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

Percentage Accurate: 99.9% → 99.9%

Time: 8.1s

Alternatives: 11

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. count-2 N/A

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

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

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

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

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

   8. +-lowering-+.f64 99.9

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

4. Applied egg-rr 99.9%

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

Alternative 2: 56.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+118}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;z \leq -7.1 \cdot 10^{+34}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(2, y, x\right)\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+61}:\\ \;\;\;\;x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5e+118)
   (+ x z)
   (if (<= z -7.1e+34)
     (* x 3.0)
     (if (<= z 6.2e-23) (fma 2.0 y x) (if (<= z 3.2e+61) (* x 3.0) (+ x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5e+118) {
		tmp = x + z;
	} else if (z <= -7.1e+34) {
		tmp = x * 3.0;
	} else if (z <= 6.2e-23) {
		tmp = fma(2.0, y, x);
	} else if (z <= 3.2e+61) {
		tmp = x * 3.0;
	} else {
		tmp = x + z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5e+118)
		tmp = Float64(x + z);
	elseif (z <= -7.1e+34)
		tmp = Float64(x * 3.0);
	elseif (z <= 6.2e-23)
		tmp = fma(2.0, y, x);
	elseif (z <= 3.2e+61)
		tmp = Float64(x * 3.0);
	else
		tmp = Float64(x + z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5e+118], N[(x + z), $MachinePrecision], If[LessEqual[z, -7.1e+34], N[(x * 3.0), $MachinePrecision], If[LessEqual[z, 6.2e-23], N[(2.0 * y + x), $MachinePrecision], If[LessEqual[z, 3.2e+61], N[(x * 3.0), $MachinePrecision], N[(x + z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.99999999999999972e118 or 3.1999999999999998e61 < 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 70.1%

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

   if -4.99999999999999972e118 < z < -7.09999999999999956e34 or 6.1999999999999998e-23 < z < 3.1999999999999998e61

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 62.7

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

   5. Simplified 62.7%

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

   if -7.09999999999999956e34 < z < 6.1999999999999998e-23

   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 91.1

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

   5. Simplified 91.1%

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

   6. Taylor expanded in y around inf

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

   8. Simplified 61.1%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+118}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;z \leq -7.1 \cdot 10^{+34}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(2, y, x\right)\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+61}:\\ \;\;\;\;x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{+117}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{+32}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-19}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+55}:\\ \;\;\;\;x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.95e+117)
   (+ x z)
   (if (<= z -5.8e+32)
     (* x 3.0)
     (if (<= z 1.7e-19) (* 2.0 y) (if (<= z 5.2e+55) (* x 3.0) (+ x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.95e+117) {
		tmp = x + z;
	} else if (z <= -5.8e+32) {
		tmp = x * 3.0;
	} else if (z <= 1.7e-19) {
		tmp = 2.0 * y;
	} else if (z <= 5.2e+55) {
		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 <= (-1.95d+117)) then
        tmp = x + z
    else if (z <= (-5.8d+32)) then
        tmp = x * 3.0d0
    else if (z <= 1.7d-19) then
        tmp = 2.0d0 * y
    else if (z <= 5.2d+55) 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 <= -1.95e+117) {
		tmp = x + z;
	} else if (z <= -5.8e+32) {
		tmp = x * 3.0;
	} else if (z <= 1.7e-19) {
		tmp = 2.0 * y;
	} else if (z <= 5.2e+55) {
		tmp = x * 3.0;
	} else {
		tmp = x + z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.95e+117:
		tmp = x + z
	elif z <= -5.8e+32:
		tmp = x * 3.0
	elif z <= 1.7e-19:
		tmp = 2.0 * y
	elif z <= 5.2e+55:
		tmp = x * 3.0
	else:
		tmp = x + z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.95e+117)
		tmp = Float64(x + z);
	elseif (z <= -5.8e+32)
		tmp = Float64(x * 3.0);
	elseif (z <= 1.7e-19)
		tmp = Float64(2.0 * y);
	elseif (z <= 5.2e+55)
		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 <= -1.95e+117)
		tmp = x + z;
	elseif (z <= -5.8e+32)
		tmp = x * 3.0;
	elseif (z <= 1.7e-19)
		tmp = 2.0 * y;
	elseif (z <= 5.2e+55)
		tmp = x * 3.0;
	else
		tmp = x + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.95e+117], N[(x + z), $MachinePrecision], If[LessEqual[z, -5.8e+32], N[(x * 3.0), $MachinePrecision], If[LessEqual[z, 1.7e-19], N[(2.0 * y), $MachinePrecision], If[LessEqual[z, 5.2e+55], N[(x * 3.0), $MachinePrecision], N[(x + z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.94999999999999995e117 or 5.2e55 < 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 70.1%

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

   if -1.94999999999999995e117 < z < -5.80000000000000006e32 or 1.7000000000000001e-19 < z < 5.2e55

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 62.7

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

   5. Simplified 62.7%

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

   if -5.80000000000000006e32 < z < 1.7000000000000001e-19

   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. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. count-2 N/A

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

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

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

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

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

      8. +-lowering-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 55.9

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

   7. Simplified 55.9%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{+117}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{+32}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-19}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+55}:\\ \;\;\;\;x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, z\right)\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(3, x, 2 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.35e+35)
   (fma x 3.0 z)
   (if (<= z 6.4e-24) (fma 3.0 x (* 2.0 y)) (fma x 3.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.35e+35) {
		tmp = fma(x, 3.0, z);
	} else if (z <= 6.4e-24) {
		tmp = fma(3.0, x, (2.0 * y));
	} else {
		tmp = fma(x, 3.0, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.35e+35)
		tmp = fma(x, 3.0, z);
	elseif (z <= 6.4e-24)
		tmp = fma(3.0, x, Float64(2.0 * y));
	else
		tmp = fma(x, 3.0, z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.35e+35], N[(x * 3.0 + z), $MachinePrecision], If[LessEqual[z, 6.4e-24], N[(3.0 * x + N[(2.0 * y), $MachinePrecision]), $MachinePrecision], N[(x * 3.0 + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.35000000000000001e35 or 6.40000000000000025e-24 < 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 84.1

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

   5. Simplified 84.1%

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

   if -1.35000000000000001e35 < z < 6.40000000000000025e-24

   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 91.8

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

   5. Simplified 91.8%

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+r+ N/A

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

      5. distribute-rgt1-in N/A

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

      6. metadata-eval N/A

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

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

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

      8. *-lowering-*.f64 91.8

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

   7. Applied egg-rr 91.8%

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

Alternative 5: 85.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.6 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, z\right)\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(2, x + y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -6.6e+33)
   (fma x 3.0 z)
   (if (<= z 6.4e-24) (fma 2.0 (+ x y) x) (fma x 3.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.6e+33) {
		tmp = fma(x, 3.0, z);
	} else if (z <= 6.4e-24) {
		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 <= -6.6e+33)
		tmp = fma(x, 3.0, z);
	elseif (z <= 6.4e-24)
		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, -6.6e+33], N[(x * 3.0 + z), $MachinePrecision], If[LessEqual[z, 6.4e-24], N[(2.0 * N[(x + y), $MachinePrecision] + x), $MachinePrecision], N[(x * 3.0 + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.59999999999999953e33 or 6.40000000000000025e-24 < 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 84.1

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

   5. Simplified 84.1%

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

   if -6.59999999999999953e33 < z < 6.40000000000000025e-24

   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 91.8

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

   5. Simplified 91.8%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.6 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, z\right)\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(2, x + y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+93}:\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right)\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.8e+93)
   (fma 2.0 y z)
   (if (<= y 2.4e+87) (fma x 3.0 z) (fma 2.0 y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.8e+93) {
		tmp = fma(2.0, y, z);
	} else if (y <= 2.4e+87) {
		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 <= -1.8e+93)
		tmp = fma(2.0, y, z);
	elseif (y <= 2.4e+87)
		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, -1.8e+93], N[(2.0 * y + z), $MachinePrecision], If[LessEqual[y, 2.4e+87], N[(x * 3.0 + z), $MachinePrecision], N[(2.0 * y + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8e93 or 2.39999999999999981e87 < 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

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

      2. accelerator-lowering-fma.f64 87.8

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

   5. Simplified 87.8%

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

   if -1.8e93 < y < 2.39999999999999981e87

   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 86.1

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

   5. Simplified 86.1%

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

Alternative 7: 79.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{+147}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+187}:\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.1e+147) (* x 3.0) (if (<= x 2.8e+187) (fma 2.0 y z) (* x 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.1e+147) {
		tmp = x * 3.0;
	} else if (x <= 2.8e+187) {
		tmp = fma(2.0, y, z);
	} else {
		tmp = x * 3.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.1e+147)
		tmp = Float64(x * 3.0);
	elseif (x <= 2.8e+187)
		tmp = fma(2.0, y, z);
	else
		tmp = Float64(x * 3.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.1e+147], N[(x * 3.0), $MachinePrecision], If[LessEqual[x, 2.8e+187], N[(2.0 * y + z), $MachinePrecision], N[(x * 3.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.09999999999999966e147 or 2.79999999999999989e187 < 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 75.9

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

   5. Simplified 75.9%

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

   if -4.09999999999999966e147 < x < 2.79999999999999989e187

   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 83.2

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

   5. Simplified 83.2%

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

Alternative 8: 57.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+59}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+82}:\\ \;\;\;\;x + z\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7e+59) (* 2.0 y) (if (<= y 1.8e+82) (+ x z) (* 2.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7e+59) {
		tmp = 2.0 * y;
	} else if (y <= 1.8e+82) {
		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 <= (-7d+59)) then
        tmp = 2.0d0 * y
    else if (y <= 1.8d+82) 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 <= -7e+59) {
		tmp = 2.0 * y;
	} else if (y <= 1.8e+82) {
		tmp = x + z;
	} else {
		tmp = 2.0 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -7e+59:
		tmp = 2.0 * y
	elif y <= 1.8e+82:
		tmp = x + z
	else:
		tmp = 2.0 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7e+59)
		tmp = Float64(2.0 * y);
	elseif (y <= 1.8e+82)
		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 <= -7e+59)
		tmp = 2.0 * y;
	elseif (y <= 1.8e+82)
		tmp = x + z;
	else
		tmp = 2.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7e+59], N[(2.0 * y), $MachinePrecision], If[LessEqual[y, 1.8e+82], N[(x + z), $MachinePrecision], N[(2.0 * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -7e59 or 1.80000000000000007e82 < y

   1. Initial program 100.0%

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

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. count-2 N/A

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

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

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

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

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

      8. +-lowering-+.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 70.3

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

   7. Simplified 70.3%

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

   if -7e59 < y < 1.80000000000000007e82

   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 51.1%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+59}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+82}:\\ \;\;\;\;x + z\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 39.2% 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 37.8%

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

6. Final simplification 37.8%

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

Alternative 10: 34.2% 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 33.0%

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

Alternative 11: 7.8% 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 37.8%

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

6. Taylor expanded in z around 0

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

8. Simplified 7.7%

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

Reproduce

herbie shell --seed 2024198 
(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))