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

Percentage Accurate: 99.9% → 99.9%

Time: 8.9s

Alternatives: 5

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: 71.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.1 \cdot 10^{-69}:\\ \;\;\;\;\mathsf{fma}\left(\left|-y\right|, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - y, 0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 4.1e-69) (fma (fabs (- y)) 0.5 x) (fma (- x y) 0.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4.1e-69) {
		tmp = fma(fabs(-y), 0.5, x);
	} else {
		tmp = fma((x - y), 0.5, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4.1e-69)
		tmp = fma(abs(Float64(-y)), 0.5, x);
	else
		tmp = fma(Float64(x - y), 0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 4.1e-69], N[(N[Abs[(-y)], $MachinePrecision] * 0.5 + x), $MachinePrecision], N[(N[(x - y), $MachinePrecision] * 0.5 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.0999999999999999e-69

   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 67.4

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

   7. Applied rewrites 67.4%

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

   if 4.0999999999999999e-69 < x

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. neg-fabs N/A

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

      8. lower-fabs.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 99.7

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

   4. Applied rewrites 99.7%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. unpow1/2 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lift-fabs.f64 N/A

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

      8. lift--.f64 N/A

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

      9. fabs-sub N/A

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

      10. lift--.f64 N/A

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

      11. lift-fabs.f64 N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 99.5

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

      14. lift-fabs.f64 N/A

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

      15. lift--.f64 N/A

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

      16. fabs-sub N/A

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

      17. lift--.f64 N/A

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

      18. lift-fabs.f64 99.5

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

   6. Applied rewrites 99.5%

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

   8. Applied rewrites 99.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\sqrt{\left|{\left(x - y\right)}^{-1}\right|}}} \cdot \sqrt{\left|y - x\right|}, 0.5, x\right) \]

   10. Applied rewrites 92.0%

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

Alternative 3: 70.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-41}:\\ \;\;\;\;\mathsf{fma}\left(x - y, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.8e-41) (fma (- x y) 0.5 x) (* (fabs (- y x)) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.8e-41) {
		tmp = fma((x - y), 0.5, x);
	} else {
		tmp = fabs((y - x)) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.8e-41)
		tmp = fma(Float64(x - y), 0.5, x);
	else
		tmp = Float64(abs(Float64(y - x)) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.8e-41], N[(N[(x - y), $MachinePrecision] * 0.5 + x), $MachinePrecision], N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.8000000000000002e-41

   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. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. unpow1/2 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lift-fabs.f64 N/A

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

      8. lift--.f64 N/A

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

      9. fabs-sub N/A

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

      10. lift--.f64 N/A

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

      11. lift-fabs.f64 N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 99.3

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

      14. lift-fabs.f64 N/A

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

      15. lift--.f64 N/A

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

      16. fabs-sub N/A

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

      17. lift--.f64 N/A

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

      18. lift-fabs.f64 99.3

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

   6. Applied rewrites 99.3%

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

   8. Applied rewrites 99.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\sqrt{\left|{\left(x - y\right)}^{-1}\right|}}} \cdot \sqrt{\left|y - x\right|}, 0.5, x\right) \]

   10. Applied rewrites 73.5%

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

   if 2.8000000000000002e-41 < y

   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 69.4

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

   5. Applied rewrites 69.4%

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

Alternative 4: 54.7% 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. unpow1 N/A

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

   2. metadata-eval N/A

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

   3. pow-prod-up N/A

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

   4. lower-*.f64 N/A

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

   5. unpow1/2 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lift-fabs.f64 N/A

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

   8. lift--.f64 N/A

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

   9. fabs-sub N/A

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

   10. lift--.f64 N/A

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

   11. lift-fabs.f64 N/A

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

   12. unpow1/2 N/A

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

   13. lower-sqrt.f64 99.3

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

   14. lift-fabs.f64 N/A

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

   15. lift--.f64 N/A

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

   16. fabs-sub N/A

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

   17. lift--.f64 N/A

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

   18. lift-fabs.f64 99.3

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

6. Applied rewrites 99.3%

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

8. Applied rewrites 99.4%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\sqrt{\left|{\left(x - y\right)}^{-1}\right|}}} \cdot \sqrt{\left|y - x\right|}, 0.5, x\right) \]

10. Applied rewrites 56.8%

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

Alternative 5: 7.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left|-x\right| \cdot 0.5 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[Abs[(-x)], $MachinePrecision] * 0.5), $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 52.9

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

5. Applied rewrites 52.9%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 7.4%

   \[\leadsto \left|-x\right| \cdot 0.5 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024268 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))