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

Average Accuracy: 90.9% → 97.4%

Time: 14.1s

Precision: binary64

Cost: 1609

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t_1 \leq -0.09 \lor \neg \left(t_1 \leq 2 \cdot 10^{-109}\right):\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

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 (or (<= t_1 -0.09) (not (<= t_1 2e-109)))
     (+ x (* (- z t) (/ y a)))
     (+ x (/ y (/ a (- 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 <= -0.09) || !(t_1 <= 2e-109)) {
		tmp = x + ((z - t) * (y / a));
	} else {
		tmp = x + (y / (a / (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 <= (-0.09d0)) .or. (.not. (t_1 <= 2d-109))) then
        tmp = x + ((z - t) * (y / a))
    else
        tmp = x + (y / (a / (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 <= -0.09) || !(t_1 <= 2e-109)) {
		tmp = x + ((z - t) * (y / a));
	} else {
		tmp = x + (y / (a / (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 <= -0.09) or not (t_1 <= 2e-109):
		tmp = x + ((z - t) * (y / a))
	else:
		tmp = x + (y / (a / (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 <= -0.09) || !(t_1 <= 2e-109))
		tmp = Float64(x + Float64(Float64(z - t) * Float64(y / a)));
	else
		tmp = Float64(x + Float64(y / Float64(a / 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 <= -0.09) || ~((t_1 <= 2e-109)))
		tmp = x + ((z - t) * (y / a));
	else
		tmp = x + (y / (a / (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[Or[LessEqual[t$95$1, -0.09], N[Not[LessEqual[t$95$1, 2e-109]], $MachinePrecision]], N[(x + N[(N[(z - t), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(a / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	90.9%
Target	99.0%
Herbie	97.4%

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -0.089999999999999997 or 2e-109 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 85.4%

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

   2. Simplified 96.2%

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

      [Start] 85.4	\[ x + \frac{y \cdot \left(z - t\right)}{a} \]
      associate-*l/ [<=] 96.2	\[ x + \color{blue}{\frac{y}{a} \cdot \left(z - t\right)} \]

   if -0.089999999999999997 < (/.f64 (*.f64 y (-.f64 z t)) a) < 2e-109

   1. Initial program 99.0%

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

   2. Simplified 99.2%

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

      [Start] 99.0	\[ x + \frac{y \cdot \left(z - t\right)}{a} \]
      associate-/l* [=>] 99.2	\[ x + \color{blue}{\frac{y}{\frac{a}{z - t}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \leq -0.09 \lor \neg \left(\frac{y \cdot \left(z - t\right)}{a} \leq 2 \cdot 10^{-109}\right):\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	50.4%
Cost	1508

\[\begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{+287}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{elif}\;z \leq -4.6 \cdot 10^{+250}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -3.4 \cdot 10^{+145}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq -4.7 \cdot 10^{+102}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1920:\\ \;\;\;\;\frac{y}{\frac{a}{z}}\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-106}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-80}:\\ \;\;\;\;t \cdot \frac{-y}{a}\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+82}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+107}:\\ \;\;\;\;\frac{z}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Accuracy	51.2%
Cost	1244

\[\begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+287}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{+250}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{+144}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq -8.8 \cdot 10^{+105}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1920:\\ \;\;\;\;\frac{y}{\frac{a}{z}}\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+82}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+107}:\\ \;\;\;\;\frac{z}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Accuracy	52.1%
Cost	982

\[\begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+144} \lor \neg \left(z \leq -4.2 \cdot 10^{+103}\right) \land \left(z \leq -1920 \lor \neg \left(z \leq 4.4 \cdot 10^{+82}\right) \land z \leq 2.8 \cdot 10^{+107}\right):\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Accuracy	51.9%
Cost	980

\[\begin{array}{l} t_1 := z \cdot \frac{y}{a}\\ \mathbf{if}\;z \leq -5 \cdot 10^{+145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.6 \cdot 10^{+102}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1920:\\ \;\;\;\;\frac{y}{\frac{a}{z}}\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+82}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+107}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Accuracy	51.8%
Cost	980

\[\begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+143}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq -3.4 \cdot 10^{+102}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1920:\\ \;\;\;\;\frac{y}{\frac{a}{z}}\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+81}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+107}:\\ \;\;\;\;\frac{z}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Accuracy	74.5%
Cost	978

\[\begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-184} \lor \neg \left(x \leq 2.15 \cdot 10^{-219} \lor \neg \left(x \leq 1.2 \cdot 10^{+55}\right) \land x \leq 4.6 \cdot 10^{+96}\right):\\ \;\;\;\;x + y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{a}\\ \end{array} \]

Alternative 7
Accuracy	69.2%
Cost	977

\[\begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{+85}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-71} \lor \neg \left(x \leq 6.5 \cdot 10^{+18}\right) \land x \leq 6.4 \cdot 10^{+96}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Accuracy	65.2%
Cost	848

\[\begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+84}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-71}:\\ \;\;\;\;y \cdot \frac{z - t}{a}\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+19}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+96}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Accuracy	78.3%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-184} \lor \neg \left(x \leq 9.5 \cdot 10^{-221}\right):\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{a}\\ \end{array} \]

Alternative 10
Accuracy	85.4%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+76} \lor \neg \left(t \leq 2 \cdot 10^{-80}\right):\\ \;\;\;\;x - t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{\frac{a}{y}}\\ \end{array} \]

Alternative 11
Accuracy	78.3%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-184}:\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-227}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{\frac{a}{y}}\\ \end{array} \]

Alternative 12
Accuracy	96.5%
Cost	576

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

Alternative 13
Accuracy	52.9%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023135 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :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)))