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

Percentage Accurate: 99.9% → 99.9%

Time: 8.1s

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, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. difference-of-squares N/A

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

   4. lift--.f64 N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 99.9

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

4. Applied rewrites 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

6. Applied rewrites 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 59.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|y - x\right|\\ \mathbf{if}\;\frac{t\_0}{2} + x \leq 5 \cdot 10^{-260}:\\ \;\;\;\;0.75 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fabs (- y x))))
   (if (<= (+ (/ t_0 2.0) x) 5e-260) (* 0.75 x) (* t_0 0.5))))

C

double code(double x, double y) {
	double t_0 = fabs((y - x));
	double tmp;
	if (((t_0 / 2.0) + x) <= 5e-260) {
		tmp = 0.75 * x;
	} else {
		tmp = t_0 * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((y - x))
    if (((t_0 / 2.0d0) + x) <= 5d-260) then
        tmp = 0.75d0 * x
    else
        tmp = t_0 * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.abs((y - x));
	double tmp;
	if (((t_0 / 2.0) + x) <= 5e-260) {
		tmp = 0.75 * x;
	} else {
		tmp = t_0 * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.fabs((y - x))
	tmp = 0
	if ((t_0 / 2.0) + x) <= 5e-260:
		tmp = 0.75 * x
	else:
		tmp = t_0 * 0.5
	return tmp

Julia

function code(x, y)
	t_0 = abs(Float64(y - x))
	tmp = 0.0
	if (Float64(Float64(t_0 / 2.0) + x) <= 5e-260)
		tmp = Float64(0.75 * x);
	else
		tmp = Float64(t_0 * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = abs((y - x));
	tmp = 0.0;
	if (((t_0 / 2.0) + x) <= 5e-260)
		tmp = 0.75 * x;
	else
		tmp = t_0 * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 / 2.0), $MachinePrecision] + x), $MachinePrecision], 5e-260], N[(0.75 * x), $MachinePrecision], N[(t$95$0 * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (fabs.f64 (-.f64 y x)) #s(literal 2 binary64))) < 5.0000000000000003e-260

   1. Initial program 100.0%

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

   4. Applied rewrites 42.1%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 20.0

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

   7. Applied rewrites 20.0%

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

   if 5.0000000000000003e-260 < (+.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. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

      9. +-commutative N/A

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

      10. lower-fabs.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

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

      14. remove-double-neg N/A

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

      15. sub-neg N/A

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

      16. lower--.f64 68.4

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

   5. Applied rewrites 68.4%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left|y - x\right|}{2} + x \leq 5 \cdot 10^{-260}:\\ \;\;\;\;0.75 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. div-inv N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-fabs.f64 N/A

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

   7. neg-fabs N/A

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

   8. lower-fabs.f64 N/A

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

   9. lift--.f64 N/A

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

   10. sub-neg N/A

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

   11. +-commutative N/A

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

   12. distribute-neg-in N/A

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

   13. remove-double-neg N/A

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

   14. sub-neg N/A

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

   15. lower--.f64 N/A

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

   16. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 4: 59.7% accurate, 1.8× 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(Float64(-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. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. div-inv N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-fabs.f64 N/A

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

   7. neg-fabs N/A

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

   8. lower-fabs.f64 N/A

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

   9. lift--.f64 N/A

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

   10. sub-neg N/A

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

   11. +-commutative N/A

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

   12. distribute-neg-in N/A

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

   13. remove-double-neg N/A

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

   14. sub-neg N/A

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

   15. lower--.f64 N/A

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

   16. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in y around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 55.6

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

7. Applied rewrites 55.6%

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

Alternative 5: 11.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* 1.0 x))

C

double code(double x, double y) {
	return 1.0 * x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 * x
end function

Java

public static double code(double x, double y) {
	return 1.0 * x;
}

Python

def code(x, y):
	return 1.0 * x

Julia

function code(x, y)
	return Float64(1.0 * x)
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 * x;
end

Wolfram

code[x_, y_] := N[(1.0 * x), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. sub-neg N/A

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

   3. mul-1-neg N/A

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

   4. lower-*.f64 N/A

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

   5. mul-1-neg N/A

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

   6. remove-double-neg N/A

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

   7. mul-1-neg N/A

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

   8. distribute-neg-in N/A

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

   9. +-commutative N/A

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

   10. lower-fabs.f64 N/A

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

   11. +-commutative N/A

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

   12. distribute-neg-in N/A

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

   13. mul-1-neg N/A

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

   14. remove-double-neg N/A

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

   15. sub-neg N/A

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

   16. lower--.f64 50.2

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

5. Applied rewrites 50.2%

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

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot \frac{\left|y - x\right|}{x}\right)} \]
7. 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. fabs-sub N/A

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

   3. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. associate-*r/ N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. lower-fma.f64 N/A

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

   9. fabs-sub N/A

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

   10. sub-neg N/A

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

   11. mul-1-neg N/A

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

   12. lower-fabs.f64 N/A

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

   13. mul-1-neg N/A

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

   14. sub-neg N/A

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

   15. lower--.f64 N/A

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

   16. lower-/.f64 86.1

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

8. Applied rewrites 86.1%

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

9. Taylor expanded in x around inf

   \[\leadsto 1 \cdot x \]
10. Step-by-step derivation

11. Applied rewrites 12.0%

    \[\leadsto 1 \cdot x \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024235 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))