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

Percentage Accurate: 99.9% → 100.0%

Time: 7.8s

Alternatives: 12

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(3.0 * x + N[(2.0 * y + z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

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

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

   4. associate-+l+ N/A

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

   5. +-commutative N/A

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

   6. count-2 N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. +-lowering-+.f64 99.9

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

4. Applied egg-rr 99.9%

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. associate-+r+ N/A

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

   5. associate-+l+ N/A

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

   6. distribute-rgt1-in N/A

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

   7. metadata-eval N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. accelerator-lowering-fma.f64 100.0

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

6. Applied egg-rr 100.0%

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

Alternative 2: 52.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+71}:\\ \;\;\;\;3 \cdot x\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-31}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+98}:\\ \;\;\;\;x + z\\ \mathbf{else}:\\ \;\;\;\;3 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.4e+71)
   (* 3.0 x)
   (if (<= x -1.35e-31)
     (+ x z)
     (if (<= x 4.8e-5) (* 2.0 y) (if (<= x 2.1e+98) (+ x z) (* 3.0 x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.4e+71) {
		tmp = 3.0 * x;
	} else if (x <= -1.35e-31) {
		tmp = x + z;
	} else if (x <= 4.8e-5) {
		tmp = 2.0 * y;
	} else if (x <= 2.1e+98) {
		tmp = x + z;
	} 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 <= (-2.4d+71)) then
        tmp = 3.0d0 * x
    else if (x <= (-1.35d-31)) then
        tmp = x + z
    else if (x <= 4.8d-5) then
        tmp = 2.0d0 * y
    else if (x <= 2.1d+98) then
        tmp = x + z
    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 <= -2.4e+71) {
		tmp = 3.0 * x;
	} else if (x <= -1.35e-31) {
		tmp = x + z;
	} else if (x <= 4.8e-5) {
		tmp = 2.0 * y;
	} else if (x <= 2.1e+98) {
		tmp = x + z;
	} else {
		tmp = 3.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.4e+71:
		tmp = 3.0 * x
	elif x <= -1.35e-31:
		tmp = x + z
	elif x <= 4.8e-5:
		tmp = 2.0 * y
	elif x <= 2.1e+98:
		tmp = x + z
	else:
		tmp = 3.0 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.4e+71)
		tmp = Float64(3.0 * x);
	elseif (x <= -1.35e-31)
		tmp = Float64(x + z);
	elseif (x <= 4.8e-5)
		tmp = Float64(2.0 * y);
	elseif (x <= 2.1e+98)
		tmp = Float64(x + z);
	else
		tmp = Float64(3.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.4e+71)
		tmp = 3.0 * x;
	elseif (x <= -1.35e-31)
		tmp = x + z;
	elseif (x <= 4.8e-5)
		tmp = 2.0 * y;
	elseif (x <= 2.1e+98)
		tmp = x + z;
	else
		tmp = 3.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.4e+71], N[(3.0 * x), $MachinePrecision], If[LessEqual[x, -1.35e-31], N[(x + z), $MachinePrecision], If[LessEqual[x, 4.8e-5], N[(2.0 * y), $MachinePrecision], If[LessEqual[x, 2.1e+98], N[(x + z), $MachinePrecision], N[(3.0 * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.39999999999999981e71 or 2.10000000000000004e98 < 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

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

      2. *-lowering-*.f64 82.7

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

   5. Simplified 82.7%

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

   if -2.39999999999999981e71 < x < -1.35000000000000007e-31 or 4.8000000000000001e-5 < x < 2.10000000000000004e98

   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 59.7%

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

   if -1.35000000000000007e-31 < x < 4.8000000000000001e-5

   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 57.1

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

   5. Simplified 57.1%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+71}:\\ \;\;\;\;3 \cdot x\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-31}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+98}:\\ \;\;\;\;x + z\\ \mathbf{else}:\\ \;\;\;\;3 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+76}:\\ \;\;\;\;x + \mathsf{fma}\left(2, y, z\right)\\ \mathbf{elif}\;z \leq 10500000000:\\ \;\;\;\;\mathsf{fma}\left(x, 3, 2 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.7e+76)
   (+ x (fma 2.0 y z))
   (if (<= z 10500000000.0) (fma x 3.0 (* 2.0 y)) (fma x 3.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.7e+76) {
		tmp = x + fma(2.0, y, z);
	} else if (z <= 10500000000.0) {
		tmp = fma(x, 3.0, (2.0 * y));
	} else {
		tmp = fma(x, 3.0, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.7e+76)
		tmp = Float64(x + fma(2.0, y, z));
	elseif (z <= 10500000000.0)
		tmp = fma(x, 3.0, Float64(2.0 * y));
	else
		tmp = fma(x, 3.0, z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.7e+76], N[(x + N[(2.0 * y + z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 10500000000.0], N[(x * 3.0 + N[(2.0 * y), $MachinePrecision]), $MachinePrecision], N[(x * 3.0 + z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.6999999999999999e76

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 95.2

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

   5. Simplified 95.2%

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

   if -3.6999999999999999e76 < z < 1.05e10

   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 92.6

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

   5. Simplified 92.6%

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

      1. distribute-lft-in N/A

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

      2. associate-+l+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 92.6

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

   7. Applied egg-rr 92.6%

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

   if 1.05e10 < 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 92.6

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

   5. Simplified 92.6%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+76}:\\ \;\;\;\;x + \mathsf{fma}\left(2, y, z\right)\\ \mathbf{elif}\;z \leq 10500000000:\\ \;\;\;\;\mathsf{fma}\left(x, 3, 2 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+79}:\\ \;\;\;\;x + \mathsf{fma}\left(2, y, z\right)\\ \mathbf{elif}\;z \leq 120000000000:\\ \;\;\;\;\mathsf{fma}\left(2, x + y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.8e+79)
   (+ x (fma 2.0 y z))
   (if (<= z 120000000000.0) (fma 2.0 (+ x y) x) (fma x 3.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.8e+79) {
		tmp = x + fma(2.0, y, z);
	} else if (z <= 120000000000.0) {
		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 <= -4.8e+79)
		tmp = Float64(x + fma(2.0, y, z));
	elseif (z <= 120000000000.0)
		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, -4.8e+79], N[(x + N[(2.0 * y + z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 120000000000.0], 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 < -4.79999999999999971e79

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 95.2

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

   5. Simplified 95.2%

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

   if -4.79999999999999971e79 < z < 1.2e11

   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 92.6

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

   5. Simplified 92.6%

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

   if 1.2e11 < 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 92.6

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

   5. Simplified 92.6%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+79}:\\ \;\;\;\;x + \mathsf{fma}\left(2, y, z\right)\\ \mathbf{elif}\;z \leq 120000000000:\\ \;\;\;\;\mathsf{fma}\left(2, x + y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, z\right)\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+20}:\\ \;\;\;\;x + \mathsf{fma}\left(2, y, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.5e+63)
   (fma x 3.0 z)
   (if (<= x 7.5e+20) (+ x (fma 2.0 y z)) (fma x 3.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.5e+63) {
		tmp = fma(x, 3.0, z);
	} else if (x <= 7.5e+20) {
		tmp = x + fma(2.0, y, z);
	} else {
		tmp = fma(x, 3.0, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.5e+63)
		tmp = fma(x, 3.0, z);
	elseif (x <= 7.5e+20)
		tmp = Float64(x + fma(2.0, y, z));
	else
		tmp = fma(x, 3.0, z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.5e+63], N[(x * 3.0 + z), $MachinePrecision], If[LessEqual[x, 7.5e+20], N[(x + N[(2.0 * y + z), $MachinePrecision]), $MachinePrecision], N[(x * 3.0 + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.50000000000000005e63 or 7.5e20 < 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 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 90.6

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

   5. Simplified 90.6%

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

   if -2.50000000000000005e63 < x < 7.5e20

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 92.1

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

   5. Simplified 92.1%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, z\right)\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+20}:\\ \;\;\;\;x + \mathsf{fma}\left(2, y, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{+40}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, z\right)\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.3e+40)
   (fma x 3.0 z)
   (if (<= x 2.6e+20) (fma 2.0 y z) (fma x 3.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.3e+40) {
		tmp = fma(x, 3.0, z);
	} else if (x <= 2.6e+20) {
		tmp = fma(2.0, y, z);
	} else {
		tmp = fma(x, 3.0, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.3e+40)
		tmp = fma(x, 3.0, z);
	elseif (x <= 2.6e+20)
		tmp = fma(2.0, y, z);
	else
		tmp = fma(x, 3.0, z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.3e+40], N[(x * 3.0 + z), $MachinePrecision], If[LessEqual[x, 2.6e+20], N[(2.0 * y + z), $MachinePrecision], N[(x * 3.0 + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.29999999999999994e40 or 2.6e20 < 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 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 89.4

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

   5. Simplified 89.4%

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

   if -2.29999999999999994e40 < x < 2.6e20

   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 91.9

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

   5. Simplified 91.9%

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

Alternative 7: 78.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.45 \cdot 10^{+71}:\\ \;\;\;\;3 \cdot x\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+92}:\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.45e+71) (* 3.0 x) (if (<= x 1.25e+92) (fma 2.0 y z) (* 3.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.45e+71) {
		tmp = 3.0 * x;
	} else if (x <= 1.25e+92) {
		tmp = fma(2.0, y, z);
	} else {
		tmp = 3.0 * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.45e+71)
		tmp = Float64(3.0 * x);
	elseif (x <= 1.25e+92)
		tmp = fma(2.0, y, z);
	else
		tmp = Float64(3.0 * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.45e+71], N[(3.0 * x), $MachinePrecision], If[LessEqual[x, 1.25e+92], N[(2.0 * y + z), $MachinePrecision], N[(3.0 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.44999999999999987e71 or 1.25000000000000005e92 < 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 82.0

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

   5. Simplified 82.0%

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

   if -3.44999999999999987e71 < x < 1.25000000000000005e92

   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 87.7

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

   5. Simplified 87.7%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.45 \cdot 10^{+71}:\\ \;\;\;\;3 \cdot x\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+92}:\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+100}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+31}:\\ \;\;\;\;2 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.4e+100) (+ x z) (if (<= z 3.8e+31) (* 2.0 y) (+ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.4e+100) {
		tmp = x + z;
	} else if (z <= 3.8e+31) {
		tmp = 2.0 * y;
	} 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 <= (-1.4d+100)) then
        tmp = x + z
    else if (z <= 3.8d+31) then
        tmp = 2.0d0 * y
    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 <= -1.4e+100) {
		tmp = x + z;
	} else if (z <= 3.8e+31) {
		tmp = 2.0 * y;
	} else {
		tmp = x + z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.4e+100:
		tmp = x + z
	elif z <= 3.8e+31:
		tmp = 2.0 * y
	else:
		tmp = x + z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.4e+100)
		tmp = Float64(x + z);
	elseif (z <= 3.8e+31)
		tmp = Float64(2.0 * y);
	else
		tmp = Float64(x + z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.4e+100)
		tmp = x + z;
	elseif (z <= 3.8e+31)
		tmp = 2.0 * y;
	else
		tmp = x + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.4e+100], N[(x + z), $MachinePrecision], If[LessEqual[z, 3.8e+31], N[(2.0 * y), $MachinePrecision], N[(x + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3999999999999999e100 or 3.8000000000000001e31 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 70.8%

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

   if -1.3999999999999999e100 < z < 3.8000000000000001e31

   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 46.9

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

   5. Simplified 46.9%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+100}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+31}:\\ \;\;\;\;2 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. associate-+l+ N/A

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

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

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

   4. associate-+l+ N/A

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

   5. +-commutative N/A

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

   6. count-2 N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. +-lowering-+.f64 99.9

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

4. Applied egg-rr 99.9%

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

Alternative 10: 39.8% 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 37.3%

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

6. Final simplification 37.3%

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

Alternative 11: 34.7% 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 32.0%

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

Alternative 12: 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 y around inf

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

   1. *-lowering-*.f64 40.4

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

5. Simplified 40.4%

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

6. Taylor expanded in y around 0

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

8. Simplified 8.2%

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

Reproduce

herbie shell --seed 2024204 
(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))