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

Percentage Accurate: 99.9% → 99.9%

Time: 5.8s

Alternatives: 6

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: 71.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.75 \cdot 10^{-218}:\\ \;\;\;\;\mathsf{fma}\left(1.5, x, -0.5 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left|-y\right|, 0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.75e-218) (fma 1.5 x (* -0.5 y)) (fma (fabs (- y)) 0.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.75e-218) {
		tmp = fma(1.5, x, (-0.5 * y));
	} else {
		tmp = fma(fabs(-y), 0.5, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.75e-218)
		tmp = fma(1.5, x, Float64(-0.5 * y));
	else
		tmp = fma(abs(Float64(-y)), 0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.75e-218], N[(1.5 * x + N[(-0.5 * y), $MachinePrecision]), $MachinePrecision], N[(N[Abs[(-y)], $MachinePrecision] * 0.5 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.75e-218

   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. rem-square-sqrt N/A

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

      5. sqrt-unprod N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. sqrt-prod N/A

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

      10. unpow1/2 N/A

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

      11. lower-fma.f64 N/A

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

   6. Applied rewrites 73.7%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot y + \frac{3}{2} \cdot x} \]
   8. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 78.7

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

   9. Applied rewrites 78.7%

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

   if 1.75e-218 < y

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. neg-fabs N/A

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

      8. lower-fabs.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\left|\color{blue}{-1 \cdot y}\right|, \frac{1}{2}, x\right) \]
   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 69.1

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

   7. Applied rewrites 69.1%

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

Alternative 3: 71.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.75 \cdot 10^{-218}:\\ \;\;\;\;\mathsf{fma}\left(x - y, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left|-y\right|, 0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.75e-218) (fma (- x y) 0.5 x) (fma (fabs (- y)) 0.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.75e-218) {
		tmp = fma((x - y), 0.5, x);
	} else {
		tmp = fma(fabs(-y), 0.5, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.75e-218)
		tmp = fma(Float64(x - y), 0.5, x);
	else
		tmp = fma(abs(Float64(-y)), 0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.75e-218], N[(N[(x - y), $MachinePrecision] * 0.5 + x), $MachinePrecision], N[(N[Abs[(-y)], $MachinePrecision] * 0.5 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.75e-218

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. neg-fabs N/A

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

      8. lower-fabs.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-fabs.f64 N/A

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

      2. unpow1 N/A

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

      3. sqr-pow N/A

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

      4. fabs-sqr N/A

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

      5. sqr-pow N/A

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

      6. unpow1 78.6

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

   6. Applied rewrites 78.6%

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

   if 1.75e-218 < y

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. neg-fabs N/A

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

      8. lower-fabs.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\left|\color{blue}{-1 \cdot y}\right|, \frac{1}{2}, x\right) \]
   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 69.1

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

   7. Applied rewrites 69.1%

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

Alternative 4: 45.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.04 \cdot 10^{-100}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1.04e-100) (* -0.5 y) (* 1.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.04e-100) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.04d-100)) then
        tmp = (-0.5d0) * y
    else
        tmp = 1.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.04e-100) {
		tmp = -0.5 * y;
	} else {
		tmp = 1.5 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.04e-100:
		tmp = -0.5 * y
	else:
		tmp = 1.5 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.04e-100)
		tmp = Float64(-0.5 * y);
	else
		tmp = Float64(1.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.04e-100)
		tmp = -0.5 * y;
	else
		tmp = 1.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.04e-100], N[(-0.5 * y), $MachinePrecision], N[(1.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.04e-100

   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. rem-square-sqrt N/A

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

      5. sqrt-unprod N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. sqrt-prod N/A

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

      10. unpow1/2 N/A

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

      11. lower-fma.f64 N/A

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

   6. Applied rewrites 87.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 73.6

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

   9. Applied rewrites 73.6%

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

   if -1.04e-100 < y

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. neg-fabs N/A

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

      8. lower-fabs.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. rem-square-sqrt N/A

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

      5. sqrt-unprod N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. sqrt-prod N/A

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

      10. unpow1/2 N/A

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

      11. lower-fma.f64 N/A

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

   6. Applied rewrites 30.5%

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

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{3}{2} \cdot x} \]
   8. Step-by-step derivation

      1. lower-*.f64 35.5

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

   9. Applied rewrites 35.5%

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

Alternative 5: 54.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[(x - y), $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. Step-by-step derivation

   1. lift-fabs.f64 N/A

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

   2. unpow1 N/A

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

   3. sqr-pow N/A

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

   4. fabs-sqr N/A

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

   5. sqr-pow N/A

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

   6. unpow1 54.3

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

6. Applied rewrites 54.3%

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

Alternative 6: 26.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. div-inv N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-fabs.f64 N/A

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

   7. neg-fabs N/A

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

   8. lower-fabs.f64 N/A

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

   9. lift--.f64 N/A

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

   10. sub-neg N/A

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

   11. +-commutative N/A

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

   12. distribute-neg-in N/A

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

   13. remove-double-neg N/A

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

   14. sub-neg N/A

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

   15. lower--.f64 N/A

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

   16. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. rem-square-sqrt N/A

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

   5. sqrt-unprod N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. sqrt-prod N/A

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

   10. unpow1/2 N/A

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

   11. lower-fma.f64 N/A

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

6. Applied rewrites 49.7%

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

7. Taylor expanded in x around 0

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

   1. lower-*.f64 26.7

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

9. Applied rewrites 26.7%

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

Reproduce

herbie shell --seed 2024296 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))