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

Percentage Accurate: 93.3% → 95.9%

Time: 7.0s

Alternatives: 9

Speedup: 1.0×

• 25.1% 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: 93.3% 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: 95.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{-87}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(z - t\right) \cdot y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= y -1.85e-87) (+ x (/ y (/ a (- z t)))) (+ x (/ (* (- z t) y) a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -1.85e-87) {
		tmp = x + (y / (a / (z - t)));
	} else {
		tmp = x + (((z - t) * 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 (y <= (-1.85d-87)) then
        tmp = x + (y / (a / (z - t)))
    else
        tmp = x + (((z - t) * 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 (y <= -1.85e-87) {
		tmp = x + (y / (a / (z - t)));
	} else {
		tmp = x + (((z - t) * y) / a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if y <= -1.85e-87:
		tmp = x + (y / (a / (z - t)))
	else:
		tmp = x + (((z - t) * y) / a)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -1.85e-87)
		tmp = Float64(x + Float64(y / Float64(a / Float64(z - t))));
	else
		tmp = Float64(x + Float64(Float64(Float64(z - t) * y) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (y <= -1.85e-87)
		tmp = x + (y / (a / (z - t)));
	else
		tmp = x + (((z - t) * y) / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, -1.85e-87], N[(x + N[(y / N[(a / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8500000000000001e-87

   1. Initial program 85.2%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x + \frac{y}{\frac{a}{z - t}}} \]
   4. Add Preprocessing

   if -1.8500000000000001e-87 < y

   1. Initial program 97.2%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{-87}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(z - t\right) \cdot y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.05 \cdot 10^{-54} \lor \neg \left(z \leq 9.2 \cdot 10^{+134}\right):\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{t}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -2.05e-54) (not (<= z 9.2e+134)))
   (+ x (* z (/ y a)))
   (- x (* y (/ t a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -2.05e-54) || !(z <= 9.2e+134)) {
		tmp = x + (z * (y / a));
	} else {
		tmp = x - (y * (t / 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 ((z <= (-2.05d-54)) .or. (.not. (z <= 9.2d+134))) then
        tmp = x + (z * (y / a))
    else
        tmp = x - (y * (t / 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 ((z <= -2.05e-54) || !(z <= 9.2e+134)) {
		tmp = x + (z * (y / a));
	} else {
		tmp = x - (y * (t / a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -2.05e-54) or not (z <= 9.2e+134):
		tmp = x + (z * (y / a))
	else:
		tmp = x - (y * (t / a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -2.05e-54) || !(z <= 9.2e+134))
		tmp = Float64(x + Float64(z * Float64(y / a)));
	else
		tmp = Float64(x - Float64(y * Float64(t / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -2.05e-54) || ~((z <= 9.2e+134)))
		tmp = x + (z * (y / a));
	else
		tmp = x - (y * (t / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -2.05e-54], N[Not[LessEqual[z, 9.2e+134]], $MachinePrecision]], N[(x + N[(z * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.05e-54 or 9.1999999999999992e134 < z

   1. Initial program 89.6%

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

      1. associate-/l* 92.6%

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

   3. Simplified 92.6%

      \[\leadsto \color{blue}{x + \frac{y}{\frac{a}{z - t}}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 76.9%

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

      1. +-commutative 76.9%

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

      2. associate-*l/ 84.1%

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

      3. *-commutative 84.1%

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

   7. Simplified 84.1%

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

   if -2.05e-54 < z < 9.1999999999999992e134

   1. Initial program 97.1%

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

      1. associate-/l* 95.7%

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

   3. Simplified 95.7%

      \[\leadsto \color{blue}{x + \frac{y}{\frac{a}{z - t}}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 89.9%

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

      1. associate-*r/ 89.1%

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

      2. associate-*r* 89.1%

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

      3. neg-mul-1 89.1%

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

      4. cancel-sign-sub-inv 89.1%

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

      5. associate-*r/ 89.9%

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

      6. associate-/l* 89.9%

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

      7. associate-/r/ 87.9%

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

   7. Simplified 87.9%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.05 \cdot 10^{-54} \lor \neg \left(z \leq 9.2 \cdot 10^{+134}\right):\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+141} \lor \neg \left(z \leq 1.6 \cdot 10^{+134}\right):\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - t \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.9e+141) (not (<= z 1.6e+134)))
   (+ x (* z (/ y a)))
   (- x (* t (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.9e+141) || !(z <= 1.6e+134)) {
		tmp = x + (z * (y / a));
	} else {
		tmp = x - (t * (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 ((z <= (-1.9d+141)) .or. (.not. (z <= 1.6d+134))) then
        tmp = x + (z * (y / a))
    else
        tmp = x - (t * (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 ((z <= -1.9e+141) || !(z <= 1.6e+134)) {
		tmp = x + (z * (y / a));
	} else {
		tmp = x - (t * (y / a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -1.9e+141) or not (z <= 1.6e+134):
		tmp = x + (z * (y / a))
	else:
		tmp = x - (t * (y / a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.9e+141) || !(z <= 1.6e+134))
		tmp = Float64(x + Float64(z * Float64(y / a)));
	else
		tmp = Float64(x - Float64(t * Float64(y / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -1.9e+141) || ~((z <= 1.6e+134)))
		tmp = x + (z * (y / a));
	else
		tmp = x - (t * (y / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.9e+141], N[Not[LessEqual[z, 1.6e+134]], $MachinePrecision]], N[(x + N[(z * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.89999999999999988e141 or 1.6e134 < z

   1. Initial program 87.5%

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

      1. associate-/l* 89.7%

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

   3. Simplified 89.7%

      \[\leadsto \color{blue}{x + \frac{y}{\frac{a}{z - t}}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 77.7%

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

      1. +-commutative 77.7%

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

      2. associate-*l/ 87.6%

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

      3. *-commutative 87.6%

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

   7. Simplified 87.6%

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

   if -1.89999999999999988e141 < z < 1.6e134

   1. Initial program 96.2%

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

      1. associate-/l* 96.2%

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

   3. Simplified 96.2%

      \[\leadsto \color{blue}{x + \frac{y}{\frac{a}{z - t}}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 86.3%

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

      1. associate-*r/ 87.3%

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

      2. associate-*r* 87.3%

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

      3. neg-mul-1 87.3%

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

      4. cancel-sign-sub-inv 87.3%

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

      5. associate-*r/ 86.3%

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

      6. *-commutative 86.3%

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

   7. Simplified 86.3%

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

      1. associate-/l* 85.4%

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

      2. associate-/r/ 87.3%

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

   9. Applied egg-rr 87.3%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+141} \lor \neg \left(z \leq 1.6 \cdot 10^{+134}\right):\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - t \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{+14} \lor \neg \left(z \leq 1.6 \cdot 10^{+134}\right):\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{t \cdot y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -5.6e+14) (not (<= z 1.6e+134)))
   (+ x (* z (/ y a)))
   (- x (/ (* t y) a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -5.6e+14) || !(z <= 1.6e+134)) {
		tmp = x + (z * (y / a));
	} else {
		tmp = x - ((t * 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 ((z <= (-5.6d+14)) .or. (.not. (z <= 1.6d+134))) then
        tmp = x + (z * (y / a))
    else
        tmp = x - ((t * 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 ((z <= -5.6e+14) || !(z <= 1.6e+134)) {
		tmp = x + (z * (y / a));
	} else {
		tmp = x - ((t * y) / a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -5.6e+14) or not (z <= 1.6e+134):
		tmp = x + (z * (y / a))
	else:
		tmp = x - ((t * y) / a)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -5.6e+14) || !(z <= 1.6e+134))
		tmp = Float64(x + Float64(z * Float64(y / a)));
	else
		tmp = Float64(x - Float64(Float64(t * y) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -5.6e+14) || ~((z <= 1.6e+134)))
		tmp = x + (z * (y / a));
	else
		tmp = x - ((t * y) / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -5.6e+14], N[Not[LessEqual[z, 1.6e+134]], $MachinePrecision]], N[(x + N[(z * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(t * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.6e14 or 1.6e134 < z

   1. Initial program 88.3%

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

      1. associate-/l* 92.5%

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

   3. Simplified 92.5%

      \[\leadsto \color{blue}{x + \frac{y}{\frac{a}{z - t}}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 77.3%

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

      1. +-commutative 77.3%

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

      2. associate-*l/ 85.4%

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

      3. *-commutative 85.4%

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

   7. Simplified 85.4%

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

   if -5.6e14 < z < 1.6e134

   1. Initial program 97.3%

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

      1. associate-/l* 95.5%

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

   3. Simplified 95.5%

      \[\leadsto \color{blue}{x + \frac{y}{\frac{a}{z - t}}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 89.5%

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

      1. associate-*r/ 88.8%

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

      2. associate-*r* 88.8%

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

      3. neg-mul-1 88.8%

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

      4. cancel-sign-sub-inv 88.8%

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

      5. associate-*r/ 89.5%

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

      6. *-commutative 89.5%

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

   7. Simplified 89.5%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{+14} \lor \neg \left(z \leq 1.6 \cdot 10^{+134}\right):\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{t \cdot y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 / (z - t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.6%

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

   1. associate-/l* 94.2%

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

3. Simplified 94.2%

   \[\leadsto \color{blue}{x + \frac{y}{\frac{a}{z - t}}} \]
4. Add Preprocessing

5. Final simplification 94.2%

   \[\leadsto x + \frac{y}{\frac{a}{z - t}} \]
6. Add Preprocessing

Alternative 6: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 - t) / (a / y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.6%

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

   1. *-commutative 93.6%

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

   2. associate-/l* 96.6%

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

3. Simplified 96.6%

   \[\leadsto \color{blue}{x + \frac{z - t}{\frac{a}{y}}} \]
4. Add Preprocessing

5. Final simplification 96.6%

   \[\leadsto x + \frac{z - t}{\frac{a}{y}} \]
6. Add Preprocessing

Alternative 7: 68.2% 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 93.6%

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

   1. associate-/l* 94.2%

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

3. Simplified 94.2%

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

   1. associate-/r/ 96.5%

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

   2. clear-num 95.8%

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

   3. associate-*l/ 96.6%

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

   4. div-inv 96.5%

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

   5. times-frac 93.5%

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

6. Applied egg-rr 93.5%

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

7. Taylor expanded in z around inf 67.4%

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

   1. associate-*r/ 68.9%

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

9. Simplified 68.9%

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

10. Final simplification 68.9%

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

Alternative 8: 71.0% 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 93.6%

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

   1. associate-/l* 94.2%

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

3. Simplified 94.2%

   \[\leadsto \color{blue}{x + \frac{y}{\frac{a}{z - t}}} \]
4. Add Preprocessing

5. Taylor expanded in t around 0 67.4%

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

   1. +-commutative 67.4%

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

   2. associate-*l/ 71.8%

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

   3. *-commutative 71.8%

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

7. Simplified 71.8%

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

8. Final simplification 71.8%

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

Alternative 9: 39.4% 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 93.6%

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

   1. associate-/l* 94.2%

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

3. Simplified 94.2%

   \[\leadsto \color{blue}{x + \frac{y}{\frac{a}{z - t}}} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 44.1%

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

6. Final simplification 44.1%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 99.2% accurate, 0.7× 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 2024010 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :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)))