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

Percentage Accurate: 99.9% → 99.9%

Time: 10.4s

Alternatives: 5

Speedup: 1.7×

Specification

Math

\[\begin{array}{l} \\ x + \frac{\left|y - x\right|}{2} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + (abs((y - x)) / 2.0d0)
end function

Java

public static double code(double x, double y) {
	return x + (Math.abs((y - x)) / 2.0);
}

Python

def code(x, y):
	return x + (math.fabs((y - x)) / 2.0)

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = x + (abs((y - x)) / 2.0);
end

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{\left|y - x\right|}{2} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + (abs((y - x)) / 2.0d0)
end function

Java

public static double code(double x, double y) {
	return x + (Math.abs((y - x)) / 2.0);
}

Python

def code(x, y):
	return x + (math.fabs((y - x)) / 2.0)

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = x + (abs((y - x)) / 2.0);
end

Wolfram

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

Alternative 1: 99.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. +-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. Add Preprocessing

Alternative 2: 54.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + \frac{\left|y - x\right|}{2} \leq -5 \cdot 10^{-198}:\\ \;\;\;\;x \cdot 0.75\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left|y\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ x (/ (fabs (- y x)) 2.0)) -5e-198) (* x 0.75) (* 0.5 (fabs y))))

C

double code(double x, double y) {
	double tmp;
	if ((x + (fabs((y - x)) / 2.0)) <= -5e-198) {
		tmp = x * 0.75;
	} 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((y - x)) / 2.0d0)) <= (-5d-198)) then
        tmp = x * 0.75d0
    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((y - x)) / 2.0)) <= -5e-198) {
		tmp = x * 0.75;
	} else {
		tmp = 0.5 * Math.abs(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x + (math.fabs((y - x)) / 2.0)) <= -5e-198:
		tmp = x * 0.75
	else:
		tmp = 0.5 * math.fabs(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x + Float64(abs(Float64(y - x)) / 2.0)) <= -5e-198)
		tmp = Float64(x * 0.75);
	else
		tmp = Float64(0.5 * abs(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x + (abs((y - x)) / 2.0)) <= -5e-198)
		tmp = x * 0.75;
	else
		tmp = 0.5 * abs(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x + N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], -5e-198], N[(x * 0.75), $MachinePrecision], 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))) < -4.9999999999999999e-198

   1. Initial program 100.0%

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

      1. flip-+ N/A

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

      2. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 52.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(y - x\right) \cdot \left(y - x\right), -0.25, x \cdot x\right)}{\mathsf{fma}\left(\left|y - x\right|, -0.5, x\right)}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{3}{4} \cdot x} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{3}{4}} \]

      2. *-lowering-*.f64 20.0

         \[\leadsto \color{blue}{x \cdot 0.75} \]

   7. Simplified 20.0%

      \[\leadsto \color{blue}{x \cdot 0.75} \]

   if -4.9999999999999999e-198 < (+.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. 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. *-lowering-*.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. fabs-lowering-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. --lowering--.f64 69.7

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

   5. Simplified 69.7%

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

   6. Taylor expanded in y around inf

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

   8. Simplified 65.2%

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

Alternative 3: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5, \left|x\right|, x\right)\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-39}:\\ \;\;\;\;\mathsf{fma}\left(\left|y\right|, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma 0.5 (fabs x) x)))
   (if (<= x -5.5e+15) t_0 (if (<= x 1.12e-39) (fma (fabs y) 0.5 x) t_0))))

C

double code(double x, double y) {
	double t_0 = fma(0.5, fabs(x), x);
	double tmp;
	if (x <= -5.5e+15) {
		tmp = t_0;
	} else if (x <= 1.12e-39) {
		tmp = fma(fabs(y), 0.5, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(0.5, abs(x), x)
	tmp = 0.0
	if (x <= -5.5e+15)
		tmp = t_0;
	elseif (x <= 1.12e-39)
		tmp = fma(abs(y), 0.5, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 * N[Abs[x], $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[x, -5.5e+15], t$95$0, If[LessEqual[x, 1.12e-39], N[(N[Abs[y], $MachinePrecision] * 0.5 + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.5e15 or 1.12e-39 < x

   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 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

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

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

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

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

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

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

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

      11. /-lowering-/.f64 99.7

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

   7. Simplified 99.7%

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

   8. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 80.7

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

   10. Simplified 80.7%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

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

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

       3. fabs-neg N/A

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

       4. fabs-lowering-fabs.f64 80.8

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

   13. Simplified 80.8%

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

   if -5.5e15 < x < 1.12e-39

   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. 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 83.9%

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

Alternative 4: 74.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5, \left|x\right|, x\right)\\ \mathbf{if}\;x \leq -1.9 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-58}:\\ \;\;\;\;0.5 \cdot \left|y\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma 0.5 (fabs x) x)))
   (if (<= x -1.9e+16) t_0 (if (<= x 1.22e-58) (* 0.5 (fabs y)) t_0))))

C

double code(double x, double y) {
	double t_0 = fma(0.5, fabs(x), x);
	double tmp;
	if (x <= -1.9e+16) {
		tmp = t_0;
	} else if (x <= 1.22e-58) {
		tmp = 0.5 * fabs(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(0.5, abs(x), x)
	tmp = 0.0
	if (x <= -1.9e+16)
		tmp = t_0;
	elseif (x <= 1.22e-58)
		tmp = Float64(0.5 * abs(y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 * N[Abs[x], $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[x, -1.9e+16], t$95$0, If[LessEqual[x, 1.22e-58], N[(0.5 * N[Abs[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.9e16 or 1.2199999999999999e-58 < x

   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 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

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

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

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

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

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

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

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

      11. /-lowering-/.f64 99.7

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

   7. Simplified 99.7%

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

   8. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 79.9

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

   10. Simplified 79.9%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

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

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

       3. fabs-neg N/A

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

       4. fabs-lowering-fabs.f64 80.0

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

   13. Simplified 80.0%

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

   if -1.9e16 < x < 1.2199999999999999e-58

   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. *-lowering-*.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. fabs-lowering-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. --lowering--.f64 82.8

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

   5. Simplified 82.8%

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

   6. Taylor expanded in y around inf

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

   8. Simplified 82.1%

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

Alternative 5: 11.4% 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 2024205 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))