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

Percentage Accurate: 99.9% → 99.9%

Time: 6.6s

Alternatives: 10

Speedup: 1.0×

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(x + N[(y + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 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. Final simplification 99.9%

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

Alternative 2: 55.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-7}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;z \leq -1.55 \cdot 10^{-287}:\\ \;\;\;\;\mathsf{fma}\left(2, y, x\right)\\ \mathbf{elif}\;z \leq 2.95 \cdot 10^{+23}:\\ \;\;\;\;x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.2e-7)
   (+ x z)
   (if (<= z -1.55e-287)
     (fma 2.0 y x)
     (if (<= z 2.95e+23) (* x 3.0) (+ x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.2e-7) {
		tmp = x + z;
	} else if (z <= -1.55e-287) {
		tmp = fma(2.0, y, x);
	} else if (z <= 2.95e+23) {
		tmp = x * 3.0;
	} else {
		tmp = x + z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.2e-7)
		tmp = Float64(x + z);
	elseif (z <= -1.55e-287)
		tmp = fma(2.0, y, x);
	elseif (z <= 2.95e+23)
		tmp = Float64(x * 3.0);
	else
		tmp = Float64(x + z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.2e-7], N[(x + z), $MachinePrecision], If[LessEqual[z, -1.55e-287], N[(2.0 * y + x), $MachinePrecision], If[LessEqual[z, 2.95e+23], N[(x * 3.0), $MachinePrecision], N[(x + z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.2000000000000001e-7 or 2.94999999999999994e23 < z

   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

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

   5. Simplified 63.3%

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

   if -2.2000000000000001e-7 < z < -1.55e-287

   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 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. accelerator-lowering-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 94.5

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

   5. Simplified 94.5%

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

   6. Taylor expanded in y around inf

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

   8. Simplified 70.3%

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

   if -1.55e-287 < z < 2.94999999999999994e23

   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. *-commutative N/A

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

      2. *-lowering-*.f64 57.8

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

   5. Simplified 57.8%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-7}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;z \leq -1.55 \cdot 10^{-287}:\\ \;\;\;\;\mathsf{fma}\left(2, y, x\right)\\ \mathbf{elif}\;z \leq 2.95 \cdot 10^{+23}:\\ \;\;\;\;x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.1 \cdot 10^{-9}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;z \leq -1.75 \cdot 10^{-286}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+23}:\\ \;\;\;\;x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -6.1e-9)
   (+ x z)
   (if (<= z -1.75e-286) (* y 2.0) (if (<= z 2.9e+23) (* x 3.0) (+ x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.1e-9) {
		tmp = x + z;
	} else if (z <= -1.75e-286) {
		tmp = y * 2.0;
	} else if (z <= 2.9e+23) {
		tmp = x * 3.0;
	} else {
		tmp = x + z;
	}
	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 (z <= (-6.1d-9)) then
        tmp = x + z
    else if (z <= (-1.75d-286)) then
        tmp = y * 2.0d0
    else if (z <= 2.9d+23) then
        tmp = x * 3.0d0
    else
        tmp = x + z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.1e-9) {
		tmp = x + z;
	} else if (z <= -1.75e-286) {
		tmp = y * 2.0;
	} else if (z <= 2.9e+23) {
		tmp = x * 3.0;
	} else {
		tmp = x + z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -6.1e-9:
		tmp = x + z
	elif z <= -1.75e-286:
		tmp = y * 2.0
	elif z <= 2.9e+23:
		tmp = x * 3.0
	else:
		tmp = x + z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -6.1e-9)
		tmp = Float64(x + z);
	elseif (z <= -1.75e-286)
		tmp = Float64(y * 2.0);
	elseif (z <= 2.9e+23)
		tmp = Float64(x * 3.0);
	else
		tmp = Float64(x + z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -6.1e-9)
		tmp = x + z;
	elseif (z <= -1.75e-286)
		tmp = y * 2.0;
	elseif (z <= 2.9e+23)
		tmp = x * 3.0;
	else
		tmp = x + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -6.1e-9], N[(x + z), $MachinePrecision], If[LessEqual[z, -1.75e-286], N[(y * 2.0), $MachinePrecision], If[LessEqual[z, 2.9e+23], N[(x * 3.0), $MachinePrecision], N[(x + z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.1e-9 or 2.90000000000000013e23 < z

   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

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

   5. Simplified 63.3%

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

   if -6.1e-9 < z < -1.74999999999999994e-286

   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 67.3

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

   5. Simplified 67.3%

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

   if -1.74999999999999994e-286 < z < 2.90000000000000013e23

   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. *-commutative N/A

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

      2. *-lowering-*.f64 57.8

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

   5. Simplified 57.8%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.1 \cdot 10^{-9}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;z \leq -1.75 \cdot 10^{-286}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+23}:\\ \;\;\;\;x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(2, x + y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.8e+33)
   (fma 2.0 y z)
   (if (<= z 1.6e+50) (fma 2.0 (+ x y) x) (fma x 3.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.8e+33) {
		tmp = fma(2.0, y, z);
	} else if (z <= 1.6e+50) {
		tmp = fma(2.0, (x + y), x);
	} else {
		tmp = fma(x, 3.0, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.8e+33)
		tmp = fma(2.0, y, z);
	elseif (z <= 1.6e+50)
		tmp = fma(2.0, Float64(x + y), x);
	else
		tmp = fma(x, 3.0, z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.8e+33], N[(2.0 * y + z), $MachinePrecision], If[LessEqual[z, 1.6e+50], N[(2.0 * N[(x + y), $MachinePrecision] + x), $MachinePrecision], N[(x * 3.0 + z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.8000000000000001e33

   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. accelerator-lowering-fma.f64 85.7

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

   5. Simplified 85.7%

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

   if -2.8000000000000001e33 < z < 1.59999999999999991e50

   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. accelerator-lowering-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 90.7

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

   5. Simplified 90.7%

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

   if 1.59999999999999991e50 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 84.5

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

   5. Simplified 84.5%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(2, x + y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right)\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+53}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.5e+137)
   (fma 2.0 y z)
   (if (<= y 3.2e+53) (fma x 3.0 z) (fma 2.0 y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.5e+137) {
		tmp = fma(2.0, y, z);
	} else if (y <= 3.2e+53) {
		tmp = fma(x, 3.0, z);
	} else {
		tmp = fma(2.0, y, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.5e+137)
		tmp = fma(2.0, y, z);
	elseif (y <= 3.2e+53)
		tmp = fma(x, 3.0, z);
	else
		tmp = fma(2.0, y, z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.5e+137], N[(2.0 * y + z), $MachinePrecision], If[LessEqual[y, 3.2e+53], N[(x * 3.0 + z), $MachinePrecision], N[(2.0 * y + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.5000000000000001e137 or 3.2e53 < 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. accelerator-lowering-fma.f64 85.6

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

   5. Simplified 85.6%

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

   if -2.5000000000000001e137 < y < 3.2e53

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 89.5

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

   5. Simplified 89.5%

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

Alternative 6: 79.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+182}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+161}:\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.6e+182) (* x 3.0) (if (<= x 9.5e+161) (fma 2.0 y z) (* x 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.6e+182) {
		tmp = x * 3.0;
	} else if (x <= 9.5e+161) {
		tmp = fma(2.0, y, z);
	} else {
		tmp = x * 3.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.6e+182)
		tmp = Float64(x * 3.0);
	elseif (x <= 9.5e+161)
		tmp = fma(2.0, y, z);
	else
		tmp = Float64(x * 3.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.6e+182], N[(x * 3.0), $MachinePrecision], If[LessEqual[x, 9.5e+161], N[(2.0 * y + z), $MachinePrecision], N[(x * 3.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.60000000000000013e182 or 9.50000000000000061e161 < x

   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. *-commutative N/A

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

      2. *-lowering-*.f64 75.2

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

   5. Simplified 75.2%

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

   if -5.60000000000000013e182 < x < 9.50000000000000061e161

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 81.6

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

   5. Simplified 81.6%

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

Alternative 7: 55.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.85 \cdot 10^{+137}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;y \leq 3.35 \cdot 10^{+53}:\\ \;\;\;\;x + z\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.85e+137) (* y 2.0) (if (<= y 3.35e+53) (+ x z) (* y 2.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.85e+137) {
		tmp = y * 2.0;
	} else if (y <= 3.35e+53) {
		tmp = x + z;
	} else {
		tmp = y * 2.0;
	}
	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 (y <= (-2.85d+137)) then
        tmp = y * 2.0d0
    else if (y <= 3.35d+53) then
        tmp = x + z
    else
        tmp = y * 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.85e+137) {
		tmp = y * 2.0;
	} else if (y <= 3.35e+53) {
		tmp = x + z;
	} else {
		tmp = y * 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.85e+137:
		tmp = y * 2.0
	elif y <= 3.35e+53:
		tmp = x + z
	else:
		tmp = y * 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.85e+137)
		tmp = Float64(y * 2.0);
	elseif (y <= 3.35e+53)
		tmp = Float64(x + z);
	else
		tmp = Float64(y * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.85e+137)
		tmp = y * 2.0;
	elseif (y <= 3.35e+53)
		tmp = x + z;
	else
		tmp = y * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.85e+137], N[(y * 2.0), $MachinePrecision], If[LessEqual[y, 3.35e+53], N[(x + z), $MachinePrecision], N[(y * 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.8499999999999999e137 or 3.3499999999999999e53 < 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. *-lowering-*.f64 66.9

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

   5. Simplified 66.9%

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

   if -2.8499999999999999e137 < y < 3.3499999999999999e53

   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

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

   5. Simplified 53.1%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.85 \cdot 10^{+137}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;y \leq 3.35 \cdot 10^{+53}:\\ \;\;\;\;x + z\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 39.0% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x + z), $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 inf

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

5. Simplified 39.5%

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

6. Final simplification 39.5%

   \[\leadsto x + z \]
7. Add Preprocessing

Alternative 9: 34.0% accurate, 16.0× 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 34.7%

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

Alternative 10: 7.9% accurate, 16.0× 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

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

5. Simplified 39.5%

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

6. Taylor expanded in z around 0

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

8. Simplified 7.5%

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

Reproduce

herbie shell --seed 2024205 
(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))