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

Percentage Accurate: 99.9% → 99.9%

Time: 9.6s

Alternatives: 8

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 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. lift-fabs.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-/.f64 N/A

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

   6. div-inv N/A

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

   7. lower-fma.f64 N/A

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

   8. metadata-eval 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 73.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + \frac{\left|y - x\right|}{2} \leq 4 \cdot 10^{-160}:\\ \;\;\;\;\mathsf{fma}\left(\left|x\right|, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - y, 0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ x (/ (fabs (- y x)) 2.0)) 4e-160)
   (fma (fabs x) 0.5 x)
   (fma (- x y) 0.5 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x + (fabs((y - x)) / 2.0)) <= 4e-160) {
		tmp = fma(fabs(x), 0.5, x);
	} else {
		tmp = fma((x - y), 0.5, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x + Float64(abs(Float64(y - x)) / 2.0)) <= 4e-160)
		tmp = fma(abs(x), 0.5, x);
	else
		tmp = fma(Float64(x - y), 0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x + N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 4e-160], N[(N[Abs[x], $MachinePrecision] * 0.5 + x), $MachinePrecision], N[(N[(x - y), $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))) < 4e-160

   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. lift-fabs.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 93.2

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

   7. Applied rewrites 93.2%

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

      1. neg-fabs N/A

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

      2. lift-fabs.f64 93.2

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

   9. Applied rewrites 93.2%

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

   if 4e-160 < (+.f64 x (/.f64 (fabs.f64 (-.f64 y x)) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-fabs.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

   7. Applied rewrites 99.8%

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

   9. Applied rewrites 98.9%

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

       1. lift-sqrt.f64 N/A

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

       2. lift--.f64 N/A

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

       3. rem-sqrt-square N/A

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

       4. sqrt-unprod N/A

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

       5. rem-square-sqrt N/A

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

       6. *-commutative N/A

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

       7. lift-sqrt.f64 N/A

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

       8. associate-*l* N/A

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

       9. lift-sqrt.f64 N/A

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

       10. lift-sqrt.f64 N/A

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

       11. rem-square-sqrt N/A

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

       12. rem-square-sqrt N/A

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

       13. lift-sqrt.f64 N/A

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

       14. lift-sqrt.f64 N/A

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

       15. lift-*.f64 N/A

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

       16. lift-fma.f64 60.6

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

   11. Applied rewrites 61.2%

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

Alternative 3: 79.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-154}:\\ \;\;\;\;\mathsf{fma}\left(x, 1.5, y \cdot -0.5\right)\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-105}:\\ \;\;\;\;\mathsf{fma}\left(\left|x\right|, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.6e-154)
   (fma x 1.5 (* y -0.5))
   (if (<= y 5.4e-105) (fma (fabs x) 0.5 x) (* (fabs (- y x)) 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.6e-154) {
		tmp = fma(x, 1.5, (y * -0.5));
	} else if (y <= 5.4e-105) {
		tmp = fma(fabs(x), 0.5, x);
	} else {
		tmp = fabs((y - x)) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.6e-154)
		tmp = fma(x, 1.5, Float64(y * -0.5));
	elseif (y <= 5.4e-105)
		tmp = fma(abs(x), 0.5, x);
	else
		tmp = Float64(abs(Float64(y - x)) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.6e-154], N[(x * 1.5 + N[(y * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.4e-105], N[(N[Abs[x], $MachinePrecision] * 0.5 + x), $MachinePrecision], N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.6e-154

   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. lift-fabs.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   6. Applied rewrites 99.3%

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

      1. lift--.f64 N/A

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

      2. lift-fabs.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-fabs.f64 N/A

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

      5. rem-square-sqrt 99.9

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

      6. lift-fabs.f64 N/A

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

      7. lift--.f64 N/A

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

      8. fabs-sub N/A

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

      9. lift--.f64 N/A

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

      10. rem-sqrt-square N/A

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

      11. sqrt-prod N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sqrt.f64 83.0

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

   8. Applied rewrites 83.0%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 86.5

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

   11. Applied rewrites 86.5%

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

   if -2.6e-154 < y < 5.39999999999999985e-105

   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. lift-fabs.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 90.3

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

   7. Applied rewrites 90.3%

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

      1. neg-fabs N/A

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

      2. lift-fabs.f64 90.3

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

   9. Applied rewrites 90.3%

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

   if 5.39999999999999985e-105 < y

   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. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-in N/A

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

      8. +-commutative N/A

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

      9. lower-fabs.f64 N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 71.3

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

   5. Applied rewrites 71.3%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-154}:\\ \;\;\;\;\mathsf{fma}\left(x, 1.5, y \cdot -0.5\right)\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-105}:\\ \;\;\;\;\mathsf{fma}\left(\left|x\right|, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-154}:\\ \;\;\;\;\mathsf{fma}\left(x - y, 0.5, x\right)\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-105}:\\ \;\;\;\;\mathsf{fma}\left(\left|x\right|, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.6e-154)
   (fma (- x y) 0.5 x)
   (if (<= y 5.4e-105) (fma (fabs x) 0.5 x) (* (fabs (- y x)) 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.6e-154) {
		tmp = fma((x - y), 0.5, x);
	} else if (y <= 5.4e-105) {
		tmp = fma(fabs(x), 0.5, x);
	} else {
		tmp = fabs((y - x)) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.6e-154)
		tmp = fma(Float64(x - y), 0.5, x);
	elseif (y <= 5.4e-105)
		tmp = fma(abs(x), 0.5, x);
	else
		tmp = Float64(abs(Float64(y - x)) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.6e-154], N[(N[(x - y), $MachinePrecision] * 0.5 + x), $MachinePrecision], If[LessEqual[y, 5.4e-105], N[(N[Abs[x], $MachinePrecision] * 0.5 + x), $MachinePrecision], N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.6e-154

   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. lift-fabs.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

   7. Applied rewrites 99.8%

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

   9. Applied rewrites 98.8%

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

       1. lift-sqrt.f64 N/A

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

       2. lift--.f64 N/A

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

       3. rem-sqrt-square N/A

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

       4. sqrt-unprod N/A

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

       5. rem-square-sqrt N/A

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

       6. *-commutative N/A

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

       7. lift-sqrt.f64 N/A

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

       8. associate-*l* N/A

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

       9. lift-sqrt.f64 N/A

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

       10. lift-sqrt.f64 N/A

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

       11. rem-square-sqrt N/A

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

       12. rem-square-sqrt N/A

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

       13. lift-sqrt.f64 N/A

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

       14. lift-sqrt.f64 N/A

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

       15. lift-*.f64 N/A

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

       16. lift-fma.f64 83.0

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

   11. Applied rewrites 86.4%

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

   if -2.6e-154 < y < 5.39999999999999985e-105

   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. lift-fabs.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 90.3

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

   7. Applied rewrites 90.3%

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

      1. neg-fabs N/A

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

      2. lift-fabs.f64 90.3

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

   9. Applied rewrites 90.3%

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

   if 5.39999999999999985e-105 < y

   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. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-in N/A

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

      8. +-commutative N/A

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

      9. lower-fabs.f64 N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 71.3

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

   5. Applied rewrites 71.3%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-154}:\\ \;\;\;\;\mathsf{fma}\left(x - y, 0.5, x\right)\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-105}:\\ \;\;\;\;\mathsf{fma}\left(\left|x\right|, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-54}:\\ \;\;\;\;\mathsf{fma}\left(-y, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left|x\right|, 0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5e-54) (fma (- y) 0.5 x) (fma (fabs x) 0.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5e-54) {
		tmp = fma(-y, 0.5, x);
	} else {
		tmp = fma(fabs(x), 0.5, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5e-54)
		tmp = fma(Float64(-y), 0.5, x);
	else
		tmp = fma(abs(x), 0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5e-54], N[((-y) * 0.5 + x), $MachinePrecision], N[(N[Abs[x], $MachinePrecision] * 0.5 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.00000000000000015e-54

   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. lift-fabs.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   6. Applied rewrites 99.3%

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

      1. lift--.f64 N/A

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

      2. lift-fabs.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-fabs.f64 N/A

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

      5. rem-square-sqrt 99.9

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

      6. lift-fabs.f64 N/A

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

      7. lift--.f64 N/A

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

      8. fabs-sub N/A

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

      9. lift--.f64 N/A

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

      10. rem-sqrt-square N/A

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

      11. sqrt-prod N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sqrt.f64 86.1

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

   8. Applied rewrites 86.1%

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

   9. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 71.7

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

   11. Applied rewrites 71.7%

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

   if -5.00000000000000015e-54 < y

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. lift-fabs.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 58.1

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

   7. Applied rewrites 58.1%

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

      1. neg-fabs N/A

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

      2. lift-fabs.f64 58.1

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

   9. Applied rewrites 58.1%

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

Alternative 6: 62.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.68 \cdot 10^{-47}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left|x\right|, 0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.68e-47) (* y -0.5) (fma (fabs x) 0.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.68e-47) {
		tmp = y * -0.5;
	} else {
		tmp = fma(fabs(x), 0.5, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.68e-47)
		tmp = Float64(y * -0.5);
	else
		tmp = fma(abs(x), 0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.68e-47], N[(y * -0.5), $MachinePrecision], N[(N[Abs[x], $MachinePrecision] * 0.5 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.67999999999999991e-47

   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. lift-fabs.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   6. Applied rewrites 99.3%

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

      1. lift--.f64 N/A

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

      2. lift-fabs.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-fabs.f64 N/A

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

      5. rem-square-sqrt 99.9

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

      6. lift-fabs.f64 N/A

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

      7. lift--.f64 N/A

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

      8. fabs-sub N/A

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

      9. lift--.f64 N/A

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

      10. rem-sqrt-square N/A

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

      11. sqrt-prod N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sqrt.f64 87.3

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

   8. Applied rewrites 87.3%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 67.3

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

   11. Applied rewrites 67.3%

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

   if -1.67999999999999991e-47 < y

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. lift-fabs.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 58.1

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

   7. Applied rewrites 58.1%

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

      1. neg-fabs N/A

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

      2. lift-fabs.f64 58.1

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

   9. Applied rewrites 58.1%

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

Alternative 7: 45.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-54}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -2.4e-54) (* y -0.5) (* x 1.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.4e-54) {
		tmp = y * -0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.4d-54)) then
        tmp = y * (-0.5d0)
    else
        tmp = x * 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.4e-54) {
		tmp = y * -0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.4e-54:
		tmp = y * -0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.4e-54)
		tmp = Float64(y * -0.5);
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.4e-54)
		tmp = y * -0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.4e-54], N[(y * -0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.40000000000000013e-54

   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. lift-fabs.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   6. Applied rewrites 99.3%

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

      1. lift--.f64 N/A

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

      2. lift-fabs.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-fabs.f64 N/A

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

      5. rem-square-sqrt 99.9

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

      6. lift-fabs.f64 N/A

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

      7. lift--.f64 N/A

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

      8. fabs-sub N/A

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

      9. lift--.f64 N/A

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

      10. rem-sqrt-square N/A

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

      11. sqrt-prod N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sqrt.f64 86.1

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

   8. Applied rewrites 86.1%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 66.7

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

   11. Applied rewrites 66.7%

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

   if -2.40000000000000013e-54 < y

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. lift-fabs.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   6. Applied rewrites 99.3%

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

      1. lift--.f64 N/A

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

      2. lift-fabs.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-fabs.f64 N/A

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

      5. rem-square-sqrt 100.0

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

      6. lift-fabs.f64 N/A

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

      7. lift--.f64 N/A

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

      8. fabs-sub N/A

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

      9. lift--.f64 N/A

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

      10. rem-sqrt-square N/A

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

      11. sqrt-prod N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sqrt.f64 32.0

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

   8. Applied rewrites 32.0%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 34.8

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

   11. Applied rewrites 34.8%

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

Alternative 8: 29.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ x \cdot 1.5 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x 1.5))

C

double code(double x, double y) {
	return x * 1.5;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * 1.5d0
end function

Java

public static double code(double x, double y) {
	return x * 1.5;
}

Python

def code(x, y):
	return x * 1.5

Julia

function code(x, y)
	return Float64(x * 1.5)
end

MATLAB

function tmp = code(x, y)
	tmp = x * 1.5;
end

Wolfram

code[x_, y_] := N[(x * 1.5), $MachinePrecision]

Derivation

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. lift-fabs.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-/.f64 N/A

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

   6. div-inv N/A

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

   7. lower-fma.f64 N/A

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

   8. metadata-eval 100.0

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

4. Applied rewrites 100.0%

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

6. Applied rewrites 99.3%

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

   1. lift--.f64 N/A

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

   2. lift-fabs.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. rem-square-sqrt 100.0

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

   6. lift-fabs.f64 N/A

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

   7. lift--.f64 N/A

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

   8. fabs-sub N/A

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

   9. lift--.f64 N/A

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

   10. rem-sqrt-square N/A

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

   11. sqrt-prod N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sqrt.f64 N/A

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

   14. lower-sqrt.f64 44.7

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

8. Applied rewrites 44.7%

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

9. Taylor expanded in x around inf

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

    1. *-commutative N/A

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

    2. lower-*.f64 31.6

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

11. Applied rewrites 31.6%

    \[\leadsto \color{blue}{x \cdot 1.5} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))