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

Percentage Accurate: 99.9% → 100.0%

Time: 3.8s

Alternatives: 7

Speedup: 1.0×

• 20.8% 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 99.5%

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

   1. +-commutative 99.5%

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

   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: 62.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.16 \cdot 10^{+67}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;x \leq -2.5 \cdot 10^{-12} \lor \neg \left(x \leq 4.2 \cdot 10^{-51}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot 5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.16e+67)
   (* z x)
   (if (or (<= x -2.5e-12) (not (<= x 4.2e-51))) (* x y) (* z 5.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.16e+67) {
		tmp = z * x;
	} else if ((x <= -2.5e-12) || !(x <= 4.2e-51)) {
		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 <= (-1.16d+67)) then
        tmp = z * x
    else if ((x <= (-2.5d-12)) .or. (.not. (x <= 4.2d-51))) 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 <= -1.16e+67) {
		tmp = z * x;
	} else if ((x <= -2.5e-12) || !(x <= 4.2e-51)) {
		tmp = x * y;
	} else {
		tmp = z * 5.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.16e+67:
		tmp = z * x
	elif (x <= -2.5e-12) or not (x <= 4.2e-51):
		tmp = x * y
	else:
		tmp = z * 5.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.16e+67)
		tmp = Float64(z * x);
	elseif ((x <= -2.5e-12) || !(x <= 4.2e-51))
		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 <= -1.16e+67)
		tmp = z * x;
	elseif ((x <= -2.5e-12) || ~((x <= 4.2e-51)))
		tmp = x * y;
	else
		tmp = z * 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.16e+67], N[(z * x), $MachinePrecision], If[Or[LessEqual[x, -2.5e-12], N[Not[LessEqual[x, 4.2e-51]], $MachinePrecision]], N[(x * y), $MachinePrecision], N[(z * 5.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.15999999999999994e67

   1. Initial program 98.2%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. +-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in z around inf 65.0%

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

   if -1.15999999999999994e67 < x < -2.49999999999999985e-12 or 4.20000000000000003e-51 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 57.4%

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

   if -2.49999999999999985e-12 < x < 4.20000000000000003e-51

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 82.9%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.16 \cdot 10^{+67}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;x \leq -2.5 \cdot 10^{-12} \lor \neg \left(x \leq 4.2 \cdot 10^{-51}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot 5\\ \end{array} \]

Alternative 3: 84.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{-13} \lor \neg \left(x \leq 8.5 \cdot 10^{-64}\right):\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot 5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -9.6e-13) (not (<= x 8.5e-64))) (* x (+ z y)) (* z 5.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -9.6e-13) || !(x <= 8.5e-64)) {
		tmp = x * (z + 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 <= (-9.6d-13)) .or. (.not. (x <= 8.5d-64))) then
        tmp = x * (z + 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 <= -9.6e-13) || !(x <= 8.5e-64)) {
		tmp = x * (z + y);
	} else {
		tmp = z * 5.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -9.6e-13) or not (x <= 8.5e-64):
		tmp = x * (z + y)
	else:
		tmp = z * 5.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -9.6e-13) || !(x <= 8.5e-64))
		tmp = Float64(x * Float64(z + y));
	else
		tmp = Float64(z * 5.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -9.6e-13) || ~((x <= 8.5e-64)))
		tmp = x * (z + y);
	else
		tmp = z * 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -9.6e-13], N[Not[LessEqual[x, 8.5e-64]], $MachinePrecision]], N[(x * N[(z + y), $MachinePrecision]), $MachinePrecision], N[(z * 5.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.5999999999999995e-13 or 8.49999999999999996e-64 < x

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf 96.4%

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

      1. +-commutative 96.4%

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

   4. Simplified 96.4%

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

   if -9.5999999999999995e-13 < x < 8.49999999999999996e-64

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 83.5%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{-13} \lor \neg \left(x \leq 8.5 \cdot 10^{-64}\right):\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot 5\\ \end{array} \]

Alternative 4: 84.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{-12} \lor \neg \left(x \leq 8 \cdot 10^{-64}\right):\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(5 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -9.6e-12) (not (<= x 8e-64))) (* x (+ z y)) (* z (+ 5.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -9.6e-12) || !(x <= 8e-64)) {
		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 <= (-9.6d-12)) .or. (.not. (x <= 8d-64))) 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 <= -9.6e-12) || !(x <= 8e-64)) {
		tmp = x * (z + y);
	} else {
		tmp = z * (5.0 + x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -9.6e-12) or not (x <= 8e-64):
		tmp = x * (z + y)
	else:
		tmp = z * (5.0 + x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -9.6e-12) || !(x <= 8e-64))
		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 <= -9.6e-12) || ~((x <= 8e-64)))
		tmp = x * (z + y);
	else
		tmp = z * (5.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -9.6e-12], N[Not[LessEqual[x, 8e-64]], $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 < -9.59999999999999948e-12 or 7.99999999999999972e-64 < x

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf 96.4%

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

      1. +-commutative 96.4%

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

   4. Simplified 96.4%

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

   if -9.59999999999999948e-12 < x < 7.99999999999999972e-64

   1. Initial program 99.7%

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

   2. Taylor expanded in y around 0 83.6%

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

      1. distribute-rgt-in 83.6%

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

   4. Simplified 83.6%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{-12} \lor \neg \left(x \leq 8 \cdot 10^{-64}\right):\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(5 + x\right)\\ \end{array} \]

Alternative 5: 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 99.5%

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

2. Final simplification 99.5%

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

Alternative 6: 61.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-12} \lor \neg \left(x \leq 4.7 \cdot 10^{-51}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot 5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.1e-12) (not (<= x 4.7e-51))) (* x y) (* z 5.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.1e-12) || !(x <= 4.7e-51)) {
		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 <= (-1.1d-12)) .or. (.not. (x <= 4.7d-51))) 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 <= -1.1e-12) || !(x <= 4.7e-51)) {
		tmp = x * y;
	} else {
		tmp = z * 5.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.1e-12) or not (x <= 4.7e-51):
		tmp = x * y
	else:
		tmp = z * 5.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.1e-12) || !(x <= 4.7e-51))
		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 <= -1.1e-12) || ~((x <= 4.7e-51)))
		tmp = x * y;
	else
		tmp = z * 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.1e-12], N[Not[LessEqual[x, 4.7e-51]], $MachinePrecision]], N[(x * y), $MachinePrecision], N[(z * 5.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.09999999999999996e-12 or 4.6999999999999997e-51 < x

   1. Initial program 99.2%

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

   2. Taylor expanded in y around inf 51.4%

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

   if -1.09999999999999996e-12 < x < 4.6999999999999997e-51

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 82.9%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-12} \lor \neg \left(x \leq 4.7 \cdot 10^{-51}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot 5\\ \end{array} \]

Alternative 7: 37.5% 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 99.5%

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

2. Taylor expanded in x around 0 40.6%

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

3. Final simplification 40.6%

   \[\leadsto z \cdot 5 \]

Developer target: 97.7% 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 2023332 
(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)))