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

Percentage Accurate: 99.9% → 99.9%

Time: 5.6s

Alternatives: 10

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(z + N[(N[(N[(y + x), $MachinePrecision] + y), $MachinePrecision] + x), $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. Final simplification 99.9%

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

Alternative 2: 85.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(3, x, z\right)\\ \mathbf{elif}\;z \leq 2.65 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(3, x, y + y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z + y\right) + \left(y + x\right)\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.1e+55)
   (fma 3.0 x z)
   (if (<= z 2.65e-30) (fma 3.0 x (+ y y)) (+ (+ (+ z y) (+ y x)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.1e+55) {
		tmp = fma(3.0, x, z);
	} else if (z <= 2.65e-30) {
		tmp = fma(3.0, x, (y + y));
	} else {
		tmp = ((z + y) + (y + x)) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.1e+55)
		tmp = fma(3.0, x, z);
	elseif (z <= 2.65e-30)
		tmp = fma(3.0, x, Float64(y + y));
	else
		tmp = Float64(Float64(Float64(z + y) + Float64(y + x)) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.1e+55], N[(3.0 * x + z), $MachinePrecision], If[LessEqual[z, 2.65e-30], N[(3.0 * x + N[(y + y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z + y), $MachinePrecision] + N[(y + x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.09999999999999981e55

   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 86.1

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

   5. Applied rewrites 86.1%

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

   if -4.09999999999999981e55 < z < 2.64999999999999987e-30

   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 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 93.6

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

   5. Applied rewrites 93.6%

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

   7. Applied rewrites 93.6%

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

   if 2.64999999999999987e-30 < z

   1. Initial program 100.0%

      \[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]
   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. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+r+ N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 100.0

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 100.0

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 95.4

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

   7. Applied rewrites 95.4%

      \[\leadsto \left(\color{blue}{\left(y + z\right)} + \left(y + x\right)\right) + x \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(3, x, z\right)\\ \mathbf{elif}\;z \leq 2.65 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(3, x, y + y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z + y\right) + \left(y + x\right)\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(3, x, z\right)\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(3, x, y + y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.1e+55)
   (fma 3.0 x z)
   (if (<= z 6e-30) (fma 3.0 x (+ y y)) (+ (fma 2.0 y z) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.1e+55) {
		tmp = fma(3.0, x, z);
	} else if (z <= 6e-30) {
		tmp = fma(3.0, x, (y + y));
	} else {
		tmp = fma(2.0, y, z) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.1e+55)
		tmp = fma(3.0, x, z);
	elseif (z <= 6e-30)
		tmp = fma(3.0, x, Float64(y + y));
	else
		tmp = Float64(fma(2.0, y, z) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.1e+55], N[(3.0 * x + z), $MachinePrecision], If[LessEqual[z, 6e-30], N[(3.0 * x + N[(y + y), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * y + z), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.09999999999999981e55

   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 86.1

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

   5. Applied rewrites 86.1%

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

   if -4.09999999999999981e55 < z < 5.9999999999999998e-30

   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 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 93.6

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

   5. Applied rewrites 93.6%

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

   7. Applied rewrites 93.6%

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

   if 5.9999999999999998e-30 < 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

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

      1. +-commutative N/A

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

      2. lower-fma.f64 95.3

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

   5. Applied rewrites 95.3%

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

Alternative 4: 84.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(3, x, z\right)\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(3, x, y + y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z + y\right) + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.1e+55)
   (fma 3.0 x z)
   (if (<= z 6e-30) (fma 3.0 x (+ y y)) (+ (+ z y) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.1e+55) {
		tmp = fma(3.0, x, z);
	} else if (z <= 6e-30) {
		tmp = fma(3.0, x, (y + y));
	} else {
		tmp = (z + y) + y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.1e+55)
		tmp = fma(3.0, x, z);
	elseif (z <= 6e-30)
		tmp = fma(3.0, x, Float64(y + y));
	else
		tmp = Float64(Float64(z + y) + y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.1e+55], N[(3.0 * x + z), $MachinePrecision], If[LessEqual[z, 6e-30], N[(3.0 * x + N[(y + y), $MachinePrecision]), $MachinePrecision], N[(N[(z + y), $MachinePrecision] + y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.09999999999999981e55

   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 86.1

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

   5. Applied rewrites 86.1%

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

   if -4.09999999999999981e55 < z < 5.9999999999999998e-30

   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 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 93.6

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

   5. Applied rewrites 93.6%

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

   7. Applied rewrites 93.6%

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

   if 5.9999999999999998e-30 < 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 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 29.0

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

   5. Applied rewrites 29.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 24.3%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 94.6

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

   11. Applied rewrites 94.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(2, y, z\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 94.6%

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

Alternative 5: 85.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.4 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right)\\ \mathbf{elif}\;y \leq 2.95 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(3, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z + y\right) + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7.4e+56)
   (fma 2.0 y z)
   (if (<= y 2.95e+54) (fma 3.0 x z) (+ (+ z y) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.4e+56) {
		tmp = fma(2.0, y, z);
	} else if (y <= 2.95e+54) {
		tmp = fma(3.0, x, z);
	} else {
		tmp = (z + y) + y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7.4e+56)
		tmp = fma(2.0, y, z);
	elseif (y <= 2.95e+54)
		tmp = fma(3.0, x, z);
	else
		tmp = Float64(Float64(z + y) + y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7.4e+56], N[(2.0 * y + z), $MachinePrecision], If[LessEqual[y, 2.95e+54], N[(3.0 * x + z), $MachinePrecision], N[(N[(z + y), $MachinePrecision] + y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.39999999999999994e56

   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 82.5

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

   5. Applied rewrites 82.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 72.1%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 89.6

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

   11. Applied rewrites 89.6%

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

   if -7.39999999999999994e56 < y < 2.9499999999999999e54

   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 92.1

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

   5. Applied rewrites 92.1%

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

   if 2.9499999999999999e54 < y

   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 86.1

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

   5. Applied rewrites 86.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 69.3%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 83.1

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

   11. Applied rewrites 83.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(2, y, z\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 83.1%

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

Alternative 6: 79.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+139}:\\ \;\;\;\;3 \cdot x\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.5e+139) (* 3.0 x) (if (<= x 1.55e+91) (fma 2.0 y z) (* 3.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.5e+139) {
		tmp = 3.0 * x;
	} else if (x <= 1.55e+91) {
		tmp = fma(2.0, y, z);
	} else {
		tmp = 3.0 * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.5e+139)
		tmp = Float64(3.0 * x);
	elseif (x <= 1.55e+91)
		tmp = fma(2.0, y, z);
	else
		tmp = Float64(3.0 * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.5e+139], N[(3.0 * x), $MachinePrecision], If[LessEqual[x, 1.55e+91], N[(2.0 * y + z), $MachinePrecision], N[(3.0 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.4999999999999996e139 or 1.54999999999999999e91 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 73.6

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

   5. Applied rewrites 73.6%

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

   if -5.4999999999999996e139 < x < 1.54999999999999999e91

   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 52.3

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

   5. Applied rewrites 52.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 39.8%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 87.5

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

   11. Applied rewrites 87.5%

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

Alternative 7: 79.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+139}:\\ \;\;\;\;3 \cdot x\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+91}:\\ \;\;\;\;\left(z + y\right) + y\\ \mathbf{else}:\\ \;\;\;\;3 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.5e+139) (* 3.0 x) (if (<= x 1.55e+91) (+ (+ z y) y) (* 3.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.5e+139) {
		tmp = 3.0 * x;
	} else if (x <= 1.55e+91) {
		tmp = (z + 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 <= (-5.5d+139)) then
        tmp = 3.0d0 * x
    else if (x <= 1.55d+91) then
        tmp = (z + 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 <= -5.5e+139) {
		tmp = 3.0 * x;
	} else if (x <= 1.55e+91) {
		tmp = (z + y) + y;
	} else {
		tmp = 3.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5.5e+139:
		tmp = 3.0 * x
	elif x <= 1.55e+91:
		tmp = (z + y) + y
	else:
		tmp = 3.0 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.5e+139)
		tmp = Float64(3.0 * x);
	elseif (x <= 1.55e+91)
		tmp = Float64(Float64(z + 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 <= -5.5e+139)
		tmp = 3.0 * x;
	elseif (x <= 1.55e+91)
		tmp = (z + y) + y;
	else
		tmp = 3.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.5e+139], N[(3.0 * x), $MachinePrecision], If[LessEqual[x, 1.55e+91], N[(N[(z + y), $MachinePrecision] + y), $MachinePrecision], N[(3.0 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.4999999999999996e139 or 1.54999999999999999e91 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 73.6

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

   5. Applied rewrites 73.6%

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

   if -5.4999999999999996e139 < x < 1.54999999999999999e91

   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 52.3

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

   5. Applied rewrites 52.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 39.8%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 87.5

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

   11. Applied rewrites 87.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(2, y, z\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 87.5%

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

Alternative 8: 54.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.4 \cdot 10^{+56}:\\ \;\;\;\;y + y\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+57}:\\ \;\;\;\;3 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7.4e+56) (+ y y) (if (<= y 3.2e+57) (* 3.0 x) (+ y y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.4e+56) {
		tmp = y + y;
	} else if (y <= 3.2e+57) {
		tmp = 3.0 * x;
	} else {
		tmp = y + y;
	}
	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 (y <= (-7.4d+56)) then
        tmp = y + y
    else if (y <= 3.2d+57) then
        tmp = 3.0d0 * x
    else
        tmp = y + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.4e+56) {
		tmp = y + y;
	} else if (y <= 3.2e+57) {
		tmp = 3.0 * x;
	} else {
		tmp = y + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -7.4e+56:
		tmp = y + y
	elif y <= 3.2e+57:
		tmp = 3.0 * x
	else:
		tmp = y + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7.4e+56)
		tmp = Float64(y + y);
	elseif (y <= 3.2e+57)
		tmp = Float64(3.0 * x);
	else
		tmp = Float64(y + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -7.4e+56)
		tmp = y + y;
	elseif (y <= 3.2e+57)
		tmp = 3.0 * x;
	else
		tmp = y + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7.4e+56], N[(y + y), $MachinePrecision], If[LessEqual[y, 3.2e+57], N[(3.0 * x), $MachinePrecision], N[(y + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.39999999999999994e56 or 3.20000000000000029e57 < y

   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 85.0

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 71.6%

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

   10. Applied rewrites 71.6%

       \[\leadsto y + y \]

   if -7.39999999999999994e56 < y < 3.20000000000000029e57

   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 44.2

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

   5. Applied rewrites 44.2%

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

Alternative 9: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. +-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. associate-+l+ N/A

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

   6. +-commutative N/A

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

   7. lift-+.f64 N/A

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

   8. associate-+r+ N/A

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

   9. lower-+.f64 N/A

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

   10. lower-+.f64 99.9

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 99.9

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 10: 34.5% 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 63.6

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

5. Applied rewrites 63.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 32.3%

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

10. Applied rewrites 32.3%

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

Reproduce

herbie shell --seed 2024296 
(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))