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

Percentage Accurate: 99.9% → 99.9%

Time: 11.8s

Alternatives: 7

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[Abs[N[(x - y), $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. +-commutative N/A

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

   2. div-inv N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. fabs-lowering-fabs.f64 N/A

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

   5. --lowering--.f64 N/A

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

   6. metadata-eval 99.9

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 55.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + \frac{\left|x - y\right|}{2} \leq -1 \cdot 10^{-238}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left|y\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ x (/ (fabs (- x y)) 2.0)) -1e-238) x (* 0.5 (fabs y))))

C

double code(double x, double y) {
	double tmp;
	if ((x + (fabs((x - y)) / 2.0)) <= -1e-238) {
		tmp = x;
	} else {
		tmp = 0.5 * fabs(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x + (abs((x - y)) / 2.0d0)) <= (-1d-238)) then
        tmp = x
    else
        tmp = 0.5d0 * abs(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x + (Math.abs((x - y)) / 2.0)) <= -1e-238) {
		tmp = x;
	} else {
		tmp = 0.5 * Math.abs(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x + (math.fabs((x - y)) / 2.0)) <= -1e-238:
		tmp = x
	else:
		tmp = 0.5 * math.fabs(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x + Float64(abs(Float64(x - y)) / 2.0)) <= -1e-238)
		tmp = x;
	else
		tmp = Float64(0.5 * abs(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x + (abs((x - y)) / 2.0)) <= -1e-238)
		tmp = x;
	else
		tmp = 0.5 * abs(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x + N[(N[Abs[N[(x - y), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], -1e-238], x, N[(0.5 * N[Abs[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (fabs.f64 (-.f64 y x)) #s(literal 2 binary64))) < -9.9999999999999999e-239

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 18.8%

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

   if -9.9999999999999999e-239 < (+.f64 x (/.f64 (fabs.f64 (-.f64 y x)) #s(literal 2 binary64)))

   1. Initial program 99.8%

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

      1. +-commutative N/A

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

      2. div-inv N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. fabs-lowering-fabs.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. metadata-eval 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 70.7%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left|y\right|} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

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

      2. fabs-lowering-fabs.f64 65.5

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

   10. Simplified 65.5%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left|x - y\right|}{2} \leq -1 \cdot 10^{-238}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left|y\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{-75}:\\ \;\;\;\;\mathsf{fma}\left(x - y, 0.5, x\right)\\ \mathbf{elif}\;y \leq 8.3 \cdot 10^{-128}:\\ \;\;\;\;\mathsf{fma}\left(\left|x\right|, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, 0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.35e-75)
   (fma (- x y) 0.5 x)
   (if (<= y 8.3e-128) (fma (fabs x) 0.5 x) (fma (- y x) 0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.35e-75) {
		tmp = fma((x - y), 0.5, x);
	} else if (y <= 8.3e-128) {
		tmp = fma(fabs(x), 0.5, x);
	} else {
		tmp = fma((y - x), 0.5, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.35e-75)
		tmp = fma(Float64(x - y), 0.5, x);
	elseif (y <= 8.3e-128)
		tmp = fma(abs(x), 0.5, x);
	else
		tmp = fma(Float64(y - x), 0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.35e-75], N[(N[(x - y), $MachinePrecision] * 0.5 + x), $MachinePrecision], If[LessEqual[y, 8.3e-128], N[(N[Abs[x], $MachinePrecision] * 0.5 + x), $MachinePrecision], N[(N[(y - x), $MachinePrecision] * 0.5 + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.3499999999999999e-75

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. div-inv N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. fabs-lowering-fabs.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. metadata-eval 99.9

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

   4. Applied egg-rr 99.9%

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

      1. flip-- N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. fabs-mul N/A

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

      5. inv-pow N/A

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

      6. sqr-pow N/A

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

      7. fabs-sqr N/A

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

      8. sqr-pow N/A

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

      9. inv-pow N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. +-lowering-+.f64 N/A

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

      13. fabs-lowering-fabs.f64 N/A

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

      14. difference-of-squares N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 N/A

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

      17. --lowering--.f64 N/A

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

      18. +-lowering-+.f64 7.5

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

   6. Applied egg-rr 7.5%

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

   8. Applied egg-rr 85.1%

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

   if -1.3499999999999999e-75 < y < 8.30000000000000015e-128

   1. Initial program 99.8%

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

      1. +-commutative N/A

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

      2. div-inv N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. fabs-lowering-fabs.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. metadata-eval 99.8

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

   4. Applied egg-rr 99.8%

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

   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. neg-lowering-neg.f64 87.8

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

   7. Simplified 87.8%

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

   if 8.30000000000000015e-128 < y

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. div-inv N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. fabs-lowering-fabs.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. metadata-eval 99.9

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

   4. Applied egg-rr 99.9%

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

      1. flip-- N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. fabs-mul N/A

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

      5. inv-pow N/A

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

      6. sqr-pow N/A

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

      7. fabs-sqr N/A

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

      8. sqr-pow N/A

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

      9. inv-pow N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. +-lowering-+.f64 N/A

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

      13. fabs-lowering-fabs.f64 N/A

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

      14. difference-of-squares N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 N/A

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

      17. --lowering--.f64 N/A

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

      18. +-lowering-+.f64 47.9

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

   6. Applied egg-rr 47.9%

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

   8. Applied egg-rr 92.6%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{-75}:\\ \;\;\;\;\mathsf{fma}\left(x - y, 0.5, x\right)\\ \mathbf{elif}\;y \leq 8.3 \cdot 10^{-128}:\\ \;\;\;\;\mathsf{fma}\left(\left|x\right|, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, 0.5, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-156}:\\ \;\;\;\;\mathsf{fma}\left(y - x, 0.5, x\right)\\ \mathbf{elif}\;x \leq 3.25 \cdot 10^{-124}:\\ \;\;\;\;\mathsf{fma}\left(\left|y\right|, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - y, 0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.5e-156)
   (fma (- y x) 0.5 x)
   (if (<= x 3.25e-124) (fma (fabs y) 0.5 x) (fma (- x y) 0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.5e-156) {
		tmp = fma((y - x), 0.5, x);
	} else if (x <= 3.25e-124) {
		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 <= -3.5e-156)
		tmp = fma(Float64(y - x), 0.5, x);
	elseif (x <= 3.25e-124)
		tmp = fma(abs(y), 0.5, x);
	else
		tmp = fma(Float64(x - y), 0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.5e-156], N[(N[(y - x), $MachinePrecision] * 0.5 + x), $MachinePrecision], If[LessEqual[x, 3.25e-124], N[(N[Abs[y], $MachinePrecision] * 0.5 + x), $MachinePrecision], N[(N[(x - y), $MachinePrecision] * 0.5 + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.4999999999999999e-156

   1. Initial program 100.0%

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

      1. +-commutative N/A

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

      2. div-inv N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. fabs-lowering-fabs.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

      1. flip-- N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. fabs-mul N/A

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

      5. inv-pow N/A

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

      6. sqr-pow N/A

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

      7. fabs-sqr N/A

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

      8. sqr-pow N/A

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

      9. inv-pow N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. +-lowering-+.f64 N/A

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

      13. fabs-lowering-fabs.f64 N/A

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

      14. difference-of-squares N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 N/A

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

      17. --lowering--.f64 N/A

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

      18. +-lowering-+.f64 18.9

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

   6. Applied egg-rr 18.9%

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

   8. Applied egg-rr 88.3%

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

   if -3.4999999999999999e-156 < x < 3.24999999999999994e-124

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. div-inv N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. fabs-lowering-fabs.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. metadata-eval 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 91.7%

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

   if 3.24999999999999994e-124 < x

   1. Initial program 99.7%

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

      1. +-commutative N/A

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

      2. div-inv N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. fabs-lowering-fabs.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. metadata-eval 99.7

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

   4. Applied egg-rr 99.7%

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

      1. flip-- N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. fabs-mul N/A

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

      5. inv-pow N/A

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

      6. sqr-pow N/A

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

      7. fabs-sqr N/A

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

      8. sqr-pow N/A

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

      9. inv-pow N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. +-lowering-+.f64 N/A

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

      13. fabs-lowering-fabs.f64 N/A

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

      14. difference-of-squares N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 N/A

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

      17. --lowering--.f64 N/A

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

      18. +-lowering-+.f64 41.9

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

   6. Applied egg-rr 41.9%

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

   8. Applied egg-rr 84.3%

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

Alternative 5: 72.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.25 \cdot 10^{-128}:\\ \;\;\;\;\mathsf{fma}\left(x - y, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left|y\right|, 0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.25e-128) (fma (- x y) 0.5 x) (fma (fabs y) 0.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.25e-128) {
		tmp = fma((x - y), 0.5, x);
	} else {
		tmp = fma(fabs(y), 0.5, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.25e-128)
		tmp = fma(Float64(x - y), 0.5, x);
	else
		tmp = fma(abs(y), 0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.25e-128], N[(N[(x - y), $MachinePrecision] * 0.5 + x), $MachinePrecision], N[(N[Abs[y], $MachinePrecision] * 0.5 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.25e-128

   1. Initial program 99.8%

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

      1. +-commutative N/A

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

      2. div-inv N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. fabs-lowering-fabs.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. metadata-eval 99.8

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

   4. Applied egg-rr 99.8%

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

      1. flip-- N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. fabs-mul N/A

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

      5. inv-pow N/A

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

      6. sqr-pow N/A

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

      7. fabs-sqr N/A

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

      8. sqr-pow N/A

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

      9. inv-pow N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. +-lowering-+.f64 N/A

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

      13. fabs-lowering-fabs.f64 N/A

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

      14. difference-of-squares N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 N/A

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

      17. --lowering--.f64 N/A

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

      18. +-lowering-+.f64 21.0

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

   6. Applied egg-rr 21.0%

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

   8. Applied egg-rr 72.8%

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

   if 2.25e-128 < y

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. div-inv N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. fabs-lowering-fabs.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. metadata-eval 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 79.8%

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

Alternative 6: 59.5% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y)
	return fma(abs(y), 0.5, x)
end

Wolfram

code[x_, y_] := N[(N[Abs[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. +-commutative N/A

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

   2. div-inv N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. fabs-lowering-fabs.f64 N/A

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

   5. --lowering--.f64 N/A

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

   6. metadata-eval 99.9

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in y around inf

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

7. Simplified 57.5%

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

Alternative 7: 11.2% accurate, 20.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Simplified 11.3%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024204 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))