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

Percentage Accurate: 99.9% → 100.0%

Time: 5.7s

Alternatives: 7

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(3.0 * x + N[(2.0 * y + 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. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. *-rgt-identity N/A

      \[\leadsto 3 \cdot x + \color{blue}{\left(z + 2 \cdot y\right) \cdot 1} \]

   4. *-inverses N/A

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

   5. associate-/l* N/A

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

   6. associate-*l/ N/A

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

   7. div-add N/A

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

   8. associate-*r/ N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. cancel-sign-sub-inv N/A

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

   12. sub-neg N/A

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

   13. mul-1-neg N/A

       \[\leadsto 3 \cdot x + \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot x\right)} \cdot \left(2 \cdot \frac{y}{x} + \frac{z}{x}\right)\right)\right) \]

   14. distribute-rgt-neg-out N/A

       \[\leadsto 3 \cdot x + \color{blue}{\left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(\left(2 \cdot \frac{y}{x} + \frac{z}{x}\right)\right)\right)} \]

   15. +-commutative N/A

       \[\leadsto 3 \cdot x + \left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{z}{x} + 2 \cdot \frac{y}{x}\right)}\right)\right) \]

   16. associate-*r/ N/A

       \[\leadsto 3 \cdot x + \left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(\left(\frac{z}{x} + \color{blue}{\frac{2 \cdot y}{x}}\right)\right)\right) \]

   17. div-add N/A

       \[\leadsto 3 \cdot x + \left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{z + 2 \cdot y}{x}}\right)\right) \]

   18. mul-1-neg N/A

       \[\leadsto 3 \cdot x + \left(-1 \cdot x\right) \cdot \color{blue}{\left(-1 \cdot \frac{z + 2 \cdot y}{x}\right)} \]

5. Applied rewrites 100.0%

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

Alternative 2: 84.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(3, x, z\right)\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(3, x, y + y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.55e-35)
   (fma 3.0 x z)
   (if (<= z 1.35e-11) (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 <= -1.55e-35) {
		tmp = fma(3.0, x, z);
	} else if (z <= 1.35e-11) {
		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 <= -1.55e-35)
		tmp = fma(3.0, x, z);
	elseif (z <= 1.35e-11)
		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, -1.55e-35], N[(3.0 * x + z), $MachinePrecision], If[LessEqual[z, 1.35e-11], 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 < -1.55000000000000006e-35

   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 82.5

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

   5. Applied rewrites 82.5%

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

   if -1.55000000000000006e-35 < z < 1.35000000000000002e-11

   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. +-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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 92.2

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

   5. Applied rewrites 92.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto 2 \cdot y + \color{blue}{3 \cdot x} \]
   7. Step-by-step derivation

   8. Applied rewrites 92.3%

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

   10. Applied rewrites 92.3%

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

   if 1.35000000000000002e-11 < 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 83.5

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

   5. Applied rewrites 83.5%

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

Alternative 3: 84.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(3, x, z\right)\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(3, x, y + y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.55e-35)
   (fma 3.0 x z)
   (if (<= z 1.35e-11) (fma 3.0 x (+ y y)) (fma 2.0 y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.55e-35) {
		tmp = fma(3.0, x, z);
	} else if (z <= 1.35e-11) {
		tmp = fma(3.0, x, (y + y));
	} else {
		tmp = fma(2.0, y, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.55e-35)
		tmp = fma(3.0, x, z);
	elseif (z <= 1.35e-11)
		tmp = fma(3.0, x, Float64(y + y));
	else
		tmp = fma(2.0, y, z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.55e-35], N[(3.0 * x + z), $MachinePrecision], If[LessEqual[z, 1.35e-11], N[(3.0 * x + N[(y + y), $MachinePrecision]), $MachinePrecision], N[(2.0 * y + z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.55000000000000006e-35

   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 82.5

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

   5. Applied rewrites 82.5%

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

   if -1.55000000000000006e-35 < z < 1.35000000000000002e-11

   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. +-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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 92.2

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

   5. Applied rewrites 92.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto 2 \cdot y + \color{blue}{3 \cdot x} \]
   7. Step-by-step derivation

   8. Applied rewrites 92.3%

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

   10. Applied rewrites 92.3%

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

   if 1.35000000000000002e-11 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-rgt-identity N/A

         \[\leadsto 3 \cdot x + \color{blue}{\left(z + 2 \cdot y\right) \cdot 1} \]

      4. *-inverses N/A

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

      5. associate-/l* N/A

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

      6. associate-*l/ N/A

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

      7. div-add N/A

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

      8. associate-*r/ N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. sub-neg N/A

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

      13. mul-1-neg N/A

          \[\leadsto 3 \cdot x + \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot x\right)} \cdot \left(2 \cdot \frac{y}{x} + \frac{z}{x}\right)\right)\right) \]

      14. distribute-rgt-neg-out N/A

          \[\leadsto 3 \cdot x + \color{blue}{\left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(\left(2 \cdot \frac{y}{x} + \frac{z}{x}\right)\right)\right)} \]

      15. +-commutative N/A

          \[\leadsto 3 \cdot x + \left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{z}{x} + 2 \cdot \frac{y}{x}\right)}\right)\right) \]

      16. associate-*r/ N/A

          \[\leadsto 3 \cdot x + \left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(\left(\frac{z}{x} + \color{blue}{\frac{2 \cdot y}{x}}\right)\right)\right) \]

      17. div-add N/A

          \[\leadsto 3 \cdot x + \left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{z + 2 \cdot y}{x}}\right)\right) \]

      18. mul-1-neg N/A

          \[\leadsto 3 \cdot x + \left(-1 \cdot x\right) \cdot \color{blue}{\left(-1 \cdot \frac{z + 2 \cdot y}{x}\right)} \]

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 81.1

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

   8. Applied rewrites 81.1%

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

Alternative 4: 84.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right)\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-47}:\\ \;\;\;\;\mathsf{fma}\left(3, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.2e+25)
   (fma 2.0 y z)
   (if (<= y 8.5e-47) (fma 3.0 x z) (fma 2.0 y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.2e+25) {
		tmp = fma(2.0, y, z);
	} else if (y <= 8.5e-47) {
		tmp = fma(3.0, x, z);
	} else {
		tmp = fma(2.0, y, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.2e+25)
		tmp = fma(2.0, y, z);
	elseif (y <= 8.5e-47)
		tmp = fma(3.0, x, z);
	else
		tmp = fma(2.0, y, z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.2e+25], N[(2.0 * y + z), $MachinePrecision], If[LessEqual[y, 8.5e-47], N[(3.0 * x + z), $MachinePrecision], N[(2.0 * y + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.1999999999999999e25 or 8.4999999999999999e-47 < 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

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-rgt-identity N/A

         \[\leadsto 3 \cdot x + \color{blue}{\left(z + 2 \cdot y\right) \cdot 1} \]

      4. *-inverses N/A

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

      5. associate-/l* N/A

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

      6. associate-*l/ N/A

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

      7. div-add N/A

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

      8. associate-*r/ N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. sub-neg N/A

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

      13. mul-1-neg N/A

          \[\leadsto 3 \cdot x + \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot x\right)} \cdot \left(2 \cdot \frac{y}{x} + \frac{z}{x}\right)\right)\right) \]

      14. distribute-rgt-neg-out N/A

          \[\leadsto 3 \cdot x + \color{blue}{\left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(\left(2 \cdot \frac{y}{x} + \frac{z}{x}\right)\right)\right)} \]

      15. +-commutative N/A

          \[\leadsto 3 \cdot x + \left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{z}{x} + 2 \cdot \frac{y}{x}\right)}\right)\right) \]

      16. associate-*r/ N/A

          \[\leadsto 3 \cdot x + \left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(\left(\frac{z}{x} + \color{blue}{\frac{2 \cdot y}{x}}\right)\right)\right) \]

      17. div-add N/A

          \[\leadsto 3 \cdot x + \left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{z + 2 \cdot y}{x}}\right)\right) \]

      18. mul-1-neg N/A

          \[\leadsto 3 \cdot x + \left(-1 \cdot x\right) \cdot \color{blue}{\left(-1 \cdot \frac{z + 2 \cdot y}{x}\right)} \]

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 80.1

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

   8. Applied rewrites 80.1%

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

   if -3.1999999999999999e25 < y < 8.4999999999999999e-47

   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 92.2

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

   5. Applied rewrites 92.2%

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

Alternative 5: 78.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{+201}:\\ \;\;\;\;3 \cdot x\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -8.6e+201) (* 3.0 x) (if (<= x 9e+47) (fma 2.0 y z) (* 3.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -8.6e+201) {
		tmp = 3.0 * x;
	} else if (x <= 9e+47) {
		tmp = fma(2.0, y, z);
	} else {
		tmp = 3.0 * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -8.6e+201)
		tmp = Float64(3.0 * x);
	elseif (x <= 9e+47)
		tmp = fma(2.0, y, z);
	else
		tmp = Float64(3.0 * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -8.6e+201], N[(3.0 * x), $MachinePrecision], If[LessEqual[x, 9e+47], N[(2.0 * y + z), $MachinePrecision], N[(3.0 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.59999999999999981e201 or 8.99999999999999958e47 < 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 76.1

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

   5. Applied rewrites 76.1%

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

   if -8.59999999999999981e201 < x < 8.99999999999999958e47

   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. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-rgt-identity N/A

         \[\leadsto 3 \cdot x + \color{blue}{\left(z + 2 \cdot y\right) \cdot 1} \]

      4. *-inverses N/A

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

      5. associate-/l* N/A

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

      6. associate-*l/ N/A

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

      7. div-add N/A

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

      8. associate-*r/ N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. sub-neg N/A

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

      13. mul-1-neg N/A

          \[\leadsto 3 \cdot x + \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot x\right)} \cdot \left(2 \cdot \frac{y}{x} + \frac{z}{x}\right)\right)\right) \]

      14. distribute-rgt-neg-out N/A

          \[\leadsto 3 \cdot x + \color{blue}{\left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(\left(2 \cdot \frac{y}{x} + \frac{z}{x}\right)\right)\right)} \]

      15. +-commutative N/A

          \[\leadsto 3 \cdot x + \left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{z}{x} + 2 \cdot \frac{y}{x}\right)}\right)\right) \]

      16. associate-*r/ N/A

          \[\leadsto 3 \cdot x + \left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(\left(\frac{z}{x} + \color{blue}{\frac{2 \cdot y}{x}}\right)\right)\right) \]

      17. div-add N/A

          \[\leadsto 3 \cdot x + \left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{z + 2 \cdot y}{x}}\right)\right) \]

      18. mul-1-neg N/A

          \[\leadsto 3 \cdot x + \left(-1 \cdot x\right) \cdot \color{blue}{\left(-1 \cdot \frac{z + 2 \cdot y}{x}\right)} \]

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 81.1

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

   8. Applied rewrites 81.1%

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

Alternative 6: 50.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+25}:\\ \;\;\;\;y + y\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{-19}:\\ \;\;\;\;3 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.2e+25) (+ y y) (if (<= y 1.22e-19) (* 3.0 x) (+ y y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.2e+25) {
		tmp = y + y;
	} else if (y <= 1.22e-19) {
		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 <= (-3.2d+25)) then
        tmp = y + y
    else if (y <= 1.22d-19) 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 <= -3.2e+25) {
		tmp = y + y;
	} else if (y <= 1.22e-19) {
		tmp = 3.0 * x;
	} else {
		tmp = y + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.2e+25:
		tmp = y + y
	elif y <= 1.22e-19:
		tmp = 3.0 * x
	else:
		tmp = y + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.2e+25)
		tmp = Float64(y + y);
	elseif (y <= 1.22e-19)
		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 <= -3.2e+25)
		tmp = y + y;
	elseif (y <= 1.22e-19)
		tmp = 3.0 * x;
	else
		tmp = y + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.2e+25], N[(y + y), $MachinePrecision], If[LessEqual[y, 1.22e-19], N[(3.0 * x), $MachinePrecision], N[(y + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.1999999999999999e25 or 1.2200000000000001e-19 < 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. +-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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 79.0

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

   5. Applied rewrites 79.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 59.1%

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

   10. Applied rewrites 59.1%

       \[\leadsto y + y \]

   if -3.1999999999999999e25 < y < 1.2200000000000001e-19

   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 49.6

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

   5. Applied rewrites 49.6%

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

Alternative 7: 33.8% 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. +-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. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 67.9

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

5. Applied rewrites 67.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 33.8%

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

10. Applied rewrites 33.8%

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

Reproduce

herbie shell --seed 2024298 
(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))