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

Percentage Accurate: 99.9% → 99.9%

Time: 8.5s

Alternatives: 4

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

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

   2. lift-fabs.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-/.f64 N/A

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

   6. div-inv N/A

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

   7. lower-fma.f64 N/A

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

   8. metadata-eval 99.9

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

4. Applied egg-rr 99.9%

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

Alternative 2: 78.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|y - x\right|\\ \mathbf{if}\;x + \frac{t\_0}{2} \leq -5 \cdot 10^{-284}:\\ \;\;\;\;\mathsf{fma}\left(\left|x\right|, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fabs (- y x))))
   (if (<= (+ x (/ t_0 2.0)) -5e-284) (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)) <= -5e-284) {
		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)) <= -5e-284)
		tmp = fma(abs(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], -5e-284], 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))) < -4.99999999999999973e-284

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. lift-fabs.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval 100.0

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

   4. Applied egg-rr 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 98.6

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

   7. Simplified 98.6%

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

      1. fabs-neg N/A

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

      2. lower-fabs.f64 98.6

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

   9. Applied egg-rr 98.6%

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

   if -4.99999999999999973e-284 < (+.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 73.8

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

   5. Simplified 73.8%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left|y - x\right|}{2} \leq -5 \cdot 10^{-284}:\\ \;\;\;\;\mathsf{fma}\left(\left|x\right|, 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.3% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[Abs[x], $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

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

   2. lift-fabs.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-/.f64 N/A

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

   6. div-inv N/A

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

   7. lower-fma.f64 N/A

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

   8. metadata-eval 99.9

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 50.8

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

7. Simplified 50.8%

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

   1. fabs-neg N/A

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

   2. lower-fabs.f64 50.8

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

9. Applied egg-rr 50.8%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\left|x\right|}, 0.5, x\right) \]
10. 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

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

   2. lift-fabs.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. flip-+ N/A

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

   5. 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 egg-rr 57.7%

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

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 11.4

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

7. Simplified 11.4%

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

Reproduce

herbie shell --seed 2024207 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))