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

Percentage Accurate: 99.9% → 99.9%

Time: 6.8s

Alternatives: 10

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, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\left|\mathsf{fma}\left(y, \frac{y}{y + x}, \left(-x\right) \cdot \frac{x}{y + x}\right)\right|}{2} + x \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ (/ (fabs (fma y (/ y (+ y x)) (* (- x) (/ x (+ y x))))) 2.0) x))

C

double code(double x, double y) {
	return (fabs(fma(y, (y / (y + x)), (-x * (x / (y + x))))) / 2.0) + x;
}

Julia

function code(x, y)
	return Float64(Float64(abs(fma(y, Float64(y / Float64(y + x)), Float64(Float64(-x) * Float64(x / Float64(y + x))))) / 2.0) + x)
end

Wolfram

code[x_, y_] := N[(N[(N[Abs[N[(y * N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] + N[((-x) * N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision] + 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

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

   2. flip-- N/A

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

   3. div-sub N/A

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

   4. sub-neg N/A

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

   5. associate-/l* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. associate-/l* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

   \[\leadsto \frac{\left|\mathsf{fma}\left(y, \frac{y}{y + x}, \left(-x\right) \cdot \frac{x}{y + x}\right)\right|}{2} + x \]
6. Add Preprocessing

Alternative 2: 74.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left|y - x\right|}{2} + x \leq -5 \cdot 10^{-201}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, -0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (/ (fabs (- y x)) 2.0) x) -5e-201) (* 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-201) {
		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-201)
		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-201], 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))) < -4.9999999999999999e-201

   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 1.0

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

   5. Applied rewrites 1.0%

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 99.4%

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

   if -4.9999999999999999e-201 < (+.f64 x (/.f64 (fabs.f64 (-.f64 y x)) #s(literal 2 binary64)))

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 67.2%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left|y - x\right|}{2} + x \leq -5 \cdot 10^{-201}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, -0.5, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left|y - x\right|}{2} + x \leq 10^{-283}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (/ (fabs (- y x)) 2.0) x) 1e-283) (* 0.5 x) (* -0.5 y)))

C

double code(double x, double y) {
	double tmp;
	if (((fabs((y - x)) / 2.0) + x) <= 1e-283) {
		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) <= 1d-283) 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) <= 1e-283) {
		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) <= 1e-283:
		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) <= 1e-283)
		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) <= 1e-283)
		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], 1e-283], 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))) < 9.99999999999999947e-284

   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 1.2

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

   5. Applied rewrites 1.2%

      \[\leadsto \color{blue}{\left|y - x\right| \cdot 0.5} \]
   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 9.99999999999999947e-284 < (+.f64 x (/.f64 (fabs.f64 (-.f64 y x)) #s(literal 2 binary64)))

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 67.1%

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

   7. Taylor expanded in y around inf

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

      1. lower-*.f64 33.9

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

   9. Applied rewrites 33.9%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left|y - x\right|}{2} + x \leq 10^{-283}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-91}:\\ \;\;\;\;-0.5 \cdot \left(y - x\right)\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(\left|-y\right|, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1.5, x, -0.5 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.8e-91)
   (* -0.5 (- y x))
   (if (<= x 1.2e-19) (fma (fabs (- y)) 0.5 x) (fma 1.5 x (* -0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.8e-91) {
		tmp = -0.5 * (y - x);
	} else if (x <= 1.2e-19) {
		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 <= -5.8e-91)
		tmp = Float64(-0.5 * Float64(y - x));
	elseif (x <= 1.2e-19)
		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, -5.8e-91], N[(-0.5 * N[(y - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.2e-19], 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 < -5.8000000000000001e-91

   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 31.0

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

   5. Applied rewrites 31.0%

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

   7. Applied rewrites 87.3%

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

   if -5.8000000000000001e-91 < x < 1.20000000000000011e-19

   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 86.6

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

   7. Applied rewrites 86.6%

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

   if 1.20000000000000011e-19 < x

   1. Initial program 99.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.6

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 93.3%

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

   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 93.8

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

   9. Applied rewrites 93.8%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-91}:\\ \;\;\;\;-0.5 \cdot \left(y - x\right)\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(\left|-y\right|, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1.5, x, -0.5 \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-91}:\\ \;\;\;\;-0.5 \cdot \left(y - x\right)\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(\left|-y\right|, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, -0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.8e-91)
   (* -0.5 (- y x))
   (if (<= x 1.2e-19) (fma (fabs (- y)) 0.5 x) (fma (- y x) -0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.8e-91) {
		tmp = -0.5 * (y - x);
	} else if (x <= 1.2e-19) {
		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 <= -5.8e-91)
		tmp = Float64(-0.5 * Float64(y - x));
	elseif (x <= 1.2e-19)
		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, -5.8e-91], N[(-0.5 * N[(y - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.2e-19], 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 < -5.8000000000000001e-91

   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 31.0

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

   5. Applied rewrites 31.0%

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

   7. Applied rewrites 87.3%

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

   if -5.8000000000000001e-91 < x < 1.20000000000000011e-19

   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 86.6

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

   7. Applied rewrites 86.6%

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

   if 1.20000000000000011e-19 < x

   1. Initial program 99.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.6

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 93.3%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-91}:\\ \;\;\;\;-0.5 \cdot \left(y - x\right)\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(\left|-y\right|, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, -0.5, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-91}:\\ \;\;\;\;-0.5 \cdot \left(y - x\right)\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-19}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, -0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.8e-91)
   (* -0.5 (- y x))
   (if (<= x 1.2e-19) (* (fabs (- y x)) 0.5) (fma (- y x) -0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.8e-91) {
		tmp = -0.5 * (y - x);
	} else if (x <= 1.2e-19) {
		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 <= -5.8e-91)
		tmp = Float64(-0.5 * Float64(y - x));
	elseif (x <= 1.2e-19)
		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, -5.8e-91], N[(-0.5 * N[(y - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.2e-19], 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 < -5.8000000000000001e-91

   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 31.0

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

   5. Applied rewrites 31.0%

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

   7. Applied rewrites 87.3%

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

   if -5.8000000000000001e-91 < x < 1.20000000000000011e-19

   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 85.6

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

   5. Applied rewrites 85.6%

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

   if 1.20000000000000011e-19 < x

   1. Initial program 99.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.6

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 93.3%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-91}:\\ \;\;\;\;-0.5 \cdot \left(y - x\right)\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-19}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, -0.5, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-144}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-132}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.4e-144) (* 0.5 x) (if (<= x 2.15e-132) (* -0.5 y) (* 1.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.4e-144) {
		tmp = 0.5 * x;
	} else if (x <= 2.15e-132) {
		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 <= (-4.4d-144)) then
        tmp = 0.5d0 * x
    else if (x <= 2.15d-132) 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 <= -4.4e-144) {
		tmp = 0.5 * x;
	} else if (x <= 2.15e-132) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.4e-144:
		tmp = 0.5 * x
	elif x <= 2.15e-132:
		tmp = -0.5 * y
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.4e-144)
		tmp = Float64(0.5 * x);
	elseif (x <= 2.15e-132)
		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 <= -4.4e-144)
		tmp = 0.5 * x;
	elseif (x <= 2.15e-132)
		tmp = -0.5 * y;
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.4e-144], N[(0.5 * x), $MachinePrecision], If[LessEqual[x, 2.15e-132], N[(-0.5 * y), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.40000000000000012e-144

   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 35.4

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

   5. Applied rewrites 35.4%

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

   7. Applied rewrites 81.3%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 64.9%

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

   if -4.40000000000000012e-144 < x < 2.1499999999999998e-132

   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

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

      2. flip-- N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   6. Applied rewrites 57.5%

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

   7. Taylor expanded in y around inf

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

      1. lower-*.f64 53.2

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

   9. Applied rewrites 53.2%

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

   if 2.1499999999999998e-132 < 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

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

      2. flip-- N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 81.5%

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

   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 72.5

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

   9. Applied rewrites 72.5%

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

Alternative 8: 67.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.15 \cdot 10^{-132}:\\ \;\;\;\;-0.5 \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.15e-132) (* -0.5 (- y x)) (* 1.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.15e-132) {
		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 <= 2.15d-132) 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 <= 2.15e-132) {
		tmp = -0.5 * (y - x);
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.15e-132:
		tmp = -0.5 * (y - x)
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.15e-132)
		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 <= 2.15e-132)
		tmp = -0.5 * (y - x);
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.15e-132], N[(-0.5 * N[(y - x), $MachinePrecision]), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.1499999999999998e-132

   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 59.4

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

   5. Applied rewrites 59.4%

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

   7. Applied rewrites 70.8%

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

   if 2.1499999999999998e-132 < 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

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

      2. flip-- N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 81.5%

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

   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 72.5

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

   9. Applied rewrites 72.5%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.15 \cdot 10^{-132}:\\ \;\;\;\;-0.5 \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 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 10: 30.5% 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 52.8

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

5. Applied rewrites 52.8%

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

7. Applied rewrites 54.1%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 31.3%

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

Reproduce

herbie shell --seed 2024244 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))