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

Average Accuracy: 90.3% → 98.1%

Time: 14.3s

Precision: binary64

Cost: 1352

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+212}:\\ \;\;\;\;x + \frac{z - t}{\frac{a}{y}}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+53}:\\ \;\;\;\;x + \frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

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))))
   (if (<= t_1 -1e+212)
     (+ x (/ (- z t) (/ a y)))
     (if (<= t_1 2e+53) (+ x (/ t_1 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);
	double tmp;
	if (t_1 <= -1e+212) {
		tmp = x + ((z - t) / (a / y));
	} else if (t_1 <= 2e+53) {
		tmp = x + (t_1 / 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)
    if (t_1 <= (-1d+212)) then
        tmp = x + ((z - t) / (a / y))
    else if (t_1 <= 2d+53) then
        tmp = x + (t_1 / 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);
	double tmp;
	if (t_1 <= -1e+212) {
		tmp = x + ((z - t) / (a / y));
	} else if (t_1 <= 2e+53) {
		tmp = x + (t_1 / 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)
	tmp = 0
	if t_1 <= -1e+212:
		tmp = x + ((z - t) / (a / y))
	elif t_1 <= 2e+53:
		tmp = x + (t_1 / 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(y * Float64(z - t))
	tmp = 0.0
	if (t_1 <= -1e+212)
		tmp = Float64(x + Float64(Float64(z - t) / Float64(a / y)));
	elseif (t_1 <= 2e+53)
		tmp = Float64(x + Float64(t_1 / 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);
	tmp = 0.0;
	if (t_1 <= -1e+212)
		tmp = x + ((z - t) / (a / y));
	elseif (t_1 <= 2e+53)
		tmp = x + (t_1 / 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[(y * N[(z - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+212], N[(x + N[(N[(z - t), $MachinePrecision] / N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+53], N[(x + N[(t$95$1 / a), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(a / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	90.3%
Target	98.9%
Herbie	98.1%

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (-.f64 z t)) < -9.9999999999999991e211

   1. Initial program 52.2%

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

   2. Simplified 98.9%

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

      [Start] 52.2	\[ x + \frac{y \cdot \left(z - t\right)}{a} \]
      associate-*l/ [<=] 98.9	\[ x + \color{blue}{\frac{y}{a} \cdot \left(z - t\right)} \]

   3. Applied egg-rr 98.9%

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

   if -9.9999999999999991e211 < (*.f64 y (-.f64 z t)) < 2e53

   1. Initial program 99.3%

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

   if 2e53 < (*.f64 y (-.f64 z t))

   1. Initial program 80.3%

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

   2. Simplified 94.2%

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

      [Start] 80.3	\[ x + \frac{y \cdot \left(z - t\right)}{a} \]
      associate-/l* [=>] 94.2	\[ x + \color{blue}{\frac{y}{\frac{a}{z - t}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -1 \cdot 10^{+212}:\\ \;\;\;\;x + \frac{z - t}{\frac{a}{y}}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq 2 \cdot 10^{+53}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	96.2%
Cost	1357

\[\begin{array}{l} \mathbf{if}\;z - t \leq -4 \cdot 10^{+142} \lor \neg \left(z - t \leq 5 \cdot 10^{+53}\right) \land z - t \leq 5 \cdot 10^{+247}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Alternative 2
Accuracy	98.0%
Cost	1353

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+179} \lor \neg \left(t_1 \leq 2 \cdot 10^{+53}\right):\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t_1}{a}\\ \end{array} \]

Alternative 3
Accuracy	70.3%
Cost	976

\[\begin{array}{l} t_1 := \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{+23}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-289}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.25 \cdot 10^{-115}:\\ \;\;\;\;y \cdot \frac{z - t}{a}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+26}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Accuracy	55.0%
Cost	848

\[\begin{array}{l} t_1 := z \cdot \frac{y}{a}\\ \mathbf{if}\;x \leq -7.8 \cdot 10^{-165}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{-284}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-126}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{+24}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Accuracy	78.5%
Cost	844

\[\begin{array}{l} t_1 := x + \frac{z}{\frac{a}{y}}\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{-75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-289}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-170}:\\ \;\;\;\;y \cdot \frac{z - t}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Accuracy	94.0%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-289} \lor \neg \left(x \leq 1.15 \cdot 10^{-168}\right):\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(z - t\right) \cdot \frac{1}{a}\right)\\ \end{array} \]

Alternative 7
Accuracy	84.9%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{-26} \lor \neg \left(z \leq 2.2 \cdot 10^{-81}\right):\\ \;\;\;\;x + \frac{z}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - t \cdot \frac{y}{a}\\ \end{array} \]

Alternative 8
Accuracy	85.0%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{-26} \lor \neg \left(z \leq 2.2 \cdot 10^{-81}\right):\\ \;\;\;\;x + \frac{z}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{t}{\frac{a}{y}}\\ \end{array} \]

Alternative 9
Accuracy	68.4%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+22}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 170000000000:\\ \;\;\;\;y \cdot \frac{z - t}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Accuracy	55.9%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-151}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-146}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Accuracy	55.4%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-157}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+22}:\\ \;\;\;\;\frac{z}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12
Accuracy	51.2%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023141 
(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)))