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

Percentage Accurate: 99.9% → 99.9%

Time: 9.4s

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: 80.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+41}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-150}:\\ \;\;\;\;x + \frac{\left|x\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5.8e+41)
   (+ x (/ (fabs y) 2.0))
   (if (<= y 6e-150) (+ x (/ (fabs x) 2.0)) (* 0.5 (+ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5.8e+41) {
		tmp = x + (fabs(y) / 2.0);
	} else if (y <= 6e-150) {
		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 <= (-5.8d+41)) then
        tmp = x + (abs(y) / 2.0d0)
    else if (y <= 6d-150) 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 <= -5.8e+41) {
		tmp = x + (Math.abs(y) / 2.0);
	} else if (y <= 6e-150) {
		tmp = x + (Math.abs(x) / 2.0);
	} else {
		tmp = 0.5 * (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5.8e+41:
		tmp = x + (math.fabs(y) / 2.0)
	elif y <= 6e-150:
		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 <= -5.8e+41)
		tmp = Float64(x + Float64(abs(y) / 2.0));
	elseif (y <= 6e-150)
		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 <= -5.8e+41)
		tmp = x + (abs(y) / 2.0);
	elseif (y <= 6e-150)
		tmp = x + (abs(x) / 2.0);
	else
		tmp = 0.5 * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5.8e+41], N[(x + N[(N[Abs[y], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6e-150], 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 < -5.79999999999999977e41

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 83.1%

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

   if -5.79999999999999977e41 < y < 6.0000000000000003e-150

   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%

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

   if 6.0000000000000003e-150 < y

   1. Initial program 99.9%

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

      1. clear-num 99.6%

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

      2. inv-pow 99.6%

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

      3. add-sqr-sqrt 79.4%

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

      4. fabs-sqr 79.4%

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

      5. add-sqr-sqrt 83.4%

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

   4. Applied egg-rr 83.4%

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

   5. Taylor expanded in x around 0 83.7%

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

      1. distribute-lft-out 83.7%

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

      2. +-commutative 83.7%

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

   7. Simplified 83.7%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+41}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-150}:\\ \;\;\;\;x + \frac{\left|x\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -62000000000000:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-174}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+37}:\\ \;\;\;\;y \cdot \left(0.5 + \frac{x \cdot 1.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -62000000000000.0)
   (* 0.5 (+ x y))
   (if (<= x 5.5e-174)
     (+ x (/ (fabs y) 2.0))
     (if (<= x 2.05e+37) (* y (+ 0.5 (/ (* x 1.5) y))) (* x 1.5)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -62000000000000.0) {
		tmp = 0.5 * (x + y);
	} else if (x <= 5.5e-174) {
		tmp = x + (fabs(y) / 2.0);
	} else if (x <= 2.05e+37) {
		tmp = y * (0.5 + ((x * 1.5) / 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 <= (-62000000000000.0d0)) then
        tmp = 0.5d0 * (x + y)
    else if (x <= 5.5d-174) then
        tmp = x + (abs(y) / 2.0d0)
    else if (x <= 2.05d+37) then
        tmp = y * (0.5d0 + ((x * 1.5d0) / 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 <= -62000000000000.0) {
		tmp = 0.5 * (x + y);
	} else if (x <= 5.5e-174) {
		tmp = x + (Math.abs(y) / 2.0);
	} else if (x <= 2.05e+37) {
		tmp = y * (0.5 + ((x * 1.5) / y));
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -62000000000000.0:
		tmp = 0.5 * (x + y)
	elif x <= 5.5e-174:
		tmp = x + (math.fabs(y) / 2.0)
	elif x <= 2.05e+37:
		tmp = y * (0.5 + ((x * 1.5) / y))
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -62000000000000.0)
		tmp = Float64(0.5 * Float64(x + y));
	elseif (x <= 5.5e-174)
		tmp = Float64(x + Float64(abs(y) / 2.0));
	elseif (x <= 2.05e+37)
		tmp = Float64(y * Float64(0.5 + Float64(Float64(x * 1.5) / y)));
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -62000000000000.0)
		tmp = 0.5 * (x + y);
	elseif (x <= 5.5e-174)
		tmp = x + (abs(y) / 2.0);
	elseif (x <= 2.05e+37)
		tmp = y * (0.5 + ((x * 1.5) / y));
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -62000000000000.0], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.5e-174], N[(x + N[(N[Abs[y], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.05e+37], N[(y * N[(0.5 + N[(N[(x * 1.5), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6.2e13

   1. Initial program 100.0%

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

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

      3. add-sqr-sqrt 88.3%

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

      4. fabs-sqr 88.3%

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

      5. add-sqr-sqrt 89.1%

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

   4. Applied egg-rr 89.1%

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

   5. Taylor expanded in x around 0 89.3%

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

      1. distribute-lft-out 89.3%

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

      2. +-commutative 89.3%

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

   7. Simplified 89.3%

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

   if -6.2e13 < x < 5.4999999999999999e-174

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 82.4%

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

   if 5.4999999999999999e-174 < x < 2.0499999999999999e37

   1. Initial program 99.8%

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

      1. add-sqr-sqrt 99.4%

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

      2. pow2 99.4%

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

      3. div-inv 99.4%

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

      4. add-sqr-sqrt 44.5%

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

      5. fabs-sqr 44.5%

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

      6. add-sqr-sqrt 44.6%

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

      7. metadata-eval 44.6%

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

   4. Applied egg-rr 44.6%

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

      1. add-sqr-sqrt 44.5%

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

      2. fabs-sqr 44.5%

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

      3. add-sqr-sqrt 99.4%

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

      4. fabs-sub 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. sqrt-prod 98.8%

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

      2. unpow-prod-down 98.6%

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

      3. pow2 98.6%

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

      4. add-sqr-sqrt 98.6%

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

      5. fabs-sub 98.6%

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

      6. add-sqr-sqrt 44.0%

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

      7. fabs-sqr 44.0%

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

      8. add-sqr-sqrt 51.8%

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

      9. sub-neg 51.8%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 81.3%

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

      12. sqr-neg 81.3%

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

      13. sqrt-unprod 83.2%

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

      14. add-sqr-sqrt 83.2%

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

      15. +-commutative 83.2%

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

   8. Applied egg-rr 83.2%

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

   9. Taylor expanded in y around -inf 83.2%

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

       1. mul-1-neg 83.2%

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

       2. distribute-rgt-neg-in 83.2%

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

       3. distribute-lft-out 83.2%

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

       4. mul-1-neg 83.2%

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

       5. remove-double-neg 83.2%

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

       6. unpow2 83.2%

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

       7. rem-square-sqrt 83.9%

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

       8. +-commutative 83.9%

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

       9. *-lft-identity 83.9%

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

       10. *-commutative 83.9%

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

       11. unpow2 83.9%

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

       12. rem-square-sqrt 84.1%

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

       13. distribute-rgt-in 84.1%

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

       14. metadata-eval 84.1%

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

   11. Simplified 84.1%

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

   if 2.0499999999999999e37 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 76.9%

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

      1. neg-mul-1 76.9%

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

   5. Simplified 76.9%

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

   6. Taylor expanded in x around 0 76.9%

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

      1. *-commutative 76.9%

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

      2. fabs-neg 76.9%

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

      3. rem-square-sqrt 77.0%

         \[\leadsto x + \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot 0.5 \]

      4. fabs-sqr 77.0%

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

      5. rem-square-sqrt 76.9%

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

      6. *-rgt-identity 76.9%

         \[\leadsto \color{blue}{x \cdot 1} + x \cdot 0.5 \]

      7. distribute-lft-out 76.9%

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

      8. metadata-eval 76.9%

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

   8. Simplified 76.9%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -62000000000000:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-174}:\\ \;\;\;\;x + \frac{\left|y\right|}{2}\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+37}:\\ \;\;\;\;y \cdot \left(0.5 + \frac{x \cdot 1.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -48000000000000:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-174}:\\ \;\;\;\;\left|y\right| \cdot 0.5\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+37}:\\ \;\;\;\;y \cdot \left(0.5 + \frac{x \cdot 1.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -48000000000000.0)
   (* 0.5 (+ x y))
   (if (<= x 5e-174)
     (* (fabs y) 0.5)
     (if (<= x 2.05e+37) (* y (+ 0.5 (/ (* x 1.5) y))) (* x 1.5)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -48000000000000.0) {
		tmp = 0.5 * (x + y);
	} else if (x <= 5e-174) {
		tmp = fabs(y) * 0.5;
	} else if (x <= 2.05e+37) {
		tmp = y * (0.5 + ((x * 1.5) / 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 <= (-48000000000000.0d0)) then
        tmp = 0.5d0 * (x + y)
    else if (x <= 5d-174) then
        tmp = abs(y) * 0.5d0
    else if (x <= 2.05d+37) then
        tmp = y * (0.5d0 + ((x * 1.5d0) / 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 <= -48000000000000.0) {
		tmp = 0.5 * (x + y);
	} else if (x <= 5e-174) {
		tmp = Math.abs(y) * 0.5;
	} else if (x <= 2.05e+37) {
		tmp = y * (0.5 + ((x * 1.5) / y));
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -48000000000000.0:
		tmp = 0.5 * (x + y)
	elif x <= 5e-174:
		tmp = math.fabs(y) * 0.5
	elif x <= 2.05e+37:
		tmp = y * (0.5 + ((x * 1.5) / y))
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -48000000000000.0)
		tmp = Float64(0.5 * Float64(x + y));
	elseif (x <= 5e-174)
		tmp = Float64(abs(y) * 0.5);
	elseif (x <= 2.05e+37)
		tmp = Float64(y * Float64(0.5 + Float64(Float64(x * 1.5) / y)));
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -48000000000000.0)
		tmp = 0.5 * (x + y);
	elseif (x <= 5e-174)
		tmp = abs(y) * 0.5;
	elseif (x <= 2.05e+37)
		tmp = y * (0.5 + ((x * 1.5) / y));
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -48000000000000.0], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5e-174], N[(N[Abs[y], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 2.05e+37], N[(y * N[(0.5 + N[(N[(x * 1.5), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.8e13

   1. Initial program 100.0%

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

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

      3. add-sqr-sqrt 88.3%

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

      4. fabs-sqr 88.3%

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

      5. add-sqr-sqrt 89.1%

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

   4. Applied egg-rr 89.1%

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

   5. Taylor expanded in x around 0 89.3%

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

      1. distribute-lft-out 89.3%

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

      2. +-commutative 89.3%

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

   7. Simplified 89.3%

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

   if -4.8e13 < x < 5.0000000000000002e-174

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 82.4%

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

   4. Taylor expanded in x around 0 79.8%

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

   if 5.0000000000000002e-174 < x < 2.0499999999999999e37

   1. Initial program 99.8%

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

      1. add-sqr-sqrt 99.4%

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

      2. pow2 99.4%

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

      3. div-inv 99.4%

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

      4. add-sqr-sqrt 44.5%

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

      5. fabs-sqr 44.5%

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

      6. add-sqr-sqrt 44.6%

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

      7. metadata-eval 44.6%

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

   4. Applied egg-rr 44.6%

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

      1. add-sqr-sqrt 44.5%

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

      2. fabs-sqr 44.5%

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

      3. add-sqr-sqrt 99.4%

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

      4. fabs-sub 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. sqrt-prod 98.8%

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

      2. unpow-prod-down 98.6%

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

      3. pow2 98.6%

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

      4. add-sqr-sqrt 98.6%

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

      5. fabs-sub 98.6%

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

      6. add-sqr-sqrt 44.0%

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

      7. fabs-sqr 44.0%

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

      8. add-sqr-sqrt 51.8%

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

      9. sub-neg 51.8%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 81.3%

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

      12. sqr-neg 81.3%

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

      13. sqrt-unprod 83.2%

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

      14. add-sqr-sqrt 83.2%

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

      15. +-commutative 83.2%

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

   8. Applied egg-rr 83.2%

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

   9. Taylor expanded in y around -inf 83.2%

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

       1. mul-1-neg 83.2%

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

       2. distribute-rgt-neg-in 83.2%

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

       3. distribute-lft-out 83.2%

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

       4. mul-1-neg 83.2%

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

       5. remove-double-neg 83.2%

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

       6. unpow2 83.2%

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

       7. rem-square-sqrt 83.9%

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

       8. +-commutative 83.9%

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

       9. *-lft-identity 83.9%

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

       10. *-commutative 83.9%

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

       11. unpow2 83.9%

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

       12. rem-square-sqrt 84.1%

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

       13. distribute-rgt-in 84.1%

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

       14. metadata-eval 84.1%

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

   11. Simplified 84.1%

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

   if 2.0499999999999999e37 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 76.9%

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

      1. neg-mul-1 76.9%

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

   5. Simplified 76.9%

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

   6. Taylor expanded in x around 0 76.9%

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

      1. *-commutative 76.9%

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

      2. fabs-neg 76.9%

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

      3. rem-square-sqrt 77.0%

         \[\leadsto x + \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot 0.5 \]

      4. fabs-sqr 77.0%

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

      5. rem-square-sqrt 76.9%

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

      6. *-rgt-identity 76.9%

         \[\leadsto \color{blue}{x \cdot 1} + x \cdot 0.5 \]

      7. distribute-lft-out 76.9%

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

      8. metadata-eval 76.9%

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

   8. Simplified 76.9%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -48000000000000:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-174}:\\ \;\;\;\;\left|y\right| \cdot 0.5\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+37}:\\ \;\;\;\;y \cdot \left(0.5 + \frac{x \cdot 1.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.7% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{-227}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+37}:\\ \;\;\;\;y \cdot \left(0.5 + \frac{x \cdot 1.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1e-227)
   (* 0.5 (+ x y))
   (if (<= x 1.55e+37) (* y (+ 0.5 (/ (* x 1.5) y))) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1e-227) {
		tmp = 0.5 * (x + y);
	} else if (x <= 1.55e+37) {
		tmp = y * (0.5 + ((x * 1.5) / 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 <= 1d-227) then
        tmp = 0.5d0 * (x + y)
    else if (x <= 1.55d+37) then
        tmp = y * (0.5d0 + ((x * 1.5d0) / 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 <= 1e-227) {
		tmp = 0.5 * (x + y);
	} else if (x <= 1.55e+37) {
		tmp = y * (0.5 + ((x * 1.5) / y));
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1e-227:
		tmp = 0.5 * (x + y)
	elif x <= 1.55e+37:
		tmp = y * (0.5 + ((x * 1.5) / y))
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1e-227)
		tmp = Float64(0.5 * Float64(x + y));
	elseif (x <= 1.55e+37)
		tmp = Float64(y * Float64(0.5 + Float64(Float64(x * 1.5) / y)));
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1e-227)
		tmp = 0.5 * (x + y);
	elseif (x <= 1.55e+37)
		tmp = y * (0.5 + ((x * 1.5) / y));
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1e-227], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.55e+37], N[(y * N[(0.5 + N[(N[(x * 1.5), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 9.99999999999999945e-228

   1. Initial program 100.0%

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

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

      3. add-sqr-sqrt 73.6%

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

      4. fabs-sqr 73.6%

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

      5. add-sqr-sqrt 74.5%

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

   4. Applied egg-rr 74.5%

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

   5. Taylor expanded in x around 0 74.8%

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

      1. distribute-lft-out 74.8%

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

      2. +-commutative 74.8%

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

   7. Simplified 74.8%

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

   if 9.99999999999999945e-228 < x < 1.5500000000000001e37

   1. Initial program 99.8%

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

      1. add-sqr-sqrt 99.4%

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

      2. pow2 99.4%

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

      3. div-inv 99.4%

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

      4. add-sqr-sqrt 34.8%

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

      5. fabs-sqr 34.8%

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

      6. add-sqr-sqrt 34.8%

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

      7. metadata-eval 34.8%

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

   4. Applied egg-rr 34.8%

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

      1. add-sqr-sqrt 34.8%

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

      2. fabs-sqr 34.8%

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

      3. add-sqr-sqrt 99.4%

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

      4. fabs-sub 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. sqrt-prod 98.8%

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

      2. unpow-prod-down 98.6%

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

      3. pow2 98.6%

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

      4. add-sqr-sqrt 98.6%

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

      5. fabs-sub 98.6%

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

      6. add-sqr-sqrt 34.4%

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

      7. fabs-sqr 34.4%

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

      8. add-sqr-sqrt 41.4%

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

      9. sub-neg 41.4%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 63.4%

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

      12. sqr-neg 63.4%

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

      13. sqrt-unprod 68.5%

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

      14. add-sqr-sqrt 68.5%

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

      15. +-commutative 68.5%

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

   8. Applied egg-rr 68.5%

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

   9. Taylor expanded in y around -inf 68.5%

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

       1. mul-1-neg 68.5%

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

       2. distribute-rgt-neg-in 68.5%

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

       3. distribute-lft-out 68.5%

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

       4. mul-1-neg 68.5%

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

       5. remove-double-neg 68.5%

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

       6. unpow2 68.5%

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

       7. rem-square-sqrt 69.1%

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

       8. +-commutative 69.1%

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

       9. *-lft-identity 69.1%

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

       10. *-commutative 69.1%

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

       11. unpow2 69.1%

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

       12. rem-square-sqrt 69.2%

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

       13. distribute-rgt-in 69.2%

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

       14. metadata-eval 69.2%

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

   11. Simplified 69.2%

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

   if 1.5500000000000001e37 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 76.9%

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

      1. neg-mul-1 76.9%

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

   5. Simplified 76.9%

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

   6. Taylor expanded in x around 0 76.9%

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

      1. *-commutative 76.9%

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

      2. fabs-neg 76.9%

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

      3. rem-square-sqrt 77.0%

         \[\leadsto x + \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot 0.5 \]

      4. fabs-sqr 77.0%

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

      5. rem-square-sqrt 76.9%

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

      6. *-rgt-identity 76.9%

         \[\leadsto \color{blue}{x \cdot 1} + x \cdot 0.5 \]

      7. distribute-lft-out 76.9%

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

      8. metadata-eval 76.9%

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

   8. Simplified 76.9%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-227}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+37}:\\ \;\;\;\;y \cdot \left(0.5 + \frac{x \cdot 1.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.2% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-78}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-103}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.35e-78) (* x 0.5) (if (<= x 8.8e-103) (* y 0.5) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.35e-78) {
		tmp = x * 0.5;
	} else if (x <= 8.8e-103) {
		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 <= (-1.35d-78)) then
        tmp = x * 0.5d0
    else if (x <= 8.8d-103) 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 <= -1.35e-78) {
		tmp = x * 0.5;
	} else if (x <= 8.8e-103) {
		tmp = y * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.35e-78:
		tmp = x * 0.5
	elif x <= 8.8e-103:
		tmp = y * 0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.35e-78)
		tmp = Float64(x * 0.5);
	elseif (x <= 8.8e-103)
		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 <= -1.35e-78)
		tmp = x * 0.5;
	elseif (x <= 8.8e-103)
		tmp = y * 0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.35e-78], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, 8.8e-103], N[(y * 0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.34999999999999997e-78

   1. Initial program 100.0%

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

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

      3. add-sqr-sqrt 81.8%

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

      4. fabs-sqr 81.8%

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

      5. add-sqr-sqrt 82.6%

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

   4. Applied egg-rr 82.6%

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

   5. Taylor expanded in x around inf 68.1%

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

   if -1.34999999999999997e-78 < x < 8.7999999999999997e-103

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 83.5%

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

   4. Taylor expanded in x around 0 81.4%

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

      1. rem-square-sqrt 44.1%

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

      2. fabs-sqr 44.1%

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

      3. rem-square-sqrt 45.7%

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

   6. Simplified 45.7%

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

   if 8.7999999999999997e-103 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 71.4%

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

      1. neg-mul-1 71.4%

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

   5. Simplified 71.4%

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

   6. Taylor expanded in x around 0 71.4%

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

      1. *-commutative 71.4%

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

      2. fabs-neg 71.4%

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

      3. rem-square-sqrt 71.5%

         \[\leadsto x + \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot 0.5 \]

      4. fabs-sqr 71.5%

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

      5. rem-square-sqrt 71.4%

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

      6. *-rgt-identity 71.4%

         \[\leadsto \color{blue}{x \cdot 1} + x \cdot 0.5 \]

      7. distribute-lft-out 71.4%

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

      8. metadata-eval 71.4%

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

   8. Simplified 71.4%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-78}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-103}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.3% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.6 \cdot 10^{-104}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 3.6e-104) (* 0.5 (+ x y)) (* x 1.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 3.6e-104) {
		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 <= 3.6d-104) 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 <= 3.6e-104) {
		tmp = 0.5 * (x + y);
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 3.6e-104:
		tmp = 0.5 * (x + y)
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 3.6e-104)
		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 <= 3.6e-104)
		tmp = 0.5 * (x + y);
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 3.6e-104], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.5999999999999998e-104

   1. Initial program 100.0%

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

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

      3. add-sqr-sqrt 68.7%

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

      4. fabs-sqr 68.7%

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

      5. add-sqr-sqrt 70.0%

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

   4. Applied egg-rr 70.0%

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

   5. Taylor expanded in x around 0 70.3%

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

      1. distribute-lft-out 70.3%

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

      2. +-commutative 70.3%

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

   7. Simplified 70.3%

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

   if 3.5999999999999998e-104 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 71.4%

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

      1. neg-mul-1 71.4%

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

   5. Simplified 71.4%

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

   6. Taylor expanded in x around 0 71.4%

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

      1. *-commutative 71.4%

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

      2. fabs-neg 71.4%

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

      3. rem-square-sqrt 71.5%

         \[\leadsto x + \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot 0.5 \]

      4. fabs-sqr 71.5%

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

      5. rem-square-sqrt 71.4%

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

      6. *-rgt-identity 71.4%

         \[\leadsto \color{blue}{x \cdot 1} + x \cdot 0.5 \]

      7. distribute-lft-out 71.4%

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

      8. metadata-eval 71.4%

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

   8. Simplified 71.4%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.6 \cdot 10^{-104}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 44.7% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.3 \cdot 10^{-137}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 5.3e-137) (* x 0.5) (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.3e-137) {
		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 <= 5.3d-137) 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 <= 5.3e-137) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.3e-137:
		tmp = x * 0.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.3e-137)
		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 <= 5.3e-137)
		tmp = x * 0.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.3e-137], N[(x * 0.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.30000000000000037e-137

   1. Initial program 99.9%

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

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

      3. add-sqr-sqrt 33.9%

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

      4. fabs-sqr 33.9%

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

      5. add-sqr-sqrt 39.3%

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

   4. Applied egg-rr 39.3%

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

   5. Taylor expanded in x around inf 37.3%

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

   if 5.30000000000000037e-137 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 68.1%

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

   4. Taylor expanded in x around 0 62.1%

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

      1. rem-square-sqrt 61.7%

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

      2. fabs-sqr 61.7%

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

      3. rem-square-sqrt 62.1%

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

   6. Simplified 62.1%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.3 \cdot 10^{-137}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 30.1% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. clear-num 99.7%

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

   2. inv-pow 99.7%

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

   3. add-sqr-sqrt 51.5%

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

   4. fabs-sqr 51.5%

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

   5. add-sqr-sqrt 56.3%

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

4. Applied egg-rr 56.3%

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

5. Taylor expanded in x around inf 31.9%

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

6. Final simplification 31.9%

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

Alternative 10: 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 11.6%

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

Reproduce

herbie shell --seed 2024138 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))