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

Percentage Accurate: 93.2% → 96.5%

Time: 12.1s

Alternatives: 10

Speedup: 1.0×

• 25.0% 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: 96.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(z - t\right) \cdot y}{a}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;x + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* (- z t) y) a)))
   (if (<= t_1 (- INFINITY)) (+ x (/ y (/ a (- z t)))) (+ x t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((z - t) * y) / a;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = x + (y / (a / (z - t)));
	} else {
		tmp = x + t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((z - t) * y) / a;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = x + (y / (a / (z - t)));
	} else {
		tmp = x + t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((z - t) * y) / a
	tmp = 0
	if t_1 <= -math.inf:
		tmp = x + (y / (a / (z - t)))
	else:
		tmp = x + t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(z - t) * y) / a)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(x + Float64(y / Float64(a / Float64(z - t))));
	else
		tmp = Float64(x + t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((z - t) * y) / a;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = x + (y / (a / (z - t)));
	else
		tmp = x + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(x + N[(y / N[(a / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -inf.0

   1. Initial program 82.1%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 82.1%

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

      1. associate-*l/ 99.9%

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

      2. associate-/r/ 100.0%

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

   7. Simplified 100.0%

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

   if -inf.0 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 97.7%

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

4. Final simplification 98.0%

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

Alternative 2: 82.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{-161} \lor \neg \left(t \leq 1.7 \cdot 10^{-44}\right):\\ \;\;\;\;x - t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{\frac{a}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -1.25e-161) (not (<= t 1.7e-44)))
   (- x (* t (/ y a)))
   (+ x (/ z (/ a y)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -1.25e-161) || !(t <= 1.7e-44)) {
		tmp = x - (t * (y / a));
	} else {
		tmp = x + (z / (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 ((t <= (-1.25d-161)) .or. (.not. (t <= 1.7d-44))) then
        tmp = x - (t * (y / a))
    else
        tmp = x + (z / (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 ((t <= -1.25e-161) || !(t <= 1.7e-44)) {
		tmp = x - (t * (y / a));
	} else {
		tmp = x + (z / (a / y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (t <= -1.25e-161) or not (t <= 1.7e-44):
		tmp = x - (t * (y / a))
	else:
		tmp = x + (z / (a / y))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -1.25e-161) || !(t <= 1.7e-44))
		tmp = Float64(x - Float64(t * Float64(y / a)));
	else
		tmp = Float64(x + Float64(z / Float64(a / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((t <= -1.25e-161) || ~((t <= 1.7e-44)))
		tmp = x - (t * (y / a));
	else
		tmp = x + (z / (a / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -1.25e-161], N[Not[LessEqual[t, 1.7e-44]], $MachinePrecision]], N[(x - N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(z / N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.25e-161 or 1.70000000000000008e-44 < t

   1. Initial program 94.6%

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

      1. +-commutative 94.6%

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

      2. associate-/l* 92.2%

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

      3. fma-define 92.2%

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

   3. Simplified 92.2%

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

      1. fma-undefine 92.2%

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

      2. associate-*r/ 94.6%

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

      3. *-commutative 94.6%

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

      4. associate-/l* 96.8%

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

   6. Applied egg-rr 96.8%

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

   7. Taylor expanded in z around 0 83.3%

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

      1. mul-1-neg 83.3%

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

      2. associate-*r/ 84.8%

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

      3. distribute-lft-neg-in 84.8%

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

      4. cancel-sign-sub-inv 84.8%

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

   9. Simplified 84.8%

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

   if -1.25e-161 < t < 1.70000000000000008e-44

   1. Initial program 96.0%

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

      1. +-commutative 96.0%

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

      2. associate-/l* 96.9%

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

      3. fma-define 97.0%

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

   3. Simplified 97.0%

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

      1. fma-undefine 96.9%

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

      2. associate-*r/ 96.0%

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

      3. *-commutative 96.0%

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

      4. associate-/l* 96.9%

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

   6. Applied egg-rr 96.9%

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

   7. Taylor expanded in z around inf 91.8%

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

      1. clear-num 91.8%

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

      2. un-div-inv 92.6%

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

   9. Applied egg-rr 92.6%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{-161} \lor \neg \left(t \leq 1.7 \cdot 10^{-44}\right):\\ \;\;\;\;x - t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{\frac{a}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{-161}:\\ \;\;\;\;x - t \cdot \frac{y}{a}\\ \mathbf{elif}\;t \leq 10^{-44}:\\ \;\;\;\;x + \frac{z}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{t}{\frac{a}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.25e-161)
   (- x (* t (/ y a)))
   (if (<= t 1e-44) (+ x (/ z (/ a y))) (- x (/ t (/ a y))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.25e-161) {
		tmp = x - (t * (y / a));
	} else if (t <= 1e-44) {
		tmp = x + (z / (a / y));
	} 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 (t <= (-1.25d-161)) then
        tmp = x - (t * (y / a))
    else if (t <= 1d-44) then
        tmp = x + (z / (a / y))
    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 (t <= -1.25e-161) {
		tmp = x - (t * (y / a));
	} else if (t <= 1e-44) {
		tmp = x + (z / (a / y));
	} else {
		tmp = x - (t / (a / y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -1.25e-161:
		tmp = x - (t * (y / a))
	elif t <= 1e-44:
		tmp = x + (z / (a / y))
	else:
		tmp = x - (t / (a / y))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.25e-161)
		tmp = Float64(x - Float64(t * Float64(y / a)));
	elseif (t <= 1e-44)
		tmp = Float64(x + Float64(z / Float64(a / y)));
	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 (t <= -1.25e-161)
		tmp = x - (t * (y / a));
	elseif (t <= 1e-44)
		tmp = x + (z / (a / y));
	else
		tmp = x - (t / (a / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.25e-161], N[(x - N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1e-44], N[(x + N[(z / N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(t / N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.25e-161

   1. Initial program 94.9%

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

      1. +-commutative 94.9%

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

      2. associate-/l* 91.7%

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

      3. fma-define 91.7%

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

   3. Simplified 91.7%

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

      1. fma-undefine 91.7%

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

      2. associate-*r/ 94.9%

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

      3. *-commutative 94.9%

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

      4. associate-/l* 96.9%

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

   6. Applied egg-rr 96.9%

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

   7. Taylor expanded in z around 0 85.1%

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

      1. mul-1-neg 85.1%

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

      2. associate-*r/ 87.0%

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

      3. distribute-lft-neg-in 87.0%

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

      4. cancel-sign-sub-inv 87.0%

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

   9. Simplified 87.0%

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

   if -1.25e-161 < t < 9.99999999999999953e-45

   1. Initial program 96.0%

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

      1. +-commutative 96.0%

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

      2. associate-/l* 96.9%

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

      3. fma-define 97.0%

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

   3. Simplified 97.0%

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

      1. fma-undefine 96.9%

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

      2. associate-*r/ 96.0%

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

      3. *-commutative 96.0%

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

      4. associate-/l* 96.9%

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

   6. Applied egg-rr 96.9%

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

   7. Taylor expanded in z around inf 91.8%

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

      1. clear-num 91.8%

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

      2. un-div-inv 92.6%

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

   9. Applied egg-rr 92.6%

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

   if 9.99999999999999953e-45 < t

   1. Initial program 94.2%

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

      1. +-commutative 94.2%

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

      2. associate-/l* 92.9%

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

      3. fma-define 92.9%

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

   3. Simplified 92.9%

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

      1. fma-undefine 92.9%

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

      2. associate-*r/ 94.2%

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

      3. *-commutative 94.2%

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

      4. associate-/l* 96.6%

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

   6. Applied egg-rr 96.6%

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

   7. Taylor expanded in z around 0 80.7%

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

      1. mul-1-neg 80.7%

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

      2. associate-*r/ 81.6%

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

      3. distribute-lft-neg-in 81.6%

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

      4. cancel-sign-sub-inv 81.6%

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

   9. Simplified 81.6%

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

       1. clear-num 81.6%

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

       2. div-inv 81.6%

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

   11. Applied egg-rr 81.6%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{-161}:\\ \;\;\;\;x - t \cdot \frac{y}{a}\\ \mathbf{elif}\;t \leq 10^{-44}:\\ \;\;\;\;x + \frac{z}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{t}{\frac{a}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.1%

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

   1. +-commutative 95.1%

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

   2. associate-/l* 94.0%

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

   3. fma-define 94.0%

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

3. Simplified 94.0%

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

   1. fma-undefine 94.0%

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

   2. associate-*r/ 95.1%

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

   3. *-commutative 95.1%

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

   4. associate-/l* 96.8%

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

6. Applied egg-rr 96.8%

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

   1. clear-num 96.5%

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

   2. inv-pow 96.5%

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

8. Applied egg-rr 96.5%

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

   1. unpow-1 96.5%

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

   2. un-div-inv 96.9%

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

10. Applied egg-rr 96.9%

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

Alternative 5: 97.1% 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 95.1%

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

   1. +-commutative 95.1%

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

   2. associate-/l* 94.0%

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

   3. fma-define 94.0%

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

3. Simplified 94.0%

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

   1. fma-undefine 94.0%

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

   2. associate-*r/ 95.1%

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

   3. *-commutative 95.1%

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

   4. associate-/l* 96.8%

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

6. Applied egg-rr 96.8%

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

7. Final simplification 96.8%

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

Alternative 6: 94.2% 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 95.1%

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

   1. associate-/l* 94.0%

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

3. Simplified 94.0%

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

5. Taylor expanded in y around 0 95.1%

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

   1. associate-*l/ 96.8%

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

   2. associate-/r/ 94.8%

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

7. Simplified 94.8%

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

Alternative 7: 93.6% accurate, 1.0× speedup

Math

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

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(y * Float64(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[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.1%

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

   1. associate-/l* 94.0%

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

3. Simplified 94.0%

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

Alternative 8: 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 95.1%

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

   1. +-commutative 95.1%

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

   2. associate-/l* 94.0%

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

   3. fma-define 94.0%

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

3. Simplified 94.0%

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

   1. fma-undefine 94.0%

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

   2. associate-*r/ 95.1%

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

   3. *-commutative 95.1%

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

   4. associate-/l* 96.8%

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

6. Applied egg-rr 96.8%

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

7. Taylor expanded in z around inf 71.2%

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

8. Final simplification 71.2%

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

Alternative 9: 67.7% accurate, 1.3× speedup

Math

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

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(Float64(z * 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[(N[(z * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.1%

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

   1. associate-/l* 94.0%

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

3. Simplified 94.0%

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

5. Taylor expanded in z around inf 68.1%

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

6. Final simplification 68.1%

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

Alternative 10: 39.0% 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 95.1%

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

   1. associate-/l* 94.0%

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

3. Simplified 94.0%

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

5. Taylor expanded in x around inf 40.2%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Developer Target 1: 99.3% 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 2024170 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -430450648655599/4000000000000000000000000) (+ x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2894426862792089/10000000000000000000000000000000000000000000000000000000000000000) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t)))))))

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