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

Percentage Accurate: 99.9% → 99.9%

Time: 5.4s

Alternatives: 8

Speedup: 1.0×

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, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (z + (((x + y) + x) + y)) + 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 = (z + (((x + y) + x) + y)) + x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. associate-+r+ N/A

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

   6. lower-+.f64 N/A

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

   7. lower-+.f64 99.9

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(3, x, z\right)\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, y\right) + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.55e+67)
   (fma 3.0 x z)
   (if (<= z 9e+15) (+ (fma x 3.0 y) y) (+ (fma 2.0 y z) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.55e+67) {
		tmp = fma(3.0, x, z);
	} else if (z <= 9e+15) {
		tmp = fma(x, 3.0, y) + y;
	} else {
		tmp = fma(2.0, y, z) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.55e+67)
		tmp = fma(3.0, x, z);
	elseif (z <= 9e+15)
		tmp = Float64(fma(x, 3.0, y) + y);
	else
		tmp = Float64(fma(2.0, y, z) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.55e+67], N[(3.0 * x + z), $MachinePrecision], If[LessEqual[z, 9e+15], N[(N[(x * 3.0 + y), $MachinePrecision] + y), $MachinePrecision], N[(N[(2.0 * y + z), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.54999999999999998e67

   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 89.0

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

   5. Applied rewrites 89.0%

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

   if -1.54999999999999998e67 < z < 9e15

   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. lower-*.f64 90.9

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

   5. Applied rewrites 90.9%

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

   7. Applied rewrites 90.9%

      \[\leadsto \mathsf{fma}\left(3, x, y + y\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 90.9%

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

   if 9e15 < z

   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 x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 84.7

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

   5. Applied rewrites 84.7%

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

Alternative 3: 85.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(3, x, z\right)\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, y\right) + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.55e+67)
   (fma 3.0 x z)
   (if (<= z 9e+15) (+ (fma x 3.0 y) y) (fma 2.0 y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.55e+67) {
		tmp = fma(3.0, x, z);
	} else if (z <= 9e+15) {
		tmp = fma(x, 3.0, y) + y;
	} else {
		tmp = fma(2.0, y, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.55e+67)
		tmp = fma(3.0, x, z);
	elseif (z <= 9e+15)
		tmp = Float64(fma(x, 3.0, y) + y);
	else
		tmp = fma(2.0, y, z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.55e+67], N[(3.0 * x + z), $MachinePrecision], If[LessEqual[z, 9e+15], N[(N[(x * 3.0 + y), $MachinePrecision] + y), $MachinePrecision], N[(2.0 * y + z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.54999999999999998e67

   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 89.0

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

   5. Applied rewrites 89.0%

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

   if -1.54999999999999998e67 < z < 9e15

   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. lower-*.f64 90.9

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

   5. Applied rewrites 90.9%

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

   7. Applied rewrites 90.9%

      \[\leadsto \mathsf{fma}\left(3, x, y + y\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 90.9%

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

   if 9e15 < z

   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 77.7

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

   5. Applied rewrites 77.7%

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

   7. Applied rewrites 77.7%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 82.5

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

   10. Applied rewrites 82.5%

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

Alternative 4: 85.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(3, x, z\right)\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+125}:\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3, x, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.4e+47)
   (fma 3.0 x z)
   (if (<= x 6.8e+125) (fma 2.0 y z) (fma 3.0 x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.4e+47) {
		tmp = fma(3.0, x, z);
	} else if (x <= 6.8e+125) {
		tmp = fma(2.0, y, z);
	} else {
		tmp = fma(3.0, x, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.4e+47)
		tmp = fma(3.0, x, z);
	elseif (x <= 6.8e+125)
		tmp = fma(2.0, y, z);
	else
		tmp = fma(3.0, x, z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.4e+47], N[(3.0 * x + z), $MachinePrecision], If[LessEqual[x, 6.8e+125], N[(2.0 * y + z), $MachinePrecision], N[(3.0 * x + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.3999999999999998e47 or 6.7999999999999998e125 < 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 86.5

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

   5. Applied rewrites 86.5%

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

   if -3.3999999999999998e47 < x < 6.7999999999999998e125

   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 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 53.8

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

   5. Applied rewrites 53.8%

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

   7. Applied rewrites 53.8%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 86.3

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

   10. Applied rewrites 86.3%

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

Alternative 5: 79.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+146}:\\ \;\;\;\;3 \cdot x\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.7e+146) (* 3.0 x) (if (<= x 2.2e+139) (fma 2.0 y z) (* 3.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.7e+146) {
		tmp = 3.0 * x;
	} else if (x <= 2.2e+139) {
		tmp = fma(2.0, y, z);
	} else {
		tmp = 3.0 * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.7e+146)
		tmp = Float64(3.0 * x);
	elseif (x <= 2.2e+139)
		tmp = fma(2.0, y, z);
	else
		tmp = Float64(3.0 * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.7e+146], N[(3.0 * x), $MachinePrecision], If[LessEqual[x, 2.2e+139], N[(2.0 * y + z), $MachinePrecision], N[(3.0 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.69999999999999995e146 or 2.1999999999999999e139 < 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 75.7

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

   5. Applied rewrites 75.7%

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

   if -1.69999999999999995e146 < x < 2.1999999999999999e139

   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 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 56.9

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

   5. Applied rewrites 56.9%

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

   7. Applied rewrites 56.8%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 83.0

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

   10. Applied rewrites 83.0%

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

Alternative 6: 52.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+59}:\\ \;\;\;\;3 \cdot x\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-19}:\\ \;\;\;\;y + y\\ \mathbf{else}:\\ \;\;\;\;3 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.2e+59) (* 3.0 x) (if (<= x 1.4e-19) (+ y y) (* 3.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.2e+59) {
		tmp = 3.0 * x;
	} else if (x <= 1.4e-19) {
		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 <= (-3.2d+59)) then
        tmp = 3.0d0 * x
    else if (x <= 1.4d-19) 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 <= -3.2e+59) {
		tmp = 3.0 * x;
	} else if (x <= 1.4e-19) {
		tmp = y + y;
	} else {
		tmp = 3.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.2e+59:
		tmp = 3.0 * x
	elif x <= 1.4e-19:
		tmp = y + y
	else:
		tmp = 3.0 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.2e+59)
		tmp = Float64(3.0 * x);
	elseif (x <= 1.4e-19)
		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 <= -3.2e+59)
		tmp = 3.0 * x;
	elseif (x <= 1.4e-19)
		tmp = y + y;
	else
		tmp = 3.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.2e+59], N[(3.0 * x), $MachinePrecision], If[LessEqual[x, 1.4e-19], N[(y + y), $MachinePrecision], N[(3.0 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.19999999999999982e59 or 1.40000000000000001e-19 < 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 x around inf

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

      1. lower-*.f64 62.7

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

   5. Applied rewrites 62.7%

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

   if -3.19999999999999982e59 < x < 1.40000000000000001e-19

   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 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. lower-*.f64 58.1

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

   5. Applied rewrites 58.1%

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

   7. Applied rewrites 58.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 48.7%

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

   12. Applied rewrites 48.7%

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

Alternative 7: 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]

Derivation

1. Initial program 99.9%

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

Alternative 8: 34.1% 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 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. lower-*.f64 68.2

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

5. Applied rewrites 68.2%

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

7. Applied rewrites 68.2%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 34.9%

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

12. Applied rewrites 34.9%

    \[\leadsto y + y \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024332 
(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))