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

Percentage Accurate: 100.0% → 98.5%

Time: 5.2s

Alternatives: 6

Speedup: 1.0×

• 21.0% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -1.8057534344232316e-111
  2. y = 3.1249438999880494e+230
  3. z = 9.402931305472895e+236

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: 100.0% 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: 98.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. distribute-rgt-in 98.4%

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

   2. associate-+l+ 98.4%

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

   3. *-commutative 98.4%

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

   4. fma-define 100.0%

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

   5. distribute-lft-out 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 59.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+107}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{-97}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-28}:\\ \;\;\;\;z \cdot 5\\ \mathbf{elif}\;x \leq 8.1 \cdot 10^{+251}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.6e+107)
   (* x z)
   (if (<= x -4.6e-97)
     (* x y)
     (if (<= x 3.7e-28) (* z 5.0) (if (<= x 8.1e+251) (* x y) (* x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.6e+107) {
		tmp = x * z;
	} else if (x <= -4.6e-97) {
		tmp = x * y;
	} else if (x <= 3.7e-28) {
		tmp = z * 5.0;
	} else if (x <= 8.1e+251) {
		tmp = x * 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 (x <= (-1.6d+107)) then
        tmp = x * z
    else if (x <= (-4.6d-97)) then
        tmp = x * y
    else if (x <= 3.7d-28) then
        tmp = z * 5.0d0
    else if (x <= 8.1d+251) then
        tmp = x * 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 (x <= -1.6e+107) {
		tmp = x * z;
	} else if (x <= -4.6e-97) {
		tmp = x * y;
	} else if (x <= 3.7e-28) {
		tmp = z * 5.0;
	} else if (x <= 8.1e+251) {
		tmp = x * y;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.6e+107:
		tmp = x * z
	elif x <= -4.6e-97:
		tmp = x * y
	elif x <= 3.7e-28:
		tmp = z * 5.0
	elif x <= 8.1e+251:
		tmp = x * y
	else:
		tmp = x * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.6e+107)
		tmp = Float64(x * z);
	elseif (x <= -4.6e-97)
		tmp = Float64(x * y);
	elseif (x <= 3.7e-28)
		tmp = Float64(z * 5.0);
	elseif (x <= 8.1e+251)
		tmp = Float64(x * y);
	else
		tmp = Float64(x * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.6e+107)
		tmp = x * z;
	elseif (x <= -4.6e-97)
		tmp = x * y;
	elseif (x <= 3.7e-28)
		tmp = z * 5.0;
	elseif (x <= 8.1e+251)
		tmp = x * y;
	else
		tmp = x * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.6e+107], N[(x * z), $MachinePrecision], If[LessEqual[x, -4.6e-97], N[(x * y), $MachinePrecision], If[LessEqual[x, 3.7e-28], N[(z * 5.0), $MachinePrecision], If[LessEqual[x, 8.1e+251], N[(x * y), $MachinePrecision], N[(x * z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.60000000000000015e107 or 8.1000000000000003e251 < x

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 100.0%

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

      1. +-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around inf 69.4%

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

   if -1.60000000000000015e107 < x < -4.59999999999999988e-97 or 3.7000000000000002e-28 < x < 8.1000000000000003e251

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 71.5%

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

   if -4.59999999999999988e-97 < x < 3.7000000000000002e-28

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 73.2%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+107}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{-97}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-28}:\\ \;\;\;\;z \cdot 5\\ \mathbf{elif}\;x \leq 8.1 \cdot 10^{+251}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{-97} \lor \neg \left(x \leq 6 \cdot 10^{-29}\right):\\ \;\;\;\;x \cdot \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot 5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -8.8e-97) (not (<= x 6e-29))) (* x (+ y z)) (* z 5.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8.8e-97) || !(x <= 6e-29)) {
		tmp = x * (y + z);
	} else {
		tmp = z * 5.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-8.8d-97)) .or. (.not. (x <= 6d-29))) then
        tmp = x * (y + z)
    else
        tmp = z * 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8.8e-97) || !(x <= 6e-29)) {
		tmp = x * (y + z);
	} else {
		tmp = z * 5.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -8.8e-97) or not (x <= 6e-29):
		tmp = x * (y + z)
	else:
		tmp = z * 5.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -8.8e-97) || !(x <= 6e-29))
		tmp = Float64(x * Float64(y + z));
	else
		tmp = Float64(z * 5.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -8.8e-97) || ~((x <= 6e-29)))
		tmp = x * (y + z);
	else
		tmp = z * 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -8.8e-97], N[Not[LessEqual[x, 6e-29]], $MachinePrecision]], N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision], N[(z * 5.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.7999999999999996e-97 or 6.0000000000000005e-29 < x

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 94.9%

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

      1. +-commutative 94.9%

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

   5. Simplified 94.9%

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

   if -8.7999999999999996e-97 < x < 6.0000000000000005e-29

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 73.2%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{-97} \lor \neg \left(x \leq 6 \cdot 10^{-29}\right):\\ \;\;\;\;x \cdot \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot 5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-99} \lor \neg \left(x \leq 2.35 \cdot 10^{-29}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot 5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.4e-99) (not (<= x 2.35e-29))) (* x y) (* z 5.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.4e-99) || !(x <= 2.35e-29)) {
		tmp = x * y;
	} else {
		tmp = z * 5.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-3.4d-99)) .or. (.not. (x <= 2.35d-29))) then
        tmp = x * y
    else
        tmp = z * 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.4e-99) || !(x <= 2.35e-29)) {
		tmp = x * y;
	} else {
		tmp = z * 5.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -3.4e-99) or not (x <= 2.35e-29):
		tmp = x * y
	else:
		tmp = z * 5.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.4e-99) || !(x <= 2.35e-29))
		tmp = Float64(x * y);
	else
		tmp = Float64(z * 5.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -3.4e-99) || ~((x <= 2.35e-29)))
		tmp = x * y;
	else
		tmp = z * 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3.4e-99], N[Not[LessEqual[x, 2.35e-29]], $MachinePrecision]], N[(x * y), $MachinePrecision], N[(z * 5.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.40000000000000007e-99 or 2.3499999999999999e-29 < x

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 62.0%

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

   if -3.40000000000000007e-99 < x < 2.3499999999999999e-29

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 73.2%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-99} \lor \neg \left(x \leq 2.35 \cdot 10^{-29}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot 5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 100.0% 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]

Derivation

1. Initial program 100.0%

   \[x \cdot \left(y + z\right) + z \cdot 5 \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto x \cdot \left(y + z\right) + z \cdot 5 \]
4. Add Preprocessing

Alternative 6: 34.0% 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 100.0%

   \[x \cdot \left(y + z\right) + z \cdot 5 \]
2. Add Preprocessing

3. Taylor expanded in x around 0 34.0%

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

4. Final simplification 34.0%

   \[\leadsto z \cdot 5 \]
5. Add Preprocessing

Developer target: 97.6% 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 2024052 
(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)))