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

Percentage Accurate: 92.9% → 98.0%

Time: 9.8s

Alternatives: 8

Speedup: 1.0×

• 25.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.2%

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

   1. associate-*l/ 97.3%

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

3. Simplified 97.3%

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

   1. *-commutative 97.3%

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

   2. clear-num 97.3%

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

   3. un-div-inv 97.3%

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

6. Applied egg-rr 97.3%

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

7. Final simplification 97.3%

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

Alternative 2: 86.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.3 \cdot 10^{+24} \lor \neg \left(x \leq 1.15 \cdot 10^{+71}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -5.3e+24) (not (<= x 1.15e+71)))
   (* x (- 1.0 (/ y t)))
   (+ x (* z (/ y t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -5.3e+24) || !(x <= 1.15e+71)) {
		tmp = x * (1.0 - (y / t));
	} else {
		tmp = x + (z * (y / t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-5.3d+24)) .or. (.not. (x <= 1.15d+71))) then
        tmp = x * (1.0d0 - (y / t))
    else
        tmp = x + (z * (y / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -5.3e+24) || !(x <= 1.15e+71)) {
		tmp = x * (1.0 - (y / t));
	} else {
		tmp = x + (z * (y / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -5.3e+24) or not (x <= 1.15e+71):
		tmp = x * (1.0 - (y / t))
	else:
		tmp = x + (z * (y / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -5.3e+24) || !(x <= 1.15e+71))
		tmp = Float64(x * Float64(1.0 - Float64(y / t)));
	else
		tmp = Float64(x + Float64(z * Float64(y / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -5.3e+24) || ~((x <= 1.15e+71)))
		tmp = x * (1.0 - (y / t));
	else
		tmp = x + (z * (y / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -5.3e+24], N[Not[LessEqual[x, 1.15e+71]], $MachinePrecision]], N[(x * N[(1.0 - N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.2999999999999999e24 or 1.1500000000000001e71 < x

   1. Initial program 87.9%

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

      1. associate-*l/ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 93.3%

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

      1. mul-1-neg 93.3%

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

      2. unsub-neg 93.3%

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

   7. Simplified 93.3%

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

   if -5.2999999999999999e24 < x < 1.1500000000000001e71

   1. Initial program 95.2%

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

      1. associate-*l/ 95.3%

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

   3. Simplified 95.3%

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

   5. Taylor expanded in z around inf 80.6%

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

      1. associate-*l/ 84.1%

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

      2. *-commutative 84.1%

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

   7. Simplified 84.1%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.3 \cdot 10^{+24} \lor \neg \left(x \leq 1.15 \cdot 10^{+71}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{+26} \lor \neg \left(x \leq 1.2 \cdot 10^{+71}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{\frac{t}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.25e+26) (not (<= x 1.2e+71)))
   (* x (- 1.0 (/ y t)))
   (+ x (/ z (/ t y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.25e+26) || !(x <= 1.2e+71)) {
		tmp = x * (1.0 - (y / t));
	} else {
		tmp = x + (z / (t / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-2.25d+26)) .or. (.not. (x <= 1.2d+71))) then
        tmp = x * (1.0d0 - (y / t))
    else
        tmp = x + (z / (t / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.25e+26) || !(x <= 1.2e+71)) {
		tmp = x * (1.0 - (y / t));
	} else {
		tmp = x + (z / (t / y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -2.25e+26) or not (x <= 1.2e+71):
		tmp = x * (1.0 - (y / t))
	else:
		tmp = x + (z / (t / y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.25e+26) || !(x <= 1.2e+71))
		tmp = Float64(x * Float64(1.0 - Float64(y / t)));
	else
		tmp = Float64(x + Float64(z / Float64(t / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -2.25e+26) || ~((x <= 1.2e+71)))
		tmp = x * (1.0 - (y / t));
	else
		tmp = x + (z / (t / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.25e+26], N[Not[LessEqual[x, 1.2e+71]], $MachinePrecision]], N[(x * N[(1.0 - N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(z / N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.24999999999999989e26 or 1.1999999999999999e71 < x

   1. Initial program 87.9%

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

      1. associate-*l/ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 93.3%

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

      1. mul-1-neg 93.3%

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

      2. unsub-neg 93.3%

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

   7. Simplified 93.3%

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

   if -2.24999999999999989e26 < x < 1.1999999999999999e71

   1. Initial program 95.2%

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

      1. associate-*l/ 95.3%

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

   3. Simplified 95.3%

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

   5. Taylor expanded in z around inf 80.6%

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

      1. associate-*l/ 84.1%

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

      2. *-commutative 84.1%

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

   7. Simplified 84.1%

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

      1. clear-num 84.1%

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

      2. div-inv 84.2%

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

   9. Applied egg-rr 84.2%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{+26} \lor \neg \left(x \leq 1.2 \cdot 10^{+71}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{\frac{t}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+26}:\\ \;\;\;\;x - \frac{x}{\frac{t}{y}}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+72}:\\ \;\;\;\;x + \frac{z}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.1e+26)
   (- x (/ x (/ t y)))
   (if (<= x 4.2e+72) (+ x (/ z (/ t y))) (* x (- 1.0 (/ y t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.1e+26) {
		tmp = x - (x / (t / y));
	} else if (x <= 4.2e+72) {
		tmp = x + (z / (t / y));
	} else {
		tmp = x * (1.0 - (y / t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-1.1d+26)) then
        tmp = x - (x / (t / y))
    else if (x <= 4.2d+72) then
        tmp = x + (z / (t / y))
    else
        tmp = x * (1.0d0 - (y / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.1e+26) {
		tmp = x - (x / (t / y));
	} else if (x <= 4.2e+72) {
		tmp = x + (z / (t / y));
	} else {
		tmp = x * (1.0 - (y / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.1e+26:
		tmp = x - (x / (t / y))
	elif x <= 4.2e+72:
		tmp = x + (z / (t / y))
	else:
		tmp = x * (1.0 - (y / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.1e+26)
		tmp = Float64(x - Float64(x / Float64(t / y)));
	elseif (x <= 4.2e+72)
		tmp = Float64(x + Float64(z / Float64(t / y)));
	else
		tmp = Float64(x * Float64(1.0 - Float64(y / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.1e+26)
		tmp = x - (x / (t / y));
	elseif (x <= 4.2e+72)
		tmp = x + (z / (t / y));
	else
		tmp = x * (1.0 - (y / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.1e+26], N[(x - N[(x / N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.2e+72], N[(x + N[(z / N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.10000000000000004e26

   1. Initial program 84.3%

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

   3. Taylor expanded in z around 0 77.2%

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

      1. mul-1-neg 77.2%

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

      2. distribute-lft-neg-out 77.2%

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

      3. *-commutative 77.2%

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

   5. Simplified 77.2%

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

      1. div-inv 77.2%

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

      2. add-sqr-sqrt 77.1%

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

      3. sqrt-unprod 56.6%

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

      4. sqr-neg 56.6%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 35.2%

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

      7. remove-double-neg 35.2%

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

      8. distribute-rgt-neg-out 35.2%

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

      9. cancel-sign-sub-inv 35.2%

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

      10. *-commutative 35.2%

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

      11. associate-*l* 42.0%

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

      12. add-sqr-sqrt 42.0%

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

      13. sqrt-unprod 15.8%

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

      14. sqr-neg 15.8%

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

      15. sqrt-unprod 0.0%

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

      16. add-sqr-sqrt 91.2%

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

      17. div-inv 91.2%

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

      18. clear-num 91.2%

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

      19. div-inv 91.2%

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

   7. Applied egg-rr 91.2%

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

   if -1.10000000000000004e26 < x < 4.2000000000000003e72

   1. Initial program 95.2%

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

      1. associate-*l/ 95.3%

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

   3. Simplified 95.3%

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

   5. Taylor expanded in z around inf 80.6%

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

      1. associate-*l/ 84.1%

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

      2. *-commutative 84.1%

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

   7. Simplified 84.1%

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

      1. clear-num 84.1%

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

      2. div-inv 84.2%

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

   9. Applied egg-rr 84.2%

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

   if 4.2000000000000003e72 < x

   1. Initial program 92.3%

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

      1. associate-*l/ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 95.9%

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

      1. mul-1-neg 95.9%

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

      2. unsub-neg 95.9%

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

   7. Simplified 95.9%

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{y}{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+26}:\\ \;\;\;\;x - \frac{x}{\frac{t}{y}}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+72}:\\ \;\;\;\;x + \frac{z}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{-27} \lor \neg \left(y \leq 7.6 \cdot 10^{+83}\right):\\ \;\;\;\;\left(-y\right) \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -3.15e-27) (not (<= y 7.6e+83))) (* (- y) (/ x t)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -3.15e-27) || !(y <= 7.6e+83)) {
		tmp = -y * (x / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y <= (-3.15d-27)) .or. (.not. (y <= 7.6d+83))) then
        tmp = -y * (x / t)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -3.15e-27) || !(y <= 7.6e+83)) {
		tmp = -y * (x / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -3.15e-27) or not (y <= 7.6e+83):
		tmp = -y * (x / t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -3.15e-27) || !(y <= 7.6e+83))
		tmp = Float64(Float64(-y) * Float64(x / t));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -3.15e-27) || ~((y <= 7.6e+83)))
		tmp = -y * (x / t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -3.15e-27], N[Not[LessEqual[y, 7.6e+83]], $MachinePrecision]], N[((-y) * N[(x / t), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -3.15000000000000005e-27 or 7.6000000000000004e83 < y

   1. Initial program 87.2%

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

   3. Taylor expanded in z around 0 54.5%

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

      1. mul-1-neg 54.5%

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

      2. distribute-lft-neg-out 54.5%

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

      3. *-commutative 54.5%

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

   5. Simplified 54.5%

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

      1. div-inv 54.4%

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

      2. add-sqr-sqrt 28.2%

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

      3. sqrt-unprod 30.0%

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

      4. sqr-neg 30.0%

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

      5. sqrt-unprod 5.8%

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

      6. add-sqr-sqrt 10.0%

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

      7. remove-double-neg 10.0%

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

      8. distribute-rgt-neg-out 10.0%

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

      9. cancel-sign-sub-inv 10.0%

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

      10. *-commutative 10.0%

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

      11. associate-*l* 17.3%

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

      12. add-sqr-sqrt 8.2%

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

      13. sqrt-unprod 27.5%

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

      14. sqr-neg 27.5%

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

      15. sqrt-unprod 27.8%

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

      16. add-sqr-sqrt 61.2%

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

      17. div-inv 61.1%

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

      18. clear-num 61.2%

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

      19. div-inv 61.2%

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

   7. Applied egg-rr 61.2%

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

      1. associate-/r/ 63.2%

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

   9. Applied egg-rr 63.2%

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

   10. Taylor expanded in t around 0 48.0%

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

       1. mul-1-neg 48.0%

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

       2. associate-*l/ 51.4%

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

       3. distribute-rgt-neg-out 51.4%

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

       4. *-commutative 51.4%

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

   12. Simplified 51.4%

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

   if -3.15000000000000005e-27 < y < 7.6000000000000004e83

   1. Initial program 96.9%

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

      1. associate-*l/ 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in y around 0 57.2%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{-27} \lor \neg \left(y \leq 7.6 \cdot 10^{+83}\right):\\ \;\;\;\;\left(-y\right) \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.2%

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

   1. associate-*l/ 97.3%

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

3. Simplified 97.3%

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

5. Final simplification 97.3%

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

Alternative 7: 65.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 - \frac{y}{t}\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* x (- 1.0 (/ y t))))

C

double code(double x, double y, double z, double t) {
	return x * (1.0 - (y / t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x * (1.0d0 - (y / t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x * (1.0 - (y / t));
}

Python

def code(x, y, z, t):
	return x * (1.0 - (y / t))

Julia

function code(x, y, z, t)
	return Float64(x * Float64(1.0 - Float64(y / t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x * (1.0 - (y / t));
end

Wolfram

code[x_, y_, z_, t_] := N[(x * N[(1.0 - N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.2%

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

   1. associate-*l/ 97.3%

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

3. Simplified 97.3%

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

5. Taylor expanded in x around inf 65.3%

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

   1. mul-1-neg 65.3%

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

   2. unsub-neg 65.3%

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

7. Simplified 65.3%

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

8. Final simplification 65.3%

   \[\leadsto x \cdot \left(1 - \frac{y}{t}\right) \]
9. Add Preprocessing

Alternative 8: 38.6% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y, z, t):
	return x

Julia

function code(x, y, z, t)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := x

Derivation

1. Initial program 92.2%

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

   1. associate-*l/ 97.3%

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

3. Simplified 97.3%

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

5. Taylor expanded in y around 0 35.8%

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

6. Final simplification 35.8%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 91.1% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024018 
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D"
  :precision binary64

  :herbie-target
  (- x (+ (* x (/ y t)) (* (- z) (/ y t))))

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