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

Percentage Accurate: 99.9% → 99.9%

Time: 9.7s

Alternatives: 10

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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((2.0d0 * y) + (x + z)) + (x * 2.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return ((2.0 * y) + (x + z)) + (x * 2.0);
}

Python

def code(x, y, z):
	return ((2.0 * y) + (x + z)) + (x * 2.0)

Julia

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

MATLAB

function tmp = code(x, y, z)
	tmp = ((2.0 * y) + (x + z)) + (x * 2.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(2.0 * y), $MachinePrecision] + N[(x + z), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $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

   \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(2 \cdot x + 2 \cdot y\right)}, z\right), x\right) \]
4. Step-by-step derivation

   1. distribute-lft-out N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \left(x + y\right)\right), z\right), x\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \left(x + y\right)\right), z\right), x\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \left(y + x\right)\right), z\right), x\right) \]

   4. +-lowering-+.f64 99.9%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(y, x\right)\right), z\right), x\right) \]

5. Simplified 99.9%

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

   1. associate-+l+ N/A

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

   2. +-commutative N/A

      \[\leadsto 2 \cdot \left(x + y\right) + \left(z + x\right) \]

   3. +-commutative N/A

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

   4. +-commutative N/A

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

   5. +-commutative N/A

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

   6. distribute-rgt-in N/A

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

   7. *-commutative N/A

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

   8. associate-+r+ N/A

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

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(x + z\right) + 2 \cdot y\right), \color{blue}{\left(x \cdot 2\right)}\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(x + z\right), \left(2 \cdot y\right)\right), \left(\color{blue}{x} \cdot 2\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(x, z\right), \left(2 \cdot y\right)\right), \left(x \cdot 2\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(x, z\right), \mathsf{*.f64}\left(2, y\right)\right), \left(x \cdot 2\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(x, z\right), \mathsf{*.f64}\left(2, y\right)\right), \left(2 \cdot \color{blue}{x}\right)\right) \]

   14. *-lowering-*.f64 99.9%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(x, z\right), \mathsf{*.f64}\left(2, y\right)\right), \mathsf{*.f64}\left(2, \color{blue}{x}\right)\right) \]

7. Applied egg-rr 99.9%

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

8. Final simplification 99.9%

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

Alternative 2: 53.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{+76}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;x \leq -1.06 \cdot 10^{-185}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-200}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{+48}:\\ \;\;\;\;2 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.02e+76)
   (* x 3.0)
   (if (<= x -1.06e-185)
     (* 2.0 y)
     (if (<= x 2.9e-200) z (if (<= x 1.06e+48) (* 2.0 y) (* x 3.0))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.02e+76) {
		tmp = x * 3.0;
	} else if (x <= -1.06e-185) {
		tmp = 2.0 * y;
	} else if (x <= 2.9e-200) {
		tmp = z;
	} else if (x <= 1.06e+48) {
		tmp = 2.0 * y;
	} else {
		tmp = x * 3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.02d+76)) then
        tmp = x * 3.0d0
    else if (x <= (-1.06d-185)) then
        tmp = 2.0d0 * y
    else if (x <= 2.9d-200) then
        tmp = z
    else if (x <= 1.06d+48) then
        tmp = 2.0d0 * y
    else
        tmp = x * 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.02e+76) {
		tmp = x * 3.0;
	} else if (x <= -1.06e-185) {
		tmp = 2.0 * y;
	} else if (x <= 2.9e-200) {
		tmp = z;
	} else if (x <= 1.06e+48) {
		tmp = 2.0 * y;
	} else {
		tmp = x * 3.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.02e+76:
		tmp = x * 3.0
	elif x <= -1.06e-185:
		tmp = 2.0 * y
	elif x <= 2.9e-200:
		tmp = z
	elif x <= 1.06e+48:
		tmp = 2.0 * y
	else:
		tmp = x * 3.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.02e+76)
		tmp = Float64(x * 3.0);
	elseif (x <= -1.06e-185)
		tmp = Float64(2.0 * y);
	elseif (x <= 2.9e-200)
		tmp = z;
	elseif (x <= 1.06e+48)
		tmp = Float64(2.0 * y);
	else
		tmp = Float64(x * 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.02e+76)
		tmp = x * 3.0;
	elseif (x <= -1.06e-185)
		tmp = 2.0 * y;
	elseif (x <= 2.9e-200)
		tmp = z;
	elseif (x <= 1.06e+48)
		tmp = 2.0 * y;
	else
		tmp = x * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.02e+76], N[(x * 3.0), $MachinePrecision], If[LessEqual[x, -1.06e-185], N[(2.0 * y), $MachinePrecision], If[LessEqual[x, 2.9e-200], z, If[LessEqual[x, 1.06e+48], N[(2.0 * y), $MachinePrecision], N[(x * 3.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.02000000000000007e76 or 1.06e48 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 68.4%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{3}\right) \]

   5. Simplified 68.4%

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

   if -1.02000000000000007e76 < x < -1.06000000000000004e-185 or 2.9e-200 < x < 1.06e48

   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. *-lowering-*.f64 54.2%

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{y}\right) \]

   5. Simplified 54.2%

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

   if -1.06000000000000004e-185 < x < 2.9e-200

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z} \]
   4. Step-by-step derivation

   5. Simplified 62.0%

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

Alternative 3: 85.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z + 2 \cdot y\\ \mathbf{if}\;y \leq -2.8 \cdot 10^{+55}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+76}:\\ \;\;\;\;z + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ z (* 2.0 y))))
   (if (<= y -2.8e+55) t_0 (if (<= y 3e+76) (+ z (* x 3.0)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z + (2.0 * y);
	double tmp;
	if (y <= -2.8e+55) {
		tmp = t_0;
	} else if (y <= 3e+76) {
		tmp = z + (x * 3.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z + (2.0d0 * y)
    if (y <= (-2.8d+55)) then
        tmp = t_0
    else if (y <= 3d+76) then
        tmp = z + (x * 3.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z + (2.0 * y);
	double tmp;
	if (y <= -2.8e+55) {
		tmp = t_0;
	} else if (y <= 3e+76) {
		tmp = z + (x * 3.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z + (2.0 * y)
	tmp = 0
	if y <= -2.8e+55:
		tmp = t_0
	elif y <= 3e+76:
		tmp = z + (x * 3.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z + Float64(2.0 * y))
	tmp = 0.0
	if (y <= -2.8e+55)
		tmp = t_0;
	elseif (y <= 3e+76)
		tmp = Float64(z + Float64(x * 3.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z + (2.0 * y);
	tmp = 0.0;
	if (y <= -2.8e+55)
		tmp = t_0;
	elseif (y <= 3e+76)
		tmp = z + (x * 3.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z + N[(2.0 * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.8e+55], t$95$0, If[LessEqual[y, 3e+76], N[(z + N[(x * 3.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.8000000000000001e55 or 2.9999999999999998e76 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(2 \cdot y\right), \color{blue}{z}\right) \]

      3. *-lowering-*.f64 89.5%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, y\right), z\right) \]

   5. Simplified 89.5%

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

   if -2.8000000000000001e55 < y < 2.9999999999999998e76

   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

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

      2. associate-+l+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto z + 3 \cdot x \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(z, \color{blue}{\left(3 \cdot x\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(z, \left(x \cdot \color{blue}{3}\right)\right) \]

      7. *-lowering-*.f64 89.9%

         \[\leadsto \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(x, \color{blue}{3}\right)\right) \]

   5. Simplified 89.9%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+55}:\\ \;\;\;\;z + 2 \cdot y\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+76}:\\ \;\;\;\;z + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;z + 2 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + 2 \cdot y\\ \mathbf{if}\;y \leq -2.4 \cdot 10^{+56}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+78}:\\ \;\;\;\;z + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (* 2.0 y))))
   (if (<= y -2.4e+56) t_0 (if (<= y 2.2e+78) (+ z (* x 3.0)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + (2.0 * y);
	double tmp;
	if (y <= -2.4e+56) {
		tmp = t_0;
	} else if (y <= 2.2e+78) {
		tmp = z + (x * 3.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (2.0d0 * y)
    if (y <= (-2.4d+56)) then
        tmp = t_0
    else if (y <= 2.2d+78) then
        tmp = z + (x * 3.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (2.0 * y);
	double tmp;
	if (y <= -2.4e+56) {
		tmp = t_0;
	} else if (y <= 2.2e+78) {
		tmp = z + (x * 3.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (2.0 * y)
	tmp = 0
	if y <= -2.4e+56:
		tmp = t_0
	elif y <= 2.2e+78:
		tmp = z + (x * 3.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(2.0 * y))
	tmp = 0.0
	if (y <= -2.4e+56)
		tmp = t_0;
	elseif (y <= 2.2e+78)
		tmp = Float64(z + Float64(x * 3.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (2.0 * y);
	tmp = 0.0;
	if (y <= -2.4e+56)
		tmp = t_0;
	elseif (y <= 2.2e+78)
		tmp = z + (x * 3.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(2.0 * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.4e+56], t$95$0, If[LessEqual[y, 2.2e+78], N[(z + N[(x * 3.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.40000000000000013e56 or 2.20000000000000014e78 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(2 \cdot y\right)}, x\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 77.5%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, y\right), x\right) \]

   5. Simplified 77.5%

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

   if -2.40000000000000013e56 < y < 2.20000000000000014e78

   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

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

      2. associate-+l+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto z + 3 \cdot x \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(z, \color{blue}{\left(3 \cdot x\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(z, \left(x \cdot \color{blue}{3}\right)\right) \]

      7. *-lowering-*.f64 89.9%

         \[\leadsto \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(x, \color{blue}{3}\right)\right) \]

   5. Simplified 89.9%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+56}:\\ \;\;\;\;x + 2 \cdot y\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+78}:\\ \;\;\;\;z + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;x + 2 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+76}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+77}:\\ \;\;\;\;z + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.2e+76) (* 2.0 y) (if (<= y 7e+77) (+ z (* x 3.0)) (* 2.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.2e+76) {
		tmp = 2.0 * y;
	} else if (y <= 7e+77) {
		tmp = z + (x * 3.0);
	} else {
		tmp = 2.0 * 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 <= (-2.2d+76)) then
        tmp = 2.0d0 * y
    else if (y <= 7d+77) then
        tmp = z + (x * 3.0d0)
    else
        tmp = 2.0d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.2e+76) {
		tmp = 2.0 * y;
	} else if (y <= 7e+77) {
		tmp = z + (x * 3.0);
	} else {
		tmp = 2.0 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.2e+76:
		tmp = 2.0 * y
	elif y <= 7e+77:
		tmp = z + (x * 3.0)
	else:
		tmp = 2.0 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.2e+76)
		tmp = Float64(2.0 * y);
	elseif (y <= 7e+77)
		tmp = Float64(z + Float64(x * 3.0));
	else
		tmp = Float64(2.0 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.2e+76)
		tmp = 2.0 * y;
	elseif (y <= 7e+77)
		tmp = z + (x * 3.0);
	else
		tmp = 2.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.2e+76], N[(2.0 * y), $MachinePrecision], If[LessEqual[y, 7e+77], N[(z + N[(x * 3.0), $MachinePrecision]), $MachinePrecision], N[(2.0 * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.2e76 or 7.0000000000000003e77 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 78.5%

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{y}\right) \]

   5. Simplified 78.5%

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

   if -2.2e76 < y < 7.0000000000000003e77

   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

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

      2. associate-+l+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto z + 3 \cdot x \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(z, \color{blue}{\left(3 \cdot x\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(z, \left(x \cdot \color{blue}{3}\right)\right) \]

      7. *-lowering-*.f64 88.1%

         \[\leadsto \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(x, \color{blue}{3}\right)\right) \]

   5. Simplified 88.1%

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

Alternative 6: 57.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+53}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+76}:\\ \;\;\;\;x + z\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.3e+53) (* 2.0 y) (if (<= y 4.1e+76) (+ x z) (* 2.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.3e+53) {
		tmp = 2.0 * y;
	} else if (y <= 4.1e+76) {
		tmp = x + z;
	} else {
		tmp = 2.0 * 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 <= (-1.3d+53)) then
        tmp = 2.0d0 * y
    else if (y <= 4.1d+76) then
        tmp = x + z
    else
        tmp = 2.0d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.3e+53) {
		tmp = 2.0 * y;
	} else if (y <= 4.1e+76) {
		tmp = x + z;
	} else {
		tmp = 2.0 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.3e+53:
		tmp = 2.0 * y
	elif y <= 4.1e+76:
		tmp = x + z
	else:
		tmp = 2.0 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.3e+53)
		tmp = Float64(2.0 * y);
	elseif (y <= 4.1e+76)
		tmp = Float64(x + z);
	else
		tmp = Float64(2.0 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.3e+53)
		tmp = 2.0 * y;
	elseif (y <= 4.1e+76)
		tmp = x + z;
	else
		tmp = 2.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.3e+53], N[(2.0 * y), $MachinePrecision], If[LessEqual[y, 4.1e+76], N[(x + z), $MachinePrecision], N[(2.0 * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.29999999999999999e53 or 4.0999999999999998e76 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 76.1%

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{y}\right) \]

   5. Simplified 76.1%

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

   if -1.29999999999999999e53 < y < 4.0999999999999998e76

   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 inf

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{z}, x\right) \]
   4. Step-by-step derivation

   5. Simplified 51.5%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+53}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+76}:\\ \;\;\;\;x + z\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+55}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+76}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.8e+55) (* 2.0 y) (if (<= y 1.3e+76) z (* 2.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.8e+55) {
		tmp = 2.0 * y;
	} else if (y <= 1.3e+76) {
		tmp = z;
	} else {
		tmp = 2.0 * 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 <= (-1.8d+55)) then
        tmp = 2.0d0 * y
    else if (y <= 1.3d+76) then
        tmp = z
    else
        tmp = 2.0d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.8e+55) {
		tmp = 2.0 * y;
	} else if (y <= 1.3e+76) {
		tmp = z;
	} else {
		tmp = 2.0 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.8e+55:
		tmp = 2.0 * y
	elif y <= 1.3e+76:
		tmp = z
	else:
		tmp = 2.0 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.8e+55)
		tmp = Float64(2.0 * y);
	elseif (y <= 1.3e+76)
		tmp = z;
	else
		tmp = Float64(2.0 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.8e+55)
		tmp = 2.0 * y;
	elseif (y <= 1.3e+76)
		tmp = z;
	else
		tmp = 2.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.8e+55], N[(2.0 * y), $MachinePrecision], If[LessEqual[y, 1.3e+76], z, N[(2.0 * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.79999999999999994e55 or 1.3e76 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 75.4%

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{y}\right) \]

   5. Simplified 75.4%

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

   if -1.79999999999999994e55 < y < 1.3e76

   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 inf

      \[\leadsto \color{blue}{z} \]
   4. Step-by-step derivation

   5. Simplified 45.0%

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

Alternative 8: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x + (z + (2.0 * (x + 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 = x + (z + (2.0d0 * (x + y)))
end function

Java

public static double code(double x, double y, double z) {
	return x + (z + (2.0 * (x + y)));
}

Python

def code(x, y, z):
	return x + (z + (2.0 * (x + y)))

Julia

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

MATLAB

function tmp = code(x, y, z)
	tmp = x + (z + (2.0 * (x + y)));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(z + N[(2.0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(2 \cdot x + 2 \cdot y\right)}, z\right), x\right) \]
4. Step-by-step derivation

   1. distribute-lft-out N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \left(x + y\right)\right), z\right), x\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \left(x + y\right)\right), z\right), x\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \left(y + x\right)\right), z\right), x\right) \]

   4. +-lowering-+.f64 99.9%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(y, x\right)\right), z\right), x\right) \]

5. Simplified 99.9%

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

6. Final simplification 99.9%

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

Alternative 9: 34.9% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ z \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := z

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 inf

   \[\leadsto \color{blue}{z} \]
4. Step-by-step derivation

5. Simplified 33.4%

   \[\leadsto \color{blue}{z} \]
6. Add Preprocessing

Alternative 10: 7.9% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

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 inf

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{z}, x\right) \]
4. Step-by-step derivation

5. Simplified 38.3%

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

6. Taylor expanded in z around 0

   \[\leadsto \color{blue}{x} \]
7. Step-by-step derivation

8. Simplified 7.6%

   \[\leadsto \color{blue}{x} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024185 
(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))