%

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

Reproduce

herbie shell --seed 2024163 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))