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

Percentage Accurate: 99.9% → 99.9%

Time: 4.6s

Alternatives: 7

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: 62.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-155}:\\ \;\;\;\;x + -0.5 \cdot y\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-77}:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left|-y\right| \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.8e-155)
   (+ x (* -0.5 y))
   (if (<= y 1.6e-77) (* 1.5 x) (* (fabs (- y)) 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.8e-155) {
		tmp = x + (-0.5 * y);
	} else if (y <= 1.6e-77) {
		tmp = 1.5 * x;
	} else {
		tmp = fabs(-y) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.8d-155)) then
        tmp = x + ((-0.5d0) * y)
    else if (y <= 1.6d-77) then
        tmp = 1.5d0 * x
    else
        tmp = abs(-y) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.8e-155) {
		tmp = x + (-0.5 * y);
	} else if (y <= 1.6e-77) {
		tmp = 1.5 * x;
	} else {
		tmp = Math.abs(-y) * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.8e-155:
		tmp = x + (-0.5 * y)
	elif y <= 1.6e-77:
		tmp = 1.5 * x
	else:
		tmp = math.fabs(-y) * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.8e-155)
		tmp = Float64(x + Float64(-0.5 * y));
	elseif (y <= 1.6e-77)
		tmp = Float64(1.5 * x);
	else
		tmp = Float64(abs(Float64(-y)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.8e-155)
		tmp = x + (-0.5 * y);
	elseif (y <= 1.6e-77)
		tmp = 1.5 * x;
	else
		tmp = abs(-y) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.8e-155], N[(x + N[(-0.5 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.6e-77], N[(1.5 * x), $MachinePrecision], N[(N[Abs[(-y)], $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.79999999999999994e-155

   1. Initial program 100.0%

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

   4. Applied rewrites 48.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 79.0

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

   7. Applied rewrites 79.0%

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

   if -1.79999999999999994e-155 < y < 1.6e-77

   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 53.0%

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

   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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 57.6

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

   7. Applied rewrites 57.6%

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

   if 1.6e-77 < y

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   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 72.3

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

   5. Applied rewrites 72.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 70.8%

      \[\leadsto \left|-y\right| \cdot 0.5 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 70.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-77}:\\ \;\;\;\;\mathsf{fma}\left(y - x, -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.5e-77) (fma (- y x) -0.5 x) (* (fabs (- y x)) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.5e-77) {
		tmp = fma((y - x), -0.5, x);
	} else {
		tmp = fabs((y - x)) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.5e-77)
		tmp = fma(Float64(y - x), -0.5, x);
	else
		tmp = Float64(abs(Float64(y - x)) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.5e-77], N[(N[(y - x), $MachinePrecision] * -0.5 + x), $MachinePrecision], N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.50000000000000008e-77

   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 72.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. rem-square-sqrt N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{-1}{2} \cdot y - \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2}}\right)\right) \cdot x \]

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

          \[\leadsto \color{blue}{\frac{-1}{2} \cdot y + \frac{3}{2} \cdot x} \]

      13. +-commutative N/A

          \[\leadsto \color{blue}{\frac{3}{2} \cdot x + \frac{-1}{2} \cdot y} \]

      14. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{3}{2}, x, \frac{-1}{2} \cdot y\right)} \]

      15. lower-*.f64 77.0

          \[\leadsto \mathsf{fma}\left(1.5, x, \color{blue}{-0.5 \cdot y}\right) \]

   7. Applied rewrites 77.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1.5, x, -0.5 \cdot y\right)} \]

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. unpow2 N/A

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

      6. rem-square-sqrt N/A

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

      7. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{3}{2}} \cdot x + y \cdot {\left(\sqrt{\frac{-1}{2}}\right)}^{2} \]

      8. metadata-eval N/A

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

      9. distribute-rgt1-in N/A

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

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

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

      11. metadata-eval N/A

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

      12. rem-square-sqrt N/A

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

      13. unpow2 N/A

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

      14. *-commutative N/A

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

      15. fp-cancel-sub-sign N/A

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

      16. mul-1-neg N/A

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

      17. associate-*r* N/A

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

   10. Applied rewrites 76.9%

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

   if 1.50000000000000008e-77 < y

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   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 72.3

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

   5. Applied rewrites 72.3%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-77}:\\ \;\;\;\;\mathsf{fma}\left(y - x, -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-77}:\\ \;\;\;\;\mathsf{fma}\left(y - x, -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left|-y\right| \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.6e-77) (fma (- y x) -0.5 x) (* (fabs (- y)) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.6e-77) {
		tmp = fma((y - x), -0.5, x);
	} else {
		tmp = fabs(-y) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.6e-77)
		tmp = fma(Float64(y - x), -0.5, x);
	else
		tmp = Float64(abs(Float64(-y)) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.6e-77], N[(N[(y - x), $MachinePrecision] * -0.5 + x), $MachinePrecision], N[(N[Abs[(-y)], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.6e-77

   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 72.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. rem-square-sqrt N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{-1}{2} \cdot y - \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2}}\right)\right) \cdot x \]

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

          \[\leadsto \color{blue}{\frac{-1}{2} \cdot y + \frac{3}{2} \cdot x} \]

      13. +-commutative N/A

          \[\leadsto \color{blue}{\frac{3}{2} \cdot x + \frac{-1}{2} \cdot y} \]

      14. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{3}{2}, x, \frac{-1}{2} \cdot y\right)} \]

      15. lower-*.f64 77.0

          \[\leadsto \mathsf{fma}\left(1.5, x, \color{blue}{-0.5 \cdot y}\right) \]

   7. Applied rewrites 77.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1.5, x, -0.5 \cdot y\right)} \]

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. unpow2 N/A

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

      6. rem-square-sqrt N/A

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

      7. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{3}{2}} \cdot x + y \cdot {\left(\sqrt{\frac{-1}{2}}\right)}^{2} \]

      8. metadata-eval N/A

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

      9. distribute-rgt1-in N/A

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

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

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

      11. metadata-eval N/A

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

      12. rem-square-sqrt N/A

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

      13. unpow2 N/A

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

      14. *-commutative N/A

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

      15. fp-cancel-sub-sign N/A

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

      16. mul-1-neg N/A

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

      17. associate-*r* N/A

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

   10. Applied rewrites 76.9%

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

   if 1.6e-77 < y

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   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 72.3

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

   5. Applied rewrites 72.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 70.8%

      \[\leadsto \left|-y\right| \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 47.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-155}:\\ \;\;\;\;x + -0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.8e-155) (+ x (* -0.5 y)) (* 1.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.8e-155) {
		tmp = x + (-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 <= (-1.8d-155)) then
        tmp = x + ((-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 <= -1.8e-155) {
		tmp = x + (-0.5 * y);
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.8e-155:
		tmp = x + (-0.5 * y)
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.8e-155)
		tmp = Float64(x + 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 <= -1.8e-155)
		tmp = x + (-0.5 * y);
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.8e-155], N[(x + N[(-0.5 * y), $MachinePrecision]), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.79999999999999994e-155

   1. Initial program 100.0%

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

   4. Applied rewrites 48.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 79.0

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

   7. Applied rewrites 79.0%

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

   if -1.79999999999999994e-155 < 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. 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 31.2%

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

   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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 35.8

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

   7. Applied rewrites 35.8%

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

Alternative 6: 45.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-155}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1.8e-155) (* -0.5 y) (* 1.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.8e-155) {
		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 <= (-1.8d-155)) 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 <= -1.8e-155) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.8e-155:
		tmp = -0.5 * y
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.8e-155)
		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 <= -1.8e-155)
		tmp = -0.5 * y;
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.8e-155], N[(-0.5 * y), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.79999999999999994e-155

   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. 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 89.9%

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

   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. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 75.2

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

   7. Applied rewrites 75.2%

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

   if -1.79999999999999994e-155 < 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. 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 31.2%

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

   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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 35.8

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

   7. Applied rewrites 35.8%

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

Alternative 7: 26.4% 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-+.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 50.5%

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

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. *-commutative N/A

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

   2. unpow2 N/A

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

   3. rem-square-sqrt N/A

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

   4. lower-*.f64 26.6

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

7. Applied rewrites 26.6%

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

Reproduce

herbie shell --seed 2024320 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))