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

Percentage Accurate: 99.9% → 99.9%

Time: 6.6s

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

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

   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 2: 59.2% 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 59.6

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

7. Applied rewrites 59.6%

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

Alternative 3: 54.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \left|y - x\right| \cdot 0.5 \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = abs((y - x)) * 0.5d0
end function

Java

public static double code(double x, double y) {
	return Math.abs((y - x)) * 0.5;
}

Python

def code(x, y):
	return math.fabs((y - x)) * 0.5

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = abs((y - x)) * 0.5;
end

Wolfram

code[x_, y_] := N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] * 0.5), $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 54.9

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

5. Applied rewrites 54.9%

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

Alternative 4: 7.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left|-x\right| \cdot 0.5 \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = abs(-x) * 0.5d0
end function

Java

public static double code(double x, double y) {
	return Math.abs(-x) * 0.5;
}

Python

def code(x, y):
	return math.fabs(-x) * 0.5

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = abs(-x) * 0.5;
end

Wolfram

code[x_, y_] := N[(N[Abs[(-x)], $MachinePrecision] * 0.5), $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 54.9

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

5. Applied rewrites 54.9%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 7.2%

   \[\leadsto \left|-x\right| \cdot 0.5 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024243 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))