Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, C

Percentage Accurate: 99.9% → 100.0%

Time: 5.0s

Alternatives: 9

Speedup: 1.0×

• 20.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot \left(y + z\right) + z \cdot 5 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x (+ y z)) (* z 5.0)))

C

double code(double x, double y, double z) {
	return (x * (y + z)) + (z * 5.0);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * (y + z)) + (z * 5.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return (x * (y + z)) + (z * 5.0);
}

Python

def code(x, y, z):
	return (x * (y + z)) + (z * 5.0)

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(y + z)) + Float64(z * 5.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * (y + z)) + (z * 5.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] + N[(z * 5.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(y + z\right) + z \cdot 5 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x (+ y z)) (* z 5.0)))

C

double code(double x, double y, double z) {
	return (x * (y + z)) + (z * 5.0);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * (y + z)) + (z * 5.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return (x * (y + z)) + (z * 5.0);
}

Python

def code(x, y, z):
	return (x * (y + z)) + (z * 5.0)

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(y + z)) + Float64(z * 5.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * (y + z)) + (z * 5.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] + N[(z * 5.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z, 5, x \cdot \left(z + y\right)\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma z 5.0 (* x (+ z y))))

C

double code(double x, double y, double z) {
	return fma(z, 5.0, (x * (z + y)));
}

Julia

function code(x, y, z)
	return fma(z, 5.0, Float64(x * Float64(z + y)))
end

Wolfram

code[x_, y_, z_] := N[(z * 5.0 + N[(x * N[(z + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x \cdot \left(y + z\right) + z \cdot 5 \]
2. Step-by-step derivation

   1. +-commutative 99.9%

      \[\leadsto \color{blue}{z \cdot 5 + x \cdot \left(y + z\right)} \]

   2. fma-def 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(z, 5, x \cdot \left(z + y\right)\right) \]

Alternative 2: 98.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, y, z \cdot \left(5 + x\right)\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma x y (* z (+ 5.0 x))))

C

double code(double x, double y, double z) {
	return fma(x, y, (z * (5.0 + x)));
}

Julia

function code(x, y, z)
	return fma(x, y, Float64(z * Float64(5.0 + x)))
end

Wolfram

code[x_, y_, z_] := N[(x * y + N[(z * N[(5.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x \cdot \left(y + z\right) + z \cdot 5 \]
2. Step-by-step derivation

   1. distribute-lft-in 99.9%

      \[\leadsto \color{blue}{\left(x \cdot y + x \cdot z\right)} + z \cdot 5 \]

   2. associate-+l+ 99.9%

      \[\leadsto \color{blue}{x \cdot y + \left(x \cdot z + z \cdot 5\right)} \]

   3. *-commutative 99.9%

      \[\leadsto x \cdot y + \left(\color{blue}{z \cdot x} + z \cdot 5\right) \]

   4. fma-def 99.9%

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

   5. distribute-lft-out 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \mathsf{fma}\left(x, y, z \cdot \left(5 + x\right)\right) \]

Alternative 3: 61.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.75 \cdot 10^{-33}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{-34}:\\ \;\;\;\;z \cdot 5\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+40}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+258}:\\ \;\;\;\;z \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.75e-33)
   (* x y)
   (if (<= x 6.4e-34)
     (* z 5.0)
     (if (<= x 1.45e+40) (* x y) (if (<= x 5.5e+258) (* z x) (* x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.75e-33) {
		tmp = x * y;
	} else if (x <= 6.4e-34) {
		tmp = z * 5.0;
	} else if (x <= 1.45e+40) {
		tmp = x * y;
	} else if (x <= 5.5e+258) {
		tmp = z * x;
	} else {
		tmp = x * 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 (x <= (-2.75d-33)) then
        tmp = x * y
    else if (x <= 6.4d-34) then
        tmp = z * 5.0d0
    else if (x <= 1.45d+40) then
        tmp = x * y
    else if (x <= 5.5d+258) then
        tmp = z * x
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.75e-33) {
		tmp = x * y;
	} else if (x <= 6.4e-34) {
		tmp = z * 5.0;
	} else if (x <= 1.45e+40) {
		tmp = x * y;
	} else if (x <= 5.5e+258) {
		tmp = z * x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.75e-33:
		tmp = x * y
	elif x <= 6.4e-34:
		tmp = z * 5.0
	elif x <= 1.45e+40:
		tmp = x * y
	elif x <= 5.5e+258:
		tmp = z * x
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.75e-33)
		tmp = Float64(x * y);
	elseif (x <= 6.4e-34)
		tmp = Float64(z * 5.0);
	elseif (x <= 1.45e+40)
		tmp = Float64(x * y);
	elseif (x <= 5.5e+258)
		tmp = Float64(z * x);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.75e-33)
		tmp = x * y;
	elseif (x <= 6.4e-34)
		tmp = z * 5.0;
	elseif (x <= 1.45e+40)
		tmp = x * y;
	elseif (x <= 5.5e+258)
		tmp = z * x;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.75e-33], N[(x * y), $MachinePrecision], If[LessEqual[x, 6.4e-34], N[(z * 5.0), $MachinePrecision], If[LessEqual[x, 1.45e+40], N[(x * y), $MachinePrecision], If[LessEqual[x, 5.5e+258], N[(z * x), $MachinePrecision], N[(x * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.75e-33 or 6.40000000000000005e-34 < x < 1.45000000000000009e40 or 5.49999999999999978e258 < x

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]

   2. Taylor expanded in y around inf 74.3%

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

   if -2.75e-33 < x < 6.40000000000000005e-34

   1. Initial program 99.8%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]

   2. Taylor expanded in x around 0 77.7%

      \[\leadsto \color{blue}{5 \cdot z} \]

   if 1.45000000000000009e40 < x < 5.49999999999999978e258

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]

   2. Taylor expanded in y around 0 66.2%

      \[\leadsto \color{blue}{z \cdot x + 5 \cdot z} \]
   3. Step-by-step derivation

      1. +-commutative 66.2%

         \[\leadsto \color{blue}{5 \cdot z + z \cdot x} \]

      2. *-commutative 66.2%

         \[\leadsto 5 \cdot z + \color{blue}{x \cdot z} \]

      3. distribute-rgt-in 66.2%

         \[\leadsto \color{blue}{z \cdot \left(5 + x\right)} \]

   4. Simplified 66.2%

      \[\leadsto \color{blue}{z \cdot \left(5 + x\right)} \]

   5. Taylor expanded in x around inf 66.2%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.75 \cdot 10^{-33}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{-34}:\\ \;\;\;\;z \cdot 5\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+40}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+258}:\\ \;\;\;\;z \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 4: 84.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-10} \lor \neg \left(x \leq 2.9 \cdot 10^{-8}\right):\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(5 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -9.5e-10) (not (<= x 2.9e-8))) (* x (+ z y)) (* z (+ 5.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -9.5e-10) || !(x <= 2.9e-8)) {
		tmp = x * (z + y);
	} else {
		tmp = z * (5.0 + 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 <= (-9.5d-10)) .or. (.not. (x <= 2.9d-8))) then
        tmp = x * (z + y)
    else
        tmp = z * (5.0d0 + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -9.5e-10) || !(x <= 2.9e-8)) {
		tmp = x * (z + y);
	} else {
		tmp = z * (5.0 + x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -9.5e-10) or not (x <= 2.9e-8):
		tmp = x * (z + y)
	else:
		tmp = z * (5.0 + x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -9.5e-10) || !(x <= 2.9e-8))
		tmp = Float64(x * Float64(z + y));
	else
		tmp = Float64(z * Float64(5.0 + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -9.5e-10) || ~((x <= 2.9e-8)))
		tmp = x * (z + y);
	else
		tmp = z * (5.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -9.5e-10], N[Not[LessEqual[x, 2.9e-8]], $MachinePrecision]], N[(x * N[(z + y), $MachinePrecision]), $MachinePrecision], N[(z * N[(5.0 + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.50000000000000028e-10 or 2.9000000000000002e-8 < x

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]

   2. Taylor expanded in x around inf 99.2%

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

      1. +-commutative 99.2%

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

   4. Simplified 99.2%

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

   if -9.50000000000000028e-10 < x < 2.9000000000000002e-8

   1. Initial program 99.8%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]

   2. Taylor expanded in y around 0 76.0%

      \[\leadsto \color{blue}{z \cdot x + 5 \cdot z} \]
   3. Step-by-step derivation

      1. +-commutative 76.0%

         \[\leadsto \color{blue}{5 \cdot z + z \cdot x} \]

      2. *-commutative 76.0%

         \[\leadsto 5 \cdot z + \color{blue}{x \cdot z} \]

      3. distribute-rgt-in 76.1%

         \[\leadsto \color{blue}{z \cdot \left(5 + x\right)} \]

   4. Simplified 76.1%

      \[\leadsto \color{blue}{z \cdot \left(5 + x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-10} \lor \neg \left(x \leq 2.9 \cdot 10^{-8}\right):\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(5 + x\right)\\ \end{array} \]

Alternative 5: 74.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-6}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-15}:\\ \;\;\;\;z \cdot \left(5 + x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2e-6) (* x y) (if (<= y 3.8e-15) (* z (+ 5.0 x)) (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e-6) {
		tmp = x * y;
	} else if (y <= 3.8e-15) {
		tmp = z * (5.0 + x);
	} else {
		tmp = x * 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 <= (-2d-6)) then
        tmp = x * y
    else if (y <= 3.8d-15) then
        tmp = z * (5.0d0 + x)
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e-6) {
		tmp = x * y;
	} else if (y <= 3.8e-15) {
		tmp = z * (5.0 + x);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2e-6:
		tmp = x * y
	elif y <= 3.8e-15:
		tmp = z * (5.0 + x)
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2e-6)
		tmp = Float64(x * y);
	elseif (y <= 3.8e-15)
		tmp = Float64(z * Float64(5.0 + x));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2e-6)
		tmp = x * y;
	elseif (y <= 3.8e-15)
		tmp = z * (5.0 + x);
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2e-6], N[(x * y), $MachinePrecision], If[LessEqual[y, 3.8e-15], N[(z * N[(5.0 + x), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.99999999999999991e-6 or 3.8000000000000002e-15 < y

   1. Initial program 99.9%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]

   2. Taylor expanded in y around inf 76.8%

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

   if -1.99999999999999991e-6 < y < 3.8000000000000002e-15

   1. Initial program 99.8%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]

   2. Taylor expanded in y around 0 83.0%

      \[\leadsto \color{blue}{z \cdot x + 5 \cdot z} \]
   3. Step-by-step derivation

      1. +-commutative 83.0%

         \[\leadsto \color{blue}{5 \cdot z + z \cdot x} \]

      2. *-commutative 83.0%

         \[\leadsto 5 \cdot z + \color{blue}{x \cdot z} \]

      3. distribute-rgt-in 83.0%

         \[\leadsto \color{blue}{z \cdot \left(5 + x\right)} \]

   4. Simplified 83.0%

      \[\leadsto \color{blue}{z \cdot \left(5 + x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-6}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-15}:\\ \;\;\;\;z \cdot \left(5 + x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 6: 83.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{-10}:\\ \;\;\;\;z \cdot x + x \cdot y\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-9}:\\ \;\;\;\;z \cdot \left(5 + x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.6e-10)
   (+ (* z x) (* x y))
   (if (<= x 1.12e-9) (* z (+ 5.0 x)) (* x (+ z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.6e-10) {
		tmp = (z * x) + (x * y);
	} else if (x <= 1.12e-9) {
		tmp = z * (5.0 + x);
	} else {
		tmp = x * (z + 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 (x <= (-5.6d-10)) then
        tmp = (z * x) + (x * y)
    else if (x <= 1.12d-9) then
        tmp = z * (5.0d0 + x)
    else
        tmp = x * (z + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.6e-10) {
		tmp = (z * x) + (x * y);
	} else if (x <= 1.12e-9) {
		tmp = z * (5.0 + x);
	} else {
		tmp = x * (z + y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5.6e-10:
		tmp = (z * x) + (x * y)
	elif x <= 1.12e-9:
		tmp = z * (5.0 + x)
	else:
		tmp = x * (z + y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.6e-10)
		tmp = Float64(Float64(z * x) + Float64(x * y));
	elseif (x <= 1.12e-9)
		tmp = Float64(z * Float64(5.0 + x));
	else
		tmp = Float64(x * Float64(z + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -5.6e-10)
		tmp = (z * x) + (x * y);
	elseif (x <= 1.12e-9)
		tmp = z * (5.0 + x);
	else
		tmp = x * (z + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.6e-10], N[(N[(z * x), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.12e-9], N[(z * N[(5.0 + x), $MachinePrecision]), $MachinePrecision], N[(x * N[(z + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.60000000000000031e-10

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]

   2. Taylor expanded in x around inf 99.7%

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

      1. +-commutative 99.7%

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

   4. Simplified 99.7%

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

   5. Taylor expanded in z around 0 99.8%

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

   if -5.60000000000000031e-10 < x < 1.12000000000000006e-9

   1. Initial program 99.8%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]

   2. Taylor expanded in y around 0 76.0%

      \[\leadsto \color{blue}{z \cdot x + 5 \cdot z} \]
   3. Step-by-step derivation

      1. +-commutative 76.0%

         \[\leadsto \color{blue}{5 \cdot z + z \cdot x} \]

      2. *-commutative 76.0%

         \[\leadsto 5 \cdot z + \color{blue}{x \cdot z} \]

      3. distribute-rgt-in 76.1%

         \[\leadsto \color{blue}{z \cdot \left(5 + x\right)} \]

   4. Simplified 76.1%

      \[\leadsto \color{blue}{z \cdot \left(5 + x\right)} \]

   if 1.12000000000000006e-9 < x

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]

   2. Taylor expanded in x around inf 98.8%

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

      1. +-commutative 98.8%

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

   4. Simplified 98.8%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{-10}:\\ \;\;\;\;z \cdot x + x \cdot y\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-9}:\\ \;\;\;\;z \cdot \left(5 + x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z + y\right)\\ \end{array} \]

Alternative 7: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(z + y\right) + z \cdot 5 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x (+ z y)) (* z 5.0)))

C

double code(double x, double y, double z) {
	return (x * (z + y)) + (z * 5.0);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * (z + y)) + (z * 5.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return (x * (z + y)) + (z * 5.0);
}

Python

def code(x, y, z):
	return (x * (z + y)) + (z * 5.0)

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(z + y)) + Float64(z * 5.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * (z + y)) + (z * 5.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(z + y), $MachinePrecision]), $MachinePrecision] + N[(z * 5.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x \cdot \left(y + z\right) + z \cdot 5 \]

2. Final simplification 99.9%

   \[\leadsto x \cdot \left(z + y\right) + z \cdot 5 \]

Alternative 8: 61.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.75 \cdot 10^{-33}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-34}:\\ \;\;\;\;z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.75e-33) (* x y) (if (<= x 1.7e-34) (* z 5.0) (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.75e-33) {
		tmp = x * y;
	} else if (x <= 1.7e-34) {
		tmp = z * 5.0;
	} else {
		tmp = x * 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 (x <= (-2.75d-33)) then
        tmp = x * y
    else if (x <= 1.7d-34) then
        tmp = z * 5.0d0
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.75e-33) {
		tmp = x * y;
	} else if (x <= 1.7e-34) {
		tmp = z * 5.0;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.75e-33:
		tmp = x * y
	elif x <= 1.7e-34:
		tmp = z * 5.0
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.75e-33)
		tmp = Float64(x * y);
	elseif (x <= 1.7e-34)
		tmp = Float64(z * 5.0);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.75e-33)
		tmp = x * y;
	elseif (x <= 1.7e-34)
		tmp = z * 5.0;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.75e-33], N[(x * y), $MachinePrecision], If[LessEqual[x, 1.7e-34], N[(z * 5.0), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.75e-33 or 1.7e-34 < x

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]

   2. Taylor expanded in y around inf 63.4%

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

   if -2.75e-33 < x < 1.7e-34

   1. Initial program 99.8%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]

   2. Taylor expanded in x around 0 77.7%

      \[\leadsto \color{blue}{5 \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.75 \cdot 10^{-33}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-34}:\\ \;\;\;\;z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 9: 35.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ z \cdot 5 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* z 5.0))

C

double code(double x, double y, double z) {
	return z * 5.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = z * 5.0d0
end function

Java

public static double code(double x, double y, double z) {
	return z * 5.0;
}

Python

def code(x, y, z):
	return z * 5.0

Julia

function code(x, y, z)
	return Float64(z * 5.0)
end

MATLAB

function tmp = code(x, y, z)
	tmp = z * 5.0;
end

Wolfram

code[x_, y_, z_] := N[(z * 5.0), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x \cdot \left(y + z\right) + z \cdot 5 \]

2. Taylor expanded in x around 0 38.0%

   \[\leadsto \color{blue}{5 \cdot z} \]

3. Final simplification 38.0%

   \[\leadsto z \cdot 5 \]

Developer target: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x + 5\right) \cdot z + x \cdot y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* (+ x 5.0) z) (* x y)))

C

double code(double x, double y, double z) {
	return ((x + 5.0) * z) + (x * y);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x + 5.0d0) * z) + (x * y)
end function

Java

public static double code(double x, double y, double z) {
	return ((x + 5.0) * z) + (x * y);
}

Python

def code(x, y, z):
	return ((x + 5.0) * z) + (x * y)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x + 5.0) * z) + Float64(x * y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x + 5.0) * z) + (x * y);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x + 5.0), $MachinePrecision] * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023238 
(FPCore (x y z)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, C"
  :precision binary64

  :herbie-target
  (+ (* (+ x 5.0) z) (* x y))

  (+ (* x (+ y z)) (* z 5.0)))