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

Percentage Accurate: 93.1% → 97.4%

Time: 10.1s

Alternatives: 9

Speedup: 1.0×

• 25.3% 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: 97.4% 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 93.3%

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

   1. associate-/l* 93.0%

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

3. Simplified 93.0%

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

5. Taylor expanded in y around 0 93.3%

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

   1. associate-*l/ 98.4%

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

   2. *-commutative 98.4%

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

7. Simplified 98.4%

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

8. Final simplification 98.4%

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

Alternative 2: 84.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+18}:\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+133}:\\ \;\;\;\;x - t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -7.8e+18)
   (+ x (* z (/ y a)))
   (if (<= z 1.6e+133) (- x (* t (/ y a))) (+ x (* y (/ z a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7.8e+18) {
		tmp = x + (z * (y / a));
	} else if (z <= 1.6e+133) {
		tmp = x - (t * (y / a));
	} else {
		tmp = x + (y * (z / 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 <= (-7.8d+18)) then
        tmp = x + (z * (y / a))
    else if (z <= 1.6d+133) then
        tmp = x - (t * (y / a))
    else
        tmp = x + (y * (z / 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 <= -7.8e+18) {
		tmp = x + (z * (y / a));
	} else if (z <= 1.6e+133) {
		tmp = x - (t * (y / a));
	} else {
		tmp = x + (y * (z / a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -7.8e+18:
		tmp = x + (z * (y / a))
	elif z <= 1.6e+133:
		tmp = x - (t * (y / a))
	else:
		tmp = x + (y * (z / a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -7.8e+18)
		tmp = Float64(x + Float64(z * Float64(y / a)));
	elseif (z <= 1.6e+133)
		tmp = Float64(x - Float64(t * Float64(y / a)));
	else
		tmp = Float64(x + Float64(y * Float64(z / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -7.8e+18)
		tmp = x + (z * (y / a));
	elseif (z <= 1.6e+133)
		tmp = x - (t * (y / a));
	else
		tmp = x + (y * (z / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -7.8e+18], N[(x + N[(z * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.6e+133], N[(x - N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.8e18

   1. Initial program 84.7%

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

      1. associate-/l* 91.5%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in y around 0 84.7%

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

      1. associate-*l/ 99.7%

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

      2. *-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in t around 0 81.4%

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

      1. +-commutative 81.4%

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

      2. associate-*l/ 93.0%

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

      3. *-commutative 93.0%

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

   10. Simplified 93.0%

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

   if -7.8e18 < z < 1.59999999999999999e133

   1. Initial program 95.4%

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

      1. associate-/l* 93.8%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in y around 0 95.4%

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

      1. associate-*l/ 98.4%

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

      2. *-commutative 98.4%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in z around 0 83.8%

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

      1. mul-1-neg 83.8%

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

      2. sub-neg 83.8%

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

      3. associate-/l* 87.5%

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

   10. Simplified 87.5%

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

   if 1.59999999999999999e133 < z

   1. Initial program 97.0%

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

      1. associate-/l* 91.4%

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

   3. Simplified 91.4%

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

   5. Taylor expanded in t around 0 85.2%

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

      1. +-commutative 85.2%

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

      2. associate-/l* 85.4%

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

   7. Simplified 85.4%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+18}:\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+133}:\\ \;\;\;\;x - t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+22}:\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+133}:\\ \;\;\;\;x - \frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.4e+22)
   (+ x (* z (/ y a)))
   (if (<= z 1.65e+133) (- x (/ t (/ a y))) (+ x (* y (/ z a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.4e+22) {
		tmp = x + (z * (y / a));
	} else if (z <= 1.65e+133) {
		tmp = x - (t / (a / y));
	} else {
		tmp = x + (y * (z / 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 <= (-1.4d+22)) then
        tmp = x + (z * (y / a))
    else if (z <= 1.65d+133) then
        tmp = x - (t / (a / y))
    else
        tmp = x + (y * (z / 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 <= -1.4e+22) {
		tmp = x + (z * (y / a));
	} else if (z <= 1.65e+133) {
		tmp = x - (t / (a / y));
	} else {
		tmp = x + (y * (z / a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.4e+22:
		tmp = x + (z * (y / a))
	elif z <= 1.65e+133:
		tmp = x - (t / (a / y))
	else:
		tmp = x + (y * (z / a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.4e+22)
		tmp = Float64(x + Float64(z * Float64(y / a)));
	elseif (z <= 1.65e+133)
		tmp = Float64(x - Float64(t / Float64(a / y)));
	else
		tmp = Float64(x + Float64(y * Float64(z / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.4e+22)
		tmp = x + (z * (y / a));
	elseif (z <= 1.65e+133)
		tmp = x - (t / (a / y));
	else
		tmp = x + (y * (z / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.4e+22], N[(x + N[(z * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.65e+133], N[(x - N[(t / N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.4e22

   1. Initial program 84.7%

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

      1. associate-/l* 91.5%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in y around 0 84.7%

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

      1. associate-*l/ 99.7%

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

      2. *-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in t around 0 81.4%

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

      1. +-commutative 81.4%

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

      2. associate-*l/ 93.0%

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

      3. *-commutative 93.0%

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

   10. Simplified 93.0%

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

   if -1.4e22 < z < 1.65e133

   1. Initial program 95.4%

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

      1. associate-/l* 93.8%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in z around 0 83.8%

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

      1. mul-1-neg 83.8%

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

      2. unsub-neg 83.8%

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

      3. *-commutative 83.8%

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

      4. associate-/l* 83.3%

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

   7. Simplified 83.3%

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

      1. associate-*r/ 83.8%

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

      2. associate-*l/ 87.5%

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

      3. clear-num 87.5%

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

      4. associate-*l/ 87.5%

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

      5. *-un-lft-identity 87.5%

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

   9. Applied egg-rr 87.5%

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

   if 1.65e133 < z

   1. Initial program 97.0%

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

      1. associate-/l* 91.4%

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

   3. Simplified 91.4%

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

   5. Taylor expanded in t around 0 85.2%

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

      1. +-commutative 85.2%

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

      2. associate-/l* 85.4%

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

   7. Simplified 85.4%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+22}:\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+133}:\\ \;\;\;\;x - \frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{+167}:\\ \;\;\;\;x - t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.9e+167) (- x (* t (/ y a))) (+ x (* y (/ (- z t) a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.9e+167) {
		tmp = x - (t * (y / a));
	} else {
		tmp = x + (y * ((z - 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 (t <= (-1.9d+167)) then
        tmp = x - (t * (y / a))
    else
        tmp = x + (y * ((z - 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 (t <= -1.9e+167) {
		tmp = x - (t * (y / a));
	} else {
		tmp = x + (y * ((z - t) / a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -1.9e+167:
		tmp = x - (t * (y / a))
	else:
		tmp = x + (y * ((z - t) / a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.9e+167)
		tmp = Float64(x - Float64(t * Float64(y / a)));
	else
		tmp = Float64(x + Float64(y * Float64(Float64(z - t) / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -1.9e+167)
		tmp = x - (t * (y / a));
	else
		tmp = x + (y * ((z - t) / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.9e+167], N[(x - N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.89999999999999997e167

   1. Initial program 82.6%

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

      1. associate-/l* 70.6%

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

   3. Simplified 70.6%

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

   5. Taylor expanded in y around 0 82.6%

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

      1. associate-*l/ 98.9%

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

      2. *-commutative 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in z around 0 78.0%

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

      1. mul-1-neg 78.0%

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

      2. sub-neg 78.0%

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

      3. associate-/l* 90.3%

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

   10. Simplified 90.3%

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

   if -1.89999999999999997e167 < t

   1. Initial program 94.3%

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

      1. associate-/l* 95.1%

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

   3. Simplified 95.1%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{+167}:\\ \;\;\;\;x - t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+82}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z \cdot y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= y -5e+82) (+ x (/ y (/ a z))) (+ x (/ (* z y) a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -5e+82) {
		tmp = x + (y / (a / z));
	} else {
		tmp = x + ((z * y) / a);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (y <= (-5d+82)) then
        tmp = x + (y / (a / z))
    else
        tmp = x + ((z * y) / a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -5e+82) {
		tmp = x + (y / (a / z));
	} else {
		tmp = x + ((z * y) / a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if y <= -5e+82:
		tmp = x + (y / (a / z))
	else:
		tmp = x + ((z * y) / a)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -5e+82)
		tmp = Float64(x + Float64(y / Float64(a / z)));
	else
		tmp = Float64(x + Float64(Float64(z * y) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (y <= -5e+82)
		tmp = x + (y / (a / z));
	else
		tmp = x + ((z * y) / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, -5e+82], N[(x + N[(y / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(z * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.00000000000000015e82

   1. Initial program 80.7%

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

      1. clear-num 99.9%

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

      2. un-div-inv 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in z around inf 71.5%

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

   if -5.00000000000000015e82 < y

   1. Initial program 96.2%

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

      1. associate-/l* 91.4%

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

   3. Simplified 91.4%

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

   5. Taylor expanded in z around inf 70.7%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+82}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z \cdot y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-74}:\\ \;\;\;\;x + y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z \cdot y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= y -2e-74) (+ x (* y (/ z a))) (+ x (/ (* z y) a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -2e-74) {
		tmp = x + (y * (z / a));
	} else {
		tmp = x + ((z * y) / a);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (y <= (-2d-74)) then
        tmp = x + (y * (z / a))
    else
        tmp = x + ((z * y) / a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -2e-74) {
		tmp = x + (y * (z / a));
	} else {
		tmp = x + ((z * y) / a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if y <= -2e-74:
		tmp = x + (y * (z / a))
	else:
		tmp = x + ((z * y) / a)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -2e-74)
		tmp = Float64(x + Float64(y * Float64(z / a)));
	else
		tmp = Float64(x + Float64(Float64(z * y) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (y <= -2e-74)
		tmp = x + (y * (z / a));
	else
		tmp = x + ((z * y) / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, -2e-74], N[(x + N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(z * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.99999999999999992e-74

   1. Initial program 87.4%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   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 t around 0 64.1%

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

      1. +-commutative 64.1%

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

      2. associate-/l* 71.9%

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

   7. Simplified 71.9%

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

   if -1.99999999999999992e-74 < y

   1. Initial program 96.0%

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

      1. associate-/l* 89.7%

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

   3. Simplified 89.7%

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

   5. Taylor expanded in z around inf 70.8%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-74}:\\ \;\;\;\;x + y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z \cdot y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

   1. associate-/l* 93.0%

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

3. Simplified 93.0%

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

   1. clear-num 93.0%

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

   2. un-div-inv 93.1%

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

6. Applied egg-rr 93.1%

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

7. Taylor expanded in z around inf 68.7%

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

8. Final simplification 68.7%

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

Alternative 8: 70.7% 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 93.3%

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

   1. associate-/l* 93.0%

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

3. Simplified 93.0%

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

5. Taylor expanded in y around 0 93.3%

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

   1. associate-*l/ 98.4%

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

   2. *-commutative 98.4%

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

7. Simplified 98.4%

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

8. Taylor expanded in t around 0 68.7%

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

   1. +-commutative 68.7%

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

   2. associate-*l/ 71.8%

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

   3. *-commutative 71.8%

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

10. Simplified 71.8%

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

11. Final simplification 71.8%

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

Alternative 9: 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 93.3%

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

   1. associate-/l* 93.0%

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

3. Simplified 93.0%

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

5. Taylor expanded in x around inf 39.3%

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

6. Final simplification 39.3%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 99.2% 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 2024059 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :precision binary64

  :alt
  (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)))