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

Percentage Accurate: 99.9% → 99.9%

Time: 7.8s

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[Abs[N[(y - x), $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. lower-fma.f64 N/A

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

   4. *-lft-identity N/A

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

   5. metadata-eval N/A

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

   6. fabs-sub N/A

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

   7. fp-cancel-sign-sub-inv N/A

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

   8. lower-fabs.f64 N/A

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

   9. fp-cancel-sign-sub-inv N/A

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

   10. metadata-eval N/A

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

   11. *-lft-identity N/A

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

   12. lower--.f64 99.9

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

5. Applied rewrites 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 49.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + \frac{\left|y - x\right|}{2} \leq 5 \cdot 10^{-252}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ x (/ (fabs (- y x)) 2.0)) 5e-252) (* 0.5 x) (* 0.5 y)))

C

double code(double x, double y) {
	double tmp;
	if ((x + (fabs((y - x)) / 2.0)) <= 5e-252) {
		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 ((x + (abs((y - x)) / 2.0d0)) <= 5d-252) 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 ((x + (Math.abs((y - x)) / 2.0)) <= 5e-252) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x + (math.fabs((y - x)) / 2.0)) <= 5e-252:
		tmp = 0.5 * x
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x + Float64(abs(Float64(y - x)) / 2.0)) <= 5e-252)
		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 ((x + (abs((y - x)) / 2.0)) <= 5e-252)
		tmp = 0.5 * x;
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x + N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 5e-252], N[(0.5 * x), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (fabs.f64 (-.f64 y x)) #s(literal 2 binary64))) < 5.00000000000000008e-252

   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. rem-sqrt-square-rev N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. lift--.f64 N/A

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

      6. flip3-- N/A

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

      7. frac-times N/A

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

      8. sqrt-div N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 3.8

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

   7. Applied rewrites 3.8%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{4} \cdot \frac{-1 \cdot y - y}{y}\right) + \frac{1}{2} \cdot y} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \frac{1}{4} \cdot \frac{-1 \cdot y - y}{y}\right) \cdot x} + \frac{1}{2} \cdot y \]

      3. div-sub N/A

         \[\leadsto \left(1 + \frac{1}{4} \cdot \color{blue}{\left(\frac{-1 \cdot y}{y} - \frac{y}{y}\right)}\right) \cdot x + \frac{1}{2} \cdot y \]

      4. associate-/l* N/A

         \[\leadsto \left(1 + \frac{1}{4} \cdot \left(\color{blue}{-1 \cdot \frac{y}{y}} - \frac{y}{y}\right)\right) \cdot x + \frac{1}{2} \cdot y \]

      5. *-inverses N/A

         \[\leadsto \left(1 + \frac{1}{4} \cdot \left(-1 \cdot \color{blue}{1} - \frac{y}{y}\right)\right) \cdot x + \frac{1}{2} \cdot y \]

      6. metadata-eval N/A

         \[\leadsto \left(1 + \frac{1}{4} \cdot \left(\color{blue}{-1} - \frac{y}{y}\right)\right) \cdot x + \frac{1}{2} \cdot y \]

      7. *-inverses N/A

         \[\leadsto \left(1 + \frac{1}{4} \cdot \left(-1 - \color{blue}{1}\right)\right) \cdot x + \frac{1}{2} \cdot y \]

      8. metadata-eval N/A

         \[\leadsto \left(1 + \frac{1}{4} \cdot \color{blue}{-2}\right) \cdot x + \frac{1}{2} \cdot y \]

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 93.9

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

   10. Applied rewrites 93.9%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 92.8%

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

   if 5.00000000000000008e-252 < (+.f64 x (/.f64 (fabs.f64 (-.f64 y x)) #s(literal 2 binary64)))

   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. rem-sqrt-square-rev N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. lift--.f64 N/A

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

      6. flip3-- N/A

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

      7. frac-times N/A

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

      8. sqrt-div N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 14.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 33.5

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

   7. Applied rewrites 33.5%

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

Alternative 3: 81.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-91}:\\ \;\;\;\;\left(y + x\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(\left|-y\right|, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.35e-91)
   (* (+ y x) 0.5)
   (if (<= x 5.1e-10) (fma (fabs (- y)) 0.5 x) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.35e-91) {
		tmp = (y + x) * 0.5;
	} else if (x <= 5.1e-10) {
		tmp = fma(fabs(-y), 0.5, x);
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.35e-91)
		tmp = Float64(Float64(y + x) * 0.5);
	elseif (x <= 5.1e-10)
		tmp = fma(abs(Float64(-y)), 0.5, x);
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.35e-91], N[(N[(y + x), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 5.1e-10], N[(N[Abs[(-y)], $MachinePrecision] * 0.5 + x), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3499999999999999e-91

   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. rem-sqrt-square-rev N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. lift--.f64 N/A

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

      6. flip3-- N/A

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

      7. frac-times N/A

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

      8. sqrt-div N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 1.4%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 18.1

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

   7. Applied rewrites 18.1%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{4} \cdot \frac{-1 \cdot y - y}{y}\right) + \frac{1}{2} \cdot y} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \frac{1}{4} \cdot \frac{-1 \cdot y - y}{y}\right) \cdot x} + \frac{1}{2} \cdot y \]

      3. div-sub N/A

         \[\leadsto \left(1 + \frac{1}{4} \cdot \color{blue}{\left(\frac{-1 \cdot y}{y} - \frac{y}{y}\right)}\right) \cdot x + \frac{1}{2} \cdot y \]

      4. associate-/l* N/A

         \[\leadsto \left(1 + \frac{1}{4} \cdot \left(\color{blue}{-1 \cdot \frac{y}{y}} - \frac{y}{y}\right)\right) \cdot x + \frac{1}{2} \cdot y \]

      5. *-inverses N/A

         \[\leadsto \left(1 + \frac{1}{4} \cdot \left(-1 \cdot \color{blue}{1} - \frac{y}{y}\right)\right) \cdot x + \frac{1}{2} \cdot y \]

      6. metadata-eval N/A

         \[\leadsto \left(1 + \frac{1}{4} \cdot \left(\color{blue}{-1} - \frac{y}{y}\right)\right) \cdot x + \frac{1}{2} \cdot y \]

      7. *-inverses N/A

         \[\leadsto \left(1 + \frac{1}{4} \cdot \left(-1 - \color{blue}{1}\right)\right) \cdot x + \frac{1}{2} \cdot y \]

      8. metadata-eval N/A

         \[\leadsto \left(1 + \frac{1}{4} \cdot \color{blue}{-2}\right) \cdot x + \frac{1}{2} \cdot y \]

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 85.6

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

   10. Applied rewrites 85.6%

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

   11. Taylor expanded in x around 0

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

       1. *-commutative N/A

          \[\leadsto \frac{1}{2} \cdot y + \color{blue}{\left(1 + \frac{1}{4} \cdot \frac{-1 \cdot y - y}{y}\right) \cdot x} \]

       2. div-sub N/A

          \[\leadsto \frac{1}{2} \cdot y + \left(1 + \frac{1}{4} \cdot \color{blue}{\left(\frac{-1 \cdot y}{y} - \frac{y}{y}\right)}\right) \cdot x \]

       3. *-inverses N/A

          \[\leadsto \frac{1}{2} \cdot y + \left(1 + \frac{1}{4} \cdot \left(\frac{-1 \cdot y}{y} - \color{blue}{1}\right)\right) \cdot x \]

       4. associate-/l* N/A

          \[\leadsto \frac{1}{2} \cdot y + \left(1 + \frac{1}{4} \cdot \left(\color{blue}{-1 \cdot \frac{y}{y}} - 1\right)\right) \cdot x \]

       5. *-inverses N/A

          \[\leadsto \frac{1}{2} \cdot y + \left(1 + \frac{1}{4} \cdot \left(-1 \cdot \color{blue}{1} - 1\right)\right) \cdot x \]

       6. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot y + \left(1 + \frac{1}{4} \cdot \left(\color{blue}{-1} - 1\right)\right) \cdot x \]

       7. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot y + \left(1 + \frac{1}{4} \cdot \color{blue}{-2}\right) \cdot x \]

       8. metadata-eval N/A

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

       9. metadata-eval N/A

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

       10. distribute-lft-out N/A

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

       11. +-commutative N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. +-commutative N/A

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

       15. lower-+.f64 85.6

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

   13. Applied rewrites 85.6%

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

   if -1.3499999999999999e-91 < x < 5.09999999999999997e-10

   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. lower-fma.f64 N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. fabs-sub N/A

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

      7. fp-cancel-sign-sub-inv N/A

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

      8. lower-fabs.f64 N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\left|-1 \cdot y\right|, \frac{1}{2}, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 77.5%

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

   if 5.09999999999999997e-10 < x

   1. Initial program 99.8%

      \[x + \frac{\left|y - x\right|}{2} \]
   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. unpow1 N/A

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

      4. metadata-eval N/A

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

      5. sqrt-pow1 N/A

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

      6. pow2 N/A

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

      7. sqrt-prod N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 84.4%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

         \[\leadsto x \cdot \color{blue}{\frac{3}{2}} \]

      7. lower-*.f64 72.9

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

   7. Applied rewrites 72.9%

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

Alternative 4: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-94}:\\ \;\;\;\;\left(y + x\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{-10}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.8e-94)
   (* (+ y x) 0.5)
   (if (<= x 5.1e-10) (* (fabs (- y x)) 0.5) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.8e-94) {
		tmp = (y + x) * 0.5;
	} else if (x <= 5.1e-10) {
		tmp = fabs((y - x)) * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-4.8d-94)) then
        tmp = (y + x) * 0.5d0
    else if (x <= 5.1d-10) then
        tmp = abs((y - x)) * 0.5d0
    else
        tmp = x * 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4.8e-94) {
		tmp = (y + x) * 0.5;
	} else if (x <= 5.1e-10) {
		tmp = Math.abs((y - x)) * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.8e-94:
		tmp = (y + x) * 0.5
	elif x <= 5.1e-10:
		tmp = math.fabs((y - x)) * 0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.8e-94)
		tmp = Float64(Float64(y + x) * 0.5);
	elseif (x <= 5.1e-10)
		tmp = Float64(abs(Float64(y - x)) * 0.5);
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.8e-94)
		tmp = (y + x) * 0.5;
	elseif (x <= 5.1e-10)
		tmp = abs((y - x)) * 0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.8e-94], N[(N[(y + x), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 5.1e-10], N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.8e-94

   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. rem-sqrt-square-rev N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. lift--.f64 N/A

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

      6. flip3-- N/A

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

      7. frac-times N/A

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

      8. sqrt-div N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 1.4%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 18.1

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

   7. Applied rewrites 18.1%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{4} \cdot \frac{-1 \cdot y - y}{y}\right) + \frac{1}{2} \cdot y} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \frac{1}{4} \cdot \frac{-1 \cdot y - y}{y}\right) \cdot x} + \frac{1}{2} \cdot y \]

      3. div-sub N/A

         \[\leadsto \left(1 + \frac{1}{4} \cdot \color{blue}{\left(\frac{-1 \cdot y}{y} - \frac{y}{y}\right)}\right) \cdot x + \frac{1}{2} \cdot y \]

      4. associate-/l* N/A

         \[\leadsto \left(1 + \frac{1}{4} \cdot \left(\color{blue}{-1 \cdot \frac{y}{y}} - \frac{y}{y}\right)\right) \cdot x + \frac{1}{2} \cdot y \]

      5. *-inverses N/A

         \[\leadsto \left(1 + \frac{1}{4} \cdot \left(-1 \cdot \color{blue}{1} - \frac{y}{y}\right)\right) \cdot x + \frac{1}{2} \cdot y \]

      6. metadata-eval N/A

         \[\leadsto \left(1 + \frac{1}{4} \cdot \left(\color{blue}{-1} - \frac{y}{y}\right)\right) \cdot x + \frac{1}{2} \cdot y \]

      7. *-inverses N/A

         \[\leadsto \left(1 + \frac{1}{4} \cdot \left(-1 - \color{blue}{1}\right)\right) \cdot x + \frac{1}{2} \cdot y \]

      8. metadata-eval N/A

         \[\leadsto \left(1 + \frac{1}{4} \cdot \color{blue}{-2}\right) \cdot x + \frac{1}{2} \cdot y \]

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 85.6

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

   10. Applied rewrites 85.6%

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

   11. Taylor expanded in x around 0

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

       1. *-commutative N/A

          \[\leadsto \frac{1}{2} \cdot y + \color{blue}{\left(1 + \frac{1}{4} \cdot \frac{-1 \cdot y - y}{y}\right) \cdot x} \]

       2. div-sub N/A

          \[\leadsto \frac{1}{2} \cdot y + \left(1 + \frac{1}{4} \cdot \color{blue}{\left(\frac{-1 \cdot y}{y} - \frac{y}{y}\right)}\right) \cdot x \]

       3. *-inverses N/A

          \[\leadsto \frac{1}{2} \cdot y + \left(1 + \frac{1}{4} \cdot \left(\frac{-1 \cdot y}{y} - \color{blue}{1}\right)\right) \cdot x \]

       4. associate-/l* N/A

          \[\leadsto \frac{1}{2} \cdot y + \left(1 + \frac{1}{4} \cdot \left(\color{blue}{-1 \cdot \frac{y}{y}} - 1\right)\right) \cdot x \]

       5. *-inverses N/A

          \[\leadsto \frac{1}{2} \cdot y + \left(1 + \frac{1}{4} \cdot \left(-1 \cdot \color{blue}{1} - 1\right)\right) \cdot x \]

       6. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot y + \left(1 + \frac{1}{4} \cdot \left(\color{blue}{-1} - 1\right)\right) \cdot x \]

       7. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot y + \left(1 + \frac{1}{4} \cdot \color{blue}{-2}\right) \cdot x \]

       8. metadata-eval N/A

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

       9. metadata-eval N/A

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

       10. distribute-lft-out N/A

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

       11. +-commutative N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. +-commutative N/A

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

       15. lower-+.f64 85.6

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

   13. Applied rewrites 85.6%

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

   if -4.8e-94 < x < 5.09999999999999997e-10

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fabs-sub N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. lower-fabs.f64 N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 76.5

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

   5. Applied rewrites 76.5%

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

   if 5.09999999999999997e-10 < x

   1. Initial program 99.8%

      \[x + \frac{\left|y - x\right|}{2} \]
   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. unpow1 N/A

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

      4. metadata-eval N/A

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

      5. sqrt-pow1 N/A

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

      6. pow2 N/A

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

      7. sqrt-prod N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 84.4%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

         \[\leadsto x \cdot \color{blue}{\frac{3}{2}} \]

      7. lower-*.f64 72.9

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

   7. Applied rewrites 72.9%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-94}:\\ \;\;\;\;\left(y + x\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{-10}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-124}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-252}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.4e-124)
   (* y -0.5)
   (if (<= y 3.9e-252) (* x 1.5) (* (+ y x) 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.4e-124) {
		tmp = y * -0.5;
	} else if (y <= 3.9e-252) {
		tmp = x * 1.5;
	} 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 <= (-2.4d-124)) then
        tmp = y * (-0.5d0)
    else if (y <= 3.9d-252) then
        tmp = x * 1.5d0
    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 <= -2.4e-124) {
		tmp = y * -0.5;
	} else if (y <= 3.9e-252) {
		tmp = x * 1.5;
	} else {
		tmp = (y + x) * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.4e-124:
		tmp = y * -0.5
	elif y <= 3.9e-252:
		tmp = x * 1.5
	else:
		tmp = (y + x) * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.4e-124)
		tmp = Float64(y * -0.5);
	elseif (y <= 3.9e-252)
		tmp = Float64(x * 1.5);
	else
		tmp = Float64(Float64(y + x) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.4e-124)
		tmp = y * -0.5;
	elseif (y <= 3.9e-252)
		tmp = x * 1.5;
	else
		tmp = (y + x) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.4e-124], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 3.9e-252], N[(x * 1.5), $MachinePrecision], N[(N[(y + x), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.39999999999999992e-124

   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. unpow1 N/A

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

      4. metadata-eval N/A

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

      5. sqrt-pow1 N/A

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

      6. pow2 N/A

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

      7. sqrt-prod N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 80.9%

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

   5. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. rem-square-sqrt N/A

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

      3. lower-*.f64 65.1

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

   7. Applied rewrites 65.1%

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

   if -2.39999999999999992e-124 < y < 3.8999999999999999e-252

   1. Initial program 99.8%

      \[x + \frac{\left|y - x\right|}{2} \]
   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. unpow1 N/A

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

      4. metadata-eval N/A

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

      5. sqrt-pow1 N/A

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

      6. pow2 N/A

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

      7. sqrt-prod N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 68.1%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

         \[\leadsto x \cdot \color{blue}{\frac{3}{2}} \]

      7. lower-*.f64 66.5

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

   7. Applied rewrites 66.5%

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

   if 3.8999999999999999e-252 < 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. rem-sqrt-square-rev N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. lift--.f64 N/A

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

      6. flip3-- N/A

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

      7. frac-times N/A

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

      8. sqrt-div N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 17.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 64.7

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

   7. Applied rewrites 64.7%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{4} \cdot \frac{-1 \cdot y - y}{y}\right) + \frac{1}{2} \cdot y} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \frac{1}{4} \cdot \frac{-1 \cdot y - y}{y}\right) \cdot x} + \frac{1}{2} \cdot y \]

      3. div-sub N/A

         \[\leadsto \left(1 + \frac{1}{4} \cdot \color{blue}{\left(\frac{-1 \cdot y}{y} - \frac{y}{y}\right)}\right) \cdot x + \frac{1}{2} \cdot y \]

      4. associate-/l* N/A

         \[\leadsto \left(1 + \frac{1}{4} \cdot \left(\color{blue}{-1 \cdot \frac{y}{y}} - \frac{y}{y}\right)\right) \cdot x + \frac{1}{2} \cdot y \]

      5. *-inverses N/A

         \[\leadsto \left(1 + \frac{1}{4} \cdot \left(-1 \cdot \color{blue}{1} - \frac{y}{y}\right)\right) \cdot x + \frac{1}{2} \cdot y \]

      6. metadata-eval N/A

         \[\leadsto \left(1 + \frac{1}{4} \cdot \left(\color{blue}{-1} - \frac{y}{y}\right)\right) \cdot x + \frac{1}{2} \cdot y \]

      7. *-inverses N/A

         \[\leadsto \left(1 + \frac{1}{4} \cdot \left(-1 - \color{blue}{1}\right)\right) \cdot x + \frac{1}{2} \cdot y \]

      8. metadata-eval N/A

         \[\leadsto \left(1 + \frac{1}{4} \cdot \color{blue}{-2}\right) \cdot x + \frac{1}{2} \cdot y \]

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 83.7

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

   10. Applied rewrites 83.7%

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

   11. Taylor expanded in x around 0

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

       1. *-commutative N/A

          \[\leadsto \frac{1}{2} \cdot y + \color{blue}{\left(1 + \frac{1}{4} \cdot \frac{-1 \cdot y - y}{y}\right) \cdot x} \]

       2. div-sub N/A

          \[\leadsto \frac{1}{2} \cdot y + \left(1 + \frac{1}{4} \cdot \color{blue}{\left(\frac{-1 \cdot y}{y} - \frac{y}{y}\right)}\right) \cdot x \]

       3. *-inverses N/A

          \[\leadsto \frac{1}{2} \cdot y + \left(1 + \frac{1}{4} \cdot \left(\frac{-1 \cdot y}{y} - \color{blue}{1}\right)\right) \cdot x \]

       4. associate-/l* N/A

          \[\leadsto \frac{1}{2} \cdot y + \left(1 + \frac{1}{4} \cdot \left(\color{blue}{-1 \cdot \frac{y}{y}} - 1\right)\right) \cdot x \]

       5. *-inverses N/A

          \[\leadsto \frac{1}{2} \cdot y + \left(1 + \frac{1}{4} \cdot \left(-1 \cdot \color{blue}{1} - 1\right)\right) \cdot x \]

       6. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot y + \left(1 + \frac{1}{4} \cdot \left(\color{blue}{-1} - 1\right)\right) \cdot x \]

       7. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot y + \left(1 + \frac{1}{4} \cdot \color{blue}{-2}\right) \cdot x \]

       8. metadata-eval N/A

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

       9. metadata-eval N/A

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

       10. distribute-lft-out N/A

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

       11. +-commutative N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. +-commutative N/A

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

       15. lower-+.f64 83.7

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

   13. Applied rewrites 83.7%

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

Alternative 6: 60.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-124}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-18}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.4e-124) (* y -0.5) (if (<= y 1.7e-18) (* x 1.5) (* 0.5 y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.4e-124) {
		tmp = y * -0.5;
	} else if (y <= 1.7e-18) {
		tmp = x * 1.5;
	} 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 <= (-2.4d-124)) then
        tmp = y * (-0.5d0)
    else if (y <= 1.7d-18) then
        tmp = x * 1.5d0
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.4e-124) {
		tmp = y * -0.5;
	} else if (y <= 1.7e-18) {
		tmp = x * 1.5;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.4e-124:
		tmp = y * -0.5
	elif y <= 1.7e-18:
		tmp = x * 1.5
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.4e-124)
		tmp = Float64(y * -0.5);
	elseif (y <= 1.7e-18)
		tmp = Float64(x * 1.5);
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.4e-124)
		tmp = y * -0.5;
	elseif (y <= 1.7e-18)
		tmp = x * 1.5;
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.4e-124], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 1.7e-18], N[(x * 1.5), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.39999999999999992e-124

   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. unpow1 N/A

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

      4. metadata-eval N/A

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

      5. sqrt-pow1 N/A

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

      6. pow2 N/A

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

      7. sqrt-prod N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 80.9%

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

   5. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. rem-square-sqrt N/A

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

      3. lower-*.f64 65.1

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

   7. Applied rewrites 65.1%

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

   if -2.39999999999999992e-124 < y < 1.70000000000000001e-18

   1. Initial program 99.8%

      \[x + \frac{\left|y - x\right|}{2} \]
   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. unpow1 N/A

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

      4. metadata-eval N/A

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

      5. sqrt-pow1 N/A

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

      6. pow2 N/A

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

      7. sqrt-prod N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 57.6%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

         \[\leadsto x \cdot \color{blue}{\frac{3}{2}} \]

      7. lower-*.f64 58.0

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

   7. Applied rewrites 58.0%

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

   if 1.70000000000000001e-18 < y

   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. rem-sqrt-square-rev N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. lift--.f64 N/A

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

      6. flip3-- N/A

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

      7. frac-times N/A

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

      8. sqrt-div N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 13.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 77.6

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

   7. Applied rewrites 77.6%

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

Alternative 7: 58.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-73}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-160}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7.2e-73) (* 0.5 x) (if (<= x 9.2e-160) (* 0.5 y) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7.2e-73) {
		tmp = 0.5 * x;
	} else if (x <= 9.2e-160) {
		tmp = 0.5 * 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 <= (-7.2d-73)) then
        tmp = 0.5d0 * x
    else if (x <= 9.2d-160) then
        tmp = 0.5d0 * 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 <= -7.2e-73) {
		tmp = 0.5 * x;
	} else if (x <= 9.2e-160) {
		tmp = 0.5 * y;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7.2e-73:
		tmp = 0.5 * x
	elif x <= 9.2e-160:
		tmp = 0.5 * y
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7.2e-73)
		tmp = Float64(0.5 * x);
	elseif (x <= 9.2e-160)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7.2e-73)
		tmp = 0.5 * x;
	elseif (x <= 9.2e-160)
		tmp = 0.5 * y;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7.2e-73], N[(0.5 * x), $MachinePrecision], If[LessEqual[x, 9.2e-160], N[(0.5 * y), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.1999999999999999e-73

   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. rem-sqrt-square-rev N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. lift--.f64 N/A

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

      6. flip3-- N/A

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

      7. frac-times N/A

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

      8. sqrt-div N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 1.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 16.1

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

   7. Applied rewrites 16.1%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{4} \cdot \frac{-1 \cdot y - y}{y}\right) + \frac{1}{2} \cdot y} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \frac{1}{4} \cdot \frac{-1 \cdot y - y}{y}\right) \cdot x} + \frac{1}{2} \cdot y \]

      3. div-sub N/A

         \[\leadsto \left(1 + \frac{1}{4} \cdot \color{blue}{\left(\frac{-1 \cdot y}{y} - \frac{y}{y}\right)}\right) \cdot x + \frac{1}{2} \cdot y \]

      4. associate-/l* N/A

         \[\leadsto \left(1 + \frac{1}{4} \cdot \left(\color{blue}{-1 \cdot \frac{y}{y}} - \frac{y}{y}\right)\right) \cdot x + \frac{1}{2} \cdot y \]

      5. *-inverses N/A

         \[\leadsto \left(1 + \frac{1}{4} \cdot \left(-1 \cdot \color{blue}{1} - \frac{y}{y}\right)\right) \cdot x + \frac{1}{2} \cdot y \]

      6. metadata-eval N/A

         \[\leadsto \left(1 + \frac{1}{4} \cdot \left(\color{blue}{-1} - \frac{y}{y}\right)\right) \cdot x + \frac{1}{2} \cdot y \]

      7. *-inverses N/A

         \[\leadsto \left(1 + \frac{1}{4} \cdot \left(-1 - \color{blue}{1}\right)\right) \cdot x + \frac{1}{2} \cdot y \]

      8. metadata-eval N/A

         \[\leadsto \left(1 + \frac{1}{4} \cdot \color{blue}{-2}\right) \cdot x + \frac{1}{2} \cdot y \]

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 84.7

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

   10. Applied rewrites 84.7%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 70.2%

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

   if -7.1999999999999999e-73 < x < 9.19999999999999939e-160

   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. rem-sqrt-square-rev N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. lift--.f64 N/A

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

      6. flip3-- N/A

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

      7. frac-times N/A

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

      8. sqrt-div N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 8.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 41.9

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

   7. Applied rewrites 41.9%

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

   if 9.19999999999999939e-160 < x

   1. Initial program 99.8%

      \[x + \frac{\left|y - x\right|}{2} \]
   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. unpow1 N/A

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

      4. metadata-eval N/A

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

      5. sqrt-pow1 N/A

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

      6. pow2 N/A

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

      7. sqrt-prod N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 79.0%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

         \[\leadsto x \cdot \color{blue}{\frac{3}{2}} \]

      7. lower-*.f64 63.2

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

   7. Applied rewrites 63.2%

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

Alternative 8: 30.2% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(0.5 * x), $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. rem-sqrt-square-rev N/A

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

   3. lift--.f64 N/A

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

   4. flip-- N/A

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

   5. lift--.f64 N/A

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

   6. flip3-- N/A

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

   7. frac-times N/A

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

   8. sqrt-div N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 11.7%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f64 27.0

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

7. Applied rewrites 27.0%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{4} \cdot \frac{-1 \cdot y - y}{y}\right) + \frac{1}{2} \cdot y} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(1 + \frac{1}{4} \cdot \frac{-1 \cdot y - y}{y}\right) \cdot x} + \frac{1}{2} \cdot y \]

   3. div-sub N/A

      \[\leadsto \left(1 + \frac{1}{4} \cdot \color{blue}{\left(\frac{-1 \cdot y}{y} - \frac{y}{y}\right)}\right) \cdot x + \frac{1}{2} \cdot y \]

   4. associate-/l* N/A

      \[\leadsto \left(1 + \frac{1}{4} \cdot \left(\color{blue}{-1 \cdot \frac{y}{y}} - \frac{y}{y}\right)\right) \cdot x + \frac{1}{2} \cdot y \]

   5. *-inverses N/A

      \[\leadsto \left(1 + \frac{1}{4} \cdot \left(-1 \cdot \color{blue}{1} - \frac{y}{y}\right)\right) \cdot x + \frac{1}{2} \cdot y \]

   6. metadata-eval N/A

      \[\leadsto \left(1 + \frac{1}{4} \cdot \left(\color{blue}{-1} - \frac{y}{y}\right)\right) \cdot x + \frac{1}{2} \cdot y \]

   7. *-inverses N/A

      \[\leadsto \left(1 + \frac{1}{4} \cdot \left(-1 - \color{blue}{1}\right)\right) \cdot x + \frac{1}{2} \cdot y \]

   8. metadata-eval N/A

      \[\leadsto \left(1 + \frac{1}{4} \cdot \color{blue}{-2}\right) \cdot x + \frac{1}{2} \cdot y \]

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-*.f64 51.6

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

10. Applied rewrites 51.6%

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

11. Taylor expanded in x around inf

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

13. Applied rewrites 27.0%

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

Reproduce

herbie shell --seed 2024329 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))