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

Percentage Accurate: 99.9% → 100.0%

Time: 6.5s

Alternatives: 8

Speedup: 1.0×

• 21.2% 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. Add Preprocessing
3. Step-by-step derivation

   1. +-commutative 99.9%

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

   2. fma-define 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 62.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-10}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-10}:\\ \;\;\;\;z \cdot 5\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+20} \lor \neg \left(x \leq 1.22 \cdot 10^{+61}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.9e-10)
   (* x y)
   (if (<= x 1.7e-10)
     (* z 5.0)
     (if (or (<= x 3e+20) (not (<= x 1.22e+61))) (* x y) (* z x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.9e-10) {
		tmp = x * y;
	} else if (x <= 1.7e-10) {
		tmp = z * 5.0;
	} else if ((x <= 3e+20) || !(x <= 1.22e+61)) {
		tmp = x * y;
	} else {
		tmp = z * 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 <= (-3.9d-10)) then
        tmp = x * y
    else if (x <= 1.7d-10) then
        tmp = z * 5.0d0
    else if ((x <= 3d+20) .or. (.not. (x <= 1.22d+61))) then
        tmp = x * y
    else
        tmp = z * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.9e-10) {
		tmp = x * y;
	} else if (x <= 1.7e-10) {
		tmp = z * 5.0;
	} else if ((x <= 3e+20) || !(x <= 1.22e+61)) {
		tmp = x * y;
	} else {
		tmp = z * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.9e-10:
		tmp = x * y
	elif x <= 1.7e-10:
		tmp = z * 5.0
	elif (x <= 3e+20) or not (x <= 1.22e+61):
		tmp = x * y
	else:
		tmp = z * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.9e-10)
		tmp = Float64(x * y);
	elseif (x <= 1.7e-10)
		tmp = Float64(z * 5.0);
	elseif ((x <= 3e+20) || !(x <= 1.22e+61))
		tmp = Float64(x * y);
	else
		tmp = Float64(z * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.9e-10)
		tmp = x * y;
	elseif (x <= 1.7e-10)
		tmp = z * 5.0;
	elseif ((x <= 3e+20) || ~((x <= 1.22e+61)))
		tmp = x * y;
	else
		tmp = z * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.9e-10], N[(x * y), $MachinePrecision], If[LessEqual[x, 1.7e-10], N[(z * 5.0), $MachinePrecision], If[Or[LessEqual[x, 3e+20], N[Not[LessEqual[x, 1.22e+61]], $MachinePrecision]], N[(x * y), $MachinePrecision], N[(z * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.9e-10 or 1.70000000000000007e-10 < x < 3e20 or 1.22e61 < 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 85.9%

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

      1. +-commutative 85.9%

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

      2. associate-/l* 88.4%

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

      3. distribute-rgt-out 91.9%

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

   5. Simplified 91.9%

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

   6. Taylor expanded in z around 0 62.2%

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

   if -3.9e-10 < x < 1.70000000000000007e-10

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 73.8%

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

      1. *-commutative 73.8%

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

   7. Simplified 73.8%

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

   if 3e20 < x < 1.22e61

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. associate-+l+ 100.0%

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

      3. *-commutative 100.0%

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

      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. Taylor expanded in x around inf 100.0%

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

   6. Taylor expanded in y around 0 92.6%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-10}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-10}:\\ \;\;\;\;z \cdot 5\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+20} \lor \neg \left(x \leq 1.22 \cdot 10^{+61}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \lor \neg \left(x \leq 0.0045\right):\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot 5 + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -5.0) (not (<= x 0.0045))) (* x (+ z y)) (+ (* z 5.0) (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.0) || !(x <= 0.0045)) {
		tmp = x * (z + y);
	} else {
		tmp = (z * 5.0) + (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 <= (-5.0d0)) .or. (.not. (x <= 0.0045d0))) then
        tmp = x * (z + y)
    else
        tmp = (z * 5.0d0) + (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.0) || !(x <= 0.0045)) {
		tmp = x * (z + y);
	} else {
		tmp = (z * 5.0) + (x * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -5.0) or not (x <= 0.0045):
		tmp = x * (z + y)
	else:
		tmp = (z * 5.0) + (x * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -5.0) || !(x <= 0.0045))
		tmp = Float64(x * Float64(z + y));
	else
		tmp = Float64(Float64(z * 5.0) + Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -5.0) || ~((x <= 0.0045)))
		tmp = x * (z + y);
	else
		tmp = (z * 5.0) + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -5.0], N[Not[LessEqual[x, 0.0045]], $MachinePrecision]], N[(x * N[(z + y), $MachinePrecision]), $MachinePrecision], N[(N[(z * 5.0), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5 or 0.00449999999999999966 < 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 99.1%

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

   if -5 < x < 0.00449999999999999966

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 99.6%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \lor \neg \left(x \leq 0.0045\right):\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot 5 + x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{+14} \lor \neg \left(x \leq 4 \cdot 10^{-7}\right):\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(5 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.05e+14) (not (<= x 4e-7))) (* x (+ z y)) (* z (+ 5.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.05e+14) || !(x <= 4e-7)) {
		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 <= (-2.05d+14)) .or. (.not. (x <= 4d-7))) 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 <= -2.05e+14) || !(x <= 4e-7)) {
		tmp = x * (z + y);
	} else {
		tmp = z * (5.0 + x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.05e+14) or not (x <= 4e-7):
		tmp = x * (z + y)
	else:
		tmp = z * (5.0 + x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.05e+14) || !(x <= 4e-7))
		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 <= -2.05e+14) || ~((x <= 4e-7)))
		tmp = x * (z + y);
	else
		tmp = z * (5.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.05e+14], N[Not[LessEqual[x, 4e-7]], $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 < -2.05e14 or 3.9999999999999998e-7 < 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 99.7%

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

   if -2.05e14 < x < 3.9999999999999998e-7

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 74.7%

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

      1. +-commutative 74.7%

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

      2. distribute-rgt-in 74.7%

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

   5. Simplified 74.7%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{+14} \lor \neg \left(x \leq 4 \cdot 10^{-7}\right):\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(5 + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-10} \lor \neg \left(x \leq 4.3 \cdot 10^{-15}\right):\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot 5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -8e-10) (not (<= x 4.3e-15))) (* x (+ z y)) (* z 5.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8e-10) || !(x <= 4.3e-15)) {
		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 <= (-8d-10)) .or. (.not. (x <= 4.3d-15))) 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 <= -8e-10) || !(x <= 4.3e-15)) {
		tmp = x * (z + y);
	} else {
		tmp = z * 5.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -8e-10) or not (x <= 4.3e-15):
		tmp = x * (z + y)
	else:
		tmp = z * 5.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -8e-10) || !(x <= 4.3e-15))
		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 <= -8e-10) || ~((x <= 4.3e-15)))
		tmp = x * (z + y);
	else
		tmp = z * 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -8e-10], N[Not[LessEqual[x, 4.3e-15]], $MachinePrecision]], N[(x * N[(z + y), $MachinePrecision]), $MachinePrecision], N[(z * 5.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.00000000000000029e-10 or 4.2999999999999997e-15 < 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 99.1%

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

   if -8.00000000000000029e-10 < x < 4.2999999999999997e-15

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 73.8%

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

      1. *-commutative 73.8%

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

   7. Simplified 73.8%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-10} \lor \neg \left(x \leq 4.3 \cdot 10^{-15}\right):\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot 5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-61} \lor \neg \left(y \leq 3.2 \cdot 10^{-44}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.2e-61) (not (<= y 3.2e-44))) (* x y) (* z x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.2e-61) || !(y <= 3.2e-44)) {
		tmp = x * y;
	} else {
		tmp = z * 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 ((y <= (-2.2d-61)) .or. (.not. (y <= 3.2d-44))) then
        tmp = x * y
    else
        tmp = z * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.2e-61) || !(y <= 3.2e-44)) {
		tmp = x * y;
	} else {
		tmp = z * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.2e-61) or not (y <= 3.2e-44):
		tmp = x * y
	else:
		tmp = z * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.2e-61) || !(y <= 3.2e-44))
		tmp = Float64(x * y);
	else
		tmp = Float64(z * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2.2e-61) || ~((y <= 3.2e-44)))
		tmp = x * y;
	else
		tmp = z * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.2e-61], N[Not[LessEqual[y, 3.2e-44]], $MachinePrecision]], N[(x * y), $MachinePrecision], N[(z * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.20000000000000009e-61 or 3.19999999999999995e-44 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 96.7%

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

      1. +-commutative 96.7%

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

      2. associate-/l* 98.6%

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

      3. distribute-rgt-out 98.6%

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

   5. Simplified 98.6%

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

   6. Taylor expanded in z around 0 64.2%

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

   if -2.20000000000000009e-61 < y < 3.19999999999999995e-44

   1. Initial program 99.9%

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

      1. distribute-rgt-in 99.9%

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

      2. associate-+l+ 99.9%

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

      3. *-commutative 99.9%

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

      4. fma-define 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. Add Preprocessing

   5. Taylor expanded in x around inf 46.6%

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

   6. Taylor expanded in y around 0 38.2%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-61} \lor \neg \left(y \leq 3.2 \cdot 10^{-44}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \]
5. Add Preprocessing

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. Add Preprocessing

3. Final simplification 99.9%

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

Alternative 8: 26.9% accurate, 3.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return 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 = z * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. distribute-rgt-in 97.6%

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

   2. associate-+l+ 97.6%

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

   3. *-commutative 97.6%

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

   4. fma-define 98.8%

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

   5. distribute-lft-out 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in x around inf 62.1%

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

6. Taylor expanded in y around 0 26.3%

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

7. Final simplification 26.3%

   \[\leadsto z \cdot x \]
8. Add Preprocessing

Developer Target 1: 98.0% 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 2024170 
(FPCore (x y z)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, C"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* (+ x 5) z) (* x y)))

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