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

Percentage Accurate: 99.9% → 99.9%

Time: 6.7s

Alternatives: 8

Speedup: 1.7×

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, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left|x - y\right|, 0.5, x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma (fabs (- x y)) 0.5 x))

C

double code(double x, double y) {
	return fma(fabs((x - y)), 0.5, x);
}

Julia

function code(x, y)
	return fma(abs(Float64(x - y)), 0.5, x)
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. sub-neg N/A

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

   4. mul-1-neg N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. mul-1-neg N/A

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

   9. distribute-neg-in N/A

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

   10. +-commutative N/A

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

   11. lower-fabs.f64 N/A

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

   12. +-commutative N/A

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

   13. distribute-neg-in N/A

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

   14. mul-1-neg N/A

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

   15. remove-double-neg N/A

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

   16. sub-neg N/A

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

   17. lower--.f64 99.9

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

5. Applied rewrites 99.9%

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

6. Final simplification 99.9%

   \[\leadsto \mathsf{fma}\left(\left|x - y\right|, 0.5, x\right) \]
7. Add Preprocessing

Alternative 2: 59.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.25 \cdot 10^{+38}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-41}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.25e+38) (* -0.5 y) (if (<= y 3.8e-41) (* 0.5 x) (* 0.5 y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.25e+38) {
		tmp = -0.5 * y;
	} else if (y <= 3.8e-41) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3.25d+38)) then
        tmp = (-0.5d0) * y
    else if (y <= 3.8d-41) then
        tmp = 0.5d0 * x
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.25e+38) {
		tmp = -0.5 * y;
	} else if (y <= 3.8e-41) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.25e+38:
		tmp = -0.5 * y
	elif y <= 3.8e-41:
		tmp = 0.5 * x
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.25e+38)
		tmp = Float64(-0.5 * y);
	elseif (y <= 3.8e-41)
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.25e+38)
		tmp = -0.5 * y;
	elseif (y <= 3.8e-41)
		tmp = 0.5 * x;
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.25e+38], N[(-0.5 * y), $MachinePrecision], If[LessEqual[y, 3.8e-41], N[(0.5 * x), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.25e38

   1. Initial program 100.0%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. rem-sqrt-square-rev N/A

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

      5. sqrt-prod N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. remove-double-neg N/A

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

      9. distribute-neg-in N/A

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

      10. +-commutative N/A

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

      11. sub-neg N/A

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

      12. lift--.f64 N/A

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

      13. rem-square-sqrt N/A

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

      14. sqrt-prod N/A

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

      15. rem-sqrt-square-rev N/A

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

      16. lift-fabs.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lift-fabs.f64 N/A

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

      19. rem-sqrt-square-rev N/A

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

      20. sqrt-prod N/A

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

      21. rem-square-sqrt N/A

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

      22. lift--.f64 N/A

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

      23. sub-neg N/A

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

      24. +-commutative N/A

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

      25. distribute-neg-in N/A

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

      26. remove-double-neg N/A

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

      27. sub-neg N/A

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

      28. lower--.f64 N/A

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

   4. Applied rewrites 93.7%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{3}{2} \cdot x} \]
   6. Step-by-step derivation

      1. lower-*.f64 8.5

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

   7. Applied rewrites 8.5%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot y} \]
   9. Step-by-step derivation

      1. lower-*.f64 86.4

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

   10. Applied rewrites 86.4%

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

   if -3.25e38 < y < 3.79999999999999979e-41

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. rem-sqrt-square-rev N/A

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

      8. sqrt-prod N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval 62.5

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

   4. Applied rewrites 62.5%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x} \]
   6. Step-by-step derivation

      1. lower-*.f64 55.4

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

   7. Applied rewrites 55.4%

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

   if 3.79999999999999979e-41 < y

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. rem-sqrt-square-rev N/A

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

      8. sqrt-prod N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval 89.7

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

   4. Applied rewrites 89.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot y} \]
   6. Step-by-step derivation

      1. lower-*.f64 75.6

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

   7. Applied rewrites 75.6%

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

Alternative 3: 58.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{-100}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-9}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7.6e-100) (* 0.5 x) (if (<= x 1.1e-9) (* -0.5 y) (* 1.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7.6e-100) {
		tmp = 0.5 * x;
	} else if (x <= 1.1e-9) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-7.6d-100)) then
        tmp = 0.5d0 * x
    else if (x <= 1.1d-9) then
        tmp = (-0.5d0) * y
    else
        tmp = 1.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -7.6e-100) {
		tmp = 0.5 * x;
	} else if (x <= 1.1e-9) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7.6e-100:
		tmp = 0.5 * x
	elif x <= 1.1e-9:
		tmp = -0.5 * y
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7.6e-100)
		tmp = Float64(0.5 * x);
	elseif (x <= 1.1e-9)
		tmp = Float64(-0.5 * y);
	else
		tmp = Float64(1.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7.6e-100)
		tmp = 0.5 * x;
	elseif (x <= 1.1e-9)
		tmp = -0.5 * y;
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7.6e-100], N[(0.5 * x), $MachinePrecision], If[LessEqual[x, 1.1e-9], N[(-0.5 * y), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.59999999999999995e-100

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. rem-sqrt-square-rev N/A

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

      8. sqrt-prod N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval 85.8

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

   4. Applied rewrites 85.8%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x} \]
   6. Step-by-step derivation

      1. lower-*.f64 68.9

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

   7. Applied rewrites 68.9%

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

   if -7.59999999999999995e-100 < x < 1.0999999999999999e-9

   1. Initial program 99.9%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. rem-sqrt-square-rev N/A

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

      5. sqrt-prod N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. remove-double-neg N/A

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

      9. distribute-neg-in N/A

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

      10. +-commutative N/A

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

      11. sub-neg N/A

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

      12. lift--.f64 N/A

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

      13. rem-square-sqrt N/A

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

      14. sqrt-prod N/A

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

      15. rem-sqrt-square-rev N/A

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

      16. lift-fabs.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lift-fabs.f64 N/A

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

      19. rem-sqrt-square-rev N/A

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

      20. sqrt-prod N/A

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

      21. rem-square-sqrt N/A

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

      22. lift--.f64 N/A

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

      23. sub-neg N/A

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

      24. +-commutative N/A

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

      25. distribute-neg-in N/A

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

      26. remove-double-neg N/A

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

      27. sub-neg N/A

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

      28. lower--.f64 N/A

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

   4. Applied rewrites 52.0%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{3}{2} \cdot x} \]
   6. Step-by-step derivation

      1. lower-*.f64 17.0

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

   7. Applied rewrites 17.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot y} \]
   9. Step-by-step derivation

      1. lower-*.f64 39.8

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

   10. Applied rewrites 39.8%

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

   if 1.0999999999999999e-9 < x

   1. Initial program 99.8%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. rem-sqrt-square-rev N/A

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

      5. sqrt-prod N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. remove-double-neg N/A

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

      9. distribute-neg-in N/A

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

      10. +-commutative N/A

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

      11. sub-neg N/A

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

      12. lift--.f64 N/A

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

      13. rem-square-sqrt N/A

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

      14. sqrt-prod N/A

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

      15. rem-sqrt-square-rev N/A

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

      16. lift-fabs.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lift-fabs.f64 N/A

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

      19. rem-sqrt-square-rev N/A

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

      20. sqrt-prod N/A

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

      21. rem-square-sqrt N/A

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

      22. lift--.f64 N/A

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

      23. sub-neg N/A

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

      24. +-commutative N/A

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

      25. distribute-neg-in N/A

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

      26. remove-double-neg N/A

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

      27. sub-neg N/A

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

      28. lower--.f64 N/A

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

   4. Applied rewrites 88.4%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{3}{2} \cdot x} \]
   6. Step-by-step derivation

      1. lower-*.f64 75.3

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

   7. Applied rewrites 75.3%

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

Alternative 4: 77.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-134}:\\ \;\;\;\;\mathsf{fma}\left(x - y, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.9e-134) (fma (- x y) 0.5 x) (* (+ y x) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.9e-134) {
		tmp = fma((x - y), 0.5, x);
	} else {
		tmp = (y + x) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.9e-134)
		tmp = fma(Float64(x - y), 0.5, x);
	else
		tmp = Float64(Float64(y + x) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.9e-134], N[(N[(x - y), $MachinePrecision] * 0.5 + x), $MachinePrecision], N[(N[(y + x), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.90000000000000001e-134

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-in N/A

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

      10. +-commutative N/A

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

      11. lower-fabs.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 82.9%

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

   if -1.90000000000000001e-134 < y

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. rem-sqrt-square-rev N/A

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

      8. sqrt-prod N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval 76.6

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

   4. Applied rewrites 76.6%

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

   5. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 76.6

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

   7. Applied rewrites 76.6%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-134}:\\ \;\;\;\;\mathsf{fma}\left(x - y, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-7}:\\ \;\;\;\;-0.5 \cdot y + x\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.5e-7) (+ (* -0.5 y) x) (* (+ y x) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.5e-7) {
		tmp = (-0.5 * y) + x;
	} else {
		tmp = (y + x) * 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 <= (-1.5d-7)) then
        tmp = ((-0.5d0) * y) + x
    else
        tmp = (y + x) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.5e-7) {
		tmp = (-0.5 * y) + x;
	} else {
		tmp = (y + x) * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.5e-7:
		tmp = (-0.5 * y) + x
	else:
		tmp = (y + x) * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.5e-7)
		tmp = Float64(Float64(-0.5 * y) + x);
	else
		tmp = Float64(Float64(y + x) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.5e-7)
		tmp = (-0.5 * y) + x;
	else
		tmp = (y + x) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.5e-7], N[(N[(-0.5 * y), $MachinePrecision] + x), $MachinePrecision], N[(N[(y + x), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.4999999999999999e-7

   1. Initial program 99.9%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. rem-sqrt-square-rev N/A

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

      5. sqrt-prod N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. remove-double-neg N/A

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

      9. distribute-neg-in N/A

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

      10. +-commutative N/A

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

      11. sub-neg N/A

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

      12. lift--.f64 N/A

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

      13. rem-square-sqrt N/A

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

      14. sqrt-prod N/A

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

      15. rem-sqrt-square-rev N/A

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

      16. lift-fabs.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lift-fabs.f64 N/A

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

      19. rem-sqrt-square-rev N/A

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

      20. sqrt-prod N/A

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

      21. rem-square-sqrt N/A

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

      22. lift--.f64 N/A

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

      23. sub-neg N/A

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

      24. +-commutative N/A

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

      25. distribute-neg-in N/A

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

      26. remove-double-neg N/A

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

      27. sub-neg N/A

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

      28. lower--.f64 N/A

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

   4. Applied rewrites 91.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 82.9

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

   7. Applied rewrites 82.9%

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

   if -1.4999999999999999e-7 < y

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. rem-sqrt-square-rev N/A

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

      8. sqrt-prod N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval 73.4

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

   4. Applied rewrites 73.4%

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

   5. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 73.4

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

   7. Applied rewrites 73.4%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-7}:\\ \;\;\;\;-0.5 \cdot y + x\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.25 \cdot 10^{+38}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.25e+38) (* -0.5 y) (* (+ y x) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.25e+38) {
		tmp = -0.5 * y;
	} else {
		tmp = (y + x) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3.25d+38)) then
        tmp = (-0.5d0) * y
    else
        tmp = (y + x) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.25e+38) {
		tmp = -0.5 * y;
	} else {
		tmp = (y + x) * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.25e+38:
		tmp = -0.5 * y
	else:
		tmp = (y + x) * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.25e+38)
		tmp = Float64(-0.5 * y);
	else
		tmp = Float64(Float64(y + x) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.25e+38)
		tmp = -0.5 * y;
	else
		tmp = (y + x) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.25e+38], N[(-0.5 * y), $MachinePrecision], N[(N[(y + x), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.25e38

   1. Initial program 100.0%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. rem-sqrt-square-rev N/A

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

      5. sqrt-prod N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. remove-double-neg N/A

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

      9. distribute-neg-in N/A

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

      10. +-commutative N/A

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

      11. sub-neg N/A

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

      12. lift--.f64 N/A

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

      13. rem-square-sqrt N/A

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

      14. sqrt-prod N/A

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

      15. rem-sqrt-square-rev N/A

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

      16. lift-fabs.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lift-fabs.f64 N/A

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

      19. rem-sqrt-square-rev N/A

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

      20. sqrt-prod N/A

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

      21. rem-square-sqrt N/A

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

      22. lift--.f64 N/A

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

      23. sub-neg N/A

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

      24. +-commutative N/A

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

      25. distribute-neg-in N/A

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

      26. remove-double-neg N/A

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

      27. sub-neg N/A

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

      28. lower--.f64 N/A

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

   4. Applied rewrites 93.7%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{3}{2} \cdot x} \]
   6. Step-by-step derivation

      1. lower-*.f64 8.5

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

   7. Applied rewrites 8.5%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot y} \]
   9. Step-by-step derivation

      1. lower-*.f64 86.4

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

   10. Applied rewrites 86.4%

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

   if -3.25e38 < y

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. rem-sqrt-square-rev N/A

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

      8. sqrt-prod N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval 72.2

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 72.2

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

   7. Applied rewrites 72.2%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.25 \cdot 10^{+38}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 44.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+28}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -5e+28) (* -0.5 y) (* 1.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5e+28) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-5d+28)) then
        tmp = (-0.5d0) * y
    else
        tmp = 1.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -5e+28) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5e+28:
		tmp = -0.5 * y
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5e+28)
		tmp = Float64(-0.5 * y);
	else
		tmp = Float64(1.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -5e+28)
		tmp = -0.5 * y;
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5e+28], N[(-0.5 * y), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.99999999999999957e28

   1. Initial program 100.0%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. rem-sqrt-square-rev N/A

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

      5. sqrt-prod N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. remove-double-neg N/A

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

      9. distribute-neg-in N/A

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

      10. +-commutative N/A

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

      11. sub-neg N/A

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

      12. lift--.f64 N/A

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

      13. rem-square-sqrt N/A

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

      14. sqrt-prod N/A

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

      15. rem-sqrt-square-rev N/A

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

      16. lift-fabs.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lift-fabs.f64 N/A

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

      19. rem-sqrt-square-rev N/A

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

      20. sqrt-prod N/A

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

      21. rem-square-sqrt N/A

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

      22. lift--.f64 N/A

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

      23. sub-neg N/A

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

      24. +-commutative N/A

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

      25. distribute-neg-in N/A

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

      26. remove-double-neg N/A

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

      27. sub-neg N/A

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

      28. lower--.f64 N/A

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

   4. Applied rewrites 92.0%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{3}{2} \cdot x} \]
   6. Step-by-step derivation

      1. lower-*.f64 8.6

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

   7. Applied rewrites 8.6%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot y} \]
   9. Step-by-step derivation

      1. lower-*.f64 85.1

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

   10. Applied rewrites 85.1%

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

   if -4.99999999999999957e28 < y

   1. Initial program 99.9%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. rem-sqrt-square-rev N/A

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

      5. sqrt-prod N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. remove-double-neg N/A

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

      9. distribute-neg-in N/A

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

      10. +-commutative N/A

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

      11. sub-neg N/A

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

      12. lift--.f64 N/A

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

      13. rem-square-sqrt N/A

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

      14. sqrt-prod N/A

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

      15. rem-sqrt-square-rev N/A

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

      16. lift-fabs.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lift-fabs.f64 N/A

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

      19. rem-sqrt-square-rev N/A

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

      20. sqrt-prod N/A

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

      21. rem-square-sqrt N/A

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

      22. lift--.f64 N/A

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

      23. sub-neg N/A

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

      24. +-commutative N/A

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

      25. distribute-neg-in N/A

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

      26. remove-double-neg N/A

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

      27. sub-neg N/A

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

      28. lower--.f64 N/A

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

   4. Applied rewrites 32.2%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{3}{2} \cdot x} \]
   6. Step-by-step derivation

      1. lower-*.f64 32.5

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

   7. Applied rewrites 32.5%

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

Alternative 8: 25.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-fabs.f64 N/A

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

   2. lift--.f64 N/A

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

   3. fabs-sub N/A

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

   4. rem-sqrt-square-rev N/A

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

   5. sqrt-prod N/A

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

   6. lower-*.f64 N/A

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

   7. sub-neg N/A

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

   8. remove-double-neg N/A

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

   9. distribute-neg-in N/A

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

   10. +-commutative N/A

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

   11. sub-neg N/A

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

   12. lift--.f64 N/A

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

   13. rem-square-sqrt N/A

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

   14. sqrt-prod N/A

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

   15. rem-sqrt-square-rev N/A

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

   16. lift-fabs.f64 N/A

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

   17. lower-sqrt.f64 N/A

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

   18. lift-fabs.f64 N/A

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

   19. rem-sqrt-square-rev N/A

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

   20. sqrt-prod N/A

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

   21. rem-square-sqrt N/A

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

   22. lift--.f64 N/A

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

   23. sub-neg N/A

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

   24. +-commutative N/A

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

   25. distribute-neg-in N/A

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

   26. remove-double-neg N/A

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

   27. sub-neg N/A

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

   28. lower--.f64 N/A

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

4. Applied rewrites 45.1%

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

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{3}{2} \cdot x} \]
6. Step-by-step derivation

   1. lower-*.f64 27.3

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

7. Applied rewrites 27.3%

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

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot y} \]
9. Step-by-step derivation

   1. lower-*.f64 25.3

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

10. Applied rewrites 25.3%

    \[\leadsto \color{blue}{-0.5 \cdot y} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024298 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))