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

Percentage Accurate: 99.9% → 99.9%

Time: 6.6s

Alternatives: 9

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: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y, z)
	return fma(2.0, Float64(x + y), Float64(z + x))
end

Wolfram

code[x_, y_, z_] := N[(2.0 * N[(x + y), $MachinePrecision] + N[(z + x), $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. 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. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. lift-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. associate-+l+ N/A

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

   7. +-commutative N/A

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

   8. lift-+.f64 N/A

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

   9. count-2 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 85.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+82}:\\ \;\;\;\;\left(3 \cdot x + y\right) + y\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(3, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z + y\right) + \left(x + y\right)\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.4e+82)
   (+ (+ (* 3.0 x) y) y)
   (if (<= y 1.8e+67) (fma 3.0 x z) (+ (+ (+ z y) (+ x y)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.4e+82) {
		tmp = ((3.0 * x) + y) + y;
	} else if (y <= 1.8e+67) {
		tmp = fma(3.0, x, z);
	} else {
		tmp = ((z + y) + (x + y)) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.4e+82)
		tmp = Float64(Float64(Float64(3.0 * x) + y) + y);
	elseif (y <= 1.8e+67)
		tmp = fma(3.0, x, z);
	else
		tmp = Float64(Float64(Float64(z + y) + Float64(x + y)) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.4e+82], N[(N[(N[(3.0 * x), $MachinePrecision] + y), $MachinePrecision] + y), $MachinePrecision], If[LessEqual[y, 1.8e+67], N[(3.0 * x + z), $MachinePrecision], N[(N[(N[(z + y), $MachinePrecision] + N[(x + y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.4e82

   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. *-commutative N/A

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

      6. lower-*.f64 84.7

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

   5. Applied rewrites 84.7%

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

   7. Applied rewrites 84.7%

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

   if -1.4e82 < y < 1.7999999999999999e67

   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 91.8

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

   5. Applied rewrites 91.8%

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

   if 1.7999999999999999e67 < y

   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. 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. +-commutative N/A

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

      2. lower-+.f64 92.3

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

   7. Applied rewrites 92.3%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+82}:\\ \;\;\;\;\left(3 \cdot x + y\right) + y\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(3, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z + y\right) + \left(x + y\right)\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+82}:\\ \;\;\;\;\left(3 \cdot x + y\right) + y\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(3, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 2, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.4e+82)
   (+ (+ (* 3.0 x) y) y)
   (if (<= y 1.8e+67) (fma 3.0 x z) (fma y 2.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.4e+82) {
		tmp = ((3.0 * x) + y) + y;
	} else if (y <= 1.8e+67) {
		tmp = fma(3.0, x, z);
	} else {
		tmp = fma(y, 2.0, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.4e+82)
		tmp = Float64(Float64(Float64(3.0 * x) + y) + y);
	elseif (y <= 1.8e+67)
		tmp = fma(3.0, x, z);
	else
		tmp = fma(y, 2.0, z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.4e+82], N[(N[(N[(3.0 * x), $MachinePrecision] + y), $MachinePrecision] + y), $MachinePrecision], If[LessEqual[y, 1.8e+67], N[(3.0 * x + z), $MachinePrecision], N[(y * 2.0 + z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.4e82

   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. *-commutative N/A

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

      6. lower-*.f64 84.7

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

   5. Applied rewrites 84.7%

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

   7. Applied rewrites 84.7%

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

   if -1.4e82 < y < 1.7999999999999999e67

   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 91.8

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

   5. Applied rewrites 91.8%

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

   if 1.7999999999999999e67 < 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. *-commutative N/A

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

      3. lower-fma.f64 90.9

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

   5. Applied rewrites 90.9%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+82}:\\ \;\;\;\;\left(3 \cdot x + y\right) + y\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(3, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 2, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(3, x, y + y\right)\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(3, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 2, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.4e+82)
   (fma 3.0 x (+ y y))
   (if (<= y 1.8e+67) (fma 3.0 x z) (fma y 2.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.4e+82) {
		tmp = fma(3.0, x, (y + y));
	} else if (y <= 1.8e+67) {
		tmp = fma(3.0, x, z);
	} else {
		tmp = fma(y, 2.0, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.4e+82)
		tmp = fma(3.0, x, Float64(y + y));
	elseif (y <= 1.8e+67)
		tmp = fma(3.0, x, z);
	else
		tmp = fma(y, 2.0, z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.4e+82], N[(3.0 * x + N[(y + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.8e+67], N[(3.0 * x + z), $MachinePrecision], N[(y * 2.0 + z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.4e82

   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. *-commutative N/A

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

      6. lower-*.f64 84.7

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

   5. Applied rewrites 84.7%

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

   7. Applied rewrites 84.7%

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

   if -1.4e82 < y < 1.7999999999999999e67

   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 91.8

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

   5. Applied rewrites 91.8%

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

   if 1.7999999999999999e67 < 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. *-commutative N/A

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

      3. lower-fma.f64 90.9

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

   5. Applied rewrites 90.9%

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

Alternative 5: 83.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+155}:\\ \;\;\;\;\left(z + y\right) + y\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(3, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 2, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.9e+155)
   (+ (+ z y) y)
   (if (<= y 1.8e+67) (fma 3.0 x z) (fma y 2.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.9e+155) {
		tmp = (z + y) + y;
	} else if (y <= 1.8e+67) {
		tmp = fma(3.0, x, z);
	} else {
		tmp = fma(y, 2.0, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.9e+155)
		tmp = Float64(Float64(z + y) + y);
	elseif (y <= 1.8e+67)
		tmp = fma(3.0, x, z);
	else
		tmp = fma(y, 2.0, z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.9e+155], N[(N[(z + y), $MachinePrecision] + y), $MachinePrecision], If[LessEqual[y, 1.8e+67], N[(3.0 * x + z), $MachinePrecision], N[(y * 2.0 + z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.8999999999999999e155

   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. *-commutative N/A

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

      3. lower-fma.f64 89.6

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

   5. Applied rewrites 89.6%

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

   7. Applied rewrites 89.6%

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

   if -2.8999999999999999e155 < y < 1.7999999999999999e67

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 89.2

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

   5. Applied rewrites 89.2%

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

   if 1.7999999999999999e67 < 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. *-commutative N/A

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

      3. lower-fma.f64 90.9

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

   5. Applied rewrites 90.9%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+155}:\\ \;\;\;\;\left(z + y\right) + y\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(3, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 2, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(z + y\right) + y\\ \mathbf{if}\;y \leq -2.9 \cdot 10^{+155}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(3, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (+ z y) y)))
   (if (<= y -2.9e+155) t_0 (if (<= y 1.8e+67) (fma 3.0 x z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (z + y) + y;
	double tmp;
	if (y <= -2.9e+155) {
		tmp = t_0;
	} else if (y <= 1.8e+67) {
		tmp = fma(3.0, x, z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z + y) + y)
	tmp = 0.0
	if (y <= -2.9e+155)
		tmp = t_0;
	elseif (y <= 1.8e+67)
		tmp = fma(3.0, x, z);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z + y), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[y, -2.9e+155], t$95$0, If[LessEqual[y, 1.8e+67], N[(3.0 * x + z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.8999999999999999e155 or 1.7999999999999999e67 < 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. *-commutative N/A

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

      3. lower-fma.f64 90.5

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

   5. Applied rewrites 90.5%

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

   7. Applied rewrites 90.5%

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

   if -2.8999999999999999e155 < y < 1.7999999999999999e67

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 89.2

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

   5. Applied rewrites 89.2%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+155}:\\ \;\;\;\;\left(z + y\right) + y\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(3, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z + y\right) + y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+154}:\\ \;\;\;\;3 \cdot x\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+235}:\\ \;\;\;\;\left(z + y\right) + y\\ \mathbf{else}:\\ \;\;\;\;3 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.6e+154) (* 3.0 x) (if (<= x 3.9e+235) (+ (+ z y) y) (* 3.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.6e+154) {
		tmp = 3.0 * x;
	} else if (x <= 3.9e+235) {
		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 <= (-1.6d+154)) then
        tmp = 3.0d0 * x
    else if (x <= 3.9d+235) 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 <= -1.6e+154) {
		tmp = 3.0 * x;
	} else if (x <= 3.9e+235) {
		tmp = (z + y) + y;
	} else {
		tmp = 3.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.6e+154:
		tmp = 3.0 * x
	elif x <= 3.9e+235:
		tmp = (z + y) + y
	else:
		tmp = 3.0 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.6e+154)
		tmp = Float64(3.0 * x);
	elseif (x <= 3.9e+235)
		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 <= -1.6e+154)
		tmp = 3.0 * x;
	elseif (x <= 3.9e+235)
		tmp = (z + y) + y;
	else
		tmp = 3.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.6e+154], N[(3.0 * x), $MachinePrecision], If[LessEqual[x, 3.9e+235], N[(N[(z + y), $MachinePrecision] + y), $MachinePrecision], N[(3.0 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6e154 or 3.9000000000000003e235 < 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 80.0

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

   5. Applied rewrites 80.0%

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

   if -1.6e154 < x < 3.9000000000000003e235

   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. *-commutative N/A

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

      3. lower-fma.f64 82.2

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

   5. Applied rewrites 82.2%

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

   7. Applied rewrites 82.1%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+154}:\\ \;\;\;\;3 \cdot x\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+235}:\\ \;\;\;\;\left(z + y\right) + y\\ \mathbf{else}:\\ \;\;\;\;3 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+155}:\\ \;\;\;\;y + y\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+67}:\\ \;\;\;\;3 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.9e+155) (+ y y) (if (<= y 1.8e+67) (* 3.0 x) (+ y y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.9e+155) {
		tmp = y + y;
	} else if (y <= 1.8e+67) {
		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 <= (-2.9d+155)) then
        tmp = y + y
    else if (y <= 1.8d+67) 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 <= -2.9e+155) {
		tmp = y + y;
	} else if (y <= 1.8e+67) {
		tmp = 3.0 * x;
	} else {
		tmp = y + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.9e+155:
		tmp = y + y
	elif y <= 1.8e+67:
		tmp = 3.0 * x
	else:
		tmp = y + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.9e+155)
		tmp = Float64(y + y);
	elseif (y <= 1.8e+67)
		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 <= -2.9e+155)
		tmp = y + y;
	elseif (y <= 1.8e+67)
		tmp = 3.0 * x;
	else
		tmp = y + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.9e+155], N[(y + y), $MachinePrecision], If[LessEqual[y, 1.8e+67], N[(3.0 * x), $MachinePrecision], N[(y + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.8999999999999999e155 or 1.7999999999999999e67 < 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 y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 75.8

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

   5. Applied rewrites 75.8%

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

   7. Applied rewrites 75.8%

      \[\leadsto y + \color{blue}{y} \]

   if -2.8999999999999999e155 < y < 1.7999999999999999e67

   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 42.9

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

   5. Applied rewrites 42.9%

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

Alternative 9: 34.6% 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. *-commutative N/A

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

   2. lower-*.f64 32.8

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

5. Applied rewrites 32.8%

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

7. Applied rewrites 32.8%

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

Reproduce

herbie shell --seed 2024276 
(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))