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

Percentage Accurate: 99.9% → 100.0%

Time: 7.8s

Alternatives: 8

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: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, 3, z + \left(y + y\right)\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma x 3.0 (+ z (+ y y))))

C

double code(double x, double y, double z) {
	return fma(x, 3.0, (z + (y + y)));
}

Julia

function code(x, y, z)
	return fma(x, 3.0, Float64(z + Float64(y + y)))
end

Wolfram

code[x_, y_, z_] := N[(x * 3.0 + N[(z + N[(y + y), $MachinePrecision]), $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. Taylor expanded in x around 0

   \[\leadsto \color{blue}{z + \left(2 \cdot y + 3 \cdot x\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(2 \cdot y + 3 \cdot x\right) + z} \]

   2. associate-+l+ N/A

      \[\leadsto \color{blue}{2 \cdot y + \left(3 \cdot x + z\right)} \]

   3. metadata-eval N/A

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

   4. distribute-rgt1-in N/A

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

   5. associate-+r+ N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. associate-+r+ N/A

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

   10. distribute-rgt1-in N/A

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

   11. metadata-eval N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 99.9

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

5. Applied rewrites 99.9%

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

7. Applied rewrites 100.0%

   \[\leadsto \mathsf{fma}\left(x, \color{blue}{3}, z + \left(y + y\right)\right) \]
8. Add Preprocessing

Alternative 2: 83.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, z\right)\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-62}:\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, y + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.1e+151)
   (fma x 3.0 z)
   (if (<= x 1.05e-62) (fma 2.0 y z) (fma x 3.0 (+ y y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.1e+151) {
		tmp = fma(x, 3.0, z);
	} else if (x <= 1.05e-62) {
		tmp = fma(2.0, y, z);
	} else {
		tmp = fma(x, 3.0, (y + y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.1e+151)
		tmp = fma(x, 3.0, z);
	elseif (x <= 1.05e-62)
		tmp = fma(2.0, y, z);
	else
		tmp = fma(x, 3.0, Float64(y + y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.1e+151], N[(x * 3.0 + z), $MachinePrecision], If[LessEqual[x, 1.05e-62], N[(2.0 * y + z), $MachinePrecision], N[(x * 3.0 + N[(y + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.10000000000000003e151

   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. *-commutative N/A

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

      6. lower-fma.f64 89.6

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

   5. Applied rewrites 89.6%

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

   if -1.10000000000000003e151 < x < 1.05e-62

   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. lower-fma.f64 90.3

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

   5. Applied rewrites 90.3%

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

   if 1.05e-62 < 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 0

      \[\leadsto \color{blue}{z + \left(2 \cdot y + 3 \cdot x\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(2 \cdot y + 3 \cdot x\right) + z} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{2 \cdot y + \left(3 \cdot x + z\right)} \]

      3. metadata-eval N/A

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

      4. distribute-rgt1-in N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+r+ N/A

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

      10. distribute-rgt1-in N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 89.4%

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

   12. Applied rewrites 89.4%

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

Alternative 3: 83.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, z\right)\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-62}:\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x + y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.1e+151)
   (fma x 3.0 z)
   (if (<= x 1.05e-62) (fma 2.0 y z) (fma 2.0 (+ x y) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.1e+151) {
		tmp = fma(x, 3.0, z);
	} else if (x <= 1.05e-62) {
		tmp = fma(2.0, y, z);
	} else {
		tmp = fma(2.0, (x + y), x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.1e+151)
		tmp = fma(x, 3.0, z);
	elseif (x <= 1.05e-62)
		tmp = fma(2.0, y, z);
	else
		tmp = fma(2.0, Float64(x + y), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.1e+151], N[(x * 3.0 + z), $MachinePrecision], If[LessEqual[x, 1.05e-62], N[(2.0 * y + z), $MachinePrecision], N[(2.0 * N[(x + y), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.10000000000000003e151

   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. *-commutative N/A

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

      6. lower-fma.f64 89.6

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

   5. Applied rewrites 89.6%

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

   if -1.10000000000000003e151 < x < 1.05e-62

   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. lower-fma.f64 90.3

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

   5. Applied rewrites 90.3%

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

   if 1.05e-62 < 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 z around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 89.3

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

   5. Applied rewrites 89.3%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, z\right)\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-62}:\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x + y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{-45}:\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right)\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -8.8e-45)
   (fma 2.0 y z)
   (if (<= y 7.8e+83) (fma x 3.0 z) (fma 2.0 y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -8.8e-45) {
		tmp = fma(2.0, y, z);
	} else if (y <= 7.8e+83) {
		tmp = fma(x, 3.0, z);
	} else {
		tmp = fma(2.0, y, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -8.8e-45)
		tmp = fma(2.0, y, z);
	elseif (y <= 7.8e+83)
		tmp = fma(x, 3.0, z);
	else
		tmp = fma(2.0, y, z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -8.8e-45], N[(2.0 * y + z), $MachinePrecision], If[LessEqual[y, 7.8e+83], N[(x * 3.0 + z), $MachinePrecision], N[(2.0 * y + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.79999999999999974e-45 or 7.8000000000000003e83 < 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 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. lower-fma.f64 82.6

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

   5. Applied rewrites 82.6%

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

   if -8.79999999999999974e-45 < y < 7.8000000000000003e83

   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. *-commutative N/A

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

      6. lower-fma.f64 88.8

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

   5. Applied rewrites 88.8%

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

Alternative 5: 79.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+151}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.1e+151) (* x 3.0) (if (<= x 1.22e+70) (fma 2.0 y z) (* x 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.1e+151) {
		tmp = x * 3.0;
	} else if (x <= 1.22e+70) {
		tmp = fma(2.0, y, z);
	} else {
		tmp = x * 3.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.1e+151)
		tmp = Float64(x * 3.0);
	elseif (x <= 1.22e+70)
		tmp = fma(2.0, y, z);
	else
		tmp = Float64(x * 3.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.1e+151], N[(x * 3.0), $MachinePrecision], If[LessEqual[x, 1.22e+70], N[(2.0 * y + z), $MachinePrecision], N[(x * 3.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1000000000000001e151 or 1.22e70 < 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. *-commutative N/A

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

      2. lower-*.f64 71.6

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

   5. Applied rewrites 71.6%

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

   if -2.1000000000000001e151 < x < 1.22e70

   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. lower-fma.f64 87.3

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

   5. Applied rewrites 87.3%

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

Alternative 6: 51.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+151}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{+68}:\\ \;\;\;\;y + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.1e+151) (* x 3.0) (if (<= x 5.7e+68) (+ y y) (* x 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.1e+151) {
		tmp = x * 3.0;
	} else if (x <= 5.7e+68) {
		tmp = y + y;
	} else {
		tmp = x * 3.0;
	}
	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 <= (-1.1d+151)) then
        tmp = x * 3.0d0
    else if (x <= 5.7d+68) then
        tmp = y + y
    else
        tmp = x * 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.1e+151) {
		tmp = x * 3.0;
	} else if (x <= 5.7e+68) {
		tmp = y + y;
	} else {
		tmp = x * 3.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.1e+151:
		tmp = x * 3.0
	elif x <= 5.7e+68:
		tmp = y + y
	else:
		tmp = x * 3.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.1e+151)
		tmp = Float64(x * 3.0);
	elseif (x <= 5.7e+68)
		tmp = Float64(y + y);
	else
		tmp = Float64(x * 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.1e+151)
		tmp = x * 3.0;
	elseif (x <= 5.7e+68)
		tmp = y + y;
	else
		tmp = x * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.1e+151], N[(x * 3.0), $MachinePrecision], If[LessEqual[x, 5.7e+68], N[(y + y), $MachinePrecision], N[(x * 3.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.10000000000000003e151 or 5.6999999999999996e68 < 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. *-commutative N/A

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

      2. lower-*.f64 71.6

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

   5. Applied rewrites 71.6%

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

   if -1.10000000000000003e151 < x < 5.6999999999999996e68

   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. lower-*.f64 46.7

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

   5. Applied rewrites 46.7%

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

   7. Applied rewrites 46.7%

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

Alternative 7: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(2, y, \mathsf{fma}\left(x, 3, z\right)\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma 2.0 y (fma x 3.0 z)))

C

double code(double x, double y, double z) {
	return fma(2.0, y, fma(x, 3.0, z));
}

Julia

function code(x, y, z)
	return fma(2.0, y, fma(x, 3.0, z))
end

Wolfram

code[x_, y_, z_] := N[(2.0 * y + N[(x * 3.0 + z), $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. Taylor expanded in x around 0

   \[\leadsto \color{blue}{z + \left(2 \cdot y + 3 \cdot x\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(2 \cdot y + 3 \cdot x\right) + z} \]

   2. associate-+l+ N/A

      \[\leadsto \color{blue}{2 \cdot y + \left(3 \cdot x + z\right)} \]

   3. metadata-eval N/A

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

   4. distribute-rgt1-in N/A

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

   5. associate-+r+ N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. associate-+r+ N/A

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

   10. distribute-rgt1-in N/A

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

   11. metadata-eval N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 99.9

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

5. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, y, \mathsf{fma}\left(x, 3, z\right)\right)} \]
6. Add Preprocessing

Alternative 8: 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. lower-*.f64 37.8

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

5. Applied rewrites 37.8%

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

7. Applied rewrites 37.8%

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

Reproduce

herbie shell --seed 2024221 
(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))