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

Percentage Accurate: 99.9% → 100.0%

Time: 5.4s

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. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 100.0

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

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: 86.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right) + x\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(3, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3, x, y + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.2e+16)
   (+ (fma 2.0 y z) x)
   (if (<= y 4e+67) (fma 3.0 x z) (fma 3.0 x (+ y y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.2e+16) {
		tmp = fma(2.0, y, z) + x;
	} else if (y <= 4e+67) {
		tmp = fma(3.0, x, z);
	} else {
		tmp = fma(3.0, x, (y + y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.2e+16)
		tmp = Float64(fma(2.0, y, z) + x);
	elseif (y <= 4e+67)
		tmp = fma(3.0, x, z);
	else
		tmp = fma(3.0, x, Float64(y + y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.2e+16], N[(N[(2.0 * y + z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 4e+67], N[(3.0 * x + z), $MachinePrecision], N[(3.0 * x + N[(y + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.2e16

   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 88.0

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

   5. Applied rewrites 88.0%

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

   if -2.2e16 < y < 3.99999999999999993e67

   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 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 94.9

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

   5. Applied rewrites 94.9%

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

   if 3.99999999999999993e67 < 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 87.3

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

   5. Applied rewrites 87.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 87.4%

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

   10. Applied rewrites 87.4%

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

Alternative 3: 85.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right)\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(3, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3, x, y + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.7e+16)
   (fma 2.0 y z)
   (if (<= y 4e+67) (fma 3.0 x z) (fma 3.0 x (+ y y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.7e+16) {
		tmp = fma(2.0, y, z);
	} else if (y <= 4e+67) {
		tmp = fma(3.0, x, z);
	} else {
		tmp = fma(3.0, x, (y + y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.7e+16)
		tmp = fma(2.0, y, z);
	elseif (y <= 4e+67)
		tmp = fma(3.0, x, z);
	else
		tmp = fma(3.0, x, Float64(y + y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.7e+16], N[(2.0 * y + z), $MachinePrecision], If[LessEqual[y, 4e+67], N[(3.0 * x + z), $MachinePrecision], N[(3.0 * x + N[(y + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.7e16

   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. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 100.0

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

   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 86.3

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

   8. Applied rewrites 86.3%

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

   if -2.7e16 < y < 3.99999999999999993e67

   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 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 94.9

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

   5. Applied rewrites 94.9%

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

   if 3.99999999999999993e67 < 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 87.3

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

   5. Applied rewrites 87.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 87.4%

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

   10. Applied rewrites 87.4%

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

Alternative 4: 85.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+16} \lor \neg \left(y \leq 3.6 \cdot 10^{+70}\right):\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3, x, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.7e+16) (not (<= y 3.6e+70))) (fma 2.0 y z) (fma 3.0 x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.7e+16) || !(y <= 3.6e+70)) {
		tmp = fma(2.0, y, z);
	} else {
		tmp = fma(3.0, x, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.7e+16) || !(y <= 3.6e+70))
		tmp = fma(2.0, y, z);
	else
		tmp = fma(3.0, x, z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.7e+16], N[Not[LessEqual[y, 3.6e+70]], $MachinePrecision]], N[(2.0 * y + z), $MachinePrecision], N[(3.0 * x + z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.7e16 or 3.6e70 < 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. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 100.0

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

   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 85.7

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

   8. Applied rewrites 85.7%

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

   if -2.7e16 < y < 3.6e70

   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 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 95.0

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

   5. Applied rewrites 95.0%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+16} \lor \neg \left(y \leq 3.6 \cdot 10^{+70}\right):\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3, x, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+134} \lor \neg \left(x \leq 9.2 \cdot 10^{+167}\right):\\ \;\;\;\;3 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.6e+134) (not (<= x 9.2e+167))) (* 3.0 x) (fma 2.0 y z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.6e+134) || !(x <= 9.2e+167)) {
		tmp = 3.0 * x;
	} else {
		tmp = fma(2.0, y, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.6e+134) || !(x <= 9.2e+167))
		tmp = Float64(3.0 * x);
	else
		tmp = fma(2.0, y, z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.6e+134], N[Not[LessEqual[x, 9.2e+167]], $MachinePrecision]], N[(3.0 * x), $MachinePrecision], N[(2.0 * y + z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6e134 or 9.19999999999999952e167 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 78.6

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

   5. Applied rewrites 78.6%

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

   if -1.6e134 < x < 9.19999999999999952e167

   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. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 100.0

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

   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 83.9

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

   8. Applied rewrites 83.9%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+134} \lor \neg \left(x \leq 9.2 \cdot 10^{+167}\right):\\ \;\;\;\;3 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+15} \lor \neg \left(y \leq 3.6 \cdot 10^{+70}\right):\\ \;\;\;\;y + y\\ \mathbf{else}:\\ \;\;\;\;3 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -6.5e+15) (not (<= y 3.6e+70))) (+ y y) (* 3.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -6.5e+15) || !(y <= 3.6e+70)) {
		tmp = y + y;
	} else {
		tmp = 3.0 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-6.5d+15)) .or. (.not. (y <= 3.6d+70))) then
        tmp = y + y
    else
        tmp = 3.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -6.5e+15) || !(y <= 3.6e+70)) {
		tmp = y + y;
	} else {
		tmp = 3.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -6.5e+15) or not (y <= 3.6e+70):
		tmp = y + y
	else:
		tmp = 3.0 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -6.5e+15) || !(y <= 3.6e+70))
		tmp = Float64(y + y);
	else
		tmp = Float64(3.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -6.5e+15) || ~((y <= 3.6e+70)))
		tmp = y + y;
	else
		tmp = 3.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -6.5e+15], N[Not[LessEqual[y, 3.6e+70]], $MachinePrecision]], N[(y + y), $MachinePrecision], N[(3.0 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.5e15 or 3.6e70 < 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 77.6

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 63.5%

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

   10. Applied rewrites 63.5%

       \[\leadsto y + y \]

   if -6.5e15 < y < 3.6e70

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 54.2

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

   5. Applied rewrites 54.2%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+15} \lor \neg \left(y \leq 3.6 \cdot 10^{+70}\right):\\ \;\;\;\;y + y\\ \mathbf{else}:\\ \;\;\;\;3 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in z around 0

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

   1. +-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 67.7

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

5. Applied rewrites 67.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 33.9%

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

10. Applied rewrites 33.9%

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

Reproduce

herbie shell --seed 2024313 
(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))