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

Average Accuracy: 89.5% → 98.9%

Time: 11.6s

Precision: binary64

Cost: 1993

Math

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

↓

\[\begin{array}{l} t_1 := x + \frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+307} \lor \neg \left(t_1 \leq 2 \cdot 10^{+306}\right):\\ \;\;\;\;x + \frac{y}{\frac{t}{z - x}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot z - x \cdot y}{t}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y (- z x)) t))))
   (if (or (<= t_1 -5e+307) (not (<= t_1 2e+306)))
     (+ x (/ y (/ t (- z x))))
     (+ x (/ (- (* y z) (* x y)) t)))))

C

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

↓

double code(double x, double y, double z, double t) {
	double t_1 = x + ((y * (z - x)) / t);
	double tmp;
	if ((t_1 <= -5e+307) || !(t_1 <= 2e+306)) {
		tmp = x + (y / (t / (z - x)));
	} else {
		tmp = x + (((y * z) - (x * 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
    code = x + ((y * (z - x)) / t)
end function

↓

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) :: t_1
    real(8) :: tmp
    t_1 = x + ((y * (z - x)) / t)
    if ((t_1 <= (-5d+307)) .or. (.not. (t_1 <= 2d+306))) then
        tmp = x + (y / (t / (z - x)))
    else
        tmp = x + (((y * z) - (x * y)) / t)
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y, double z, double t) {
	double t_1 = x + ((y * (z - x)) / t);
	double tmp;
	if ((t_1 <= -5e+307) || !(t_1 <= 2e+306)) {
		tmp = x + (y / (t / (z - x)));
	} else {
		tmp = x + (((y * z) - (x * y)) / t);
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t):
	t_1 = x + ((y * (z - x)) / t)
	tmp = 0
	if (t_1 <= -5e+307) or not (t_1 <= 2e+306):
		tmp = x + (y / (t / (z - x)))
	else:
		tmp = x + (((y * z) - (x * y)) / t)
	return tmp

Julia

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

↓

function code(x, y, z, t)
	t_1 = Float64(x + Float64(Float64(y * Float64(z - x)) / t))
	tmp = 0.0
	if ((t_1 <= -5e+307) || !(t_1 <= 2e+306))
		tmp = Float64(x + Float64(y / Float64(t / Float64(z - x))));
	else
		tmp = Float64(x + Float64(Float64(Float64(y * z) - Float64(x * y)) / t));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z, t)
	t_1 = x + ((y * (z - x)) / t);
	tmp = 0.0;
	if ((t_1 <= -5e+307) || ~((t_1 <= 2e+306)))
		tmp = x + (y / (t / (z - x)));
	else
		tmp = x + (((y * z) - (x * y)) / t);
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x + N[(N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e+307], N[Not[LessEqual[t$95$1, 2e+306]], $MachinePrecision]], N[(x + N[(y / N[(t / N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(y * z), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Target

Original	89.5%
Target	97.2%
Herbie	98.9%

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t)) < -5e307 or 2.00000000000000003e306 < (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t))

   1. Initial program 1.9%

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

   2. Simplified 98.9%

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

      [Start] 1.9	\[ x + \frac{y \cdot \left(z - x\right)}{t} \]
      associate-/l* [=>] 98.9	\[ x + \color{blue}{\frac{y}{\frac{t}{z - x}}} \]

   if -5e307 < (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t)) < 2.00000000000000003e306

   1. Initial program 98.9%

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

   2. Applied egg-rr 98.9%

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

      [Start] 98.9	\[ x + \frac{y \cdot \left(z - x\right)}{t} \]
      sub-neg [=>] 98.9	\[ x + \frac{y \cdot \color{blue}{\left(z + \left(-x\right)\right)}}{t} \]
      distribute-rgt-in [=>] 98.9	\[ x + \frac{\color{blue}{z \cdot y + \left(-x\right) \cdot y}}{t} \]

   3. Applied egg-rr 98.9%

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

      [Start] 98.9	\[ x + \frac{z \cdot y + \left(-x\right) \cdot y}{t} \]
      distribute-lft-neg-out [=>] 98.9	\[ x + \frac{z \cdot y + \color{blue}{\left(-x \cdot y\right)}}{t} \]
      unsub-neg [=>] 98.9	\[ x + \frac{\color{blue}{z \cdot y - x \cdot y}}{t} \]
      *-commutative [=>] 98.9	\[ x + \frac{z \cdot y - \color{blue}{y \cdot x}}{t} \]

3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(z - x\right)}{t} \leq -5 \cdot 10^{+307} \lor \neg \left(x + \frac{y \cdot \left(z - x\right)}{t} \leq 2 \cdot 10^{+306}\right):\\ \;\;\;\;x + \frac{y}{\frac{t}{z - x}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot z - x \cdot y}{t}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.9%
Cost	1865

\[\begin{array}{l} t_1 := x + \frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+307} \lor \neg \left(t_1 \leq 2 \cdot 10^{+306}\right):\\ \;\;\;\;x + \frac{y}{\frac{t}{z - x}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Accuracy	96.3%
Cost	1105

\[\begin{array}{l} t_1 := x + \left(z - x\right) \cdot \frac{y}{t}\\ \mathbf{if}\;t \leq -3.6 \cdot 10^{-201}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-195} \lor \neg \left(t \leq 4 \cdot 10^{-180}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - x}{t}\\ \end{array} \]

Alternative 3
Accuracy	65.9%
Cost	978

\[\begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-37} \lor \neg \left(y \leq 4.4 \cdot 10^{-13} \lor \neg \left(y \leq 8 \cdot 10^{+24}\right) \land y \leq 1.7 \cdot 10^{+62}\right):\\ \;\;\;\;y \cdot \frac{z - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Accuracy	81.9%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{-37} \lor \neg \left(y \leq 10^{+172}\right):\\ \;\;\;\;y \cdot \frac{z - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \end{array} \]

Alternative 5
Accuracy	81.3%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{-37} \lor \neg \left(y \leq 2.1 \cdot 10^{+170}\right):\\ \;\;\;\;y \cdot \frac{z - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \end{array} \]

Alternative 6
Accuracy	97.1%
Cost	708

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-112}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z - x}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - x\right) \cdot \frac{y}{t}\\ \end{array} \]

Alternative 7
Accuracy	59.1%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -8.3 \cdot 10^{-122}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.26 \cdot 10^{-102}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Accuracy	60.2%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-153}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{-93}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Accuracy	60.1%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-154}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-99}:\\ \;\;\;\;\frac{z}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Accuracy	59.2%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-122}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-99}:\\ \;\;\;\;\frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Accuracy	51.0%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023151 
(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)))