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

Percentage Accurate: 93.2% → 97.6%

Time: 9.1s

Alternatives: 8

Speedup: 1.0×

• 25.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x - \frac{y \cdot \left(z - t\right)}{a} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (- x (/ (* y (- z t)) a)))

C

double code(double x, double y, double z, double t, double a) {
	return x - ((y * (z - t)) / a);
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x - ((y * (z - t)) / a)
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x - ((y * (z - t)) / a);
}

Python

def code(x, y, z, t, a):
	return x - ((y * (z - t)) / a)

Julia

function code(x, y, z, t, a)
	return Float64(x - Float64(Float64(y * Float64(z - t)) / a))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x - ((y * (z - t)) / a);
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x - N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]

Initial Program: 93.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - \frac{y \cdot \left(z - t\right)}{a} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (- x (/ (* y (- z t)) a)))

C

double code(double x, double y, double z, double t, double a) {
	return x - ((y * (z - t)) / a);
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x - ((y * (z - t)) / a)
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x - ((y * (z - t)) / a);
}

Python

def code(x, y, z, t, a):
	return x - ((y * (z - t)) / a)

Julia

function code(x, y, z, t, a)
	return Float64(x - Float64(Float64(y * Float64(z - t)) / a))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x - ((y * (z - t)) / a);
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x - N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 97.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-67}:\\ \;\;\;\;x + y \cdot \frac{t - z}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{a} \cdot \left(t - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= y -5e-67) (+ x (* y (/ (- t z) a))) (+ x (* (/ y a) (- t z)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -5e-67) {
		tmp = x + (y * ((t - z) / a));
	} else {
		tmp = x + ((y / a) * (t - z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (y <= (-5d-67)) then
        tmp = x + (y * ((t - z) / a))
    else
        tmp = x + ((y / a) * (t - z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -5e-67) {
		tmp = x + (y * ((t - z) / a));
	} else {
		tmp = x + ((y / a) * (t - z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if y <= -5e-67:
		tmp = x + (y * ((t - z) / a))
	else:
		tmp = x + ((y / a) * (t - z))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -5e-67)
		tmp = Float64(x + Float64(y * Float64(Float64(t - z) / a)));
	else
		tmp = Float64(x + Float64(Float64(y / a) * Float64(t - z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (y <= -5e-67)
		tmp = x + (y * ((t - z) / a));
	else
		tmp = x + ((y / a) * (t - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, -5e-67], N[(x + N[(y * N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y / a), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.9999999999999999e-67

   1. Initial program 84.6%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   2. Step-by-step derivation

      1. associate-/l* 99.7%

         \[\leadsto x - \color{blue}{\frac{y}{\frac{a}{z - t}}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x - \frac{y}{\frac{a}{z - t}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 99.8%

         \[\leadsto x - \color{blue}{\frac{1}{\frac{\frac{a}{z - t}}{y}}} \]

      2. associate-/r/ 99.6%

         \[\leadsto x - \color{blue}{\frac{1}{\frac{a}{z - t}} \cdot y} \]

      3. clear-num 99.8%

         \[\leadsto x - \color{blue}{\frac{z - t}{a}} \cdot y \]

   6. Applied egg-rr 99.8%

      \[\leadsto x - \color{blue}{\frac{z - t}{a} \cdot y} \]

   if -4.9999999999999999e-67 < y

   1. Initial program 95.2%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   2. Step-by-step derivation

      1. associate-*l/ 98.7%

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

   3. Simplified 98.7%

      \[\leadsto \color{blue}{x - \frac{y}{a} \cdot \left(z - t\right)} \]
   4. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-67}:\\ \;\;\;\;x + y \cdot \frac{t - z}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{a} \cdot \left(t - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{+96} \lor \neg \left(t \leq 5.1 \cdot 10^{+61}\right):\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -3.2e+96) (not (<= t 5.1e+61)))
   (+ x (* y (/ t a)))
   (- x (* z (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -3.2e+96) || !(t <= 5.1e+61)) {
		tmp = x + (y * (t / a));
	} else {
		tmp = x - (z * (y / a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((t <= (-3.2d+96)) .or. (.not. (t <= 5.1d+61))) then
        tmp = x + (y * (t / a))
    else
        tmp = x - (z * (y / a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -3.2e+96) || !(t <= 5.1e+61)) {
		tmp = x + (y * (t / a));
	} else {
		tmp = x - (z * (y / a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (t <= -3.2e+96) or not (t <= 5.1e+61):
		tmp = x + (y * (t / a))
	else:
		tmp = x - (z * (y / a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -3.2e+96) || !(t <= 5.1e+61))
		tmp = Float64(x + Float64(y * Float64(t / a)));
	else
		tmp = Float64(x - Float64(z * Float64(y / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((t <= -3.2e+96) || ~((t <= 5.1e+61)))
		tmp = x + (y * (t / a));
	else
		tmp = x - (z * (y / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -3.2e+96], N[Not[LessEqual[t, 5.1e+61]], $MachinePrecision]], N[(x + N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(z * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.20000000000000006e96 or 5.1000000000000001e61 < t

   1. Initial program 90.9%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   2. Step-by-step derivation

      1. associate-/l* 93.3%

         \[\leadsto x - \color{blue}{\frac{y}{\frac{a}{z - t}}} \]

   3. Simplified 93.3%

      \[\leadsto \color{blue}{x - \frac{y}{\frac{a}{z - t}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 93.2%

         \[\leadsto x - \color{blue}{\frac{1}{\frac{\frac{a}{z - t}}{y}}} \]

      2. associate-/r/ 91.9%

         \[\leadsto x - \color{blue}{\frac{1}{\frac{a}{z - t}} \cdot y} \]

      3. clear-num 92.0%

         \[\leadsto x - \color{blue}{\frac{z - t}{a}} \cdot y \]

   6. Applied egg-rr 92.0%

      \[\leadsto x - \color{blue}{\frac{z - t}{a} \cdot y} \]

   7. Taylor expanded in z around 0 83.8%

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

      1. neg-mul-1 83.8%

         \[\leadsto x - \color{blue}{\left(-\frac{t}{a}\right)} \cdot y \]

   9. Simplified 83.8%

      \[\leadsto x - \color{blue}{\left(-\frac{t}{a}\right)} \cdot y \]

   if -3.20000000000000006e96 < t < 5.1000000000000001e61

   1. Initial program 92.8%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   2. Step-by-step derivation

      1. associate-*l/ 96.9%

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

   3. Simplified 96.9%

      \[\leadsto \color{blue}{x - \frac{y}{a} \cdot \left(z - t\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 85.7%

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

      1. associate-*l/ 92.0%

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

      2. *-commutative 92.0%

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

   7. Simplified 92.0%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{+96} \lor \neg \left(t \leq 5.1 \cdot 10^{+61}\right):\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{+96} \lor \neg \left(t \leq 6.2 \cdot 10^{+60}\right):\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -2.8e+96) (not (<= t 6.2e+60)))
   (+ x (* t (/ y a)))
   (- x (* z (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -2.8e+96) || !(t <= 6.2e+60)) {
		tmp = x + (t * (y / a));
	} else {
		tmp = x - (z * (y / a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((t <= (-2.8d+96)) .or. (.not. (t <= 6.2d+60))) then
        tmp = x + (t * (y / a))
    else
        tmp = x - (z * (y / a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -2.8e+96) || !(t <= 6.2e+60)) {
		tmp = x + (t * (y / a));
	} else {
		tmp = x - (z * (y / a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (t <= -2.8e+96) or not (t <= 6.2e+60):
		tmp = x + (t * (y / a))
	else:
		tmp = x - (z * (y / a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -2.8e+96) || !(t <= 6.2e+60))
		tmp = Float64(x + Float64(t * Float64(y / a)));
	else
		tmp = Float64(x - Float64(z * Float64(y / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((t <= -2.8e+96) || ~((t <= 6.2e+60)))
		tmp = x + (t * (y / a));
	else
		tmp = x - (z * (y / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -2.8e+96], N[Not[LessEqual[t, 6.2e+60]], $MachinePrecision]], N[(x + N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(z * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.8e96 or 6.2000000000000001e60 < t

   1. Initial program 90.9%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   2. Step-by-step derivation

      1. associate-*l/ 97.7%

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

   3. Simplified 97.7%

      \[\leadsto \color{blue}{x - \frac{y}{a} \cdot \left(z - t\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 82.1%

      \[\leadsto x - \color{blue}{-1 \cdot \frac{t \cdot y}{a}} \]
   6. Step-by-step derivation

      1. *-commutative 82.1%

         \[\leadsto x - -1 \cdot \frac{\color{blue}{y \cdot t}}{a} \]

      2. associate-*l/ 87.7%

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

      3. neg-mul-1 87.7%

         \[\leadsto x - \color{blue}{\left(-\frac{y}{a} \cdot t\right)} \]

      4. distribute-rgt-neg-out 87.7%

         \[\leadsto x - \color{blue}{\frac{y}{a} \cdot \left(-t\right)} \]

   7. Simplified 87.7%

      \[\leadsto x - \color{blue}{\frac{y}{a} \cdot \left(-t\right)} \]

   if -2.8e96 < t < 6.2000000000000001e60

   1. Initial program 92.8%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   2. Step-by-step derivation

      1. associate-*l/ 96.9%

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

   3. Simplified 96.9%

      \[\leadsto \color{blue}{x - \frac{y}{a} \cdot \left(z - t\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 85.7%

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

      1. associate-*l/ 92.0%

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

      2. *-commutative 92.0%

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

   7. Simplified 92.0%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{+96} \lor \neg \left(t \leq 6.2 \cdot 10^{+60}\right):\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (* (/ y a) (- t z))))

C

double code(double x, double y, double z, double t, double a) {
	return x + ((y / a) * (t - z));
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + ((y / a) * (t - z))
end function

Java

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

Python

def code(x, y, z, t, a):
	return x + ((y / a) * (t - z))

Julia

function code(x, y, z, t, a)
	return Float64(x + Float64(Float64(y / a) * Float64(t - z)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x + ((y / a) * (t - z));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + N[(N[(y / a), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.1%

   \[x - \frac{y \cdot \left(z - t\right)}{a} \]
2. Step-by-step derivation

   1. associate-*l/ 97.2%

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

3. Simplified 97.2%

   \[\leadsto \color{blue}{x - \frac{y}{a} \cdot \left(z - t\right)} \]
4. Add Preprocessing

5. Final simplification 97.2%

   \[\leadsto x + \frac{y}{a} \cdot \left(t - z\right) \]
6. Add Preprocessing

Alternative 5: 40.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ x + y \cdot \frac{z}{a} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (* y (/ z a))))

C

double code(double x, double y, double z, double t, double a) {
	return x + (y * (z / a));
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + (y * (z / a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.1%

   \[x - \frac{y \cdot \left(z - t\right)}{a} \]
2. Step-by-step derivation

   1. associate-*l/ 97.2%

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

3. Simplified 97.2%

   \[\leadsto \color{blue}{x - \frac{y}{a} \cdot \left(z - t\right)} \]
4. Add Preprocessing

5. Taylor expanded in z around inf 69.4%

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

   1. associate-*l/ 75.6%

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

   2. *-commutative 75.6%

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

7. Simplified 75.6%

   \[\leadsto x - \color{blue}{z \cdot \frac{y}{a}} \]
8. Step-by-step derivation

   1. frac-2neg 75.6%

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

   2. associate-*r/ 69.4%

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

   3. add-sqr-sqrt 34.6%

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

   4. sqrt-unprod 52.5%

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

   5. sqr-neg 52.5%

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

   6. sqrt-unprod 22.7%

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

   7. add-sqr-sqrt 41.4%

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

9. Applied egg-rr 41.4%

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

    1. associate-/l* 42.1%

       \[\leadsto x - \color{blue}{\frac{z}{\frac{a}{-y}}} \]

11. Simplified 42.1%

    \[\leadsto x - \color{blue}{\frac{z}{\frac{a}{-y}}} \]

12. Taylor expanded in x around 0 41.4%

    \[\leadsto \color{blue}{x + \frac{y \cdot z}{a}} \]
13. Step-by-step derivation

    1. +-commutative 41.4%

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

    2. associate-*r/ 40.6%

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

14. Simplified 40.6%

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

15. Final simplification 40.6%

    \[\leadsto x + y \cdot \frac{z}{a} \]
16. Add Preprocessing

Alternative 6: 40.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ x + \frac{y \cdot z}{a} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (/ (* y z) a)))

C

double code(double x, double y, double z, double t, double a) {
	return x + ((y * z) / a);
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + ((y * z) / a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.1%

   \[x - \frac{y \cdot \left(z - t\right)}{a} \]
2. Step-by-step derivation

   1. associate-*l/ 97.2%

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

3. Simplified 97.2%

   \[\leadsto \color{blue}{x - \frac{y}{a} \cdot \left(z - t\right)} \]
4. Add Preprocessing

5. Taylor expanded in z around inf 69.4%

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

   1. associate-*l/ 75.6%

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

   2. *-commutative 75.6%

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

7. Simplified 75.6%

   \[\leadsto x - \color{blue}{z \cdot \frac{y}{a}} \]
8. Step-by-step derivation

   1. frac-2neg 75.6%

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

   2. associate-*r/ 69.4%

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

   3. add-sqr-sqrt 34.6%

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

   4. sqrt-unprod 52.5%

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

   5. sqr-neg 52.5%

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

   6. sqrt-unprod 22.7%

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

   7. add-sqr-sqrt 41.4%

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

9. Applied egg-rr 41.4%

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

    1. associate-/l* 42.1%

       \[\leadsto x - \color{blue}{\frac{z}{\frac{a}{-y}}} \]

11. Simplified 42.1%

    \[\leadsto x - \color{blue}{\frac{z}{\frac{a}{-y}}} \]
12. Step-by-step derivation

    1. sub-neg 42.1%

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

    2. associate-/r/ 40.6%

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

    3. *-un-lft-identity 40.6%

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

    4. associate-*l/ 40.6%

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

    5. distribute-rgt-neg-in 40.6%

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

    6. remove-double-neg 40.6%

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

    7. *-commutative 40.6%

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

    8. associate-*r* 42.1%

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

    9. div-inv 42.1%

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

    10. associate-/r/ 40.9%

        \[\leadsto x + \color{blue}{\frac{y}{\frac{a}{z}}} \]

    11. +-commutative 40.9%

        \[\leadsto \color{blue}{\frac{y}{\frac{a}{z}} + x} \]

    12. associate-/r/ 42.1%

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

    13. associate-*l/ 41.4%

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

13. Applied egg-rr 41.4%

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

14. Final simplification 41.4%

    \[\leadsto x + \frac{y \cdot z}{a} \]
15. Add Preprocessing

Alternative 7: 71.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ x - z \cdot \frac{y}{a} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (- x (* z (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	return x - (z * (y / a));
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x - (z * (y / a))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x - (z * (y / a));
}

Python

def code(x, y, z, t, a):
	return x - (z * (y / a))

Julia

function code(x, y, z, t, a)
	return Float64(x - Float64(z * Float64(y / a)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x - (z * (y / a));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x - N[(z * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.1%

   \[x - \frac{y \cdot \left(z - t\right)}{a} \]
2. Step-by-step derivation

   1. associate-*l/ 97.2%

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

3. Simplified 97.2%

   \[\leadsto \color{blue}{x - \frac{y}{a} \cdot \left(z - t\right)} \]
4. Add Preprocessing

5. Taylor expanded in z around inf 69.4%

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

   1. associate-*l/ 75.6%

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

   2. *-commutative 75.6%

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

7. Simplified 75.6%

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

8. Final simplification 75.6%

   \[\leadsto x - z \cdot \frac{y}{a} \]
9. Add Preprocessing

Alternative 8: 39.2% accurate, 9.0× speedup

Math

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

FPCore

(FPCore (x y z t a) :precision binary64 x)

C

double code(double x, double y, double z, double t, double a) {
	return x;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x
end function

Java

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

Python

def code(x, y, z, t, a):
	return x

Julia

function code(x, y, z, t, a)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := x

Derivation

1. Initial program 92.1%

   \[x - \frac{y \cdot \left(z - t\right)}{a} \]
2. Step-by-step derivation

   1. associate-*l/ 97.2%

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

3. Simplified 97.2%

   \[\leadsto \color{blue}{x - \frac{y}{a} \cdot \left(z - t\right)} \]
4. Add Preprocessing

5. Taylor expanded in z around inf 69.4%

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

   1. associate-*l/ 75.6%

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

   2. *-commutative 75.6%

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

7. Simplified 75.6%

   \[\leadsto x - \color{blue}{z \cdot \frac{y}{a}} \]
8. Step-by-step derivation

   1. frac-2neg 75.6%

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

   2. associate-*r/ 69.4%

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

   3. add-sqr-sqrt 34.6%

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

   4. sqrt-unprod 52.5%

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

   5. sqr-neg 52.5%

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

   6. sqrt-unprod 22.7%

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

   7. add-sqr-sqrt 41.4%

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

9. Applied egg-rr 41.4%

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

    1. associate-/l* 42.1%

       \[\leadsto x - \color{blue}{\frac{z}{\frac{a}{-y}}} \]

11. Simplified 42.1%

    \[\leadsto x - \color{blue}{\frac{z}{\frac{a}{-y}}} \]

12. Taylor expanded in x around inf 40.2%

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

13. Final simplification 40.2%

    \[\leadsto x \]
14. Add Preprocessing

Developer target: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{z - t}\\ \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{t\_1}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ a (- z t))))
   (if (< y -1.0761266216389975e-10)
     (- x (/ 1.0 (/ t_1 y)))
     (if (< y 2.894426862792089e-49)
       (- x (/ (* y (- z t)) a))
       (- x (/ y t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a / (z - t);
	double tmp;
	if (y < -1.0761266216389975e-10) {
		tmp = x - (1.0 / (t_1 / y));
	} else if (y < 2.894426862792089e-49) {
		tmp = x - ((y * (z - t)) / a);
	} else {
		tmp = x - (y / t_1);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a / (z - t)
    if (y < (-1.0761266216389975d-10)) then
        tmp = x - (1.0d0 / (t_1 / y))
    else if (y < 2.894426862792089d-49) then
        tmp = x - ((y * (z - t)) / a)
    else
        tmp = x - (y / t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a / (z - t);
	double tmp;
	if (y < -1.0761266216389975e-10) {
		tmp = x - (1.0 / (t_1 / y));
	} else if (y < 2.894426862792089e-49) {
		tmp = x - ((y * (z - t)) / a);
	} else {
		tmp = x - (y / t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a / (z - t)
	tmp = 0
	if y < -1.0761266216389975e-10:
		tmp = x - (1.0 / (t_1 / y))
	elif y < 2.894426862792089e-49:
		tmp = x - ((y * (z - t)) / a)
	else:
		tmp = x - (y / t_1)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a / Float64(z - t))
	tmp = 0.0
	if (y < -1.0761266216389975e-10)
		tmp = Float64(x - Float64(1.0 / Float64(t_1 / y)));
	elseif (y < 2.894426862792089e-49)
		tmp = Float64(x - Float64(Float64(y * Float64(z - t)) / a));
	else
		tmp = Float64(x - Float64(y / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a / (z - t);
	tmp = 0.0;
	if (y < -1.0761266216389975e-10)
		tmp = x - (1.0 / (t_1 / y));
	elseif (y < 2.894426862792089e-49)
		tmp = x - ((y * (z - t)) / a);
	else
		tmp = x - (y / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a / N[(z - t), $MachinePrecision]), $MachinePrecision]}, If[Less[y, -1.0761266216389975e-10], N[(x - N[(1.0 / N[(t$95$1 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Less[y, 2.894426862792089e-49], N[(x - N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2024026 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"
  :precision binary64

  :herbie-target
  (if (< y -1.0761266216389975e-10) (- x (/ 1.0 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (- x (/ (* y (- z t)) a)) (- x (/ y (/ a (- z t))))))

  (- x (/ (* y (- z t)) a)))