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

Percentage Accurate: 100.0% → 100.0%

Time: 5.3s

Alternatives: 9

Speedup: 1.0×

• 21.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

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) * (1.0d0 - z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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) * (1.0d0 - z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (1.0 - 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 = (1.0d0 - z) * (x + y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 75.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(-z\right)\\ t_1 := z \cdot \left(-x\right)\\ \mathbf{if}\;z \leq -5 \cdot 10^{+232}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{+148}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -6.3 \cdot 10^{+72}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -29000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+190}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- z))) (t_1 (* z (- x))))
   (if (<= z -5e+232)
     t_0
     (if (<= z -2.2e+148)
       t_1
       (if (<= z -6.3e+72)
         t_0
         (if (<= z -29000000.0)
           t_1
           (if (<= z 1.0) (+ x y) (if (<= z 6.2e+190) t_1 t_0))))))))

C

double code(double x, double y, double z) {
	double t_0 = y * -z;
	double t_1 = z * -x;
	double tmp;
	if (z <= -5e+232) {
		tmp = t_0;
	} else if (z <= -2.2e+148) {
		tmp = t_1;
	} else if (z <= -6.3e+72) {
		tmp = t_0;
	} else if (z <= -29000000.0) {
		tmp = t_1;
	} else if (z <= 1.0) {
		tmp = x + y;
	} else if (z <= 6.2e+190) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = y * -z
    t_1 = z * -x
    if (z <= (-5d+232)) then
        tmp = t_0
    else if (z <= (-2.2d+148)) then
        tmp = t_1
    else if (z <= (-6.3d+72)) then
        tmp = t_0
    else if (z <= (-29000000.0d0)) then
        tmp = t_1
    else if (z <= 1.0d0) then
        tmp = x + y
    else if (z <= 6.2d+190) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * -z;
	double t_1 = z * -x;
	double tmp;
	if (z <= -5e+232) {
		tmp = t_0;
	} else if (z <= -2.2e+148) {
		tmp = t_1;
	} else if (z <= -6.3e+72) {
		tmp = t_0;
	} else if (z <= -29000000.0) {
		tmp = t_1;
	} else if (z <= 1.0) {
		tmp = x + y;
	} else if (z <= 6.2e+190) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * -z
	t_1 = z * -x
	tmp = 0
	if z <= -5e+232:
		tmp = t_0
	elif z <= -2.2e+148:
		tmp = t_1
	elif z <= -6.3e+72:
		tmp = t_0
	elif z <= -29000000.0:
		tmp = t_1
	elif z <= 1.0:
		tmp = x + y
	elif z <= 6.2e+190:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(-z))
	t_1 = Float64(z * Float64(-x))
	tmp = 0.0
	if (z <= -5e+232)
		tmp = t_0;
	elseif (z <= -2.2e+148)
		tmp = t_1;
	elseif (z <= -6.3e+72)
		tmp = t_0;
	elseif (z <= -29000000.0)
		tmp = t_1;
	elseif (z <= 1.0)
		tmp = Float64(x + y);
	elseif (z <= 6.2e+190)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * -z;
	t_1 = z * -x;
	tmp = 0.0;
	if (z <= -5e+232)
		tmp = t_0;
	elseif (z <= -2.2e+148)
		tmp = t_1;
	elseif (z <= -6.3e+72)
		tmp = t_0;
	elseif (z <= -29000000.0)
		tmp = t_1;
	elseif (z <= 1.0)
		tmp = x + y;
	elseif (z <= 6.2e+190)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * (-z)), $MachinePrecision]}, Block[{t$95$1 = N[(z * (-x)), $MachinePrecision]}, If[LessEqual[z, -5e+232], t$95$0, If[LessEqual[z, -2.2e+148], t$95$1, If[LessEqual[z, -6.3e+72], t$95$0, If[LessEqual[z, -29000000.0], t$95$1, If[LessEqual[z, 1.0], N[(x + y), $MachinePrecision], If[LessEqual[z, 6.2e+190], t$95$1, t$95$0]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.99999999999999987e232 or -2.1999999999999999e148 < z < -6.29999999999999963e72 or 6.2000000000000003e190 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 55.8%

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

   4. Taylor expanded in z around inf 55.8%

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

      1. mul-1-neg 55.8%

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

      2. *-commutative 55.8%

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

      3. distribute-rgt-neg-in 55.8%

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

   6. Simplified 55.8%

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

   if -4.99999999999999987e232 < z < -2.1999999999999999e148 or -6.29999999999999963e72 < z < -2.9e7 or 1 < z < 6.2000000000000003e190

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 96.5%

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

      1. associate-*r* 96.5%

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

      2. neg-mul-1 96.5%

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

      3. *-commutative 96.5%

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

      4. +-commutative 96.5%

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

   5. Simplified 96.5%

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

   6. Taylor expanded in y around 0 52.7%

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

      1. associate-*r* 52.7%

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

      2. neg-mul-1 52.7%

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

   8. Simplified 52.7%

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

   if -2.9e7 < z < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 96.4%

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

      1. +-commutative 96.4%

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

   5. Simplified 96.4%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+232}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{+148}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq -6.3 \cdot 10^{+72}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq -29000000:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+190}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - z \leq -5 \cdot 10^{+191}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{elif}\;1 - z \leq -500:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{elif}\;1 - z \leq 2:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (- 1.0 z) -5e+191)
   (* y (- z))
   (if (<= (- 1.0 z) -500.0)
     (* z (- x))
     (if (<= (- 1.0 z) 2.0) (+ x y) (* y (- 1.0 z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((1.0 - z) <= -5e+191) {
		tmp = y * -z;
	} else if ((1.0 - z) <= -500.0) {
		tmp = z * -x;
	} else if ((1.0 - z) <= 2.0) {
		tmp = x + y;
	} else {
		tmp = y * (1.0 - 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 ((1.0d0 - z) <= (-5d+191)) then
        tmp = y * -z
    else if ((1.0d0 - z) <= (-500.0d0)) then
        tmp = z * -x
    else if ((1.0d0 - z) <= 2.0d0) then
        tmp = x + y
    else
        tmp = y * (1.0d0 - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((1.0 - z) <= -5e+191) {
		tmp = y * -z;
	} else if ((1.0 - z) <= -500.0) {
		tmp = z * -x;
	} else if ((1.0 - z) <= 2.0) {
		tmp = x + y;
	} else {
		tmp = y * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (1.0 - z) <= -5e+191:
		tmp = y * -z
	elif (1.0 - z) <= -500.0:
		tmp = z * -x
	elif (1.0 - z) <= 2.0:
		tmp = x + y
	else:
		tmp = y * (1.0 - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(1.0 - z) <= -5e+191)
		tmp = Float64(y * Float64(-z));
	elseif (Float64(1.0 - z) <= -500.0)
		tmp = Float64(z * Float64(-x));
	elseif (Float64(1.0 - z) <= 2.0)
		tmp = Float64(x + y);
	else
		tmp = Float64(y * Float64(1.0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((1.0 - z) <= -5e+191)
		tmp = y * -z;
	elseif ((1.0 - z) <= -500.0)
		tmp = z * -x;
	elseif ((1.0 - z) <= 2.0)
		tmp = x + y;
	else
		tmp = y * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(1.0 - z), $MachinePrecision], -5e+191], N[(y * (-z)), $MachinePrecision], If[LessEqual[N[(1.0 - z), $MachinePrecision], -500.0], N[(z * (-x)), $MachinePrecision], If[LessEqual[N[(1.0 - z), $MachinePrecision], 2.0], N[(x + y), $MachinePrecision], N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 1 z) < -5.0000000000000002e191

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 45.6%

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

   4. Taylor expanded in z around inf 45.6%

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

      1. mul-1-neg 45.6%

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

      2. *-commutative 45.6%

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

      3. distribute-rgt-neg-in 45.6%

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

   6. Simplified 45.6%

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

   if -5.0000000000000002e191 < (-.f64 1 z) < -500

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 93.8%

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

      1. associate-*r* 93.8%

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

      2. neg-mul-1 93.8%

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

      3. *-commutative 93.8%

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

      4. +-commutative 93.8%

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

   5. Simplified 93.8%

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

   6. Taylor expanded in y around 0 46.1%

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

      1. associate-*r* 46.1%

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

      2. neg-mul-1 46.1%

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

   8. Simplified 46.1%

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

   if -500 < (-.f64 1 z) < 2

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(1 - z\right) \]
   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 2 < (-.f64 1 z)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 53.4%

      \[\leadsto \color{blue}{y \cdot \left(1 - z\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - z \leq -5 \cdot 10^{+191}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{elif}\;1 - z \leq -500:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{elif}\;1 - z \leq 2:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - z \leq -500 \lor \neg \left(1 - z \leq 2\right):\\ \;\;\;\;z \cdot \left(\left(-y\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (- 1.0 z) -500.0) (not (<= (- 1.0 z) 2.0)))
   (* z (- (- y) x))
   (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((1.0 - z) <= -500.0) || !((1.0 - z) <= 2.0)) {
		tmp = z * (-y - 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 (((1.0d0 - z) <= (-500.0d0)) .or. (.not. ((1.0d0 - z) <= 2.0d0))) then
        tmp = z * (-y - 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 (((1.0 - z) <= -500.0) || !((1.0 - z) <= 2.0)) {
		tmp = z * (-y - x);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((1.0 - z) <= -500.0) or not ((1.0 - z) <= 2.0):
		tmp = z * (-y - x)
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(1.0 - z) <= -500.0) || !(Float64(1.0 - z) <= 2.0))
		tmp = Float64(z * Float64(Float64(-y) - x));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((1.0 - z) <= -500.0) || ~(((1.0 - z) <= 2.0)))
		tmp = z * (-y - x);
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(1.0 - z), $MachinePrecision], -500.0], N[Not[LessEqual[N[(1.0 - z), $MachinePrecision], 2.0]], $MachinePrecision]], N[(z * N[((-y) - x), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 z) < -500 or 2 < (-.f64 1 z)

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 97.2%

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

      1. associate-*r* 97.2%

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

      2. neg-mul-1 97.2%

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

      3. *-commutative 97.2%

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

      4. +-commutative 97.2%

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

   5. Simplified 97.2%

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

   if -500 < (-.f64 1 z) < 2

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(1 - z\right) \]
   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%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - z \leq -500 \lor \neg \left(1 - z \leq 2\right):\\ \;\;\;\;z \cdot \left(\left(-y\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.5 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -6.5) (not (<= z 1.0))) (* y (- z)) (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6.5) || !(z <= 1.0)) {
		tmp = y * -z;
	} 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 <= (-6.5d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = y * -z
    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 <= -6.5) || !(z <= 1.0)) {
		tmp = y * -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -6.5) or not (z <= 1.0):
		tmp = y * -z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -6.5) || !(z <= 1.0))
		tmp = Float64(y * Float64(-z));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -6.5) || ~((z <= 1.0)))
		tmp = y * -z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -6.5], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(y * (-z)), $MachinePrecision], N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.5 or 1 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 52.2%

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

   4. Taylor expanded in z around inf 50.3%

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

      1. mul-1-neg 50.3%

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

      2. *-commutative 50.3%

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

      3. distribute-rgt-neg-in 50.3%

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

   6. Simplified 50.3%

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

   if -6.5 < z < 1

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(1 - z\right) \]
   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%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{-84}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.4e-84) (* x (- 1.0 z)) (* y (- 1.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.4e-84) {
		tmp = x * (1.0 - z);
	} else {
		tmp = y * (1.0 - 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 (y <= 1.4d-84) then
        tmp = x * (1.0d0 - z)
    else
        tmp = y * (1.0d0 - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.4e-84) {
		tmp = x * (1.0 - z);
	} else {
		tmp = y * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.4e-84:
		tmp = x * (1.0 - z)
	else:
		tmp = y * (1.0 - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.4e-84)
		tmp = Float64(x * Float64(1.0 - z));
	else
		tmp = Float64(y * Float64(1.0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.4e-84)
		tmp = x * (1.0 - z);
	else
		tmp = y * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.4e-84], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.39999999999999991e-84

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 64.2%

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

      1. *-commutative 64.2%

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

   5. Simplified 64.2%

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

   if 1.39999999999999991e-84 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 66.8%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{-84}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 32.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-85}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= y 2.4e-85) x y))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.4e-85) {
		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 <= 2.4d-85) 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 <= 2.4e-85) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.4e-85:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.4e-85)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.4e-85)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.4e-85], x, y]

Derivation

1. Split input into 2 regimes

2. if y < 2.4000000000000001e-85

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 64.2%

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

      1. *-commutative 64.2%

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

   5. Simplified 64.2%

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

   6. Taylor expanded in z around 0 32.5%

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

   if 2.4000000000000001e-85 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 66.8%

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

   4. Taylor expanded in z around 0 33.5%

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

4. Final simplification 32.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-85}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.1% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in z around 0 49.1%

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

   1. +-commutative 49.1%

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

5. Simplified 49.1%

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

6. Final simplification 49.1%

   \[\leadsto x + y \]
7. Add Preprocessing

Alternative 9: 27.0% 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%

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

3. Taylor expanded in x around inf 53.8%

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

   1. *-commutative 53.8%

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

5. Simplified 53.8%

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

6. Taylor expanded in z around 0 28.1%

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

7. Final simplification 28.1%

   \[\leadsto x \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(FPCore (x y z)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, H"
  :precision binary64
  (* (+ x y) (- 1.0 z)))