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

Percentage Accurate: 99.9% → 99.9%

Time: 12.6s

Alternatives: 10

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: 81.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{-94}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-35}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+88}:\\ \;\;\;\;x + \frac{y \cdot \left(\frac{x}{y} + 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left|x\right|}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7.4e-94)
   (* 0.5 (+ x y))
   (if (<= x 2.7e-35)
     (+ x (/ (fabs y) 2.0))
     (if (<= x 1.25e+88)
       (+ x (/ (* y (+ (/ x y) 1.0)) 2.0))
       (+ x (/ (fabs x) 2.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7.4e-94) {
		tmp = 0.5 * (x + y);
	} else if (x <= 2.7e-35) {
		tmp = x + (fabs(y) / 2.0);
	} else if (x <= 1.25e+88) {
		tmp = x + ((y * ((x / y) + 1.0)) / 2.0);
	} else {
		tmp = x + (fabs(x) / 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-7.4d-94)) then
        tmp = 0.5d0 * (x + y)
    else if (x <= 2.7d-35) then
        tmp = x + (abs(y) / 2.0d0)
    else if (x <= 1.25d+88) then
        tmp = x + ((y * ((x / y) + 1.0d0)) / 2.0d0)
    else
        tmp = x + (abs(x) / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -7.4e-94) {
		tmp = 0.5 * (x + y);
	} else if (x <= 2.7e-35) {
		tmp = x + (Math.abs(y) / 2.0);
	} else if (x <= 1.25e+88) {
		tmp = x + ((y * ((x / y) + 1.0)) / 2.0);
	} else {
		tmp = x + (Math.abs(x) / 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7.4e-94:
		tmp = 0.5 * (x + y)
	elif x <= 2.7e-35:
		tmp = x + (math.fabs(y) / 2.0)
	elif x <= 1.25e+88:
		tmp = x + ((y * ((x / y) + 1.0)) / 2.0)
	else:
		tmp = x + (math.fabs(x) / 2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7.4e-94)
		tmp = Float64(0.5 * Float64(x + y));
	elseif (x <= 2.7e-35)
		tmp = Float64(x + Float64(abs(y) / 2.0));
	elseif (x <= 1.25e+88)
		tmp = Float64(x + Float64(Float64(y * Float64(Float64(x / y) + 1.0)) / 2.0));
	else
		tmp = Float64(x + Float64(abs(x) / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7.4e-94)
		tmp = 0.5 * (x + y);
	elseif (x <= 2.7e-35)
		tmp = x + (abs(y) / 2.0);
	elseif (x <= 1.25e+88)
		tmp = x + ((y * ((x / y) + 1.0)) / 2.0);
	else
		tmp = x + (abs(x) / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7.4e-94], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.7e-35], N[(x + N[(N[Abs[y], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.25e+88], N[(x + N[(N[(y * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Abs[x], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.3999999999999996e-94

   1. Initial program 100.0%

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

   3. Taylor expanded in y around -inf 100.0%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-sub 100.0%

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

      5. rem-square-sqrt 94.7%

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

      6. fabs-sqr 94.7%

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

      7. rem-square-sqrt 95.7%

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

   5. Simplified 95.7%

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

   6. Taylor expanded in x around 0 95.7%

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

      1. distribute-lft-out 95.7%

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

   8. Simplified 95.7%

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

   if -7.3999999999999996e-94 < x < 2.6999999999999997e-35

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 82.3%

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

   if 2.6999999999999997e-35 < x < 1.24999999999999999e88

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

   5. Simplified 99.8%

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

      1. add-sqr-sqrt 23.0%

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

      2. fabs-sqr 23.0%

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

      3. add-sqr-sqrt 34.1%

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

      4. *-commutative 34.1%

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

      5. div-inv 34.1%

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

      6. cancel-sign-sub-inv 34.1%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 83.4%

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

      9. sqr-neg 83.4%

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

      10. sqrt-unprod 83.3%

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

      11. add-sqr-sqrt 83.4%

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

      12. div-inv 83.4%

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

   7. Applied egg-rr 83.4%

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

   if 1.24999999999999999e88 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 86.0%

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

      1. neg-mul-1 86.0%

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

   5. Simplified 86.0%

      \[\leadsto x + \frac{\left|\color{blue}{-x}\right|}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{-94}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-35}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+88}:\\ \;\;\;\;x + \frac{y \cdot \left(\frac{x}{y} + 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left|x\right|}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-94}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-36}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+85}:\\ \;\;\;\;x + \frac{y \cdot \left(\frac{x}{y} + 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.1e-94)
   (* 0.5 (+ x y))
   (if (<= x 6.2e-36)
     (+ x (/ (fabs y) 2.0))
     (if (<= x 1.55e+85) (+ x (/ (* y (+ (/ x y) 1.0)) 2.0)) (* x 1.5)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.1e-94) {
		tmp = 0.5 * (x + y);
	} else if (x <= 6.2e-36) {
		tmp = x + (fabs(y) / 2.0);
	} else if (x <= 1.55e+85) {
		tmp = x + ((y * ((x / y) + 1.0)) / 2.0);
	} 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 <= (-1.1d-94)) then
        tmp = 0.5d0 * (x + y)
    else if (x <= 6.2d-36) then
        tmp = x + (abs(y) / 2.0d0)
    else if (x <= 1.55d+85) then
        tmp = x + ((y * ((x / y) + 1.0d0)) / 2.0d0)
    else
        tmp = x * 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.1e-94) {
		tmp = 0.5 * (x + y);
	} else if (x <= 6.2e-36) {
		tmp = x + (Math.abs(y) / 2.0);
	} else if (x <= 1.55e+85) {
		tmp = x + ((y * ((x / y) + 1.0)) / 2.0);
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.1e-94:
		tmp = 0.5 * (x + y)
	elif x <= 6.2e-36:
		tmp = x + (math.fabs(y) / 2.0)
	elif x <= 1.55e+85:
		tmp = x + ((y * ((x / y) + 1.0)) / 2.0)
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.1e-94)
		tmp = Float64(0.5 * Float64(x + y));
	elseif (x <= 6.2e-36)
		tmp = Float64(x + Float64(abs(y) / 2.0));
	elseif (x <= 1.55e+85)
		tmp = Float64(x + Float64(Float64(y * Float64(Float64(x / y) + 1.0)) / 2.0));
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.1e-94)
		tmp = 0.5 * (x + y);
	elseif (x <= 6.2e-36)
		tmp = x + (abs(y) / 2.0);
	elseif (x <= 1.55e+85)
		tmp = x + ((y * ((x / y) + 1.0)) / 2.0);
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.1e-94], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.2e-36], N[(x + N[(N[Abs[y], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.55e+85], N[(x + N[(N[(y * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.10000000000000001e-94

   1. Initial program 100.0%

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

   3. Taylor expanded in y around -inf 100.0%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-sub 100.0%

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

      5. rem-square-sqrt 94.7%

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

      6. fabs-sqr 94.7%

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

      7. rem-square-sqrt 95.7%

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

   5. Simplified 95.7%

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

   6. Taylor expanded in x around 0 95.7%

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

      1. distribute-lft-out 95.7%

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

   8. Simplified 95.7%

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

   if -1.10000000000000001e-94 < x < 6.1999999999999997e-36

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 82.3%

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

   if 6.1999999999999997e-36 < x < 1.55000000000000006e85

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

   5. Simplified 99.8%

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

      1. add-sqr-sqrt 23.0%

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

      2. fabs-sqr 23.0%

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

      3. add-sqr-sqrt 34.1%

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

      4. *-commutative 34.1%

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

      5. div-inv 34.1%

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

      6. cancel-sign-sub-inv 34.1%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 83.4%

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

      9. sqr-neg 83.4%

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

      10. sqrt-unprod 83.3%

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

      11. add-sqr-sqrt 83.4%

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

      12. div-inv 83.4%

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

   7. Applied egg-rr 83.4%

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

   if 1.55000000000000006e85 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 86.0%

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

      1. neg-mul-1 86.0%

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

   5. Simplified 86.0%

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

      1. fabs-neg 86.0%

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

      2. add-sqr-sqrt 85.8%

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

      3. fabs-sqr 85.8%

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

      4. add-sqr-sqrt 86.0%

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

      5. add-cube-cbrt 85.3%

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

      6. associate-/l* 85.3%

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

      7. pow2 85.3%

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

   7. Applied egg-rr 85.3%

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

      1. associate-*r/ 85.3%

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

      2. unpow2 85.3%

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

      3. rem-3cbrt-lft 86.0%

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

   9. Simplified 86.0%

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

   10. Taylor expanded in x around 0 86.0%

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

       1. *-commutative 86.0%

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

   12. Simplified 86.0%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-94}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-36}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+85}:\\ \;\;\;\;x + \frac{y \cdot \left(\frac{x}{y} + 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.4% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-281}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+86}:\\ \;\;\;\;x + \frac{y \cdot \left(\frac{x}{y} + 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5e-281)
   (* 0.5 (+ x y))
   (if (<= x 6.5e+86) (+ x (/ (* y (+ (/ x y) 1.0)) 2.0)) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5e-281) {
		tmp = 0.5 * (x + y);
	} else if (x <= 6.5e+86) {
		tmp = x + ((y * ((x / y) + 1.0)) / 2.0);
	} 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 <= (-5d-281)) then
        tmp = 0.5d0 * (x + y)
    else if (x <= 6.5d+86) then
        tmp = x + ((y * ((x / y) + 1.0d0)) / 2.0d0)
    else
        tmp = x * 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5e-281) {
		tmp = 0.5 * (x + y);
	} else if (x <= 6.5e+86) {
		tmp = x + ((y * ((x / y) + 1.0)) / 2.0);
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5e-281:
		tmp = 0.5 * (x + y)
	elif x <= 6.5e+86:
		tmp = x + ((y * ((x / y) + 1.0)) / 2.0)
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5e-281)
		tmp = Float64(0.5 * Float64(x + y));
	elseif (x <= 6.5e+86)
		tmp = Float64(x + Float64(Float64(y * Float64(Float64(x / y) + 1.0)) / 2.0));
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5e-281)
		tmp = 0.5 * (x + y);
	elseif (x <= 6.5e+86)
		tmp = x + ((y * ((x / y) + 1.0)) / 2.0);
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5e-281], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.5e+86], N[(x + N[(N[(y * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.9999999999999998e-281

   1. Initial program 100.0%

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

   3. Taylor expanded in y around -inf 100.0%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-sub 100.0%

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

      5. rem-square-sqrt 77.0%

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

      6. fabs-sqr 77.0%

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

      7. rem-square-sqrt 78.0%

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

   5. Simplified 78.0%

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

   6. Taylor expanded in x around 0 78.0%

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

      1. distribute-lft-out 78.0%

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

   8. Simplified 78.0%

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

   if -4.9999999999999998e-281 < x < 6.49999999999999996e86

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. unsub-neg 99.9%

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

   5. Simplified 99.9%

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

      1. add-sqr-sqrt 33.4%

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

      2. fabs-sqr 33.4%

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

      3. add-sqr-sqrt 40.1%

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

      4. *-commutative 40.1%

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

      5. div-inv 40.1%

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

      6. cancel-sign-sub-inv 40.1%

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

      7. add-sqr-sqrt 2.4%

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

      8. sqrt-unprod 62.3%

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

      9. sqr-neg 62.3%

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

      10. sqrt-unprod 63.9%

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

      11. add-sqr-sqrt 66.3%

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

      12. div-inv 66.3%

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

   7. Applied egg-rr 66.3%

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

   if 6.49999999999999996e86 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 86.0%

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

      1. neg-mul-1 86.0%

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

   5. Simplified 86.0%

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

      1. fabs-neg 86.0%

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

      2. add-sqr-sqrt 85.8%

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

      3. fabs-sqr 85.8%

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

      4. add-sqr-sqrt 86.0%

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

      5. add-cube-cbrt 85.3%

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

      6. associate-/l* 85.3%

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

      7. pow2 85.3%

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

   7. Applied egg-rr 85.3%

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

      1. associate-*r/ 85.3%

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

      2. unpow2 85.3%

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

      3. rem-3cbrt-lft 86.0%

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

   9. Simplified 86.0%

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

   10. Taylor expanded in x around 0 86.0%

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

       1. *-commutative 86.0%

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

   12. Simplified 86.0%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-281}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+86}:\\ \;\;\;\;x + \frac{y \cdot \left(\frac{x}{y} + 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.4% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-283}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+85}:\\ \;\;\;\;y \cdot \left(0.5 + \frac{x}{y} \cdot 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2e-283)
   (* 0.5 (+ x y))
   (if (<= x 2e+85) (* y (+ 0.5 (* (/ x y) 1.5))) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2e-283) {
		tmp = 0.5 * (x + y);
	} else if (x <= 2e+85) {
		tmp = y * (0.5 + ((x / y) * 1.5));
	} 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 <= (-2d-283)) then
        tmp = 0.5d0 * (x + y)
    else if (x <= 2d+85) then
        tmp = y * (0.5d0 + ((x / y) * 1.5d0))
    else
        tmp = x * 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2e-283) {
		tmp = 0.5 * (x + y);
	} else if (x <= 2e+85) {
		tmp = y * (0.5 + ((x / y) * 1.5));
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2e-283:
		tmp = 0.5 * (x + y)
	elif x <= 2e+85:
		tmp = y * (0.5 + ((x / y) * 1.5))
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2e-283)
		tmp = Float64(0.5 * Float64(x + y));
	elseif (x <= 2e+85)
		tmp = Float64(y * Float64(0.5 + Float64(Float64(x / y) * 1.5)));
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2e-283)
		tmp = 0.5 * (x + y);
	elseif (x <= 2e+85)
		tmp = y * (0.5 + ((x / y) * 1.5));
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2e-283], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2e+85], N[(y * N[(0.5 + N[(N[(x / y), $MachinePrecision] * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.99999999999999989e-283

   1. Initial program 100.0%

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

   3. Taylor expanded in y around -inf 100.0%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-sub 100.0%

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

      5. rem-square-sqrt 77.0%

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

      6. fabs-sqr 77.0%

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

      7. rem-square-sqrt 78.0%

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

   5. Simplified 78.0%

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

   6. Taylor expanded in x around 0 78.0%

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

      1. distribute-lft-out 78.0%

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

   8. Simplified 78.0%

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

   if -1.99999999999999989e-283 < x < 2e85

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. unsub-neg 99.9%

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

   5. Simplified 99.9%

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

      1. clear-num 99.5%

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

      2. inv-pow 99.5%

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

      3. add-sqr-sqrt 33.4%

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

      4. fabs-sqr 33.4%

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

      5. add-sqr-sqrt 39.9%

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

      6. associate-/r* 39.9%

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

      7. div-inv 39.9%

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

      8. cancel-sign-sub-inv 39.9%

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

      9. add-sqr-sqrt 2.4%

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

      10. sqrt-unprod 62.1%

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

      11. sqr-neg 62.1%

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

      12. sqrt-unprod 63.7%

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

      13. add-sqr-sqrt 66.1%

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

      14. div-inv 66.1%

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

   7. Applied egg-rr 66.1%

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

      1. unpow-1 66.1%

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

   9. Simplified 66.1%

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

   10. Taylor expanded in y around inf 66.1%

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

       1. distribute-lft1-in 66.1%

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

       2. metadata-eval 66.1%

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

   12. Simplified 66.1%

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

   if 2e85 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 86.0%

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

      1. neg-mul-1 86.0%

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

   5. Simplified 86.0%

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

      1. fabs-neg 86.0%

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

      2. add-sqr-sqrt 85.8%

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

      3. fabs-sqr 85.8%

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

      4. add-sqr-sqrt 86.0%

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

      5. add-cube-cbrt 85.3%

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

      6. associate-/l* 85.3%

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

      7. pow2 85.3%

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

   7. Applied egg-rr 85.3%

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

      1. associate-*r/ 85.3%

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

      2. unpow2 85.3%

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

      3. rem-3cbrt-lft 86.0%

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

   9. Simplified 86.0%

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

   10. Taylor expanded in x around 0 86.0%

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

       1. *-commutative 86.0%

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

   12. Simplified 86.0%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-283}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+85}:\\ \;\;\;\;y \cdot \left(0.5 + \frac{x}{y} \cdot 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.8% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-51}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-184}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.55e-51) (* x 0.5) (if (<= x 8.2e-184) (* y 0.5) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.55e-51) {
		tmp = x * 0.5;
	} else if (x <= 8.2e-184) {
		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 <= (-1.55d-51)) then
        tmp = x * 0.5d0
    else if (x <= 8.2d-184) 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 <= -1.55e-51) {
		tmp = x * 0.5;
	} else if (x <= 8.2e-184) {
		tmp = y * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.55e-51:
		tmp = x * 0.5
	elif x <= 8.2e-184:
		tmp = y * 0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.55e-51)
		tmp = Float64(x * 0.5);
	elseif (x <= 8.2e-184)
		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 <= -1.55e-51)
		tmp = x * 0.5;
	elseif (x <= 8.2e-184)
		tmp = y * 0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.55e-51], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, 8.2e-184], N[(y * 0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5499999999999999e-51

   1. Initial program 100.0%

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

   3. Taylor expanded in y around -inf 100.0%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-sub 100.0%

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

      5. rem-square-sqrt 94.2%

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

      6. fabs-sqr 94.2%

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

      7. rem-square-sqrt 95.2%

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

   5. Simplified 95.2%

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

   6. Taylor expanded in x around inf 75.4%

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

      1. *-commutative 75.4%

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

   8. Simplified 75.4%

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

   if -1.5499999999999999e-51 < x < 8.2e-184

   1. Initial program 100.0%

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

   3. Taylor expanded in y around -inf 100.0%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-sub 100.0%

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

      5. rem-square-sqrt 55.9%

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

      6. fabs-sqr 55.9%

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

      7. rem-square-sqrt 57.4%

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

   5. Simplified 57.4%

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

   6. Taylor expanded in x around 0 41.8%

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

      1. *-commutative 41.8%

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

   8. Simplified 41.8%

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

   if 8.2e-184 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 63.9%

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

      1. neg-mul-1 63.9%

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

   5. Simplified 63.9%

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

      1. fabs-neg 63.9%

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

      2. add-sqr-sqrt 63.8%

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

      3. fabs-sqr 63.8%

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

      4. add-sqr-sqrt 63.9%

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

      5. add-cube-cbrt 63.4%

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

      6. associate-/l* 63.4%

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

      7. pow2 63.4%

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

   7. Applied egg-rr 63.4%

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

      1. associate-*r/ 63.4%

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

      2. unpow2 63.4%

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

      3. rem-3cbrt-lft 63.9%

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

   9. Simplified 63.9%

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

   10. Taylor expanded in x around 0 63.9%

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

       1. *-commutative 63.9%

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

   12. Simplified 63.9%

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

Alternative 7: 67.4% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{-45}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.1e-45) (* 0.5 (+ x y)) (* x 1.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.1e-45) {
		tmp = 0.5 * (x + y);
	} 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 <= 2.1d-45) then
        tmp = 0.5d0 * (x + y)
    else
        tmp = x * 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2.1e-45) {
		tmp = 0.5 * (x + y);
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.1e-45:
		tmp = 0.5 * (x + y)
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.1e-45)
		tmp = Float64(0.5 * Float64(x + y));
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.1e-45)
		tmp = 0.5 * (x + y);
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.1e-45], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.09999999999999995e-45

   1. Initial program 100.0%

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

   3. Taylor expanded in y around -inf 100.0%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-sub 100.0%

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

      5. rem-square-sqrt 64.5%

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

      6. fabs-sqr 64.5%

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

      7. rem-square-sqrt 66.8%

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

   5. Simplified 66.8%

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

   6. Taylor expanded in x around 0 66.8%

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

      1. distribute-lft-out 66.8%

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

   8. Simplified 66.8%

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

   if 2.09999999999999995e-45 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 76.4%

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

      1. neg-mul-1 76.4%

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

   5. Simplified 76.4%

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

      1. fabs-neg 76.4%

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

      2. add-sqr-sqrt 76.3%

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

      3. fabs-sqr 76.3%

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

      4. add-sqr-sqrt 76.4%

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

      5. add-cube-cbrt 75.9%

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

      6. associate-/l* 75.9%

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

      7. pow2 75.9%

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

   7. Applied egg-rr 75.9%

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

      1. associate-*r/ 75.9%

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

      2. unpow2 75.9%

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

      3. rem-3cbrt-lft 76.4%

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

   9. Simplified 76.4%

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

   10. Taylor expanded in x around 0 76.4%

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

       1. *-commutative 76.4%

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

   12. Simplified 76.4%

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

Alternative 8: 51.2% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -1e-309) (* x 0.5) (* x 1.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1e-309) {
		tmp = x * 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-309)) then
        tmp = x * 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-309) {
		tmp = x * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1e-309:
		tmp = x * 0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1e-309)
		tmp = Float64(x * 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-309)
		tmp = x * 0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1e-309], N[(x * 0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.000000000000002e-309

   1. Initial program 100.0%

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

   3. Taylor expanded in y around -inf 100.0%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-sub 100.0%

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

      5. rem-square-sqrt 74.9%

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

      6. fabs-sqr 74.9%

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

      7. rem-square-sqrt 75.9%

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

   5. Simplified 75.9%

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

   6. Taylor expanded in x around inf 48.0%

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

      1. *-commutative 48.0%

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

   8. Simplified 48.0%

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

   if -1.000000000000002e-309 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 57.3%

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

      1. neg-mul-1 57.3%

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

   5. Simplified 57.3%

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

      1. fabs-neg 57.3%

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

      2. add-sqr-sqrt 57.3%

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

      3. fabs-sqr 57.3%

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

      4. add-sqr-sqrt 57.3%

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

      5. add-cube-cbrt 56.9%

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

      6. associate-/l* 56.9%

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

      7. pow2 56.9%

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

   7. Applied egg-rr 56.9%

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

      1. associate-*r/ 56.9%

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

      2. unpow2 56.9%

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

      3. rem-3cbrt-lft 57.3%

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

   9. Simplified 57.3%

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

   10. Taylor expanded in x around 0 57.3%

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

       1. *-commutative 57.3%

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

   12. Simplified 57.3%

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

Alternative 9: 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. Taylor expanded in y around -inf 99.9%

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

   1. fabs-neg 99.9%

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

   2. mul-1-neg 99.9%

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

   3. sub-neg 99.9%

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

   4. fabs-sub 99.9%

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

   5. rem-square-sqrt 47.6%

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

   6. fabs-sqr 47.6%

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

   7. rem-square-sqrt 53.5%

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

5. Simplified 53.5%

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

6. Taylor expanded in x around inf 29.5%

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

   1. *-commutative 29.5%

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

8. Simplified 29.5%

   \[\leadsto \color{blue}{x \cdot 0.5} \]
9. Add Preprocessing

Alternative 10: 11.5% 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 11.9%

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

Reproduce

herbie shell --seed 2024110 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))