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

Percentage Accurate: 100.0% → 100.0%

Time: 4.5s

Alternatives: 10

Speedup: 1.0×

• 20.9% 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, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(x + y\right) \cdot \left(z + 1\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. +-commutative 100.0%

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

   2. distribute-lft-in 100.0%

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

   3. *-rgt-identity 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 51.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq -1.16 \cdot 10^{-58}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.22 \cdot 10^{-214}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-82}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.36 \cdot 10^{-8}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+157}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0)
   (* x z)
   (if (<= z -1.16e-58)
     x
     (if (<= z -1.22e-214)
       y
       (if (<= z 8.5e-82)
         x
         (if (<= z 1.36e-8) y (if (<= z 9e+157) (* y z) (* x z))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = x * z;
	} else if (z <= -1.16e-58) {
		tmp = x;
	} else if (z <= -1.22e-214) {
		tmp = y;
	} else if (z <= 8.5e-82) {
		tmp = x;
	} else if (z <= 1.36e-8) {
		tmp = y;
	} else if (z <= 9e+157) {
		tmp = y * z;
	} 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 <= (-1.0d0)) then
        tmp = x * z
    else if (z <= (-1.16d-58)) then
        tmp = x
    else if (z <= (-1.22d-214)) then
        tmp = y
    else if (z <= 8.5d-82) then
        tmp = x
    else if (z <= 1.36d-8) then
        tmp = y
    else if (z <= 9d+157) then
        tmp = y * z
    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 <= -1.0) {
		tmp = x * z;
	} else if (z <= -1.16e-58) {
		tmp = x;
	} else if (z <= -1.22e-214) {
		tmp = y;
	} else if (z <= 8.5e-82) {
		tmp = x;
	} else if (z <= 1.36e-8) {
		tmp = y;
	} else if (z <= 9e+157) {
		tmp = y * z;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = x * z
	elif z <= -1.16e-58:
		tmp = x
	elif z <= -1.22e-214:
		tmp = y
	elif z <= 8.5e-82:
		tmp = x
	elif z <= 1.36e-8:
		tmp = y
	elif z <= 9e+157:
		tmp = y * z
	else:
		tmp = x * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(x * z);
	elseif (z <= -1.16e-58)
		tmp = x;
	elseif (z <= -1.22e-214)
		tmp = y;
	elseif (z <= 8.5e-82)
		tmp = x;
	elseif (z <= 1.36e-8)
		tmp = y;
	elseif (z <= 9e+157)
		tmp = Float64(y * z);
	else
		tmp = Float64(x * 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 <= -1.16e-58)
		tmp = x;
	elseif (z <= -1.22e-214)
		tmp = y;
	elseif (z <= 8.5e-82)
		tmp = x;
	elseif (z <= 1.36e-8)
		tmp = y;
	elseif (z <= 9e+157)
		tmp = y * z;
	else
		tmp = x * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.0], N[(x * z), $MachinePrecision], If[LessEqual[z, -1.16e-58], x, If[LessEqual[z, -1.22e-214], y, If[LessEqual[z, 8.5e-82], x, If[LessEqual[z, 1.36e-8], y, If[LessEqual[z, 9e+157], N[(y * z), $MachinePrecision], N[(x * z), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1 or 8.9999999999999997e157 < 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 44.5%

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

      1. +-commutative 44.5%

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

      2. distribute-lft-in 44.5%

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

      3. *-rgt-identity 44.5%

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

   5. Applied egg-rr 44.5%

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

   6. Taylor expanded in z around inf 44.3%

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

   if -1 < z < -1.16000000000000007e-58 or -1.22000000000000004e-214 < z < 8.4999999999999997e-82

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 52.0%

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

   4. Taylor expanded in z around 0 49.9%

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

   if -1.16000000000000007e-58 < z < -1.22000000000000004e-214 or 8.4999999999999997e-82 < z < 1.3599999999999999e-8

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 52.5%

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

   4. Taylor expanded in z around 0 51.5%

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

   if 1.3599999999999999e-8 < z < 8.9999999999999997e157

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 58.4%

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

      1. +-commutative 58.4%

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

      2. distribute-lft-in 58.4%

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

      3. *-rgt-identity 58.4%

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

   5. Applied egg-rr 58.4%

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

   6. Taylor expanded in z around inf 46.8%

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

      1. *-commutative 46.8%

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

   8. Simplified 46.8%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq -1.16 \cdot 10^{-58}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.22 \cdot 10^{-214}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-82}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.36 \cdot 10^{-8}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+157}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq -1.35 \cdot 10^{-58}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -5.4 \cdot 10^{-215}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-82}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.36 \cdot 10^{-8}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0)
   (* x z)
   (if (<= z -1.35e-58)
     x
     (if (<= z -5.4e-215)
       y
       (if (<= z 3.8e-82) x (if (<= z 1.36e-8) y (* x z)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = x * z;
	} else if (z <= -1.35e-58) {
		tmp = x;
	} else if (z <= -5.4e-215) {
		tmp = y;
	} else if (z <= 3.8e-82) {
		tmp = x;
	} else if (z <= 1.36e-8) {
		tmp = y;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.0d0)) then
        tmp = x * z
    else if (z <= (-1.35d-58)) then
        tmp = x
    else if (z <= (-5.4d-215)) then
        tmp = y
    else if (z <= 3.8d-82) then
        tmp = x
    else if (z <= 1.36d-8) then
        tmp = y
    else
        tmp = x * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = x * z;
	} else if (z <= -1.35e-58) {
		tmp = x;
	} else if (z <= -5.4e-215) {
		tmp = y;
	} else if (z <= 3.8e-82) {
		tmp = x;
	} else if (z <= 1.36e-8) {
		tmp = y;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = x * z
	elif z <= -1.35e-58:
		tmp = x
	elif z <= -5.4e-215:
		tmp = y
	elif z <= 3.8e-82:
		tmp = x
	elif z <= 1.36e-8:
		tmp = y
	else:
		tmp = x * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(x * z);
	elseif (z <= -1.35e-58)
		tmp = x;
	elseif (z <= -5.4e-215)
		tmp = y;
	elseif (z <= 3.8e-82)
		tmp = x;
	elseif (z <= 1.36e-8)
		tmp = y;
	else
		tmp = Float64(x * 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 <= -1.35e-58)
		tmp = x;
	elseif (z <= -5.4e-215)
		tmp = y;
	elseif (z <= 3.8e-82)
		tmp = x;
	elseif (z <= 1.36e-8)
		tmp = y;
	else
		tmp = x * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.0], N[(x * z), $MachinePrecision], If[LessEqual[z, -1.35e-58], x, If[LessEqual[z, -5.4e-215], y, If[LessEqual[z, 3.8e-82], x, If[LessEqual[z, 1.36e-8], y, N[(x * z), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1 or 1.3599999999999999e-8 < 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 44.3%

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

      1. +-commutative 44.3%

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

      2. distribute-lft-in 44.3%

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

      3. *-rgt-identity 44.3%

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

   5. Applied egg-rr 44.3%

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

   6. Taylor expanded in z around inf 42.4%

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

   if -1 < z < -1.3499999999999999e-58 or -5.40000000000000035e-215 < z < 3.8000000000000002e-82

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 52.0%

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

   4. Taylor expanded in z around 0 49.9%

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

   if -1.3499999999999999e-58 < z < -5.40000000000000035e-215 or 3.8000000000000002e-82 < z < 1.3599999999999999e-8

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 52.5%

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

   4. Taylor expanded in z around 0 51.5%

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

Alternative 4: 75.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(z + 1\right)\\ \mathbf{if}\;z \leq -1.5 \cdot 10^{-13}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{-10}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+45} \lor \neg \left(z \leq 5.3 \cdot 10^{+157}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (+ z 1.0))))
   (if (<= z -1.5e-13)
     t_0
     (if (<= z 8.8e-10)
       (+ x y)
       (if (or (<= z 1.2e+45) (not (<= z 5.3e+157))) t_0 (* y z))))))

C

double code(double x, double y, double z) {
	double t_0 = x * (z + 1.0);
	double tmp;
	if (z <= -1.5e-13) {
		tmp = t_0;
	} else if (z <= 8.8e-10) {
		tmp = x + y;
	} else if ((z <= 1.2e+45) || !(z <= 5.3e+157)) {
		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 <= (-1.5d-13)) then
        tmp = t_0
    else if (z <= 8.8d-10) then
        tmp = x + y
    else if ((z <= 1.2d+45) .or. (.not. (z <= 5.3d+157))) 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 <= -1.5e-13) {
		tmp = t_0;
	} else if (z <= 8.8e-10) {
		tmp = x + y;
	} else if ((z <= 1.2e+45) || !(z <= 5.3e+157)) {
		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 <= -1.5e-13:
		tmp = t_0
	elif z <= 8.8e-10:
		tmp = x + y
	elif (z <= 1.2e+45) or not (z <= 5.3e+157):
		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 <= -1.5e-13)
		tmp = t_0;
	elseif (z <= 8.8e-10)
		tmp = Float64(x + y);
	elseif ((z <= 1.2e+45) || !(z <= 5.3e+157))
		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 <= -1.5e-13)
		tmp = t_0;
	elseif (z <= 8.8e-10)
		tmp = x + y;
	elseif ((z <= 1.2e+45) || ~((z <= 5.3e+157)))
		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, -1.5e-13], t$95$0, If[LessEqual[z, 8.8e-10], N[(x + y), $MachinePrecision], If[Or[LessEqual[z, 1.2e+45], N[Not[LessEqual[z, 5.3e+157]], $MachinePrecision]], t$95$0, N[(y * z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.49999999999999992e-13 or 8.7999999999999996e-10 < z < 1.19999999999999995e45 or 5.2999999999999998e157 < 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 45.3%

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

   if -1.49999999999999992e-13 < z < 8.7999999999999996e-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 99.3%

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

      1. +-commutative 99.3%

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

   5. Simplified 99.3%

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

   if 1.19999999999999995e45 < z < 5.2999999999999998e157

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 60.8%

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

      1. +-commutative 60.8%

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

      2. distribute-lft-in 60.8%

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

      3. *-rgt-identity 60.8%

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

   5. Applied egg-rr 60.8%

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

   6. Taylor expanded in z around inf 60.8%

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

      1. *-commutative 60.8%

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

   8. Simplified 60.8%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-13}:\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{-10}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+45} \lor \neg \left(z \leq 5.3 \cdot 10^{+157}\right):\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(x + y\right)\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-9}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 32000000:\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (+ x y))))
   (if (<= z -1.0)
     t_0
     (if (<= z 2.3e-9) (+ x y) (if (<= z 32000000.0) (* x (+ z 1.0)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = z * (x + y);
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 2.3e-9) {
		tmp = x + y;
	} else if (z <= 32000000.0) {
		tmp = x * (z + 1.0);
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = z * (x + y)
    if (z <= (-1.0d0)) then
        tmp = t_0
    else if (z <= 2.3d-9) then
        tmp = x + y
    else if (z <= 32000000.0d0) then
        tmp = x * (z + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (x + y);
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 2.3e-9) {
		tmp = x + y;
	} else if (z <= 32000000.0) {
		tmp = x * (z + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (x + y)
	tmp = 0
	if z <= -1.0:
		tmp = t_0
	elif z <= 2.3e-9:
		tmp = x + y
	elif z <= 32000000.0:
		tmp = x * (z + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(x + y))
	tmp = 0.0
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 2.3e-9)
		tmp = Float64(x + y);
	elseif (z <= 32000000.0)
		tmp = Float64(x * Float64(z + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (x + y);
	tmp = 0.0;
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 2.3e-9)
		tmp = x + y;
	elseif (z <= 32000000.0)
		tmp = x * (z + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.0], t$95$0, If[LessEqual[z, 2.3e-9], N[(x + y), $MachinePrecision], If[LessEqual[z, 32000000.0], N[(x * N[(z + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1 or 3.2e7 < 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 98.7%

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

      1. +-commutative 98.7%

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

   5. Simplified 98.7%

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

   if -1 < z < 2.2999999999999999e-9

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 98.0%

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

      1. +-commutative 98.0%

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

   5. Simplified 98.0%

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

   if 2.2999999999999999e-9 < z < 3.2e7

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 43.2%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;z \cdot \left(x + y\right)\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-9}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 32000000:\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq 4500000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{+163}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0)
   (* x z)
   (if (<= z 4500000.0) (+ x y) (if (<= z 8.6e+163) (* y z) (* x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = x * z;
	} else if (z <= 4500000.0) {
		tmp = x + y;
	} else if (z <= 8.6e+163) {
		tmp = y * z;
	} 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 <= (-1.0d0)) then
        tmp = x * z
    else if (z <= 4500000.0d0) then
        tmp = x + y
    else if (z <= 8.6d+163) then
        tmp = y * z
    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 <= -1.0) {
		tmp = x * z;
	} else if (z <= 4500000.0) {
		tmp = x + y;
	} else if (z <= 8.6e+163) {
		tmp = y * z;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = x * z
	elif z <= 4500000.0:
		tmp = x + y
	elif z <= 8.6e+163:
		tmp = y * z
	else:
		tmp = x * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(x * z);
	elseif (z <= 4500000.0)
		tmp = Float64(x + y);
	elseif (z <= 8.6e+163)
		tmp = Float64(y * z);
	else
		tmp = Float64(x * 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 <= 4500000.0)
		tmp = x + y;
	elseif (z <= 8.6e+163)
		tmp = y * z;
	else
		tmp = x * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.0], N[(x * z), $MachinePrecision], If[LessEqual[z, 4500000.0], N[(x + y), $MachinePrecision], If[LessEqual[z, 8.6e+163], N[(y * z), $MachinePrecision], N[(x * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1 or 8.6000000000000004e163 < 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 44.5%

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

      1. +-commutative 44.5%

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

      2. distribute-lft-in 44.5%

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

      3. *-rgt-identity 44.5%

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

   5. Applied egg-rr 44.5%

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

   6. Taylor expanded in z around inf 44.3%

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

   if -1 < z < 4.5e6

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 94.3%

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

      1. +-commutative 94.3%

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

   5. Simplified 94.3%

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

   if 4.5e6 < z < 8.6000000000000004e163

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 58.7%

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

      1. +-commutative 58.7%

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

      2. distribute-lft-in 58.7%

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

      3. *-rgt-identity 58.7%

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

   5. Applied egg-rr 58.7%

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

   6. Taylor expanded in z around inf 57.1%

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

      1. *-commutative 57.1%

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

   8. Simplified 57.1%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq 4500000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{+163}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-179}:\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.2e-179) (* x (+ z 1.0)) (* y (+ z 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.2e-179) {
		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 (x <= (-5.2d-179)) 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 (x <= -5.2e-179) {
		tmp = x * (z + 1.0);
	} else {
		tmp = y * (z + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5.2e-179:
		tmp = x * (z + 1.0)
	else:
		tmp = y * (z + 1.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.2e-179)
		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 (x <= -5.2e-179)
		tmp = x * (z + 1.0);
	else
		tmp = y * (z + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.2e-179], 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 x < -5.20000000000000011e-179

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 63.3%

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

   if -5.20000000000000011e-179 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 66.7%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-179}:\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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. Add Preprocessing

Alternative 9: 32.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-63}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= y 1.1e-63) x y))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.1e-63) {
		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 <= 1.1d-63) 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 <= 1.1e-63) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.1e-63:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.1e-63)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.1e-63)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.1e-63], x, y]

Derivation

1. Split input into 2 regimes

2. if y < 1.1e-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 53.4%

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

   4. Taylor expanded in z around 0 28.8%

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

   if 1.1e-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 72.2%

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

   4. Taylor expanded in z around 0 36.4%

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

Alternative 10: 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 47.0%

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

4. Taylor expanded in z around 0 24.4%

   \[\leadsto \color{blue}{x} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024091 
(FPCore (x y z)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, G"
  :precision binary64
  (* (+ x y) (+ z 1.0)))