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

Percentage Accurate: 100.0% → 100.0%

Time: 8.3s

Alternatives: 7

Speedup: 1.0×

• 21.0% 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

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

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]

Derivation

1. Initial program 100.0%

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

Alternative 2: 73.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -y \cdot z\\ t_1 := -x \cdot z\\ \mathbf{if}\;z \leq -3.8 \cdot 10^{+207}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{+30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+264}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* y z))) (t_1 (- (* x z))))
   (if (<= z -3.8e+207)
     t_0
     (if (<= z -2.9e+30)
       t_1
       (if (<= z 1.0) (+ x y) (if (<= z 1.6e+264) t_1 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = -(y * z);
	double t_1 = -(x * z);
	double tmp;
	if (z <= -3.8e+207) {
		tmp = t_0;
	} else if (z <= -2.9e+30) {
		tmp = t_1;
	} else if (z <= 1.0) {
		tmp = x + y;
	} else if (z <= 1.6e+264) {
		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 = -(x * z)
    if (z <= (-3.8d+207)) then
        tmp = t_0
    else if (z <= (-2.9d+30)) then
        tmp = t_1
    else if (z <= 1.0d0) then
        tmp = x + y
    else if (z <= 1.6d+264) 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 = -(x * z);
	double tmp;
	if (z <= -3.8e+207) {
		tmp = t_0;
	} else if (z <= -2.9e+30) {
		tmp = t_1;
	} else if (z <= 1.0) {
		tmp = x + y;
	} else if (z <= 1.6e+264) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -(y * z)
	t_1 = -(x * z)
	tmp = 0
	if z <= -3.8e+207:
		tmp = t_0
	elif z <= -2.9e+30:
		tmp = t_1
	elif z <= 1.0:
		tmp = x + y
	elif z <= 1.6e+264:
		tmp = t_1
	else:
		tmp = t_0
	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 <= -3.8e+207)
		tmp = t_0;
	elseif (z <= -2.9e+30)
		tmp = t_1;
	elseif (z <= 1.0)
		tmp = Float64(x + y);
	elseif (z <= 1.6e+264)
		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 = -(x * z);
	tmp = 0.0;
	if (z <= -3.8e+207)
		tmp = t_0;
	elseif (z <= -2.9e+30)
		tmp = t_1;
	elseif (z <= 1.0)
		tmp = x + y;
	elseif (z <= 1.6e+264)
		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[(x * z), $MachinePrecision])}, If[LessEqual[z, -3.8e+207], t$95$0, If[LessEqual[z, -2.9e+30], t$95$1, If[LessEqual[z, 1.0], N[(x + y), $MachinePrecision], If[LessEqual[z, 1.6e+264], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.79999999999999986e207 or 1.60000000000000003e264 < 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

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

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

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 51.9

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

   5. Applied rewrites 51.9%

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

   6. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 51.9

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

   8. Applied rewrites 51.9%

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

   if -3.79999999999999986e207 < z < -2.8999999999999998e30 or 1 < z < 1.60000000000000003e264

   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. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 49.5

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

   5. Applied rewrites 49.5%

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

   6. Taylor expanded in z 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. lower-*.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. lower-neg.f64 49.5

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

   8. Applied rewrites 49.5%

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

   if -2.8999999999999998e30 < 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. lower-+.f64 94.6

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

   5. Applied rewrites 94.6%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+207}:\\ \;\;\;\;-y \cdot z\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{+30}:\\ \;\;\;\;-x \cdot z\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+264}:\\ \;\;\;\;-x \cdot z\\ \mathbf{else}:\\ \;\;\;\;-y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 42.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-257}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{elif}\;x + y \leq 2 \cdot 10^{-17}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ x y) -2e-257)
   (- x (* x z))
   (if (<= (+ x y) 2e-17) (+ x y) (- (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= -2e-257) {
		tmp = x - (x * z);
	} else if ((x + y) <= 2e-17) {
		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 ((x + y) <= (-2d-257)) then
        tmp = x - (x * z)
    else if ((x + y) <= 2d-17) 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 ((x + y) <= -2e-257) {
		tmp = x - (x * z);
	} else if ((x + y) <= 2e-17) {
		tmp = x + y;
	} else {
		tmp = -(y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x + y) <= -2e-257:
		tmp = x - (x * z)
	elif (x + y) <= 2e-17:
		tmp = x + y
	else:
		tmp = -(y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x + y) <= -2e-257)
		tmp = Float64(x - Float64(x * z));
	elseif (Float64(x + y) <= 2e-17)
		tmp = Float64(x + y);
	else
		tmp = Float64(-Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x + y) <= -2e-257)
		tmp = x - (x * z);
	elseif ((x + y) <= 2e-17)
		tmp = x + y;
	else
		tmp = -(y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x + y), $MachinePrecision], -2e-257], N[(x - N[(x * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 2e-17], N[(x + y), $MachinePrecision], (-N[(y * z), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -2e-257

   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. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 45.8

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

   5. Applied rewrites 45.8%

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

   if -2e-257 < (+.f64 x y) < 2.00000000000000014e-17

   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. lower-+.f64 67.7

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

   5. Applied rewrites 67.7%

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

   if 2.00000000000000014e-17 < (+.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

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

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

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 51.4

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

   5. Applied rewrites 51.4%

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

   6. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 35.1

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

   8. Applied rewrites 35.1%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-257}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{elif}\;x + y \leq 2 \cdot 10^{-17}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -y \cdot z\\ \mathbf{if}\;z \leq -116:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* y z)))) (if (<= z -116.0) t_0 (if (<= z 1.0) (+ x y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -(y * z);
	double tmp;
	if (z <= -116.0) {
		tmp = t_0;
	} else if (z <= 1.0) {
		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 = -(y * z)
    if (z <= (-116.0d0)) then
        tmp = t_0
    else if (z <= 1.0d0) 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 = -(y * z);
	double tmp;
	if (z <= -116.0) {
		tmp = t_0;
	} else if (z <= 1.0) {
		tmp = x + y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -(y * z)
	tmp = 0
	if z <= -116.0:
		tmp = t_0
	elif z <= 1.0:
		tmp = x + y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-Float64(y * z))
	tmp = 0.0
	if (z <= -116.0)
		tmp = t_0;
	elseif (z <= 1.0)
		tmp = Float64(x + y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -(y * z);
	tmp = 0.0;
	if (z <= -116.0)
		tmp = t_0;
	elseif (z <= 1.0)
		tmp = x + y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = (-N[(y * z), $MachinePrecision])}, If[LessEqual[z, -116.0], t$95$0, If[LessEqual[z, 1.0], N[(x + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -116 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

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

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

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

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 53.5

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

   5. Applied rewrites 53.5%

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

   6. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 53.1

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

   8. Applied rewrites 53.1%

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

   if -116 < 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. lower-+.f64 96.9

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

   5. Applied rewrites 96.9%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -116:\\ \;\;\;\;-y \cdot z\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-304}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ x y) -2e-304) (- x (* x z)) (fma (- z) y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= -2e-304) {
		tmp = x - (x * z);
	} else {
		tmp = fma(-z, y, y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x + y) <= -2e-304)
		tmp = Float64(x - Float64(x * z));
	else
		tmp = fma(Float64(-z), y, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x + y), $MachinePrecision], -2e-304], N[(x - N[(x * z), $MachinePrecision]), $MachinePrecision], N[((-z) * y + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -1.99999999999999994e-304

   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. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 45.6

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

   5. Applied rewrites 45.6%

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

   if -1.99999999999999994e-304 < (+.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

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

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

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 47.8

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

   5. Applied rewrites 47.8%

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

      1. *-commutative N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-rgt1-in N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-fma.f64 47.8

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

   7. Applied rewrites 47.8%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-304}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-304}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ x y) -2e-304) (- x (* x z)) (- y (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= -2e-304) {
		tmp = x - (x * z);
	} else {
		tmp = y - (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 ((x + y) <= (-2d-304)) then
        tmp = x - (x * z)
    else
        tmp = y - (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= -2e-304) {
		tmp = x - (x * z);
	} else {
		tmp = y - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x + y) <= -2e-304:
		tmp = x - (x * z)
	else:
		tmp = y - (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x + y) <= -2e-304)
		tmp = Float64(x - Float64(x * z));
	else
		tmp = Float64(y - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x + y) <= -2e-304)
		tmp = x - (x * z);
	else
		tmp = y - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x + y), $MachinePrecision], -2e-304], N[(x - N[(x * z), $MachinePrecision]), $MachinePrecision], N[(y - N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -1.99999999999999994e-304

   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. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 45.6

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

   5. Applied rewrites 45.6%

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

   if -1.99999999999999994e-304 < (+.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

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

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

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 47.8

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

   5. Applied rewrites 47.8%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-304}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.1% 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. lower-+.f64 43.6

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

5. Applied rewrites 43.6%

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

6. Final simplification 43.6%

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

Reproduce

herbie shell --seed 2024214 
(FPCore (x y z)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, H"
  :precision binary64
  (* (+ x y) (- 1.0 z)))