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

Percentage Accurate: 93.3% → 97.2%

Time: 9.9s

Alternatives: 8

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: 93.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.2% 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 94.1%

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

   1. associate-/l* 91.8%

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

3. Simplified 91.8%

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

5. Taylor expanded in y around 0 94.1%

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

   1. *-commutative 94.1%

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

   2. associate-*r/ 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: 92.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{+176} \lor \neg \left(z \leq 2.2 \cdot 10^{+112}\right):\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -2.25e+176) (not (<= z 2.2e+112)))
   (+ x (* z (/ y a)))
   (+ x (* y (/ (- z t) a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -2.25e+176) || !(z <= 2.2e+112)) {
		tmp = x + (z * (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 ((z <= (-2.25d+176)) .or. (.not. (z <= 2.2d+112))) then
        tmp = x + (z * (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 ((z <= -2.25e+176) || !(z <= 2.2e+112)) {
		tmp = x + (z * (y / a));
	} else {
		tmp = x + (y * ((z - t) / a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -2.25e+176) or not (z <= 2.2e+112):
		tmp = x + (z * (y / a))
	else:
		tmp = x + (y * ((z - t) / a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -2.25e+176) || !(z <= 2.2e+112))
		tmp = Float64(x + Float64(z * 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 ((z <= -2.25e+176) || ~((z <= 2.2e+112)))
		tmp = x + (z * (y / a));
	else
		tmp = x + (y * ((z - t) / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -2.25e+176], N[Not[LessEqual[z, 2.2e+112]], $MachinePrecision]], N[(x + N[(z * 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 z < -2.25000000000000002e176 or 2.1999999999999999e112 < z

   1. Initial program 94.6%

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

      1. associate-/l* 80.8%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in y around 0 94.6%

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

      1. *-commutative 94.6%

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

      2. associate-*r/ 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in z around inf 93.2%

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

      1. associate-*l/ 97.1%

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

      2. *-commutative 97.1%

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

   10. Simplified 97.1%

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

   if -2.25000000000000002e176 < z < 2.1999999999999999e112

   1. Initial program 94.0%

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

      1. associate-/l* 95.8%

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

   3. Simplified 95.8%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{+176} \lor \neg \left(z \leq 2.2 \cdot 10^{+112}\right):\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{+65} \lor \neg \left(z \leq 8.2 \cdot 10^{-16}\right):\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - t \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -2.9e+65) (not (<= z 8.2e-16)))
   (+ 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 <= -2.9e+65) || !(z <= 8.2e-16)) {
		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 <= (-2.9d+65)) .or. (.not. (z <= 8.2d-16))) 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 <= -2.9e+65) || !(z <= 8.2e-16)) {
		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 <= -2.9e+65) or not (z <= 8.2e-16):
		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 <= -2.9e+65) || !(z <= 8.2e-16))
		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 <= -2.9e+65) || ~((z <= 8.2e-16)))
		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, -2.9e+65], N[Not[LessEqual[z, 8.2e-16]], $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 < -2.9e65 or 8.20000000000000012e-16 < z

   1. Initial program 93.2%

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

      1. associate-/l* 88.5%

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

   3. Simplified 88.5%

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

   5. Taylor expanded in y around 0 93.2%

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

      1. *-commutative 93.2%

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

      2. associate-*r/ 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in z around inf 87.5%

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

      1. associate-*l/ 92.7%

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

      2. *-commutative 92.7%

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

   10. Simplified 92.7%

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

   if -2.9e65 < z < 8.20000000000000012e-16

   1. Initial program 95.0%

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

      1. associate-/l* 94.9%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in y around 0 95.0%

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

      1. *-commutative 95.0%

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

      2. associate-*r/ 97.0%

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

   7. Simplified 97.0%

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

   8. Taylor expanded in z around 0 88.5%

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

      1. associate-*r/ 88.5%

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

      2. neg-mul-1 88.5%

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

      3. distribute-rgt-neg-in 88.5%

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

      4. associate-*r/ 91.9%

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

   10. Simplified 91.9%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{+65} \lor \neg \left(z \leq 8.2 \cdot 10^{-16}\right):\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - t \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 50.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-52} \lor \neg \left(y \leq 4.4 \cdot 10^{+26}\right):\\ \;\;\;\;\frac{y}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= y -2.4e-52) (not (<= y 4.4e+26))) (/ y (/ a z)) x))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((y <= -2.4e-52) || !(y <= 4.4e+26)) {
		tmp = y / (a / z);
	} else {
		tmp = x;
	}
	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 <= (-2.4d-52)) .or. (.not. (y <= 4.4d+26))) then
        tmp = y / (a / z)
    else
        tmp = x
    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 <= -2.4e-52) || !(y <= 4.4e+26)) {
		tmp = y / (a / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (y <= -2.4e-52) or not (y <= 4.4e+26):
		tmp = y / (a / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((y <= -2.4e-52) || !(y <= 4.4e+26))
		tmp = Float64(y / Float64(a / z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((y <= -2.4e-52) || ~((y <= 4.4e+26)))
		tmp = y / (a / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[y, -2.4e-52], N[Not[LessEqual[y, 4.4e+26]], $MachinePrecision]], N[(y / N[(a / z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -2.4000000000000002e-52 or 4.40000000000000014e26 < y

   1. Initial program 89.4%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around inf 56.5%

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

      1. associate-/l* 61.4%

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

   7. Simplified 61.4%

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

      1. *-commutative 61.4%

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

      2. frac-2neg 61.4%

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

      3. distribute-frac-neg2 61.4%

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

      4. add-sqr-sqrt 27.8%

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

      5. sqrt-unprod 37.4%

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

      6. sqr-neg 37.4%

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

      7. sqrt-unprod 7.4%

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

      8. add-sqr-sqrt 15.5%

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

      9. frac-2neg 15.5%

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

      10. cancel-sign-sub-inv 15.5%

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

      11. div-inv 15.5%

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

      12. associate-*l* 19.0%

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

      13. associate-/r/ 19.0%

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

      14. clear-num 19.0%

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

   9. Applied egg-rr 19.0%

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

       1. *-commutative 19.0%

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

       2. associate-*l/ 14.2%

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

       3. associate-/l* 15.5%

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

   11. Simplified 15.5%

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

   12. Taylor expanded in x around 0 3.1%

       \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot z}{a}} \]
   13. Step-by-step derivation

       1. mul-1-neg 3.1%

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

       2. distribute-neg-frac2 3.1%

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

       3. associate-*r/ 3.1%

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

   14. Simplified 3.1%

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

       1. clear-num 3.1%

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

       2. un-div-inv 3.1%

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

       3. add-sqr-sqrt 1.5%

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

       4. sqrt-unprod 24.8%

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

       5. sqr-neg 24.8%

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

       6. sqrt-unprod 21.2%

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

       7. add-sqr-sqrt 49.0%

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

   16. Applied egg-rr 49.0%

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

   if -2.4000000000000002e-52 < y < 4.40000000000000014e26

   1. Initial program 99.2%

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

      1. associate-/l* 83.1%

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

   3. Simplified 83.1%

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

   5. Taylor expanded in z around inf 80.2%

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

      1. associate-/l* 68.8%

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

   7. Simplified 68.8%

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

   8. Taylor expanded in x around inf 62.9%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-52} \lor \neg \left(y \leq 4.4 \cdot 10^{+26}\right):\\ \;\;\;\;\frac{y}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.65 \cdot 10^{-56} \lor \neg \left(y \leq 3.8 \cdot 10^{+26}\right):\\ \;\;\;\;\frac{z}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= y -3.65e-56) (not (<= y 3.8e+26))) (/ z (/ a y)) x))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((y <= -3.65e-56) || !(y <= 3.8e+26)) {
		tmp = z / (a / y);
	} else {
		tmp = x;
	}
	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 <= (-3.65d-56)) .or. (.not. (y <= 3.8d+26))) then
        tmp = z / (a / y)
    else
        tmp = x
    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 <= -3.65e-56) || !(y <= 3.8e+26)) {
		tmp = z / (a / y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (y <= -3.65e-56) or not (y <= 3.8e+26):
		tmp = z / (a / y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((y <= -3.65e-56) || !(y <= 3.8e+26))
		tmp = Float64(z / Float64(a / y));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((y <= -3.65e-56) || ~((y <= 3.8e+26)))
		tmp = z / (a / y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[y, -3.65e-56], N[Not[LessEqual[y, 3.8e+26]], $MachinePrecision]], N[(z / N[(a / y), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -3.65000000000000022e-56 or 3.8000000000000002e26 < y

   1. Initial program 89.4%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around inf 56.5%

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

      1. associate-/l* 61.4%

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

   7. Simplified 61.4%

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

      1. *-commutative 61.4%

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

      2. frac-2neg 61.4%

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

      3. distribute-frac-neg2 61.4%

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

      4. add-sqr-sqrt 27.8%

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

      5. sqrt-unprod 37.4%

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

      6. sqr-neg 37.4%

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

      7. sqrt-unprod 7.4%

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

      8. add-sqr-sqrt 15.5%

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

      9. frac-2neg 15.5%

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

      10. cancel-sign-sub-inv 15.5%

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

      11. div-inv 15.5%

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

      12. associate-*l* 19.0%

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

      13. associate-/r/ 19.0%

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

      14. clear-num 19.0%

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

   9. Applied egg-rr 19.0%

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

       1. *-commutative 19.0%

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

       2. associate-*l/ 14.2%

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

       3. associate-/l* 15.5%

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

   11. Simplified 15.5%

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

   12. Taylor expanded in x around 0 3.1%

       \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot z}{a}} \]
   13. Step-by-step derivation

       1. mul-1-neg 3.1%

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

       2. distribute-neg-frac2 3.1%

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

       3. associate-*r/ 3.1%

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

   14. Simplified 3.1%

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

       1. associate-*r/ 3.1%

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

       2. add-sqr-sqrt 1.5%

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

       3. sqrt-unprod 24.0%

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

       4. sqr-neg 24.0%

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

       5. sqrt-unprod 19.8%

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

       6. add-sqr-sqrt 45.5%

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

       7. associate-*l/ 51.1%

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

       8. *-commutative 51.1%

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

       9. clear-num 51.0%

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

       10. div-inv 51.0%

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

   16. Applied egg-rr 51.0%

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

   if -3.65000000000000022e-56 < y < 3.8000000000000002e26

   1. Initial program 99.2%

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

      1. associate-/l* 83.1%

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

   3. Simplified 83.1%

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

   5. Taylor expanded in z around inf 80.2%

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

      1. associate-/l* 68.8%

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

   7. Simplified 68.8%

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

   8. Taylor expanded in x around inf 62.9%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.65 \cdot 10^{-56} \lor \neg \left(y \leq 3.8 \cdot 10^{+26}\right):\\ \;\;\;\;\frac{z}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

   1. associate-/l* 91.8%

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

3. Simplified 91.8%

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

5. Taylor expanded in z around inf 67.9%

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

   1. associate-/l* 65.0%

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

7. Simplified 65.0%

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

8. Final simplification 65.0%

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

Alternative 7: 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 94.1%

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

   1. associate-/l* 91.8%

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

3. Simplified 91.8%

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

5. Taylor expanded in y around 0 94.1%

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

   1. *-commutative 94.1%

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

   2. associate-*r/ 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 inf 67.9%

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

   1. associate-*l/ 71.5%

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

   2. *-commutative 71.5%

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

10. Simplified 71.5%

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

11. Final simplification 71.5%

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

Alternative 8: 39.7% 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 94.1%

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

   1. associate-/l* 91.8%

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

3. Simplified 91.8%

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

5. Taylor expanded in z around inf 67.9%

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

   1. associate-/l* 65.0%

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

7. Simplified 65.0%

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

8. Taylor expanded in x around inf 38.0%

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

9. Final simplification 38.0%

   \[\leadsto x \]
10. Add Preprocessing

Developer target: 99.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024077 
(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)))