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

Average Accuracy: 91.1% → 99.4%

Time: 16.1s

Precision: binary64

Cost: 1481

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -4 \cdot 10^{+244} \lor \neg \left(t_1 \leq 10^{+247}\right):\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot t - y \cdot z}{a}\\ \end{array} \]

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))))
   (if (or (<= t_1 -4e+244) (not (<= t_1 1e+247)))
     (- x (/ y (/ a (- z t))))
     (+ x (/ (- (* y t) (* y z)) a)))))

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);
	double tmp;
	if ((t_1 <= -4e+244) || !(t_1 <= 1e+247)) {
		tmp = x - (y / (a / (z - t)));
	} else {
		tmp = x + (((y * t) - (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
    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)
    if ((t_1 <= (-4d+244)) .or. (.not. (t_1 <= 1d+247))) then
        tmp = x - (y / (a / (z - t)))
    else
        tmp = x + (((y * t) - (y * z)) / a)
    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);
	double tmp;
	if ((t_1 <= -4e+244) || !(t_1 <= 1e+247)) {
		tmp = x - (y / (a / (z - t)));
	} else {
		tmp = x + (((y * t) - (y * z)) / a);
	}
	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)
	tmp = 0
	if (t_1 <= -4e+244) or not (t_1 <= 1e+247):
		tmp = x - (y / (a / (z - t)))
	else:
		tmp = x + (((y * t) - (y * z)) / a)
	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(y * Float64(z - t))
	tmp = 0.0
	if ((t_1 <= -4e+244) || !(t_1 <= 1e+247))
		tmp = Float64(x - Float64(y / Float64(a / Float64(z - t))));
	else
		tmp = Float64(x + Float64(Float64(Float64(y * t) - Float64(y * z)) / a));
	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);
	tmp = 0.0;
	if ((t_1 <= -4e+244) || ~((t_1 <= 1e+247)))
		tmp = x - (y / (a / (z - t)));
	else
		tmp = x + (((y * t) - (y * z)) / a);
	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[(y * N[(z - t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -4e+244], N[Not[LessEqual[t$95$1, 1e+247]], $MachinePrecision]], N[(x - N[(y / N[(a / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(y * t), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]

Target

Original	91.1%
Target	98.7%
Herbie	99.4%

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (-.f64 z t)) < -4.0000000000000003e244 or 9.99999999999999952e246 < (*.f64 y (-.f64 z t))

   1. Initial program 41.1%

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

   2. Simplified 99.4%

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

      [Start] 41.1	\[ x - \frac{y \cdot \left(z - t\right)}{a} \]
      associate-/l* [=>] 99.4	\[ x - \color{blue}{\frac{y}{\frac{a}{z - t}}} \]

   if -4.0000000000000003e244 < (*.f64 y (-.f64 z t)) < 9.99999999999999952e246

   1. Initial program 99.4%

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

   2. Applied egg-rr 99.4%

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

      [Start] 99.4	\[ x - \frac{y \cdot \left(z - t\right)}{a} \]
      sub-neg [=>] 99.4	\[ x - \frac{y \cdot \color{blue}{\left(z + \left(-t\right)\right)}}{a} \]
      distribute-rgt-in [=>] 99.4	\[ x - \frac{\color{blue}{z \cdot y + \left(-t\right) \cdot y}}{a} \]

   3. Applied egg-rr 99.4%

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

      [Start] 99.4	\[ x - \frac{z \cdot y + \left(-t\right) \cdot y}{a} \]
      distribute-lft-neg-out [=>] 99.4	\[ x - \frac{z \cdot y + \color{blue}{\left(-t \cdot y\right)}}{a} \]
      unsub-neg [=>] 99.4	\[ x - \frac{\color{blue}{z \cdot y - t \cdot y}}{a} \]
      *-commutative [=>] 99.4	\[ x - \frac{z \cdot y - \color{blue}{y \cdot t}}{a} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -4 \cdot 10^{+244} \lor \neg \left(y \cdot \left(z - t\right) \leq 10^{+247}\right):\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot t - y \cdot z}{a}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	49.5%
Cost	1376

\[\begin{array}{l} t_1 := \frac{-z}{\frac{a}{y}}\\ \mathbf{if}\;t \leq -2.7 \cdot 10^{+134}:\\ \;\;\;\;\frac{y}{\frac{a}{t}}\\ \mathbf{elif}\;t \leq -1.02 \cdot 10^{-94}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -1.85 \cdot 10^{-140}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.4 \cdot 10^{-270}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -9 \cdot 10^{-288}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{-222}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-170}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{+56}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \end{array} \]

Alternative 2
Accuracy	99.4%
Cost	1353

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -4 \cdot 10^{+244} \lor \neg \left(t_1 \leq 10^{+247}\right):\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(t - z\right)}{a}\\ \end{array} \]

Alternative 3
Accuracy	56.7%
Cost	980

\[\begin{array}{l} t_1 := y \cdot \frac{-z}{a}\\ \mathbf{if}\;x \leq -1.1 \cdot 10^{-70}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-279}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6.3 \cdot 10^{-232}:\\ \;\;\;\;\frac{y}{\frac{a}{t}}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-213}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-91}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Accuracy	56.8%
Cost	980

\[\begin{array}{l} t_1 := \frac{-y}{\frac{a}{z}}\\ \mathbf{if}\;x \leq -6.6 \cdot 10^{-71}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-284}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-235}:\\ \;\;\;\;\frac{y}{\frac{a}{t}}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-212}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-91}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Accuracy	68.0%
Cost	977

\[\begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+42}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{+19} \lor \neg \left(x \leq -3.2 \cdot 10^{-36}\right) \land x \leq 3.4 \cdot 10^{-90}:\\ \;\;\;\;y \cdot \frac{t - z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Accuracy	80.6%
Cost	977

\[\begin{array}{l} t_1 := x - \frac{y \cdot z}{a}\\ \mathbf{if}\;z \leq -4.8 \cdot 10^{+189}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{+122}:\\ \;\;\;\;y \cdot \frac{t - z}{a}\\ \mathbf{elif}\;z \leq -7 \cdot 10^{-32} \lor \neg \left(z \leq 1.8 \cdot 10^{-73}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \end{array} \]

Alternative 7
Accuracy	89.6%
Cost	841

\[\begin{array}{l} \mathbf{if}\;a \leq 6.8 \cdot 10^{-307} \lor \neg \left(a \leq 6.6 \cdot 10^{-115}\right):\\ \;\;\;\;x - y \cdot \frac{z - t}{a}\\ \mathbf{else}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \end{array} \]

Alternative 8
Accuracy	95.8%
Cost	841

\[\begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-106} \lor \neg \left(a \leq 3 \cdot 10^{-85}\right):\\ \;\;\;\;x - y \cdot \frac{z - t}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \left(z - t\right) \cdot \frac{y}{a}\\ \end{array} \]

Alternative 9
Accuracy	96.3%
Cost	840

\[\begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-80}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;a \leq 2.35 \cdot 10^{-85}:\\ \;\;\;\;x - \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z - t}{a}\\ \end{array} \]

Alternative 10
Accuracy	77.2%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-71} \lor \neg \left(x \leq 1.55 \cdot 10^{-148}\right):\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t - z}{a}\\ \end{array} \]

Alternative 11
Accuracy	77.3%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-71}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-148}:\\ \;\;\;\;y \cdot \frac{t - z}{a}\\ \mathbf{else}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \end{array} \]

Alternative 12
Accuracy	76.1%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-70}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-168}:\\ \;\;\;\;y \cdot \frac{t - z}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \end{array} \]

Alternative 13
Accuracy	56.2%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-69}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-91}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 14
Accuracy	56.3%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-70}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-91}:\\ \;\;\;\;\frac{y}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 15
Accuracy	51.7%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023151 
(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)))