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

Percentage Accurate: 99.9% → 100.0%

Time: 4.5s

Alternatives: 8

Speedup: 1.0×

• 20.5% 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: 61.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+120}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{+50}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;x \leq -1.95 \cdot 10^{-19}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-12}:\\ \;\;\;\;z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.2e+120)
   (* x y)
   (if (<= x -1.2e+50)
     (* z x)
     (if (<= x -1.95e-19) (* x y) (if (<= x 1.55e-12) (* z 5.0) (* x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.2e+120) {
		tmp = x * y;
	} else if (x <= -1.2e+50) {
		tmp = z * x;
	} else if (x <= -1.95e-19) {
		tmp = x * y;
	} else if (x <= 1.55e-12) {
		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 <= (-1.2d+120)) then
        tmp = x * y
    else if (x <= (-1.2d+50)) then
        tmp = z * x
    else if (x <= (-1.95d-19)) then
        tmp = x * y
    else if (x <= 1.55d-12) 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 <= -1.2e+120) {
		tmp = x * y;
	} else if (x <= -1.2e+50) {
		tmp = z * x;
	} else if (x <= -1.95e-19) {
		tmp = x * y;
	} else if (x <= 1.55e-12) {
		tmp = z * 5.0;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.2e+120:
		tmp = x * y
	elif x <= -1.2e+50:
		tmp = z * x
	elif x <= -1.95e-19:
		tmp = x * y
	elif x <= 1.55e-12:
		tmp = z * 5.0
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.2e+120)
		tmp = Float64(x * y);
	elseif (x <= -1.2e+50)
		tmp = Float64(z * x);
	elseif (x <= -1.95e-19)
		tmp = Float64(x * y);
	elseif (x <= 1.55e-12)
		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 <= -1.2e+120)
		tmp = x * y;
	elseif (x <= -1.2e+50)
		tmp = z * x;
	elseif (x <= -1.95e-19)
		tmp = x * y;
	elseif (x <= 1.55e-12)
		tmp = z * 5.0;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.2e+120], N[(x * y), $MachinePrecision], If[LessEqual[x, -1.2e+50], N[(z * x), $MachinePrecision], If[LessEqual[x, -1.95e-19], N[(x * y), $MachinePrecision], If[LessEqual[x, 1.55e-12], N[(z * 5.0), $MachinePrecision], N[(x * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.2e120 or -1.2000000000000001e50 < x < -1.94999999999999998e-19 or 1.5500000000000001e-12 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 61.4%

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

   if -1.2e120 < x < -1.2000000000000001e50

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 72.5%

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

      1. +-commutative 72.5%

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

      2. *-commutative 72.5%

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

      3. distribute-rgt-in 72.5%

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

   4. Simplified 72.5%

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

   5. Taylor expanded in x around inf 72.5%

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

   if -1.94999999999999998e-19 < x < 1.5500000000000001e-12

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 79.4%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+120}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{+50}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;x \leq -1.95 \cdot 10^{-19}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-12}:\\ \;\;\;\;z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 3: 99.0% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -5.0) (not (<= x 5.0))) (* x (+ z y)) (- (* x y) (* z -5.0))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -5 or 5 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 97.9%

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

      1. +-commutative 97.9%

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

   4. Simplified 97.9%

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

   if -5 < x < 5

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      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. Taylor expanded in z around -inf 99.8%

      \[\leadsto \color{blue}{-1 \cdot \left(z \cdot \left(-1 \cdot x - 5\right)\right) + y \cdot x} \]
   5. Step-by-step derivation

      1. +-commutative 99.8%

         \[\leadsto \color{blue}{y \cdot x + -1 \cdot \left(z \cdot \left(-1 \cdot x - 5\right)\right)} \]

      2. fma-def 99.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, -1 \cdot \left(z \cdot \left(-1 \cdot x - 5\right)\right)\right)} \]

      3. mul-1-neg 99.8%

         \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{-z \cdot \left(-1 \cdot x - 5\right)}\right) \]

      4. fma-neg 99.8%

         \[\leadsto \color{blue}{y \cdot x - z \cdot \left(-1 \cdot x - 5\right)} \]

      5. sub-neg 99.8%

         \[\leadsto y \cdot x - z \cdot \color{blue}{\left(-1 \cdot x + \left(-5\right)\right)} \]

      6. +-commutative 99.8%

         \[\leadsto y \cdot x - z \cdot \color{blue}{\left(\left(-5\right) + -1 \cdot x\right)} \]

      7. mul-1-neg 99.8%

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

      8. unsub-neg 99.8%

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

      9. metadata-eval 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in x around 0 99.2%

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

      1. *-commutative 99.2%

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

   9. Simplified 99.2%

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

4. Final simplification 98.5%

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

Alternative 4: 74.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{-117} \lor \neg \left(z \leq 3.8 \cdot 10^{-155}\right):\\ \;\;\;\;z \cdot \left(5 + x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.85e-117) (not (<= z 3.8e-155))) (* z (+ 5.0 x)) (* x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.85e-117) || !(z <= 3.8e-155)) {
		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 ((z <= (-1.85d-117)) .or. (.not. (z <= 3.8d-155))) 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 ((z <= -1.85e-117) || !(z <= 3.8e-155)) {
		tmp = z * (5.0 + x);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.85e-117) or not (z <= 3.8e-155):
		tmp = z * (5.0 + x)
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.85e-117) || !(z <= 3.8e-155))
		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 ((z <= -1.85e-117) || ~((z <= 3.8e-155)))
		tmp = z * (5.0 + x);
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.85e-117], N[Not[LessEqual[z, 3.8e-155]], $MachinePrecision]], N[(z * N[(5.0 + x), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.8500000000000001e-117 or 3.7999999999999998e-155 < z

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 79.2%

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

      1. +-commutative 79.2%

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

      2. *-commutative 79.2%

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

      3. distribute-rgt-in 79.2%

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

   4. Simplified 79.2%

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

   if -1.8500000000000001e-117 < z < 3.7999999999999998e-155

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 74.0%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{-117} \lor \neg \left(z \leq 3.8 \cdot 10^{-155}\right):\\ \;\;\;\;z \cdot \left(5 + x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 5: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3800 \lor \neg \left(x \leq 9 \cdot 10^{-10}\right):\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(5 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3800.0) (not (<= x 9e-10))) (* x (+ z y)) (* z (+ 5.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3800.0) || !(x <= 9e-10)) {
		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 <= (-3800.0d0)) .or. (.not. (x <= 9d-10))) 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 <= -3800.0) || !(x <= 9e-10)) {
		tmp = x * (z + y);
	} else {
		tmp = z * (5.0 + x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -3800.0) or not (x <= 9e-10):
		tmp = x * (z + y)
	else:
		tmp = z * (5.0 + x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3800.0) || !(x <= 9e-10))
		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 <= -3800.0) || ~((x <= 9e-10)))
		tmp = x * (z + y);
	else
		tmp = z * (5.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3800.0], N[Not[LessEqual[x, 9e-10]], $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 < -3800 or 8.9999999999999999e-10 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 98.9%

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

      1. +-commutative 98.9%

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

   4. Simplified 98.9%

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

   if -3800 < x < 8.9999999999999999e-10

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 78.8%

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

      1. +-commutative 78.8%

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

      2. *-commutative 78.8%

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

      3. distribute-rgt-in 78.8%

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

   4. Simplified 78.8%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3800 \lor \neg \left(x \leq 9 \cdot 10^{-10}\right):\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(5 + x\right)\\ \end{array} \]

Alternative 6: 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 7: 62.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-20}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-10}:\\ \;\;\;\;z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1e-20) (* x y) (if (<= x 1.05e-10) (* z 5.0) (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1e-20) {
		tmp = x * y;
	} else if (x <= 1.05e-10) {
		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 <= (-1d-20)) then
        tmp = x * y
    else if (x <= 1.05d-10) 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 <= -1e-20) {
		tmp = x * y;
	} else if (x <= 1.05e-10) {
		tmp = z * 5.0;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1e-20:
		tmp = x * y
	elif x <= 1.05e-10:
		tmp = z * 5.0
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1e-20)
		tmp = Float64(x * y);
	elseif (x <= 1.05e-10)
		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 <= -1e-20)
		tmp = x * y;
	elseif (x <= 1.05e-10)
		tmp = z * 5.0;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1e-20], N[(x * y), $MachinePrecision], If[LessEqual[x, 1.05e-10], N[(z * 5.0), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.99999999999999945e-21 or 1.05e-10 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 57.9%

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

   if -9.99999999999999945e-21 < x < 1.05e-10

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 79.4%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-20}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-10}:\\ \;\;\;\;z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 8: 36.6% 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 40.9%

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

3. Final simplification 40.9%

   \[\leadsto z \cdot 5 \]

Developer target: 97.5% 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 2023257 
(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)))