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

Percentage Accurate: 99.9% → 99.9%

Time: 6.7s

Alternatives: 9

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: 74.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left|y - x\right|}{2} + x \leq 5 \cdot 10^{-225}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, -0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (/ (fabs (- y x)) 2.0) x) 5e-225) (* 0.5 x) (fma (- y x) -0.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (((fabs((y - x)) / 2.0) + x) <= 5e-225) {
		tmp = 0.5 * x;
	} else {
		tmp = fma((y - x), -0.5, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(abs(Float64(y - x)) / 2.0) + x) <= 5e-225)
		tmp = Float64(0.5 * x);
	else
		tmp = fma(Float64(y - x), -0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision] + x), $MachinePrecision], 5e-225], N[(0.5 * x), $MachinePrecision], N[(N[(y - x), $MachinePrecision] * -0.5 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (fabs.f64 (-.f64 y x)) #s(literal 2 binary64))) < 5.0000000000000001e-225

   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 2.7

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

   5. Applied rewrites 2.7%

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

   7. Applied rewrites 98.1%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 98.2%

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

   if 5.0000000000000001e-225 < (+.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. metadata-eval N/A

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

      3. div-inv N/A

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

      4. frac-2neg N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-fabs.f64 N/A

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

      10. unpow1 N/A

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

      11. sqr-pow N/A

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

      12. fabs-sqr N/A

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

      13. sqr-pow N/A

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

      14. unpow1 N/A

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

      15. lift--.f64 N/A

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

      16. sub-neg N/A

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

      17. distribute-neg-in N/A

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

      18. remove-double-neg N/A

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

      19. +-commutative N/A

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

      20. sub-neg N/A

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

      21. lift--.f64 69.8

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

   6. Applied rewrites 69.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, -0.5, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left|y - x\right|}{2} + x \leq 5 \cdot 10^{-225}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, -0.5, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left|y - x\right|}{2} + x \leq 5 \cdot 10^{-225}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (/ (fabs (- y x)) 2.0) x) 5e-225) (* 0.5 x) (* -0.5 y)))

C

double code(double x, double y) {
	double tmp;
	if (((fabs((y - x)) / 2.0) + x) <= 5e-225) {
		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) <= 5d-225) 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) <= 5e-225) {
		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) <= 5e-225:
		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) <= 5e-225)
		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) <= 5e-225)
		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], 5e-225], N[(0.5 * x), $MachinePrecision], N[(-0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (fabs.f64 (-.f64 y x)) #s(literal 2 binary64))) < 5.0000000000000001e-225

   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 2.7

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

   5. Applied rewrites 2.7%

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

   7. Applied rewrites 98.1%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 98.2%

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

   if 5.0000000000000001e-225 < (+.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. metadata-eval N/A

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

      3. div-inv N/A

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

      4. frac-2neg N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-fabs.f64 N/A

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

      10. unpow1 N/A

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

      11. sqr-pow N/A

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

      12. fabs-sqr N/A

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

      13. sqr-pow N/A

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

      14. unpow1 N/A

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

      15. lift--.f64 N/A

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

      16. sub-neg N/A

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

      17. distribute-neg-in N/A

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

      18. remove-double-neg N/A

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

      19. +-commutative N/A

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

      20. sub-neg N/A

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

      21. lift--.f64 69.8

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

   6. Applied rewrites 69.8%

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

   7. Taylor expanded in y around inf

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

      1. lower-*.f64 34.9

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

   9. Applied rewrites 34.9%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left|y - x\right|}{2} + x \leq 5 \cdot 10^{-225}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-39}:\\ \;\;\;\;-0.5 \cdot \left(y - x\right)\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-41}:\\ \;\;\;\;\mathsf{fma}\left(\left|-y\right|, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1.5, x, -0.5 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2e-39)
   (* -0.5 (- y x))
   (if (<= x 5.2e-41) (fma (fabs (- y)) 0.5 x) (fma 1.5 x (* -0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2e-39) {
		tmp = -0.5 * (y - x);
	} else if (x <= 5.2e-41) {
		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 <= -2e-39)
		tmp = Float64(-0.5 * Float64(y - x));
	elseif (x <= 5.2e-41)
		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, -2e-39], N[(-0.5 * N[(y - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.2e-41], 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 < -1.99999999999999986e-39

   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 28.5

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

   5. Applied rewrites 28.5%

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

   7. Applied rewrites 84.8%

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

   if -1.99999999999999986e-39 < x < 5.1999999999999999e-41

   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 82.6

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

   7. Applied rewrites 82.6%

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

   if 5.1999999999999999e-41 < x

   1. Initial program 99.7%

      \[x + \frac{\left|y - x\right|}{2} \]
   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 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-fma.f64 N/A

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

      2. metadata-eval N/A

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

      3. div-inv N/A

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

      4. frac-2neg N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-fabs.f64 N/A

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

      10. unpow1 N/A

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

      11. sqr-pow N/A

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

      12. fabs-sqr N/A

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

      13. sqr-pow N/A

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

      14. unpow1 N/A

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

      15. lift--.f64 N/A

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

      16. sub-neg N/A

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

      17. distribute-neg-in N/A

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

      18. remove-double-neg N/A

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

      19. +-commutative N/A

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

      20. sub-neg N/A

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

      21. lift--.f64 86.6

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

   6. Applied rewrites 86.6%

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

   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 86.9

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

   9. Applied rewrites 86.9%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-39}:\\ \;\;\;\;-0.5 \cdot \left(y - x\right)\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-41}:\\ \;\;\;\;\mathsf{fma}\left(\left|-y\right|, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1.5, x, -0.5 \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-39}:\\ \;\;\;\;-0.5 \cdot \left(y - x\right)\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-41}:\\ \;\;\;\;\mathsf{fma}\left(\left|-y\right|, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, -0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2e-39)
   (* -0.5 (- y x))
   (if (<= x 5.2e-41) (fma (fabs (- y)) 0.5 x) (fma (- y x) -0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2e-39) {
		tmp = -0.5 * (y - x);
	} else if (x <= 5.2e-41) {
		tmp = fma(fabs(-y), 0.5, x);
	} else {
		tmp = fma((y - x), -0.5, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2e-39)
		tmp = Float64(-0.5 * Float64(y - x));
	elseif (x <= 5.2e-41)
		tmp = fma(abs(Float64(-y)), 0.5, x);
	else
		tmp = fma(Float64(y - x), -0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2e-39], N[(-0.5 * N[(y - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.2e-41], N[(N[Abs[(-y)], $MachinePrecision] * 0.5 + x), $MachinePrecision], N[(N[(y - x), $MachinePrecision] * -0.5 + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.99999999999999986e-39

   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 28.5

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

   5. Applied rewrites 28.5%

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

   7. Applied rewrites 84.8%

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

   if -1.99999999999999986e-39 < x < 5.1999999999999999e-41

   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 82.6

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

   7. Applied rewrites 82.6%

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

   if 5.1999999999999999e-41 < x

   1. Initial program 99.7%

      \[x + \frac{\left|y - x\right|}{2} \]
   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 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-fma.f64 N/A

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

      2. metadata-eval N/A

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

      3. div-inv N/A

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

      4. frac-2neg N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-fabs.f64 N/A

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

      10. unpow1 N/A

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

      11. sqr-pow N/A

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

      12. fabs-sqr N/A

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

      13. sqr-pow N/A

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

      14. unpow1 N/A

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

      15. lift--.f64 N/A

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

      16. sub-neg N/A

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

      17. distribute-neg-in N/A

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

      18. remove-double-neg N/A

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

      19. +-commutative N/A

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

      20. sub-neg N/A

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

      21. lift--.f64 86.6

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

   6. Applied rewrites 86.6%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-39}:\\ \;\;\;\;-0.5 \cdot \left(y - x\right)\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-41}:\\ \;\;\;\;\mathsf{fma}\left(\left|-y\right|, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, -0.5, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-40}:\\ \;\;\;\;-0.5 \cdot \left(y - x\right)\\ \mathbf{elif}\;x \leq 10^{-211}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, -0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.2e-40)
   (* -0.5 (- y x))
   (if (<= x 1e-211) (* (fabs (- y x)) 0.5) (fma (- y x) -0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.2e-40) {
		tmp = -0.5 * (y - x);
	} else if (x <= 1e-211) {
		tmp = fabs((y - x)) * 0.5;
	} else {
		tmp = fma((y - x), -0.5, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.2e-40)
		tmp = Float64(-0.5 * Float64(y - x));
	elseif (x <= 1e-211)
		tmp = Float64(abs(Float64(y - x)) * 0.5);
	else
		tmp = fma(Float64(y - x), -0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.2e-40], N[(-0.5 * N[(y - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1e-211], N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(y - x), $MachinePrecision] * -0.5 + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.20000000000000009e-40

   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 28.5

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

   5. Applied rewrites 28.5%

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

   7. Applied rewrites 84.8%

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

   if -2.20000000000000009e-40 < x < 1.00000000000000009e-211

   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 86.2

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

   5. Applied rewrites 86.2%

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

   if 1.00000000000000009e-211 < x

   1. Initial program 99.7%

      \[x + \frac{\left|y - x\right|}{2} \]
   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 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-fma.f64 N/A

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

      2. metadata-eval N/A

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

      3. div-inv N/A

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

      4. frac-2neg N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-fabs.f64 N/A

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

      10. unpow1 N/A

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

      11. sqr-pow N/A

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

      12. fabs-sqr N/A

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

      13. sqr-pow N/A

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

      14. unpow1 N/A

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

      15. lift--.f64 N/A

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

      16. sub-neg N/A

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

      17. distribute-neg-in N/A

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

      18. remove-double-neg N/A

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

      19. +-commutative N/A

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

      20. sub-neg N/A

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

      21. lift--.f64 81.2

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

   6. Applied rewrites 81.2%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-40}:\\ \;\;\;\;-0.5 \cdot \left(y - x\right)\\ \mathbf{elif}\;x \leq 10^{-211}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, -0.5, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{-74}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-220}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -9.2e-74) (* 0.5 x) (if (<= x 1.35e-220) (* -0.5 y) (* 1.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -9.2e-74) {
		tmp = 0.5 * x;
	} else if (x <= 1.35e-220) {
		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 <= (-9.2d-74)) then
        tmp = 0.5d0 * x
    else if (x <= 1.35d-220) 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 <= -9.2e-74) {
		tmp = 0.5 * x;
	} else if (x <= 1.35e-220) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -9.2e-74:
		tmp = 0.5 * x
	elif x <= 1.35e-220:
		tmp = -0.5 * y
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -9.2e-74)
		tmp = Float64(0.5 * x);
	elseif (x <= 1.35e-220)
		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 <= -9.2e-74)
		tmp = 0.5 * x;
	elseif (x <= 1.35e-220)
		tmp = -0.5 * y;
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -9.2e-74], N[(0.5 * x), $MachinePrecision], If[LessEqual[x, 1.35e-220], N[(-0.5 * y), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.19999999999999922e-74

   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 32.7

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

   5. Applied rewrites 32.7%

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

   7. Applied rewrites 80.4%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 66.2%

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

   if -9.19999999999999922e-74 < x < 1.35e-220

   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. metadata-eval N/A

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

      3. div-inv N/A

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

      4. frac-2neg N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-fabs.f64 N/A

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

      10. unpow1 N/A

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

      11. sqr-pow N/A

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

      12. fabs-sqr N/A

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

      13. sqr-pow N/A

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

      14. unpow1 N/A

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

      15. lift--.f64 N/A

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

      16. sub-neg N/A

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

      17. distribute-neg-in N/A

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

      18. remove-double-neg N/A

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

      19. +-commutative N/A

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

      20. sub-neg N/A

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

      21. lift--.f64 58.7

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

   6. Applied rewrites 58.7%

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

   7. Taylor expanded in y around inf

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

      1. lower-*.f64 57.4

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

   9. Applied rewrites 57.4%

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

   if 1.35e-220 < x

   1. Initial program 99.7%

      \[x + \frac{\left|y - x\right|}{2} \]
   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 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-fma.f64 N/A

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

      2. metadata-eval N/A

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

      3. div-inv N/A

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

      4. frac-2neg N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-fabs.f64 N/A

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

      10. unpow1 N/A

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

      11. sqr-pow N/A

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

      12. fabs-sqr N/A

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

      13. sqr-pow N/A

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

      14. unpow1 N/A

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

      15. lift--.f64 N/A

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

      16. sub-neg N/A

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

      17. distribute-neg-in N/A

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

      18. remove-double-neg N/A

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

      19. +-commutative N/A

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

      20. sub-neg N/A

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

      21. lift--.f64 78.9

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

   6. Applied rewrites 78.9%

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

   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 64.0

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

   9. Applied rewrites 64.0%

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

Alternative 8: 65.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{-220}:\\ \;\;\;\;-0.5 \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.35e-220) (* -0.5 (- y x)) (* 1.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.35e-220) {
		tmp = -0.5 * (y - x);
	} 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 <= 1.35d-220) then
        tmp = (-0.5d0) * (y - x)
    else
        tmp = 1.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.35e-220) {
		tmp = -0.5 * (y - x);
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.35e-220:
		tmp = -0.5 * (y - x)
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.35e-220)
		tmp = Float64(-0.5 * Float64(y - x));
	else
		tmp = Float64(1.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.35e-220)
		tmp = -0.5 * (y - x);
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.35e-220], N[(-0.5 * N[(y - x), $MachinePrecision]), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35e-220

   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 56.5

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

   5. Applied rewrites 56.5%

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

   7. Applied rewrites 75.1%

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

   if 1.35e-220 < x

   1. Initial program 99.7%

      \[x + \frac{\left|y - x\right|}{2} \]
   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 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-fma.f64 N/A

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

      2. metadata-eval N/A

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

      3. div-inv N/A

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

      4. frac-2neg N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-fabs.f64 N/A

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

      10. unpow1 N/A

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

      11. sqr-pow N/A

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

      12. fabs-sqr N/A

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

      13. sqr-pow N/A

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

      14. unpow1 N/A

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

      15. lift--.f64 N/A

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

      16. sub-neg N/A

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

      17. distribute-neg-in N/A

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

      18. remove-double-neg N/A

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

      19. +-commutative N/A

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

      20. sub-neg N/A

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

      21. lift--.f64 78.9

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

   6. Applied rewrites 78.9%

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

   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 64.0

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

   9. Applied rewrites 64.0%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{-220}:\\ \;\;\;\;-0.5 \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 30.7% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. 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 52.1

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

5. Applied rewrites 52.1%

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

7. Applied rewrites 54.8%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 31.1%

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

Reproduce

herbie shell --seed 2024277 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))