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

Percentage Accurate: 99.9% → 99.9%

Time: 7.8s

Alternatives: 11

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. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. div-inv N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-fabs.f64 N/A

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

   7. neg-fabs N/A

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

   8. lower-fabs.f64 N/A

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

   9. lift--.f64 N/A

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

   10. sub-neg N/A

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

   11. +-commutative N/A

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

   12. distribute-neg-in N/A

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

   13. remove-double-neg N/A

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

   14. sub-neg N/A

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

   15. lower--.f64 N/A

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

   16. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 49.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left|y - x\right|}{2} + x \leq 4 \cdot 10^{-240}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (/ (fabs (- y x)) 2.0) x) 4e-240) (* 0.5 x) (* 0.5 y)))

C

double code(double x, double y) {
	double tmp;
	if (((fabs((y - x)) / 2.0) + x) <= 4e-240) {
		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 (((abs((y - x)) / 2.0d0) + x) <= 4d-240) 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 (((Math.abs((y - x)) / 2.0) + x) <= 4e-240) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((math.fabs((y - x)) / 2.0) + x) <= 4e-240:
		tmp = 0.5 * x
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(abs(Float64(y - x)) / 2.0) + x) <= 4e-240)
		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 (((abs((y - x)) / 2.0) + x) <= 4e-240)
		tmp = 0.5 * x;
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision] + x), $MachinePrecision], 4e-240], 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))) < 3.9999999999999999e-240

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. neg-fabs N/A

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

      8. lower-fabs.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. unpow1 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-pow N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

   6. Applied rewrites 46.4%

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

   7. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 89.8

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

   9. Applied rewrites 89.8%

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

   if 3.9999999999999999e-240 < (+.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-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. neg-fabs N/A

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

      8. lower-fabs.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-fma.f64 N/A

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

      2. unpow1 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-pow N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

   6. Applied rewrites 49.0%

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

   7. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

   9. Applied rewrites 31.4%

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

   10. Taylor expanded in y around inf

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

   12. Applied rewrites 32.4%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left|y - x\right|}{2} + x \leq 4 \cdot 10^{-240}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 58.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-129}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 10^{-262}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-35}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.8e-129)
   (* 0.5 x)
   (if (<= x 1e-262) (* -0.5 y) (if (<= x 2.5e-35) (* 0.5 y) (* 1.5 x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.8e-129) {
		tmp = 0.5 * x;
	} else if (x <= 1e-262) {
		tmp = -0.5 * y;
	} else if (x <= 2.5e-35) {
		tmp = 0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.8d-129)) then
        tmp = 0.5d0 * x
    else if (x <= 1d-262) then
        tmp = (-0.5d0) * y
    else if (x <= 2.5d-35) then
        tmp = 0.5d0 * y
    else
        tmp = 1.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.8e-129) {
		tmp = 0.5 * x;
	} else if (x <= 1e-262) {
		tmp = -0.5 * y;
	} else if (x <= 2.5e-35) {
		tmp = 0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.8e-129:
		tmp = 0.5 * x
	elif x <= 1e-262:
		tmp = -0.5 * y
	elif x <= 2.5e-35:
		tmp = 0.5 * y
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.8e-129)
		tmp = Float64(0.5 * x);
	elseif (x <= 1e-262)
		tmp = Float64(-0.5 * y);
	elseif (x <= 2.5e-35)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(1.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.8e-129)
		tmp = 0.5 * x;
	elseif (x <= 1e-262)
		tmp = -0.5 * y;
	elseif (x <= 2.5e-35)
		tmp = 0.5 * y;
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.8e-129], N[(0.5 * x), $MachinePrecision], If[LessEqual[x, 1e-262], N[(-0.5 * y), $MachinePrecision], If[LessEqual[x, 2.5e-35], N[(0.5 * y), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.7999999999999999e-129

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. neg-fabs N/A

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

      8. lower-fabs.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. unpow1 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-pow N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

   6. Applied rewrites 43.0%

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

   7. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 75.7

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

   9. Applied rewrites 75.7%

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

   if -2.7999999999999999e-129 < x < 1.00000000000000001e-262

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. neg-fabs N/A

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

      8. lower-fabs.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fabs.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqr-pow N/A

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

      5. fabs-sqr N/A

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

      6. sqr-pow N/A

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

      7. metadata-eval N/A

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

      8. unpow1 59.8

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

   6. Applied rewrites 59.8%

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

   7. Taylor expanded in y around inf

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

      1. lower-*.f64 56.7

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

   9. Applied rewrites 56.7%

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

   if 1.00000000000000001e-262 < x < 2.49999999999999982e-35

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. neg-fabs N/A

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

      8. lower-fabs.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. unpow1 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-pow N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

   6. Applied rewrites 58.9%

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

   7. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

   9. Applied rewrites 35.3%

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

   10. Taylor expanded in y around inf

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

   12. Applied rewrites 48.4%

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

   if 2.49999999999999982e-35 < 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. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. neg-fabs N/A

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

      8. lower-fabs.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-fabs.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqr-pow N/A

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

      5. fabs-sqr N/A

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

      6. sqr-pow N/A

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

      7. metadata-eval N/A

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

      8. unpow1 88.0

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

   6. Applied rewrites 88.0%

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

   7. Taylor expanded in y around 0

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 65.6

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

   9. Applied rewrites 65.6%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-129}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 10^{-262}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-35}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-127}:\\ \;\;\;\;\left(y + x\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(\left|-y\right|, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1.5, x, -0.5 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.3e-127)
   (* (+ y x) 0.5)
   (if (<= x 4.6e-35) (fma (fabs (- y)) 0.5 x) (fma 1.5 x (* -0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.3e-127) {
		tmp = (y + x) * 0.5;
	} else if (x <= 4.6e-35) {
		tmp = fma(fabs(-y), 0.5, x);
	} else {
		tmp = fma(1.5, x, (-0.5 * y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.3e-127)
		tmp = Float64(Float64(y + x) * 0.5);
	elseif (x <= 4.6e-35)
		tmp = fma(abs(Float64(-y)), 0.5, x);
	else
		tmp = fma(1.5, x, Float64(-0.5 * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.3e-127], N[(N[(y + x), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 4.6e-35], N[(N[Abs[(-y)], $MachinePrecision] * 0.5 + x), $MachinePrecision], N[(1.5 * x + N[(-0.5 * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.30000000000000019e-127

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. neg-fabs N/A

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

      8. lower-fabs.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. unpow1 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-pow N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

   6. Applied rewrites 43.0%

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

   7. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 75.7

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

   9. Applied rewrites 75.7%

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

   10. Taylor expanded in x around -inf

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

   12. Applied rewrites 89.8%

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

   if -2.30000000000000019e-127 < x < 4.5999999999999998e-35

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. neg-fabs N/A

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

      8. lower-fabs.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 88.4

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

   7. Applied rewrites 88.4%

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

   if 4.5999999999999998e-35 < 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. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. neg-fabs N/A

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

      8. lower-fabs.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-fabs.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqr-pow N/A

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

      5. fabs-sqr N/A

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

      6. sqr-pow N/A

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

      7. metadata-eval N/A

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

      8. unpow1 88.0

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

   6. Applied rewrites 88.0%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 88.2

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

   9. Applied rewrites 88.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1.5, x, -0.5 \cdot y\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 84.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-127}:\\ \;\;\;\;\left(y + x\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(\left|-y\right|, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - y, 0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.3e-127)
   (* (+ y x) 0.5)
   (if (<= x 4.6e-35) (fma (fabs (- y)) 0.5 x) (fma (- x y) 0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.3e-127) {
		tmp = (y + x) * 0.5;
	} else if (x <= 4.6e-35) {
		tmp = fma(fabs(-y), 0.5, x);
	} else {
		tmp = fma((x - y), 0.5, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.3e-127)
		tmp = Float64(Float64(y + x) * 0.5);
	elseif (x <= 4.6e-35)
		tmp = fma(abs(Float64(-y)), 0.5, x);
	else
		tmp = fma(Float64(x - y), 0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.3e-127], N[(N[(y + x), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 4.6e-35], N[(N[Abs[(-y)], $MachinePrecision] * 0.5 + x), $MachinePrecision], N[(N[(x - y), $MachinePrecision] * 0.5 + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.30000000000000019e-127

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. neg-fabs N/A

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

      8. lower-fabs.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. unpow1 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-pow N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

   6. Applied rewrites 43.0%

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

   7. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 75.7

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

   9. Applied rewrites 75.7%

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

   10. Taylor expanded in x around -inf

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

   12. Applied rewrites 89.8%

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

   if -2.30000000000000019e-127 < x < 4.5999999999999998e-35

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. neg-fabs N/A

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

      8. lower-fabs.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 88.4

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

   7. Applied rewrites 88.4%

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

   if 4.5999999999999998e-35 < 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. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. neg-fabs N/A

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

      8. lower-fabs.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-fabs.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqr-pow N/A

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

      5. fabs-sqr N/A

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

      6. sqr-pow N/A

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

      7. metadata-eval N/A

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

      8. unpow1 88.0

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

   6. Applied rewrites 88.0%

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

Alternative 6: 83.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-129}:\\ \;\;\;\;\left(y + x\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-35}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - y, 0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.8e-129)
   (* (+ y x) 0.5)
   (if (<= x 4.6e-35) (* (fabs (- y x)) 0.5) (fma (- x y) 0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.8e-129) {
		tmp = (y + x) * 0.5;
	} else if (x <= 4.6e-35) {
		tmp = fabs((y - x)) * 0.5;
	} else {
		tmp = fma((x - y), 0.5, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.8e-129)
		tmp = Float64(Float64(y + x) * 0.5);
	elseif (x <= 4.6e-35)
		tmp = Float64(abs(Float64(y - x)) * 0.5);
	else
		tmp = fma(Float64(x - y), 0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.8e-129], N[(N[(y + x), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 4.6e-35], N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(x - y), $MachinePrecision] * 0.5 + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.7999999999999999e-129

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. neg-fabs N/A

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

      8. lower-fabs.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. unpow1 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-pow N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

   6. Applied rewrites 43.0%

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

   7. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 75.7

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

   9. Applied rewrites 75.7%

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

   10. Taylor expanded in x around -inf

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

   12. Applied rewrites 89.8%

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

   if -2.7999999999999999e-129 < x < 4.5999999999999998e-35

   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. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

      9. +-commutative N/A

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

      10. lower-fabs.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

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

      14. remove-double-neg N/A

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

      15. sub-neg N/A

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

      16. lower--.f64 87.5

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

   5. Applied rewrites 87.5%

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

   if 4.5999999999999998e-35 < 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. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. neg-fabs N/A

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

      8. lower-fabs.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-fabs.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqr-pow N/A

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

      5. fabs-sqr N/A

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

      6. sqr-pow N/A

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

      7. metadata-eval N/A

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

      8. unpow1 88.0

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

   6. Applied rewrites 88.0%

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

Alternative 7: 78.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-254}:\\ \;\;\;\;0.5 \cdot \left(x - y\right)\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-271}:\\ \;\;\;\;\mathsf{fma}\left(x - y, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.2e-254)
   (* 0.5 (- x y))
   (if (<= y 1.7e-271) (fma (- x y) 0.5 x) (* (+ y x) 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.2e-254) {
		tmp = 0.5 * (x - y);
	} else if (y <= 1.7e-271) {
		tmp = fma((x - y), 0.5, x);
	} else {
		tmp = (y + x) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.2e-254)
		tmp = Float64(0.5 * Float64(x - y));
	elseif (y <= 1.7e-271)
		tmp = fma(Float64(x - y), 0.5, x);
	else
		tmp = Float64(Float64(y + x) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.2e-254], N[(0.5 * N[(x - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.7e-271], N[(N[(x - y), $MachinePrecision] * 0.5 + x), $MachinePrecision], N[(N[(y + x), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.20000000000000001e-254

   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. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

      9. +-commutative N/A

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

      10. lower-fabs.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

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

      14. remove-double-neg N/A

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

      15. sub-neg N/A

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

      16. lower--.f64 58.3

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

   5. Applied rewrites 58.3%

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

   7. Applied rewrites 83.1%

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

   if -1.20000000000000001e-254 < y < 1.7e-271

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. neg-fabs N/A

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

      8. lower-fabs.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-fabs.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqr-pow N/A

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

      5. fabs-sqr N/A

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

      6. sqr-pow N/A

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

      7. metadata-eval N/A

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

      8. unpow1 72.5

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

   6. Applied rewrites 72.5%

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

   if 1.7e-271 < y

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. neg-fabs N/A

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

      8. lower-fabs.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-fma.f64 N/A

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

      2. unpow1 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-pow N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

   6. Applied rewrites 46.5%

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

   7. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 34.7

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

   9. Applied rewrites 34.7%

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

   10. Taylor expanded in x around -inf

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

   12. Applied rewrites 81.9%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-254}:\\ \;\;\;\;0.5 \cdot \left(x - y\right)\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-271}:\\ \;\;\;\;\mathsf{fma}\left(x - y, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{-257}:\\ \;\;\;\;0.5 \cdot \left(x - y\right)\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-295}:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.6e-257)
   (* 0.5 (- x y))
   (if (<= y 1.25e-295) (* 1.5 x) (* (+ y x) 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.6e-257) {
		tmp = 0.5 * (x - y);
	} else if (y <= 1.25e-295) {
		tmp = 1.5 * x;
	} else {
		tmp = (y + x) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-4.6d-257)) then
        tmp = 0.5d0 * (x - y)
    else if (y <= 1.25d-295) then
        tmp = 1.5d0 * x
    else
        tmp = (y + x) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.6e-257) {
		tmp = 0.5 * (x - y);
	} else if (y <= 1.25e-295) {
		tmp = 1.5 * x;
	} else {
		tmp = (y + x) * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.6e-257:
		tmp = 0.5 * (x - y)
	elif y <= 1.25e-295:
		tmp = 1.5 * x
	else:
		tmp = (y + x) * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.6e-257)
		tmp = Float64(0.5 * Float64(x - y));
	elseif (y <= 1.25e-295)
		tmp = Float64(1.5 * x);
	else
		tmp = Float64(Float64(y + x) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.6e-257)
		tmp = 0.5 * (x - y);
	elseif (y <= 1.25e-295)
		tmp = 1.5 * x;
	else
		tmp = (y + x) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.6e-257], N[(0.5 * N[(x - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.25e-295], N[(1.5 * x), $MachinePrecision], N[(N[(y + x), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.6e-257

   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. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

      9. +-commutative N/A

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

      10. lower-fabs.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

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

      14. remove-double-neg N/A

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

      15. sub-neg N/A

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

      16. lower--.f64 58.3

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

   5. Applied rewrites 58.3%

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

   7. Applied rewrites 83.1%

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

   if -4.6e-257 < y < 1.25000000000000002e-295

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. neg-fabs N/A

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

      8. lower-fabs.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-fabs.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqr-pow N/A

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

      5. fabs-sqr N/A

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

      6. sqr-pow N/A

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

      7. metadata-eval N/A

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

      8. unpow1 80.9

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

   6. Applied rewrites 80.9%

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

   7. Taylor expanded in y around 0

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 73.8

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

   9. Applied rewrites 73.8%

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

   if 1.25000000000000002e-295 < y

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. neg-fabs N/A

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

      8. lower-fabs.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-fma.f64 N/A

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

      2. unpow1 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-pow N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

   6. Applied rewrites 46.9%

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

   7. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 36.2

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

   9. Applied rewrites 36.2%

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

   10. Taylor expanded in x around -inf

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

   12. Applied rewrites 80.4%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{-257}:\\ \;\;\;\;0.5 \cdot \left(x - y\right)\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-295}:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-72}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{+27}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.2e-72) (* -0.5 y) (if (<= y 2.45e+27) (* 0.5 x) (* 0.5 y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.2e-72) {
		tmp = -0.5 * y;
	} else if (y <= 2.45e+27) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-7.2d-72)) then
        tmp = (-0.5d0) * y
    else if (y <= 2.45d+27) then
        tmp = 0.5d0 * x
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7.2e-72) {
		tmp = -0.5 * y;
	} else if (y <= 2.45e+27) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.2e-72:
		tmp = -0.5 * y
	elif y <= 2.45e+27:
		tmp = 0.5 * x
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.2e-72)
		tmp = Float64(-0.5 * y);
	elseif (y <= 2.45e+27)
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7.2e-72)
		tmp = -0.5 * y;
	elseif (y <= 2.45e+27)
		tmp = 0.5 * x;
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.2e-72], N[(-0.5 * y), $MachinePrecision], If[LessEqual[y, 2.45e+27], N[(0.5 * x), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.2e-72

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. neg-fabs N/A

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

      8. lower-fabs.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-fabs.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqr-pow N/A

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

      5. fabs-sqr N/A

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

      6. sqr-pow N/A

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

      7. metadata-eval N/A

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

      8. unpow1 85.2

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

   6. Applied rewrites 85.2%

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

   7. Taylor expanded in y around inf

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

      1. lower-*.f64 68.6

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

   9. Applied rewrites 68.6%

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

   if -7.2e-72 < y < 2.45000000000000007e27

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. neg-fabs N/A

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

      8. lower-fabs.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-fma.f64 N/A

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

      2. unpow1 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-pow N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

   6. Applied rewrites 57.6%

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

   7. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 46.6

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

   9. Applied rewrites 46.6%

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

   if 2.45000000000000007e27 < y

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. neg-fabs N/A

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

      8. lower-fabs.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. unpow1 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-pow N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

   6. Applied rewrites 31.8%

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

   7. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

   9. Applied rewrites 70.1%

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

   10. Taylor expanded in y around inf

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

   12. Applied rewrites 68.2%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-72}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{+27}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.45 \cdot 10^{+27}:\\ \;\;\;\;0.5 \cdot \left(x - y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.45e+27) (* 0.5 (- x y)) (* 0.5 y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.45e+27) {
		tmp = 0.5 * (x - y);
	} 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 <= 2.45d+27) then
        tmp = 0.5d0 * (x - y)
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.45e+27) {
		tmp = 0.5 * (x - y);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.45e+27:
		tmp = 0.5 * (x - y)
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.45e+27)
		tmp = Float64(0.5 * Float64(x - y));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.45e+27)
		tmp = 0.5 * (x - y);
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.45e+27], N[(0.5 * N[(x - y), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.45000000000000007e27

   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. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

      9. +-commutative N/A

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

      10. lower-fabs.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

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

      14. remove-double-neg N/A

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

      15. sub-neg N/A

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

      16. lower--.f64 46.0

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

   5. Applied rewrites 46.0%

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

   7. Applied rewrites 67.6%

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

   if 2.45000000000000007e27 < y

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. neg-fabs N/A

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

      8. lower-fabs.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. unpow1 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-pow N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

   6. Applied rewrites 31.8%

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

   7. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

   9. Applied rewrites 70.1%

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

   10. Taylor expanded in y around inf

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

   12. Applied rewrites 68.2%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.45 \cdot 10^{+27}:\\ \;\;\;\;0.5 \cdot \left(x - y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 27.1% 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. lift-/.f64 N/A

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

   4. div-inv N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-fabs.f64 N/A

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

   7. neg-fabs N/A

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

   8. lower-fabs.f64 N/A

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

   9. lift--.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left|\mathsf{neg}\left(\color{blue}{\left(y - x\right)}\right)\right|, \frac{1}{2}, x\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(\left|\mathsf{neg}\left(\color{blue}{\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)}\right)\right|, \frac{1}{2}, x\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\left|\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)}\right)\right|, \frac{1}{2}, x\right) \]

   12. distribute-neg-in N/A

       \[\leadsto \mathsf{fma}\left(\left|\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}\right|, \frac{1}{2}, x\right) \]

   13. remove-double-neg N/A

       \[\leadsto \mathsf{fma}\left(\left|\color{blue}{x} + \left(\mathsf{neg}\left(y\right)\right)\right|, \frac{1}{2}, x\right) \]

   14. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(\left|\color{blue}{x - y}\right|, \frac{1}{2}, x\right) \]

   15. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left|\color{blue}{x - y}\right|, \frac{1}{2}, x\right) \]

   16. metadata-eval 99.9

       \[\leadsto \mathsf{fma}\left(\left|x - y\right|, \color{blue}{0.5}, x\right) \]

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left|x - y\right|, 0.5, x\right)} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto \color{blue}{\left|x - y\right| \cdot \frac{1}{2} + x} \]

   2. unpow1 N/A

      \[\leadsto \color{blue}{{\left(\left|x - y\right| \cdot \frac{1}{2}\right)}^{1}} + x \]

   3. *-commutative N/A

      \[\leadsto {\color{blue}{\left(\frac{1}{2} \cdot \left|x - y\right|\right)}}^{1} + x \]

   4. lift-*.f64 N/A

      \[\leadsto {\color{blue}{\left(\frac{1}{2} \cdot \left|x - y\right|\right)}}^{1} + x \]

   5. metadata-eval N/A

      \[\leadsto {\left(\frac{1}{2} \cdot \left|x - y\right|\right)}^{\color{blue}{\left(\frac{1}{2} \cdot 2\right)}} + x \]

   6. pow-pow N/A

      \[\leadsto \color{blue}{{\left({\left(\frac{1}{2} \cdot \left|x - y\right|\right)}^{\frac{1}{2}}\right)}^{2}} + x \]

   7. lift-pow.f64 N/A

      \[\leadsto {\color{blue}{\left({\left(\frac{1}{2} \cdot \left|x - y\right|\right)}^{\frac{1}{2}}\right)}}^{2} + x \]

   8. unpow2 N/A

      \[\leadsto \color{blue}{{\left(\frac{1}{2} \cdot \left|x - y\right|\right)}^{\frac{1}{2}} \cdot {\left(\frac{1}{2} \cdot \left|x - y\right|\right)}^{\frac{1}{2}}} + x \]

6. Applied rewrites 48.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\left(0.5 \cdot \left(x - y\right)\right) \cdot \left(x - y\right)}, \sqrt{0.5}, x\right)} \]

7. Taylor expanded in x around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\left(-1 \cdot \frac{y \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{2}}{x} + {\left(\sqrt{\frac{1}{2}}\right)}^{2}\right) - 1\right)\right)} \]
8. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(\left(-1 \cdot \frac{y \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{2}}{x} + {\left(\sqrt{\frac{1}{2}}\right)}^{2}\right) - 1\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\left(-1 \cdot \frac{y \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{2}}{x} + {\left(\sqrt{\frac{1}{2}}\right)}^{2}\right) - 1\right) \cdot x}\right) \]

   3. distribute-lft-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\left(-1 \cdot \frac{y \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{2}}{x} + {\left(\sqrt{\frac{1}{2}}\right)}^{2}\right) - 1\right)\right)\right) \cdot x} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\left(-1 \cdot \frac{y \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{2}}{x} + {\left(\sqrt{\frac{1}{2}}\right)}^{2}\right) - 1\right)\right)\right) \cdot x} \]

9. Applied rewrites 50.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.5}{x}, y, 0.5\right) \cdot x} \]

10. Taylor expanded in y around inf

    \[\leadsto \frac{1}{2} \cdot \color{blue}{y} \]
11. Step-by-step derivation

12. Applied rewrites 23.5%

    \[\leadsto y \cdot \color{blue}{0.5} \]

13. Final simplification 23.5%

    \[\leadsto 0.5 \cdot y \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024270 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))