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

Percentage Accurate: 93.1% → 97.1%

Time: 7.1s

Alternatives: 7

Speedup: 1.0×

• 24.8% 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: 93.1% 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: 97.1% 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 91.8%

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

   1. *-commutative 91.8%

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

   2. associate-/l* 97.0%

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

3. Simplified 97.0%

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

5. Final simplification 97.0%

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

Alternative 2: 84.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{+86} \lor \neg \left(z \leq 1.1 \cdot 10^{+46}\right):\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{t}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -4.1e+86) (not (<= z 1.1e+46)))
   (+ 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 <= -4.1e+86) || !(z <= 1.1e+46)) {
		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 <= (-4.1d+86)) .or. (.not. (z <= 1.1d+46))) 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 <= -4.1e+86) || !(z <= 1.1e+46)) {
		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 <= -4.1e+86) or not (z <= 1.1e+46):
		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 <= -4.1e+86) || !(z <= 1.1e+46))
		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 <= -4.1e+86) || ~((z <= 1.1e+46)))
		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, -4.1e+86], N[Not[LessEqual[z, 1.1e+46]], $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 < -4.0999999999999999e86 or 1.1e46 < z

   1. Initial program 91.3%

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

      1. associate-/l* 91.8%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in z around inf 84.6%

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

      1. associate-/r/ 89.4%

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

   7. Applied egg-rr 89.4%

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

   if -4.0999999999999999e86 < z < 1.1e46

   1. Initial program 92.2%

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

      1. +-commutative 92.2%

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

      2. associate-*l/ 96.1%

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

      3. fma-def 96.1%

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

   3. Simplified 96.1%

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

   5. Taylor expanded in z around 0 85.6%

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

      1. mul-1-neg 85.6%

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

      2. associate-*l/ 88.7%

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

      3. unsub-neg 88.7%

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

      4. *-commutative 88.7%

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

   7. Simplified 88.7%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{+86} \lor \neg \left(z \leq 1.1 \cdot 10^{+46}\right):\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{+86} \lor \neg \left(z \leq 7 \cdot 10^{+41}\right):\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - t \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -4.1e+86) (not (<= z 7e+41)))
   (+ 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 <= -4.1e+86) || !(z <= 7e+41)) {
		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 <= (-4.1d+86)) .or. (.not. (z <= 7d+41))) 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 <= -4.1e+86) || !(z <= 7e+41)) {
		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 <= -4.1e+86) or not (z <= 7e+41):
		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 <= -4.1e+86) || !(z <= 7e+41))
		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 <= -4.1e+86) || ~((z <= 7e+41)))
		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, -4.1e+86], N[Not[LessEqual[z, 7e+41]], $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 < -4.0999999999999999e86 or 6.9999999999999998e41 < z

   1. Initial program 91.3%

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

      1. associate-/l* 91.8%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in z around inf 84.6%

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

      1. associate-/r/ 89.4%

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

   7. Applied egg-rr 89.4%

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

   if -4.0999999999999999e86 < z < 6.9999999999999998e41

   1. Initial program 92.2%

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

      1. +-commutative 92.2%

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

      2. associate-*l/ 96.1%

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

      3. fma-def 96.1%

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

   3. Simplified 96.1%

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

   5. Taylor expanded in z around 0 85.6%

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

      1. mul-1-neg 85.6%

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

      2. associate-*l/ 88.7%

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

      3. unsub-neg 88.7%

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

      4. *-commutative 88.7%

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

   7. Simplified 88.7%

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

      1. *-commutative 88.7%

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

      2. associate-/r/ 91.3%

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

   9. Applied egg-rr 91.3%

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

       1. *-un-lft-identity 91.3%

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

       2. associate-*l/ 90.6%

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

       3. clear-num 90.7%

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

   11. Applied egg-rr 90.7%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{+86} \lor \neg \left(z \leq 7 \cdot 10^{+41}\right):\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - t \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+87} \lor \neg \left(z \leq 8.2 \cdot 10^{+43}\right):\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{t}{\frac{a}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -2.7e+87) (not (<= z 8.2e+43)))
   (+ x (* z (/ y a)))
   (- x (/ t (/ a y)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -2.7e+87) || !(z <= 8.2e+43)) {
		tmp = x + (z * (y / a));
	} else {
		tmp = x - (t / (a / y));
	}
	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 <= (-2.7d+87)) .or. (.not. (z <= 8.2d+43))) then
        tmp = x + (z * (y / a))
    else
        tmp = x - (t / (a / y))
    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 <= -2.7e+87) || !(z <= 8.2e+43)) {
		tmp = x + (z * (y / a));
	} else {
		tmp = x - (t / (a / y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -2.7e+87) or not (z <= 8.2e+43):
		tmp = x + (z * (y / a))
	else:
		tmp = x - (t / (a / y))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -2.7e+87) || !(z <= 8.2e+43))
		tmp = Float64(x + Float64(z * Float64(y / a)));
	else
		tmp = Float64(x - Float64(t / Float64(a / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -2.7e+87) || ~((z <= 8.2e+43)))
		tmp = x + (z * (y / a));
	else
		tmp = x - (t / (a / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -2.7e+87], N[Not[LessEqual[z, 8.2e+43]], $MachinePrecision]], N[(x + N[(z * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(t / N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.70000000000000007e87 or 8.2000000000000001e43 < z

   1. Initial program 91.3%

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

      1. associate-/l* 91.8%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in z around inf 84.6%

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

      1. associate-/r/ 89.4%

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

   7. Applied egg-rr 89.4%

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

   if -2.70000000000000007e87 < z < 8.2000000000000001e43

   1. Initial program 92.2%

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

      1. +-commutative 92.2%

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

      2. associate-*l/ 96.1%

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

      3. fma-def 96.1%

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

   3. Simplified 96.1%

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

   5. Taylor expanded in z around 0 85.6%

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

      1. mul-1-neg 85.6%

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

      2. associate-*l/ 88.7%

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

      3. unsub-neg 88.7%

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

      4. *-commutative 88.7%

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

   7. Simplified 88.7%

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

      1. *-commutative 88.7%

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

      2. associate-/r/ 91.3%

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

   9. Applied egg-rr 91.3%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+87} \lor \neg \left(z \leq 8.2 \cdot 10^{+43}\right):\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{t}{\frac{a}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.8%

   \[x + \frac{y \cdot \left(z - t\right)}{a} \]
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/ 96.6%

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

6. Applied egg-rr 96.6%

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

7. Final simplification 96.6%

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

Alternative 6: 71.3% 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 91.8%

   \[x + \frac{y \cdot \left(z - t\right)}{a} \]
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 z around inf 63.7%

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

   1. associate-/r/ 66.6%

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

7. Applied egg-rr 66.6%

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

8. Final simplification 66.6%

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

Alternative 7: 40.0% 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 91.8%

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

   1. +-commutative 91.8%

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

   2. associate-*l/ 96.6%

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

   3. fma-def 96.6%

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

3. Simplified 96.6%

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

5. Taylor expanded in y around 0 34.8%

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

6. Final simplification 34.8%

   \[\leadsto x \]
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 2024011 
(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)))