Optimisation.CirclePacking:place from circle-packing-0.1.0.4, G

Percentage Accurate: 100.0% → 100.0%

Time: 5.6s

Alternatives: 9

Speedup: 1.0×

• 20.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x + y\right) \cdot \left(z + 1\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* (+ x y) (+ z 1.0)))

C

double code(double x, double y, double z) {
	return (x + y) * (z + 1.0);
}

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 + 1.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return (x + y) * (z + 1.0);
}

Python

def code(x, y, z):
	return (x + y) * (z + 1.0)

Julia

function code(x, y, z)
	return Float64(Float64(x + y) * Float64(z + 1.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x + y) * (z + 1.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(x + y), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x + y\right) \cdot \left(z + 1\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* (+ x y) (+ z 1.0)))

C

double code(double x, double y, double z) {
	return (x + y) * (z + 1.0);
}

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 + 1.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return (x + y) * (z + 1.0);
}

Python

def code(x, y, z):
	return (x + y) * (z + 1.0)

Julia

function code(x, y, z)
	return Float64(Float64(x + y) * Float64(z + 1.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x + y) * (z + 1.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(x + y), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x + y\right) \cdot \left(z + 1\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* (+ x y) (+ z 1.0)))

C

double code(double x, double y, double z) {
	return (x + y) * (z + 1.0);
}

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 + 1.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return (x + y) * (z + 1.0);
}

Python

def code(x, y, z):
	return (x + y) * (z + 1.0)

Julia

function code(x, y, z)
	return Float64(Float64(x + y) * Float64(z + 1.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x + y) * (z + 1.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(x + y), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(x + y\right) \cdot \left(z + 1\right) \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto \left(x + y\right) \cdot \left(z + 1\right) \]
4. Add Preprocessing

Alternative 2: 50.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-124}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-75}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 10000:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+77} \lor \neg \left(z \leq 1.03 \cdot 10^{+130}\right):\\ \;\;\;\;x \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -6.0)
   (* x z)
   (if (<= z 1.8e-124)
     x
     (if (<= z 2e-75)
       y
       (if (<= z 10000.0)
         x
         (if (or (<= z 4e+77) (not (<= z 1.03e+130))) (* x z) (* y z)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.0) {
		tmp = x * z;
	} else if (z <= 1.8e-124) {
		tmp = x;
	} else if (z <= 2e-75) {
		tmp = y;
	} else if (z <= 10000.0) {
		tmp = x;
	} else if ((z <= 4e+77) || !(z <= 1.03e+130)) {
		tmp = x * z;
	} else {
		tmp = y * 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 (z <= (-6.0d0)) then
        tmp = x * z
    else if (z <= 1.8d-124) then
        tmp = x
    else if (z <= 2d-75) then
        tmp = y
    else if (z <= 10000.0d0) then
        tmp = x
    else if ((z <= 4d+77) .or. (.not. (z <= 1.03d+130))) then
        tmp = x * z
    else
        tmp = y * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.0) {
		tmp = x * z;
	} else if (z <= 1.8e-124) {
		tmp = x;
	} else if (z <= 2e-75) {
		tmp = y;
	} else if (z <= 10000.0) {
		tmp = x;
	} else if ((z <= 4e+77) || !(z <= 1.03e+130)) {
		tmp = x * z;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -6.0:
		tmp = x * z
	elif z <= 1.8e-124:
		tmp = x
	elif z <= 2e-75:
		tmp = y
	elif z <= 10000.0:
		tmp = x
	elif (z <= 4e+77) or not (z <= 1.03e+130):
		tmp = x * z
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -6.0)
		tmp = Float64(x * z);
	elseif (z <= 1.8e-124)
		tmp = x;
	elseif (z <= 2e-75)
		tmp = y;
	elseif (z <= 10000.0)
		tmp = x;
	elseif ((z <= 4e+77) || !(z <= 1.03e+130))
		tmp = Float64(x * z);
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -6.0)
		tmp = x * z;
	elseif (z <= 1.8e-124)
		tmp = x;
	elseif (z <= 2e-75)
		tmp = y;
	elseif (z <= 10000.0)
		tmp = x;
	elseif ((z <= 4e+77) || ~((z <= 1.03e+130)))
		tmp = x * z;
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -6.0], N[(x * z), $MachinePrecision], If[LessEqual[z, 1.8e-124], x, If[LessEqual[z, 2e-75], y, If[LessEqual[z, 10000.0], x, If[Or[LessEqual[z, 4e+77], N[Not[LessEqual[z, 1.03e+130]], $MachinePrecision]], N[(x * z), $MachinePrecision], N[(y * z), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -6 or 1e4 < z < 3.99999999999999993e77 or 1.03e130 < z

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 57.7%

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

      1. +-commutative 57.7%

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

      2. distribute-lft-in 57.7%

         \[\leadsto \color{blue}{x \cdot z + x \cdot 1} \]

      3. *-rgt-identity 57.7%

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

   5. Applied egg-rr 57.7%

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

   6. Taylor expanded in z around inf 57.2%

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

   if -6 < z < 1.80000000000000005e-124 or 1.9999999999999999e-75 < z < 1e4

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 54.1%

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

   4. Taylor expanded in z around 0 51.7%

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

   if 1.80000000000000005e-124 < z < 1.9999999999999999e-75

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 72.0%

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

   4. Taylor expanded in z around 0 72.0%

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

   if 3.99999999999999993e77 < z < 1.03e130

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 69.4%

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

      1. +-commutative 69.4%

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

      2. distribute-lft-in 69.4%

         \[\leadsto \color{blue}{y \cdot z + y \cdot 1} \]

      3. *-rgt-identity 69.4%

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

   5. Applied egg-rr 69.4%

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

   6. Taylor expanded in z around inf 69.4%

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

      1. *-commutative 69.4%

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

   8. Simplified 69.4%

      \[\leadsto \color{blue}{z \cdot y} \]
3. Recombined 4 regimes into one program.

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-124}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-75}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 10000:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+77} \lor \neg \left(z \leq 1.03 \cdot 10^{+130}\right):\\ \;\;\;\;x \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(z + 1\right)\\ \mathbf{if}\;z \leq -1.45 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-10}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+77} \lor \neg \left(z \leq 5.8 \cdot 10^{+126}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (+ z 1.0))))
   (if (<= z -1.45e-9)
     t_0
     (if (<= z 8.5e-10)
       (+ x y)
       (if (or (<= z 4.1e+77) (not (<= z 5.8e+126))) t_0 (* y z))))))

C

double code(double x, double y, double z) {
	double t_0 = x * (z + 1.0);
	double tmp;
	if (z <= -1.45e-9) {
		tmp = t_0;
	} else if (z <= 8.5e-10) {
		tmp = x + y;
	} else if ((z <= 4.1e+77) || !(z <= 5.8e+126)) {
		tmp = t_0;
	} else {
		tmp = y * 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) :: t_0
    real(8) :: tmp
    t_0 = x * (z + 1.0d0)
    if (z <= (-1.45d-9)) then
        tmp = t_0
    else if (z <= 8.5d-10) then
        tmp = x + y
    else if ((z <= 4.1d+77) .or. (.not. (z <= 5.8d+126))) then
        tmp = t_0
    else
        tmp = y * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (z + 1.0);
	double tmp;
	if (z <= -1.45e-9) {
		tmp = t_0;
	} else if (z <= 8.5e-10) {
		tmp = x + y;
	} else if ((z <= 4.1e+77) || !(z <= 5.8e+126)) {
		tmp = t_0;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (z + 1.0)
	tmp = 0
	if z <= -1.45e-9:
		tmp = t_0
	elif z <= 8.5e-10:
		tmp = x + y
	elif (z <= 4.1e+77) or not (z <= 5.8e+126):
		tmp = t_0
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(z + 1.0))
	tmp = 0.0
	if (z <= -1.45e-9)
		tmp = t_0;
	elseif (z <= 8.5e-10)
		tmp = Float64(x + y);
	elseif ((z <= 4.1e+77) || !(z <= 5.8e+126))
		tmp = t_0;
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (z + 1.0);
	tmp = 0.0;
	if (z <= -1.45e-9)
		tmp = t_0;
	elseif (z <= 8.5e-10)
		tmp = x + y;
	elseif ((z <= 4.1e+77) || ~((z <= 5.8e+126)))
		tmp = t_0;
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.45e-9], t$95$0, If[LessEqual[z, 8.5e-10], N[(x + y), $MachinePrecision], If[Or[LessEqual[z, 4.1e+77], N[Not[LessEqual[z, 5.8e+126]], $MachinePrecision]], t$95$0, N[(y * z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.44999999999999996e-9 or 8.4999999999999996e-10 < z < 4.1000000000000001e77 or 5.79999999999999971e126 < z

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 57.4%

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

   if -1.44999999999999996e-9 < z < 8.4999999999999996e-10

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 99.6%

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

      1. +-commutative 99.6%

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

   5. Simplified 99.6%

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

   if 4.1000000000000001e77 < z < 5.79999999999999971e126

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 69.4%

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

      1. +-commutative 69.4%

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

      2. distribute-lft-in 69.4%

         \[\leadsto \color{blue}{y \cdot z + y \cdot 1} \]

      3. *-rgt-identity 69.4%

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

   5. Applied egg-rr 69.4%

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

   6. Taylor expanded in z around inf 69.4%

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

      1. *-commutative 69.4%

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

   8. Simplified 69.4%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-10}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+77} \lor \neg \left(z \leq 5.8 \cdot 10^{+126}\right):\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq 10000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 4.35 \cdot 10^{+77} \lor \neg \left(z \leq 8.5 \cdot 10^{+133}\right):\\ \;\;\;\;x \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0)
   (* x z)
   (if (<= z 10000.0)
     (+ x y)
     (if (or (<= z 4.35e+77) (not (<= z 8.5e+133))) (* x z) (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = x * z;
	} else if (z <= 10000.0) {
		tmp = x + y;
	} else if ((z <= 4.35e+77) || !(z <= 8.5e+133)) {
		tmp = x * z;
	} else {
		tmp = y * 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 (z <= (-1.0d0)) then
        tmp = x * z
    else if (z <= 10000.0d0) then
        tmp = x + y
    else if ((z <= 4.35d+77) .or. (.not. (z <= 8.5d+133))) then
        tmp = x * z
    else
        tmp = y * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = x * z;
	} else if (z <= 10000.0) {
		tmp = x + y;
	} else if ((z <= 4.35e+77) || !(z <= 8.5e+133)) {
		tmp = x * z;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = x * z
	elif z <= 10000.0:
		tmp = x + y
	elif (z <= 4.35e+77) or not (z <= 8.5e+133):
		tmp = x * z
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(x * z);
	elseif (z <= 10000.0)
		tmp = Float64(x + y);
	elseif ((z <= 4.35e+77) || !(z <= 8.5e+133))
		tmp = Float64(x * z);
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = x * z;
	elseif (z <= 10000.0)
		tmp = x + y;
	elseif ((z <= 4.35e+77) || ~((z <= 8.5e+133)))
		tmp = x * z;
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.0], N[(x * z), $MachinePrecision], If[LessEqual[z, 10000.0], N[(x + y), $MachinePrecision], If[Or[LessEqual[z, 4.35e+77], N[Not[LessEqual[z, 8.5e+133]], $MachinePrecision]], N[(x * z), $MachinePrecision], N[(y * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1 or 1e4 < z < 4.3500000000000003e77 or 8.50000000000000044e133 < z

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 58.5%

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

      1. +-commutative 58.5%

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

      2. distribute-lft-in 58.5%

         \[\leadsto \color{blue}{x \cdot z + x \cdot 1} \]

      3. *-rgt-identity 58.5%

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

   5. Applied egg-rr 58.5%

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

   6. Taylor expanded in z around inf 56.5%

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

   if -1 < z < 1e4

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 97.8%

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

      1. +-commutative 97.8%

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

   5. Simplified 97.8%

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

   if 4.3500000000000003e77 < z < 8.50000000000000044e133

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 69.4%

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

      1. +-commutative 69.4%

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

      2. distribute-lft-in 69.4%

         \[\leadsto \color{blue}{y \cdot z + y \cdot 1} \]

      3. *-rgt-identity 69.4%

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

   5. Applied egg-rr 69.4%

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

   6. Taylor expanded in z around inf 69.4%

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

      1. *-commutative 69.4%

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

   8. Simplified 69.4%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq 10000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 4.35 \cdot 10^{+77} \lor \neg \left(z \leq 8.5 \cdot 10^{+133}\right):\\ \;\;\;\;x \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{-124}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-72}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 10000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -6.0)
   (* x z)
   (if (<= z 2.25e-124)
     x
     (if (<= z 3.2e-72) y (if (<= z 10000.0) x (* x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.0) {
		tmp = x * z;
	} else if (z <= 2.25e-124) {
		tmp = x;
	} else if (z <= 3.2e-72) {
		tmp = y;
	} else if (z <= 10000.0) {
		tmp = x;
	} 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 (z <= (-6.0d0)) then
        tmp = x * z
    else if (z <= 2.25d-124) then
        tmp = x
    else if (z <= 3.2d-72) then
        tmp = y
    else if (z <= 10000.0d0) then
        tmp = x
    else
        tmp = x * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.0) {
		tmp = x * z;
	} else if (z <= 2.25e-124) {
		tmp = x;
	} else if (z <= 3.2e-72) {
		tmp = y;
	} else if (z <= 10000.0) {
		tmp = x;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -6.0:
		tmp = x * z
	elif z <= 2.25e-124:
		tmp = x
	elif z <= 3.2e-72:
		tmp = y
	elif z <= 10000.0:
		tmp = x
	else:
		tmp = x * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -6.0)
		tmp = Float64(x * z);
	elseif (z <= 2.25e-124)
		tmp = x;
	elseif (z <= 3.2e-72)
		tmp = y;
	elseif (z <= 10000.0)
		tmp = x;
	else
		tmp = Float64(x * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -6.0)
		tmp = x * z;
	elseif (z <= 2.25e-124)
		tmp = x;
	elseif (z <= 3.2e-72)
		tmp = y;
	elseif (z <= 10000.0)
		tmp = x;
	else
		tmp = x * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -6.0], N[(x * z), $MachinePrecision], If[LessEqual[z, 2.25e-124], x, If[LessEqual[z, 3.2e-72], y, If[LessEqual[z, 10000.0], x, N[(x * z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6 or 1e4 < z

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 54.8%

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

      1. +-commutative 54.8%

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

      2. distribute-lft-in 54.8%

         \[\leadsto \color{blue}{x \cdot z + x \cdot 1} \]

      3. *-rgt-identity 54.8%

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

   5. Applied egg-rr 54.8%

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

   6. Taylor expanded in z around inf 54.4%

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

   if -6 < z < 2.2499999999999998e-124 or 3.19999999999999999e-72 < z < 1e4

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 54.1%

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

   4. Taylor expanded in z around 0 51.7%

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

   if 2.2499999999999998e-124 < z < 3.19999999999999999e-72

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 72.0%

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

   4. Taylor expanded in z around 0 72.0%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{-124}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-72}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 10000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;z \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 1.0))) (* z (+ x y)) (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.0)) {
		tmp = z * (x + y);
	} 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 <= (-1.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = z * (x + y)
    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 <= -1.0) || !(z <= 1.0)) {
		tmp = z * (x + y);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 1.0):
		tmp = z * (x + y)
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 1.0))
		tmp = Float64(z * Float64(x + y));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 1.0)))
		tmp = z * (x + y);
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 97.0%

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

      1. +-commutative 97.0%

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

   5. Simplified 97.0%

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

   if -1 < z < 1

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 98.5%

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

      1. +-commutative 98.5%

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

   5. Simplified 98.5%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;z \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{-63}:\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 8.2e-63) (* x (+ z 1.0)) (* y (+ z 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.2e-63) {
		tmp = x * (z + 1.0);
	} else {
		tmp = y * (z + 1.0);
	}
	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 (y <= 8.2d-63) then
        tmp = x * (z + 1.0d0)
    else
        tmp = y * (z + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.2e-63) {
		tmp = x * (z + 1.0);
	} else {
		tmp = y * (z + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 8.2e-63:
		tmp = x * (z + 1.0)
	else:
		tmp = y * (z + 1.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 8.2e-63)
		tmp = Float64(x * Float64(z + 1.0));
	else
		tmp = Float64(y * Float64(z + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 8.2e-63)
		tmp = x * (z + 1.0);
	else
		tmp = y * (z + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 8.2e-63], N[(x * N[(z + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.1999999999999995e-63

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 63.0%

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

   if 8.1999999999999995e-63 < y

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 76.7%

      \[\leadsto \color{blue}{y \cdot \left(1 + z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{-63}:\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 32.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4500000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= y 4500000.0) x y))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4500000.0) {
		tmp = x;
	} else {
		tmp = 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 <= 4500000.0d0) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4500000.0) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4500000.0:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4500000.0)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4500000.0)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4500000.0], x, y]

Derivation

1. Split input into 2 regimes

2. if y < 4.5e6

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 62.4%

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

   4. Taylor expanded in z around 0 30.4%

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

   if 4.5e6 < y

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 82.1%

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

   4. Taylor expanded in z around 0 36.7%

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

4. Final simplification 31.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4500000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 26.4% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 100.0%

   \[\left(x + y\right) \cdot \left(z + 1\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf 53.8%

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

4. Taylor expanded in z around 0 26.9%

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

5. Final simplification 26.9%

   \[\leadsto x \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024018 
(FPCore (x y z)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, G"
  :precision binary64
  (* (+ x y) (+ z 1.0)))