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

Percentage Accurate: 100.0% → 100.0%

Time: 8.0s

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: 74.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -y \cdot z\\ t_1 := -x \cdot z\\ \mathbf{if}\;z \leq -2.45 \cdot 10^{+156}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{+80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -65:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{+261}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* y z))) (t_1 (- (* x z))))
   (if (<= z -2.45e+156)
     t_0
     (if (<= z -1.4e+80)
       t_1
       (if (<= z -65.0)
         t_0
         (if (<= z 1.0) (+ x y) (if (<= z 4.9e+261) t_0 t_1)))))))

C

double code(double x, double y, double z) {
	double t_0 = -(y * z);
	double t_1 = -(x * z);
	double tmp;
	if (z <= -2.45e+156) {
		tmp = t_0;
	} else if (z <= -1.4e+80) {
		tmp = t_1;
	} else if (z <= -65.0) {
		tmp = t_0;
	} else if (z <= 1.0) {
		tmp = x + y;
	} else if (z <= 4.9e+261) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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 = -(x * z)
    if (z <= (-2.45d+156)) then
        tmp = t_0
    else if (z <= (-1.4d+80)) then
        tmp = t_1
    else if (z <= (-65.0d0)) then
        tmp = t_0
    else if (z <= 1.0d0) then
        tmp = x + y
    else if (z <= 4.9d+261) then
        tmp = t_0
    else
        tmp = t_1
    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 = -(x * z);
	double tmp;
	if (z <= -2.45e+156) {
		tmp = t_0;
	} else if (z <= -1.4e+80) {
		tmp = t_1;
	} else if (z <= -65.0) {
		tmp = t_0;
	} else if (z <= 1.0) {
		tmp = x + y;
	} else if (z <= 4.9e+261) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -(y * z)
	t_1 = -(x * z)
	tmp = 0
	if z <= -2.45e+156:
		tmp = t_0
	elif z <= -1.4e+80:
		tmp = t_1
	elif z <= -65.0:
		tmp = t_0
	elif z <= 1.0:
		tmp = x + y
	elif z <= 4.9e+261:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-Float64(y * z))
	t_1 = Float64(-Float64(x * z))
	tmp = 0.0
	if (z <= -2.45e+156)
		tmp = t_0;
	elseif (z <= -1.4e+80)
		tmp = t_1;
	elseif (z <= -65.0)
		tmp = t_0;
	elseif (z <= 1.0)
		tmp = Float64(x + y);
	elseif (z <= 4.9e+261)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -(y * z);
	t_1 = -(x * z);
	tmp = 0.0;
	if (z <= -2.45e+156)
		tmp = t_0;
	elseif (z <= -1.4e+80)
		tmp = t_1;
	elseif (z <= -65.0)
		tmp = t_0;
	elseif (z <= 1.0)
		tmp = x + y;
	elseif (z <= 4.9e+261)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = (-N[(y * z), $MachinePrecision])}, Block[{t$95$1 = (-N[(x * z), $MachinePrecision])}, If[LessEqual[z, -2.45e+156], t$95$0, If[LessEqual[z, -1.4e+80], t$95$1, If[LessEqual[z, -65.0], t$95$0, If[LessEqual[z, 1.0], N[(x + y), $MachinePrecision], If[LessEqual[z, 4.9e+261], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.44999999999999984e156 or -1.39999999999999992e80 < z < -65 or 1 < z < 4.9e261

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 53.7%

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

   6. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. neg-lowering-neg.f64 53.7

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

   8. Simplified 53.7%

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

   if -2.44999999999999984e156 < z < -1.39999999999999992e80 or 4.9e261 < 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

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. neg-lowering-neg.f64 65.2

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

   8. Simplified 65.2%

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

   if -65 < 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

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 99.4

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

   5. Simplified 99.4%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{+156}:\\ \;\;\;\;-y \cdot z\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{+80}:\\ \;\;\;\;-x \cdot z\\ \mathbf{elif}\;z \leq -65:\\ \;\;\;\;-y \cdot z\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{+261}:\\ \;\;\;\;-y \cdot z\\ \mathbf{else}:\\ \;\;\;\;-x \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 42.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-5}:\\ \;\;\;\;-x \cdot z\\ \mathbf{elif}\;x + y \leq -2 \cdot 10^{-228}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ x y) -2e-5)
   (- (* x z))
   (if (<= (+ x y) -2e-228) (+ x y) (* y (- 1.0 z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= -2e-5) {
		tmp = -(x * z);
	} else if ((x + y) <= -2e-228) {
		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 ((x + y) <= (-2d-5)) then
        tmp = -(x * z)
    else if ((x + y) <= (-2d-228)) 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 ((x + y) <= -2e-5) {
		tmp = -(x * z);
	} else if ((x + y) <= -2e-228) {
		tmp = x + y;
	} else {
		tmp = y * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x + y) <= -2e-5:
		tmp = -(x * z)
	elif (x + y) <= -2e-228:
		tmp = x + y
	else:
		tmp = y * (1.0 - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x + y) <= -2e-5)
		tmp = Float64(-Float64(x * z));
	elseif (Float64(x + y) <= -2e-228)
		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 ((x + y) <= -2e-5)
		tmp = -(x * z);
	elseif ((x + y) <= -2e-228)
		tmp = x + y;
	else
		tmp = y * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x + y), $MachinePrecision], -2e-5], (-N[(x * z), $MachinePrecision]), If[LessEqual[N[(x + y), $MachinePrecision], -2e-228], N[(x + y), $MachinePrecision], N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -2.00000000000000016e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 54.9

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

   5. Simplified 54.9%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. neg-lowering-neg.f64 30.6

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

   8. Simplified 30.6%

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

   if -2.00000000000000016e-5 < (+.f64 x y) < -2.00000000000000007e-228

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 76.5

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

   5. Simplified 76.5%

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

   if -2.00000000000000007e-228 < (+.f64 x y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 52.5%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-5}:\\ \;\;\;\;-x \cdot z\\ \mathbf{elif}\;x + y \leq -2 \cdot 10^{-228}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -x \cdot z\\ \mathbf{if}\;1 - z \leq -0.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;1 - z \leq 3 \cdot 10^{+19}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* x z))))
   (if (<= (- 1.0 z) -0.5) t_0 (if (<= (- 1.0 z) 3e+19) (+ x y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -(x * z);
	double tmp;
	if ((1.0 - z) <= -0.5) {
		tmp = t_0;
	} else if ((1.0 - z) <= 3e+19) {
		tmp = x + y;
	} 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 = -(x * z)
    if ((1.0d0 - z) <= (-0.5d0)) then
        tmp = t_0
    else if ((1.0d0 - z) <= 3d+19) then
        tmp = x + y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -(x * z);
	double tmp;
	if ((1.0 - z) <= -0.5) {
		tmp = t_0;
	} else if ((1.0 - z) <= 3e+19) {
		tmp = x + y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -(x * z)
	tmp = 0
	if (1.0 - z) <= -0.5:
		tmp = t_0
	elif (1.0 - z) <= 3e+19:
		tmp = x + y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-Float64(x * z))
	tmp = 0.0
	if (Float64(1.0 - z) <= -0.5)
		tmp = t_0;
	elseif (Float64(1.0 - z) <= 3e+19)
		tmp = Float64(x + y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -(x * z);
	tmp = 0.0;
	if ((1.0 - z) <= -0.5)
		tmp = t_0;
	elseif ((1.0 - z) <= 3e+19)
		tmp = x + y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = (-N[(x * z), $MachinePrecision])}, If[LessEqual[N[(1.0 - z), $MachinePrecision], -0.5], t$95$0, If[LessEqual[N[(1.0 - z), $MachinePrecision], 3e+19], N[(x + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) z) < -0.5 or 3e19 < (-.f64 #s(literal 1 binary64) 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

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 98.5

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

   5. Simplified 98.5%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. neg-lowering-neg.f64 53.3

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

   8. Simplified 53.3%

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

   if -0.5 < (-.f64 #s(literal 1 binary64) z) < 3e19

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 98.7

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

   5. Simplified 98.7%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - z \leq -0.5:\\ \;\;\;\;-x \cdot z\\ \mathbf{elif}\;1 - z \leq 3 \cdot 10^{+19}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-x \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-228}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ x y) -2e-228) (- x (* x z)) (* y (- 1.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= -2e-228) {
		tmp = x - (x * 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 ((x + y) <= (-2d-228)) then
        tmp = x - (x * 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 ((x + y) <= -2e-228) {
		tmp = x - (x * z);
	} else {
		tmp = y * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x + y) <= -2e-228:
		tmp = x - (x * z)
	else:
		tmp = y * (1.0 - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x + y) <= -2e-228)
		tmp = Float64(x - Float64(x * 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 ((x + y) <= -2e-228)
		tmp = x - (x * z);
	else
		tmp = y * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x + y), $MachinePrecision], -2e-228], N[(x - N[(x * z), $MachinePrecision]), $MachinePrecision], N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -2.00000000000000007e-228

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-out-- N/A

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

      2. *-rgt-identity N/A

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

      3. --lowering--.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 45.2

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

   5. Simplified 45.2%

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

   if -2.00000000000000007e-228 < (+.f64 x y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 52.5%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-228}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 26.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - z\right) \cdot \left(x + y\right) \leq -2 \cdot 10^{-228}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* (- 1.0 z) (+ x y)) -2e-228) x y))

C

double code(double x, double y, double z) {
	double tmp;
	if (((1.0 - z) * (x + y)) <= -2e-228) {
		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 (((1.0d0 - z) * (x + y)) <= (-2d-228)) 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 (((1.0 - z) * (x + y)) <= -2e-228) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((1.0 - z) * (x + y)) <= -2e-228:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(1.0 - z) * Float64(x + y)) <= -2e-228)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((1.0 - z) * (x + y)) <= -2e-228)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(1.0 - z), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision], -2e-228], x, y]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) z)) < -2.00000000000000007e-228

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 51.6

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

   5. Simplified 51.6%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x} \]
   7. Step-by-step derivation

   8. Simplified 20.0%

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

   if -2.00000000000000007e-228 < (*.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) 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

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

   5. Simplified 53.3%

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

   6. Taylor expanded in z around 0

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

   8. Simplified 28.1%

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

4. Final simplification 24.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - z\right) \cdot \left(x + y\right) \leq -2 \cdot 10^{-228}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.8:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 1.8) (+ x y) (* y z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.8) {
		tmp = x + y;
	} 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 <= 1.8d0) then
        tmp = x + y
    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 <= 1.8) {
		tmp = x + y;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.8:
		tmp = x + y
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.8)
		tmp = Float64(x + y);
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.8)
		tmp = x + y;
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.8], N[(x + y), $MachinePrecision], N[(y * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.80000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 65.2

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

   5. Simplified 65.2%

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

   if 1.80000000000000004 < 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

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

   5. Simplified 57.1%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. neg-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - z\right)} \cdot y + 1 \cdot y \]

      5. flip3-- N/A

         \[\leadsto \color{blue}{\frac{{0}^{3} - {z}^{3}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}} \cdot y + 1 \cdot y \]

      6. metadata-eval N/A

         \[\leadsto \frac{{0}^{3} - {z}^{3}}{\color{blue}{0} + \left(z \cdot z + 0 \cdot z\right)} \cdot y + 1 \cdot y \]

      7. +-lft-identity N/A

         \[\leadsto \frac{{0}^{3} - {z}^{3}}{\color{blue}{z \cdot z + 0 \cdot z}} \cdot y + 1 \cdot y \]

      8. distribute-rgt-out N/A

         \[\leadsto \frac{{0}^{3} - {z}^{3}}{\color{blue}{z \cdot \left(z + 0\right)}} \cdot y + 1 \cdot y \]

      9. +-commutative N/A

         \[\leadsto \frac{{0}^{3} - {z}^{3}}{z \cdot \color{blue}{\left(0 + z\right)}} \cdot y + 1 \cdot y \]

      10. +-lft-identity N/A

          \[\leadsto \frac{{0}^{3} - {z}^{3}}{z \cdot \color{blue}{z}} \cdot y + 1 \cdot y \]

      11. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{0} - {z}^{3}}{z \cdot z} \cdot y + 1 \cdot y \]

      12. neg-sub0 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{neg}\left({z}^{3}\right)}}{z \cdot z} \cdot y + 1 \cdot y \]

      13. cube-neg N/A

          \[\leadsto \frac{\color{blue}{{\left(\mathsf{neg}\left(z\right)\right)}^{3}}}{z \cdot z} \cdot y + 1 \cdot y \]

      14. sqr-pow N/A

          \[\leadsto \frac{\color{blue}{{\left(\mathsf{neg}\left(z\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(z\right)\right)}^{\left(\frac{3}{2}\right)}}}{z \cdot z} \cdot y + 1 \cdot y \]

      15. pow-prod-down N/A

          \[\leadsto \frac{\color{blue}{{\left(\left(\mathsf{neg}\left(z\right)\right) \cdot \left(\mathsf{neg}\left(z\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{z \cdot z} \cdot y + 1 \cdot y \]

      16. sqr-neg N/A

          \[\leadsto \frac{{\color{blue}{\left(z \cdot z\right)}}^{\left(\frac{3}{2}\right)}}{z \cdot z} \cdot y + 1 \cdot y \]

      17. pow-prod-down N/A

          \[\leadsto \frac{\color{blue}{{z}^{\left(\frac{3}{2}\right)} \cdot {z}^{\left(\frac{3}{2}\right)}}}{z \cdot z} \cdot y + 1 \cdot y \]

      18. sqr-pow N/A

          \[\leadsto \frac{\color{blue}{{z}^{3}}}{z \cdot z} \cdot y + 1 \cdot y \]

      19. pow2 N/A

          \[\leadsto \frac{{z}^{3}}{\color{blue}{{z}^{2}}} \cdot y + 1 \cdot y \]

      20. pow-div N/A

          \[\leadsto \color{blue}{{z}^{\left(3 - 2\right)}} \cdot y + 1 \cdot y \]

      21. metadata-eval N/A

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

      22. unpow1 N/A

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

      23. *-lft-identity N/A

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

   7. Applied egg-rr 5.2%

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

   8. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 5.2

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

   10. Simplified 5.2%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.8:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.4% accurate, 3.0× 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

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

   1. +-commutative N/A

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

   2. +-lowering-+.f64 50.8

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

5. Simplified 50.8%

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

6. Final simplification 50.8%

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

Alternative 9: 25.5% accurate, 12.0× 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 z around 0

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

   1. +-commutative N/A

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

   2. +-lowering-+.f64 50.8

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

5. Simplified 50.8%

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

6. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x} \]
7. Step-by-step derivation

8. Simplified 21.5%

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

Reproduce

herbie shell --seed 2024198 
(FPCore (x y z)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, H"
  :precision binary64
  (* (+ x y) (- 1.0 z)))