Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendInside from plot-0.2.3.4

Percentage Accurate: 99.9% → 99.9%

Time: 4.9s

Alternatives: 6

Speedup: 1.2×

Specification

Math

\[\begin{array}{l} \\ \left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (+ (+ (+ (+ x y) y) x) z) x))

C

double code(double x, double y, double z) {
	return ((((x + y) + y) + x) + z) + x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((((x + y) + y) + x) + z) + x
end function

Java

public static double code(double x, double y, double z) {
	return ((((x + y) + y) + x) + z) + x;
}

Python

def code(x, y, z):
	return ((((x + y) + y) + x) + z) + x

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(Float64(x + y) + y) + x) + z) + x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((((x + y) + y) + x) + z) + x;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(N[(x + y), $MachinePrecision] + y), $MachinePrecision] + x), $MachinePrecision] + z), $MachinePrecision] + x), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (+ (+ (+ (+ x y) y) x) z) x))

C

double code(double x, double y, double z) {
	return ((((x + y) + y) + x) + z) + x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((((x + y) + y) + x) + z) + x
end function

Java

public static double code(double x, double y, double z) {
	return ((((x + y) + y) + x) + z) + x;
}

Python

def code(x, y, z):
	return ((((x + y) + y) + x) + z) + x

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(Float64(x + y) + y) + x) + z) + x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((((x + y) + y) + x) + z) + x;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(N[(x + y), $MachinePrecision] + y), $MachinePrecision] + x), $MachinePrecision] + z), $MachinePrecision] + x), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(2, x + y, z + x\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma 2.0 (+ x y) (+ z x)))

C

double code(double x, double y, double z) {
	return fma(2.0, (x + y), (z + x));
}

Julia

function code(x, y, z)
	return fma(2.0, Float64(x + y), Float64(z + x))
end

Wolfram

code[x_, y_, z_] := N[(2.0 * N[(x + y), $MachinePrecision] + N[(z + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x} \]

   2. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right)} + x \]

   3. associate-+l+ N/A

      \[\leadsto \color{blue}{\left(\left(\left(x + y\right) + y\right) + x\right) + \left(z + x\right)} \]

   4. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\left(x + y\right) + y\right) + x\right)} + \left(z + x\right) \]

   5. lift-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\left(x + y\right) + y\right)} + x\right) + \left(z + x\right) \]

   6. associate-+l+ N/A

      \[\leadsto \color{blue}{\left(\left(x + y\right) + \left(y + x\right)\right)} + \left(z + x\right) \]

   7. +-commutative N/A

      \[\leadsto \left(\left(x + y\right) + \color{blue}{\left(x + y\right)}\right) + \left(z + x\right) \]

   8. lift-+.f64 N/A

      \[\leadsto \left(\left(x + y\right) + \color{blue}{\left(x + y\right)}\right) + \left(z + x\right) \]

   9. count-2 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(2, x + y, x + z\right)} \]

   12. lift-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(2, \color{blue}{x + y}, x + z\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(2, \color{blue}{y + x}, x + z\right) \]

   14. lower-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(2, \color{blue}{y + x}, x + z\right) \]

   15. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(2, y + x, \color{blue}{z + x}\right) \]

   16. lower-+.f64 99.9

       \[\leadsto \mathsf{fma}\left(2, y + x, \color{blue}{z + x}\right) \]

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, y + x, z + x\right)} \]

5. Final simplification 99.9%

   \[\leadsto \mathsf{fma}\left(2, x + y, z + x\right) \]
6. Add Preprocessing

Alternative 2: 85.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(3, x, z\right)\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(3, x, y + y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 2, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.7e+82)
   (fma 3.0 x z)
   (if (<= z 1.35e+81) (fma 3.0 x (+ y y)) (fma y 2.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.7e+82) {
		tmp = fma(3.0, x, z);
	} else if (z <= 1.35e+81) {
		tmp = fma(3.0, x, (y + y));
	} else {
		tmp = fma(y, 2.0, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.7e+82)
		tmp = fma(3.0, x, z);
	elseif (z <= 1.35e+81)
		tmp = fma(3.0, x, Float64(y + y));
	else
		tmp = fma(y, 2.0, z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.7e+82], N[(3.0 * x + z), $MachinePrecision], If[LessEqual[z, 1.35e+81], N[(3.0 * x + N[(y + y), $MachinePrecision]), $MachinePrecision], N[(y * 2.0 + z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.6999999999999999e82

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \left(z + 2 \cdot x\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \color{blue}{\left(2 \cdot x + z\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + 2 \cdot x\right) + z} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(2 + 1\right) \cdot x} + z \]

      4. metadata-eval N/A

         \[\leadsto \color{blue}{3} \cdot x + z \]

      5. lower-fma.f64 91.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(3, x, z\right)} \]

   5. Applied rewrites 91.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3, x, z\right)} \]

   if -2.6999999999999999e82 < z < 1.35e81

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \color{blue}{3} \cdot x + 2 \cdot y \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(3, x, 2 \cdot y\right)} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(3, x, \color{blue}{y \cdot 2}\right) \]

      6. lower-*.f64 90.2

         \[\leadsto \mathsf{fma}\left(3, x, \color{blue}{y \cdot 2}\right) \]

   5. Applied rewrites 90.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3, x, y \cdot 2\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 90.2%

      \[\leadsto \mathsf{fma}\left(3, x, y + y\right) \]

   if 1.35e81 < z

   1. Initial program 100.0%

      \[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{z + 2 \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{2 \cdot y + z} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot 2} + z \]

      3. lower-fma.f64 96.6

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 2, z\right)} \]

   5. Applied rewrites 96.6%

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

Alternative 3: 85.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(3, x, z\right)\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(y, 2, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3, x, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.15e+69)
   (fma 3.0 x z)
   (if (<= x 2.5e-46) (fma y 2.0 z) (fma 3.0 x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.15e+69) {
		tmp = fma(3.0, x, z);
	} else if (x <= 2.5e-46) {
		tmp = fma(y, 2.0, z);
	} else {
		tmp = fma(3.0, x, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.15e+69)
		tmp = fma(3.0, x, z);
	elseif (x <= 2.5e-46)
		tmp = fma(y, 2.0, z);
	else
		tmp = fma(3.0, x, z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.15e+69], N[(3.0 * x + z), $MachinePrecision], If[LessEqual[x, 2.5e-46], N[(y * 2.0 + z), $MachinePrecision], N[(3.0 * x + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.15000000000000008e69 or 2.49999999999999996e-46 < x

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \left(z + 2 \cdot x\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \color{blue}{\left(2 \cdot x + z\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + 2 \cdot x\right) + z} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(2 + 1\right) \cdot x} + z \]

      4. metadata-eval N/A

         \[\leadsto \color{blue}{3} \cdot x + z \]

      5. lower-fma.f64 86.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(3, x, z\right)} \]

   5. Applied rewrites 86.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3, x, z\right)} \]

   if -1.15000000000000008e69 < x < 2.49999999999999996e-46

   1. Initial program 100.0%

      \[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{z + 2 \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{2 \cdot y + z} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot 2} + z \]

      3. lower-fma.f64 94.5

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 2, z\right)} \]

   5. Applied rewrites 94.5%

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

Alternative 4: 79.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+197}:\\ \;\;\;\;y + y\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(3, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;y + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.2e+197) (+ y y) (if (<= y 1.35e+116) (fma 3.0 x z) (+ y y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.2e+197) {
		tmp = y + y;
	} else if (y <= 1.35e+116) {
		tmp = fma(3.0, x, z);
	} else {
		tmp = y + y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.2e+197)
		tmp = Float64(y + y);
	elseif (y <= 1.35e+116)
		tmp = fma(3.0, x, z);
	else
		tmp = Float64(y + y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.2e+197], N[(y + y), $MachinePrecision], If[LessEqual[y, 1.35e+116], N[(3.0 * x + z), $MachinePrecision], N[(y + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.19999999999999989e197 or 1.35e116 < y

   1. Initial program 100.0%

      \[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{2 \cdot y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot 2} \]

      2. lower-*.f64 78.2

         \[\leadsto \color{blue}{y \cdot 2} \]

   5. Applied rewrites 78.2%

      \[\leadsto \color{blue}{y \cdot 2} \]
   6. Step-by-step derivation

   7. Applied rewrites 78.2%

      \[\leadsto y + \color{blue}{y} \]

   if -2.19999999999999989e197 < y < 1.35e116

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \left(z + 2 \cdot x\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \color{blue}{\left(2 \cdot x + z\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + 2 \cdot x\right) + z} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(2 + 1\right) \cdot x} + z \]

      4. metadata-eval N/A

         \[\leadsto \color{blue}{3} \cdot x + z \]

      5. lower-fma.f64 81.7

         \[\leadsto \color{blue}{\mathsf{fma}\left(3, x, z\right)} \]

   5. Applied rewrites 81.7%

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

Alternative 5: 52.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+69}:\\ \;\;\;\;3 \cdot x\\ \mathbf{elif}\;x \leq 750000000:\\ \;\;\;\;y + y\\ \mathbf{else}:\\ \;\;\;\;3 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.15e+69) (* 3.0 x) (if (<= x 750000000.0) (+ y y) (* 3.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.15e+69) {
		tmp = 3.0 * x;
	} else if (x <= 750000000.0) {
		tmp = y + y;
	} else {
		tmp = 3.0 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.15d+69)) then
        tmp = 3.0d0 * x
    else if (x <= 750000000.0d0) then
        tmp = y + y
    else
        tmp = 3.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.15e+69) {
		tmp = 3.0 * x;
	} else if (x <= 750000000.0) {
		tmp = y + y;
	} else {
		tmp = 3.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.15e+69:
		tmp = 3.0 * x
	elif x <= 750000000.0:
		tmp = y + y
	else:
		tmp = 3.0 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.15e+69)
		tmp = Float64(3.0 * x);
	elseif (x <= 750000000.0)
		tmp = Float64(y + y);
	else
		tmp = Float64(3.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.15e+69)
		tmp = 3.0 * x;
	elseif (x <= 750000000.0)
		tmp = y + y;
	else
		tmp = 3.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.15e+69], N[(3.0 * x), $MachinePrecision], If[LessEqual[x, 750000000.0], N[(y + y), $MachinePrecision], N[(3.0 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.15000000000000008e69 or 7.5e8 < x

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 66.9

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

   5. Applied rewrites 66.9%

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

   if -1.15000000000000008e69 < x < 7.5e8

   1. Initial program 100.0%

      \[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{2 \cdot y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot 2} \]

      2. lower-*.f64 53.7

         \[\leadsto \color{blue}{y \cdot 2} \]

   5. Applied rewrites 53.7%

      \[\leadsto \color{blue}{y \cdot 2} \]
   6. Step-by-step derivation

   7. Applied rewrites 53.7%

      \[\leadsto y + \color{blue}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 34.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ y + y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ y y))

C

double code(double x, double y, double z) {
	return y + y;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = y + y
end function

Java

public static double code(double x, double y, double z) {
	return y + y;
}

Python

def code(x, y, z):
	return y + y

Julia

function code(x, y, z)
	return Float64(y + y)
end

MATLAB

function tmp = code(x, y, z)
	tmp = y + y;
end

Wolfram

code[x_, y_, z_] := N[(y + y), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]
2. Add Preprocessing

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{2 \cdot y} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{y \cdot 2} \]

   2. lower-*.f64 35.6

      \[\leadsto \color{blue}{y \cdot 2} \]

5. Applied rewrites 35.6%

   \[\leadsto \color{blue}{y \cdot 2} \]
6. Step-by-step derivation

7. Applied rewrites 35.6%

   \[\leadsto y + \color{blue}{y} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024255 
(FPCore (x y z)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendInside from plot-0.2.3.4"
  :precision binary64
  (+ (+ (+ (+ (+ x y) y) x) z) x))