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

Percentage Accurate: 93.3% → 99.6%

Time: 17.1s

Precision: binary64

Cost: 1736

• 24.5% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+307}:\\ \;\;\;\;x + \frac{t - z}{\frac{a}{y}}\\ \mathbf{elif}\;t_1 \leq 10^{+306}:\\ \;\;\;\;x - t_1\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{\frac{\frac{a}{y}}{z - t}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)))
   (if (<= t_1 -5e+307)
     (+ x (/ (- t z) (/ a y)))
     (if (<= t_1 1e+306) (- x t_1) (+ x (/ -1.0 (/ (/ a y) (- z t))))))))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double tmp;
	if (t_1 <= -5e+307) {
		tmp = x + ((t - z) / (a / y));
	} else if (t_1 <= 1e+306) {
		tmp = x - t_1;
	} else {
		tmp = x + (-1.0 / ((a / y) / (z - t)));
	}
	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
    code = x - ((y * (z - t)) / a)
end function

↓

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 = (y * (z - t)) / a
    if (t_1 <= (-5d+307)) then
        tmp = x + ((t - z) / (a / y))
    else if (t_1 <= 1d+306) then
        tmp = x - t_1
    else
        tmp = x + ((-1.0d0) / ((a / y) / (z - t)))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double tmp;
	if (t_1 <= -5e+307) {
		tmp = x + ((t - z) / (a / y));
	} else if (t_1 <= 1e+306) {
		tmp = x - t_1;
	} else {
		tmp = x + (-1.0 / ((a / y) / (z - t)));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t, a):
	t_1 = (y * (z - t)) / a
	tmp = 0
	if t_1 <= -5e+307:
		tmp = x + ((t - z) / (a / y))
	elif t_1 <= 1e+306:
		tmp = x - t_1
	else:
		tmp = x + (-1.0 / ((a / y) / (z - t)))
	return tmp

Julia

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

↓

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * Float64(z - t)) / a)
	tmp = 0.0
	if (t_1 <= -5e+307)
		tmp = Float64(x + Float64(Float64(t - z) / Float64(a / y)));
	elseif (t_1 <= 1e+306)
		tmp = Float64(x - t_1);
	else
		tmp = Float64(x + Float64(-1.0 / Float64(Float64(a / y) / Float64(z - t))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y * (z - t)) / a;
	tmp = 0.0;
	if (t_1 <= -5e+307)
		tmp = x + ((t - z) / (a / y));
	elseif (t_1 <= 1e+306)
		tmp = x - t_1;
	else
		tmp = x + (-1.0 / ((a / y) / (z - t)));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+307], N[(x + N[(N[(t - z), $MachinePrecision] / N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+306], N[(x - t$95$1), $MachinePrecision], N[(x + N[(-1.0 / N[(N[(a / y), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	93.3%
Target	99.2%
Herbie	99.6%

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -5e307

   1. Initial program 81.9%

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

   2. Simplified 98.2%

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

      [Start] 81.9%	\[ x - \frac{y \cdot \left(z - t\right)}{a} \]
      associate-*r/ [<=] 98.2%	\[ x - \color{blue}{y \cdot \frac{z - t}{a}} \]

   3. Applied egg-rr 100.0%

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

      [Start] 98.2%	\[ x - y \cdot \frac{z - t}{a} \]
      *-commutative [=>] 98.2%	\[ x - \color{blue}{\frac{z - t}{a} \cdot y} \]
      associate-*l/ [=>] 81.9%	\[ x - \color{blue}{\frac{\left(z - t\right) \cdot y}{a}} \]
      associate-/l* [=>] 100.0%	\[ x - \color{blue}{\frac{z - t}{\frac{a}{y}}} \]

   if -5e307 < (/.f64 (*.f64 y (-.f64 z t)) a) < 1.00000000000000002e306

   1. Initial program 99.8%

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

   if 1.00000000000000002e306 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 79.4%

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

   2. Simplified 98.0%

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

      [Start] 79.4%	\[ x - \frac{y \cdot \left(z - t\right)}{a} \]
      associate-*r/ [<=] 98.0%	\[ x - \color{blue}{y \cdot \frac{z - t}{a}} \]

   3. Applied egg-rr 99.9%

      \[\leadsto x - \color{blue}{\frac{1}{\frac{\frac{a}{y}}{z - t}}} \]
      Step-by-step derivation

      [Start] 98.0%	\[ x - y \cdot \frac{z - t}{a} \]
      associate-*r/ [=>] 79.4%	\[ x - \color{blue}{\frac{y \cdot \left(z - t\right)}{a}} \]
      clear-num [=>] 79.4%	\[ x - \color{blue}{\frac{1}{\frac{a}{y \cdot \left(z - t\right)}}} \]
      associate-/r* [=>] 99.9%	\[ x - \frac{1}{\color{blue}{\frac{\frac{a}{y}}{z - t}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \leq -5 \cdot 10^{+307}:\\ \;\;\;\;x + \frac{t - z}{\frac{a}{y}}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 10^{+306}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{\frac{\frac{a}{y}}{z - t}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.6%
Cost	1736

\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+307}:\\ \;\;\;\;x + \frac{t - z}{\frac{a}{y}}\\ \mathbf{elif}\;t_1 \leq 10^{+306}:\\ \;\;\;\;x - t_1\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{\frac{\frac{a}{y}}{z - t}}\\ \end{array} \]

Alternative 2
Accuracy	99.6%
Cost	1608

\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+307}:\\ \;\;\;\;x - \frac{z - t}{\frac{a}{y}}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;x - t_1\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{a} \cdot \left(t - z\right)\\ \end{array} \]

Alternative 3
Accuracy	99.6%
Cost	1353

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+240} \lor \neg \left(t_1 \leq 2 \cdot 10^{+302}\right):\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{t_1}{a}\\ \end{array} \]

Alternative 4
Accuracy	66.1%
Cost	977

\[\begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+209}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1 \cdot 10^{+108} \lor \neg \left(x \leq -1.7 \cdot 10^{+91}\right) \land x \leq 5.2 \cdot 10^{+33}:\\ \;\;\;\;\frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Accuracy	92.4%
Cost	841

\[\begin{array}{l} \mathbf{if}\;a \leq -1.4 \cdot 10^{-99} \lor \neg \left(a \leq 5 \cdot 10^{-179}\right):\\ \;\;\;\;x - y \cdot \frac{z - t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(t - z\right)\\ \end{array} \]

Alternative 6
Accuracy	48.9%
Cost	780

\[\begin{array}{l} \mathbf{if}\;t \leq -1060000000000:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-242}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-36}:\\ \;\;\;\;y \cdot \frac{-z}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \end{array} \]

Alternative 7
Accuracy	79.6%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -4200000000000 \lor \neg \left(z \leq 4.4 \cdot 10^{+70}\right):\\ \;\;\;\;\frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{else}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \end{array} \]

Alternative 8
Accuracy	84.2%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -8.8 \cdot 10^{-21} \lor \neg \left(t \leq 1.6 \cdot 10^{-37}\right):\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z}{a}\\ \end{array} \]

Alternative 9
Accuracy	84.4%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{+21} \lor \neg \left(t \leq 1.06 \cdot 10^{-36}\right):\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z}}\\ \end{array} \]

Alternative 10
Accuracy	84.1%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -1.14 \cdot 10^{+24} \lor \neg \left(t \leq 2.6 \cdot 10^{-37}\right):\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot z}{a}\\ \end{array} \]

Alternative 11
Accuracy	47.7%
Cost	648

\[\begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{+23}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-36}:\\ \;\;\;\;z \cdot \left(-\frac{y}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \end{array} \]

Alternative 12
Accuracy	47.5%
Cost	648

\[\begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{+24}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{-38}:\\ \;\;\;\;\frac{-z}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \end{array} \]

Alternative 13
Accuracy	50.9%
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -4300000000000 \lor \neg \left(t \leq 4800\right):\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 14
Accuracy	50.8%
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -19000000000000:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;t \leq 0.063:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \end{array} \]

Alternative 15
Accuracy	97.3%
Cost	576

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

Alternative 16
Accuracy	40.1%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023263 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"
  :precision binary64

  :herbie-target
  (if (< y -1.0761266216389975e-10) (- x (/ 1.0 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (- x (/ (* y (- z t)) a)) (- x (/ y (/ a (- z t))))))

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