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

Percentage Accurate: 99.9% → 99.9%

Time: 10.3s

Alternatives: 5

Speedup: 1.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. div-inv N/A

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

   5. lower-fma.f64 N/A

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

   6. 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. Add Preprocessing

Alternative 2: 77.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|y - x\right|\\ \mathbf{if}\;x + \frac{t\_0}{2} \leq 10^{-272}:\\ \;\;\;\;\mathsf{fma}\left(\left|-x\right|, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fabs (- y x))))
   (if (<= (+ x (/ t_0 2.0)) 1e-272) (fma (fabs (- x)) 0.5 x) (* t_0 0.5))))

C

double code(double x, double y) {
	double t_0 = fabs((y - x));
	double tmp;
	if ((x + (t_0 / 2.0)) <= 1e-272) {
		tmp = fma(fabs(-x), 0.5, 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(x + Float64(t_0 / 2.0)) <= 1e-272)
		tmp = fma(abs(Float64(-x)), 0.5, x);
	else
		tmp = Float64(t_0 * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(x + N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision], 1e-272], N[(N[Abs[(-x)], $MachinePrecision] * 0.5 + 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))) < 9.9999999999999993e-273

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. 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 99.3

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

   7. Applied rewrites 99.3%

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

   if 9.9999999999999993e-273 < (+.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. 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 73.2

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

   5. Applied rewrites 73.2%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left|y - x\right|}{2} \leq 10^{-272}:\\ \;\;\;\;\mathsf{fma}\left(\left|-x\right|, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|y - x\right|\\ \mathbf{if}\;x + \frac{t\_0}{2} \leq 10^{-272}:\\ \;\;\;\;x \cdot 0.75\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fabs (- y x))))
   (if (<= (+ x (/ t_0 2.0)) 1e-272) (* x 0.75) (* t_0 0.5))))

C

double code(double x, double y) {
	double t_0 = fabs((y - x));
	double tmp;
	if ((x + (t_0 / 2.0)) <= 1e-272) {
		tmp = x * 0.75;
	} 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 ((x + (t_0 / 2.0d0)) <= 1d-272) then
        tmp = x * 0.75d0
    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 ((x + (t_0 / 2.0)) <= 1e-272) {
		tmp = x * 0.75;
	} else {
		tmp = t_0 * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.fabs((y - x))
	tmp = 0
	if (x + (t_0 / 2.0)) <= 1e-272:
		tmp = x * 0.75
	else:
		tmp = t_0 * 0.5
	return tmp

Julia

function code(x, y)
	t_0 = abs(Float64(y - x))
	tmp = 0.0
	if (Float64(x + Float64(t_0 / 2.0)) <= 1e-272)
		tmp = Float64(x * 0.75);
	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 ((x + (t_0 / 2.0)) <= 1e-272)
		tmp = x * 0.75;
	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[(x + N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision], 1e-272], N[(x * 0.75), $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))) < 9.9999999999999993e-273

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.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 rewrites 58.1%

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

   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. lower-*.f64 19.9

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

   7. Applied rewrites 19.9%

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

   if 9.9999999999999993e-273 < (+.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. 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 73.2

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

   5. Applied rewrites 73.2%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left|y - x\right|}{2} \leq 10^{-272}:\\ \;\;\;\;x \cdot 0.75\\ \mathbf{else}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 11.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.75 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x 0.75))

C

double code(double x, double y) {
	return x * 0.75;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * 0.75d0
end function

Java

public static double code(double x, double y) {
	return x * 0.75;
}

Python

def code(x, y):
	return x * 0.75

Julia

function code(x, y)
	return Float64(x * 0.75)
end

MATLAB

function tmp = code(x, y)
	tmp = x * 0.75;
end

Wolfram

code[x_, y_] := N[(x * 0.75), $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. flip-+ N/A

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

   3. lower-/.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 rewrites 53.9%

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

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. lower-*.f64 11.3

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

7. Applied rewrites 11.3%

   \[\leadsto \color{blue}{x \cdot 0.75} \]
8. Add Preprocessing

Alternative 5: 11.5% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(x * 1.0), $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. flip3-+ N/A

      \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{\left|y - x\right|}{2}\right)}^{3}}{x \cdot x + \left(\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot \frac{\left|y - x\right|}{2}\right)}} \]

   3. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot \frac{\left|y - x\right|}{2}\right)}{{x}^{3} + {\left(\frac{\left|y - x\right|}{2}\right)}^{3}}}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot \frac{\left|y - x\right|}{2}\right)}{{x}^{3} + {\left(\frac{\left|y - x\right|}{2}\right)}^{3}}}} \]

   5. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{x}^{3} + {\left(\frac{\left|y - x\right|}{2}\right)}^{3}}{x \cdot x + \left(\frac{\left|y - x\right|}{2} \cdot \frac{\left|y - x\right|}{2} - x \cdot \frac{\left|y - x\right|}{2}\right)}}}} \]

   6. flip3-+ N/A

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

   7. lift-+.f64 N/A

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

   8. lower-/.f64 99.7

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-/.f64 N/A

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

   12. div-inv N/A

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

   13. lower-fma.f64 N/A

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

4. Applied rewrites 99.7%

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

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. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-fabs.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-/.f64 90.2

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

7. Applied rewrites 90.2%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 11.2%

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

Reproduce

herbie shell --seed 2024233 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))