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

Percentage Accurate: 100.0% → 100.0%

Time: 5.1s

Alternatives: 9

Speedup: 1.0×

• 20.7% 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. Final simplification 100.0%

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

Alternative 2: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - z \leq -20 \lor \neg \left(1 - z \leq 2\right):\\ \;\;\;\;z \cdot \left(\left(-y\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (- 1.0 z) -20.0) (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) <= -20.0) || !((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) <= (-20.0d0)) .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) <= -20.0) || !((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) <= -20.0) 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) <= -20.0) || !(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) <= -20.0) || ~(((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], -20.0], 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) < -20 or 2 < (-.f64 1 z)

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 96.6%

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

      1. mul-1-neg 96.6%

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

      2. +-commutative 96.6%

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

      3. distribute-rgt-neg-out 96.6%

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

      4. +-commutative 96.6%

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

   4. Simplified 96.6%

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

   if -20 < (-.f64 1 z) < 2

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 98.1%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - z \leq -20 \lor \neg \left(1 - z \leq 2\right):\\ \;\;\;\;z \cdot \left(\left(-y\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 3: 75.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 - z\right)\\ \mathbf{if}\;z \leq -4.2 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-8}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- 1.0 z))))
   (if (<= z -4.2e-7)
     t_0
     (if (<= z 1.05e-8) (+ x y) (if (<= z 5.8e+15) t_0 (* x (- z)))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (1.0 - z);
	double tmp;
	if (z <= -4.2e-7) {
		tmp = t_0;
	} else if (z <= 1.05e-8) {
		tmp = x + y;
	} else if (z <= 5.8e+15) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = y * (1.0d0 - z)
    if (z <= (-4.2d-7)) then
        tmp = t_0
    else if (z <= 1.05d-8) then
        tmp = x + y
    else if (z <= 5.8d+15) then
        tmp = t_0
    else
        tmp = x * -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (1.0 - z);
	double tmp;
	if (z <= -4.2e-7) {
		tmp = t_0;
	} else if (z <= 1.05e-8) {
		tmp = x + y;
	} else if (z <= 5.8e+15) {
		tmp = t_0;
	} else {
		tmp = x * -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (1.0 - z)
	tmp = 0
	if z <= -4.2e-7:
		tmp = t_0
	elif z <= 1.05e-8:
		tmp = x + y
	elif z <= 5.8e+15:
		tmp = t_0
	else:
		tmp = x * -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 - z))
	tmp = 0.0
	if (z <= -4.2e-7)
		tmp = t_0;
	elseif (z <= 1.05e-8)
		tmp = Float64(x + y);
	elseif (z <= 5.8e+15)
		tmp = t_0;
	else
		tmp = Float64(x * Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (1.0 - z);
	tmp = 0.0;
	if (z <= -4.2e-7)
		tmp = t_0;
	elseif (z <= 1.05e-8)
		tmp = x + y;
	elseif (z <= 5.8e+15)
		tmp = t_0;
	else
		tmp = x * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.2e-7], t$95$0, If[LessEqual[z, 1.05e-8], N[(x + y), $MachinePrecision], If[LessEqual[z, 5.8e+15], t$95$0, N[(x * (-z)), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.2e-7 or 1.04999999999999997e-8 < z < 5.8e15

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 56.5%

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

   if -4.2e-7 < z < 1.04999999999999997e-8

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 99.6%

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

   if 5.8e15 < z

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 98.1%

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

   3. Applied egg-rr 98.1%

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

   4. Taylor expanded in z around inf 98.1%

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

      1. associate-*r* 98.1%

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

      2. mul-1-neg 98.1%

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

   6. Simplified 98.1%

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

   7. Taylor expanded in y around 0 49.4%

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

      1. mul-1-neg 49.4%

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

      2. *-commutative 49.4%

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

   9. Simplified 49.4%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-7}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-8}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{+15}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \end{array} \]

Alternative 4: 75.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.5 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -7.5) (not (<= z 1.0))) (* x (- z)) (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -7.5) || !(z <= 1.0)) {
		tmp = x * -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 <= (-7.5d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = x * -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 <= -7.5) || !(z <= 1.0)) {
		tmp = x * -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -7.5) or not (z <= 1.0):
		tmp = x * -z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -7.5) || !(z <= 1.0))
		tmp = Float64(x * Float64(-z));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -7.5) || ~((z <= 1.0)))
		tmp = x * -z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -7.5], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(x * (-z)), $MachinePrecision], N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.5 or 1 < z

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 98.4%

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

   3. Applied egg-rr 98.4%

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

   4. Taylor expanded in z around inf 97.1%

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

      1. associate-*r* 97.1%

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

      2. mul-1-neg 97.1%

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

   6. Simplified 97.1%

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

   7. Taylor expanded in y around 0 45.9%

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

      1. mul-1-neg 45.9%

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

      2. *-commutative 45.9%

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

   9. Simplified 45.9%

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

   if -7.5 < z < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 98.1%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.5 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 5: 74.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -116000:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -116000.0) (* y (- z)) (if (<= z 1.0) (+ x y) (* x (- z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -116000.0) {
		tmp = y * -z;
	} else if (z <= 1.0) {
		tmp = x + 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 <= (-116000.0d0)) then
        tmp = y * -z
    else if (z <= 1.0d0) then
        tmp = x + 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 <= -116000.0) {
		tmp = y * -z;
	} else if (z <= 1.0) {
		tmp = x + y;
	} else {
		tmp = x * -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -116000.0:
		tmp = y * -z
	elif z <= 1.0:
		tmp = x + y
	else:
		tmp = x * -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -116000.0)
		tmp = Float64(y * Float64(-z));
	elseif (z <= 1.0)
		tmp = Float64(x + y);
	else
		tmp = Float64(x * Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -116000.0)
		tmp = y * -z;
	elseif (z <= 1.0)
		tmp = x + y;
	else
		tmp = x * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -116000.0], N[(y * (-z)), $MachinePrecision], If[LessEqual[z, 1.0], N[(x + y), $MachinePrecision], N[(x * (-z)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -116000

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 98.5%

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

   3. Applied egg-rr 98.5%

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

   4. Taylor expanded in z around inf 98.0%

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

      1. associate-*r* 98.0%

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

      2. mul-1-neg 98.0%

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

   6. Simplified 98.0%

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

   7. Taylor expanded in z around inf 97.3%

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

      1. associate-*r* 97.3%

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

      2. mul-1-neg 97.3%

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

   9. Simplified 97.3%

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

   10. Taylor expanded in y around inf 56.7%

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

       1. associate-*r* 56.7%

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

       2. mul-1-neg 56.7%

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

   12. Simplified 56.7%

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

   if -116000 < z < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 96.8%

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

   if 1 < z

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 98.2%

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

   3. Applied egg-rr 98.2%

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

   4. Taylor expanded in z around inf 97.9%

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

      1. associate-*r* 97.9%

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

      2. mul-1-neg 97.9%

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

   6. Simplified 97.9%

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

   7. Taylor expanded in y around 0 46.6%

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

      1. mul-1-neg 46.6%

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

      2. *-commutative 46.6%

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

   9. Simplified 46.6%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -116000:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \end{array} \]

Alternative 6: 62.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-121}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 8e-121) (- x (* x z)) (* y (- 1.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 8e-121) {
		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 (y <= 8d-121) 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 (y <= 8e-121) {
		tmp = x - (x * z);
	} else {
		tmp = y * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 8e-121:
		tmp = x - (x * z)
	else:
		tmp = y * (1.0 - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 8e-121)
		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 (y <= 8e-121)
		tmp = x - (x * z);
	else
		tmp = y * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 8e-121], N[(x - N[(x * z), $MachinePrecision]), $MachinePrecision], N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.9999999999999998e-121

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 63.0%

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

      1. sub-neg 63.0%

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

      2. +-commutative 63.0%

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

      3. distribute-rgt1-in 63.0%

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

      4. distribute-lft-neg-out 63.0%

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

      5. unsub-neg 63.0%

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

   4. Simplified 63.0%

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

   if 7.9999999999999998e-121 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 71.2%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-121}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \end{array} \]

Alternative 7: 32.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{-84}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= x -9.2e-84) x y))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -9.2e-84) {
		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 (x <= (-9.2d-84)) 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 (x <= -9.2e-84) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -9.2e-84:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -9.2e-84)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -9.2e-84)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -9.2e-84], x, y]

Derivation

1. Split input into 2 regimes

2. if x < -9.19999999999999922e-84

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. flip-- 92.2%

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

      3. associate-*l/ 88.3%

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

      4. metadata-eval 88.3%

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

   3. Applied egg-rr 88.3%

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

   4. Taylor expanded in x around inf 69.0%

      \[\leadsto \frac{\color{blue}{\left(1 - {z}^{2}\right) \cdot x}}{1 + z} \]
   5. Step-by-step derivation

      1. *-commutative 69.0%

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

      2. unpow2 69.0%

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

   6. Simplified 69.0%

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

   7. Taylor expanded in z around 0 44.9%

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

   if -9.19999999999999922e-84 < x

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 99.3%

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

   3. Applied egg-rr 99.3%

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

   4. Taylor expanded in z around inf 79.7%

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

      1. associate-*r* 79.7%

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

      2. mul-1-neg 79.7%

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

   6. Simplified 79.7%

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

   7. Taylor expanded in z around 0 30.0%

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

4. Final simplification 35.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{-84}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 8: 51.0% 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. Taylor expanded in z around 0 51.7%

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

3. Final simplification 51.7%

   \[\leadsto x + y \]

Alternative 9: 26.2% 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. Step-by-step derivation

   1. *-commutative 100.0%

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

   2. flip-- 88.5%

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

   3. associate-*l/ 85.3%

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

   4. metadata-eval 85.3%

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

3. Applied egg-rr 85.3%

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

4. Taylor expanded in x around inf 49.0%

   \[\leadsto \frac{\color{blue}{\left(1 - {z}^{2}\right) \cdot x}}{1 + z} \]
5. Step-by-step derivation

   1. *-commutative 49.0%

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

   2. unpow2 49.0%

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

6. Simplified 49.0%

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

7. Taylor expanded in z around 0 30.0%

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

8. Final simplification 30.0%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023257 
(FPCore (x y z)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, H"
  :precision binary64
  (* (+ x y) (- 1.0 z)))