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

Average Accuracy: 89.7% → 97.7%

Time: 9.9s

Precision: binary64

Cost: 1864

Math

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

↓

\[\begin{array}{l} t_1 := x + \frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{if}\;t_1 \leq -5.5 \cdot 10^{+307}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z - x}}\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-241}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - x\right) \cdot \frac{y}{t}\\ \end{array} \]

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 (<= t_1 -5.5e+307)
     (+ x (/ y (/ t (- z x))))
     (if (<= t_1 -1e-241) t_1 (+ x (* (- 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 <= -5.5e+307) {
		tmp = x + (y / (t / (z - x)));
	} else if (t_1 <= -1e-241) {
		tmp = t_1;
	} else {
		tmp = x + ((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 <= (-5.5d+307)) then
        tmp = x + (y / (t / (z - x)))
    else if (t_1 <= (-1d-241)) then
        tmp = t_1
    else
        tmp = x + ((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 <= -5.5e+307) {
		tmp = x + (y / (t / (z - x)));
	} else if (t_1 <= -1e-241) {
		tmp = t_1;
	} else {
		tmp = x + ((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 <= -5.5e+307:
		tmp = x + (y / (t / (z - x)))
	elif t_1 <= -1e-241:
		tmp = t_1
	else:
		tmp = x + ((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 <= -5.5e+307)
		tmp = Float64(x + Float64(y / Float64(t / Float64(z - x))));
	elseif (t_1 <= -1e-241)
		tmp = t_1;
	else
		tmp = Float64(x + Float64(Float64(z - x) * Float64(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 <= -5.5e+307)
		tmp = x + (y / (t / (z - x)));
	elseif (t_1 <= -1e-241)
		tmp = t_1;
	else
		tmp = x + ((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[LessEqual[t$95$1, -5.5e+307], N[(x + N[(y / N[(t / N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1e-241], t$95$1, N[(x + N[(N[(z - x), $MachinePrecision] * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	89.7%
Target	96.6%
Herbie	97.7%

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

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t)) < -5.49999999999999986e307

   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 -5.49999999999999986e307 < (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t)) < -9.9999999999999997e-242

   1. Initial program 99.2%

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

   if -9.9999999999999997e-242 < (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t))

   1. Initial program 89.5%

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

   2. Simplified 96.4%

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

      [Start] 89.5	\[ x + \frac{y \cdot \left(z - x\right)}{t} \]
      associate-*l/ [<=] 96.4	\[ x + \color{blue}{\frac{y}{t} \cdot \left(z - x\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(z - x\right)}{t} \leq -5.5 \cdot 10^{+307}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z - x}}\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \leq -1 \cdot 10^{-241}:\\ \;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - x\right) \cdot \frac{y}{t}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	62.0%
Cost	1176

\[\begin{array}{l} t_1 := y \cdot \frac{z - x}{t}\\ \mathbf{if}\;x \leq -13500000000000:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-211}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.22 \cdot 10^{-281}:\\ \;\;\;\;\frac{y \cdot z}{t}\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{+50}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+69}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+85}:\\ \;\;\;\;\frac{y}{t} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Accuracy	56.4%
Cost	912

\[\begin{array}{l} \mathbf{if}\;x \leq -48000000000000:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+45}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+70}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+85}:\\ \;\;\;\;\frac{y}{t} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Accuracy	85.6%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{-86} \lor \neg \left(z \leq 3.7 \cdot 10^{-236}\right):\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot \frac{y}{t}\\ \end{array} \]

Alternative 4
Accuracy	85.7%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{-86} \lor \neg \left(z \leq 3.7 \cdot 10^{-236}\right):\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{\frac{t}{y}}\\ \end{array} \]

Alternative 5
Accuracy	57.4%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+16}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-148}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Accuracy	56.8%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -11500000000000:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+45}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Accuracy	82.2%
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+88}:\\ \;\;\;\;y \cdot \frac{z - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \end{array} \]

Alternative 8
Accuracy	96.6%
Cost	576

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

Alternative 9
Accuracy	50.2%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023141 
(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)))