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

Average Error: 6.2 → 0.7

Time: 11.6s

Precision: binary64

Cost: 1864

Math

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

↓

\[\begin{array}{l} t_1 := x + \frac{y}{\frac{t}{z - x}}\\ t_2 := x + \frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (/ t (- z x))))) (t_2 (+ x (/ (* y (- z x)) t))))
   (if (<= t_2 (- INFINITY)) t_1 (if (<= t_2 5e+307) t_2 t_1))))

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 / (t / (z - x)));
	double t_2 = x + ((y * (z - x)) / t);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 5e+307) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

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 / (t / (z - x)));
	double t_2 = x + ((y * (z - x)) / t);
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= 5e+307) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	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 / (t / (z - x)))
	t_2 = x + ((y * (z - x)) / t)
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= 5e+307:
		tmp = t_2
	else:
		tmp = t_1
	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(y / Float64(t / Float64(z - x))))
	t_2 = Float64(x + Float64(Float64(y * Float64(z - x)) / t))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 5e+307)
		tmp = t_2;
	else
		tmp = t_1;
	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 / (t / (z - x)));
	t_2 = x + ((y * (z - x)) / t);
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= 5e+307)
		tmp = t_2;
	else
		tmp = t_1;
	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[(y / N[(t / N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 5e+307], t$95$2, t$95$1]]]]

Target

Original	6.2
Target	2.0
Herbie	0.7

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t)) < -inf.0 or 5e307 < (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t))

   1. Initial program 63.6

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

   2. Taylor expanded in z around 0 63.6

      \[\leadsto x + \color{blue}{\left(\frac{y \cdot z}{t} + -1 \cdot \frac{y \cdot x}{t}\right)} \]

   3. Simplified 0.2

      \[\leadsto x + \color{blue}{\frac{y}{\frac{t}{z - x}}} \]
      Proof
      (/.f64 y (/.f64 t (-.f64 z x))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 y t) (-.f64 z x))): 52 points increase in error, 75 points decrease in error
      (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 (/.f64 y t) z) (*.f64 (/.f64 y t) x))): 3 points increase in error, 2 points decrease in error
      (-.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 y z) t)) (*.f64 (/.f64 y t) x)): 40 points increase in error, 19 points decrease in error
      (-.f64 (/.f64 (*.f64 y z) t) (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 y x) t))): 38 points increase in error, 30 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 (*.f64 y z) t) (neg.f64 (/.f64 (*.f64 y x) t)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (*.f64 y z) t) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 y x) t)))): 0 points increase in error, 0 points decrease in error

   if -inf.0 < (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t)) < 5e307

   1. Initial program 0.7

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.7

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

Alternatives

Alternative 1
Error	18.9
Cost	1240

\[\begin{array}{l} t_1 := x - \frac{x}{\frac{t}{y}}\\ \mathbf{if}\;x \leq -3.733380478835523 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.1088060373500884 \cdot 10^{-24}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;x \leq -9.367692764829999 \cdot 10^{-97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.836244289550675 \cdot 10^{-187}:\\ \;\;\;\;\frac{z}{\frac{t}{y}}\\ \mathbf{elif}\;x \leq -4.2913794126638345 \cdot 10^{-230}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.5909742138466043 \cdot 10^{-159}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	18.9
Cost	1240

\[\begin{array}{l} t_1 := x - \frac{x}{\frac{t}{y}}\\ t_2 := x - x \cdot \frac{y}{t}\\ \mathbf{if}\;x \leq -3.733380478835523 \cdot 10^{-5}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.1088060373500884 \cdot 10^{-24}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;x \leq -9.367692764829999 \cdot 10^{-97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.836244289550675 \cdot 10^{-187}:\\ \;\;\;\;\frac{z}{\frac{t}{y}}\\ \mathbf{elif}\;x \leq -4.2913794126638345 \cdot 10^{-230}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.5909742138466043 \cdot 10^{-159}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	27.7
Cost	1112

\[\begin{array}{l} t_1 := \frac{z}{\frac{t}{y}}\\ \mathbf{if}\;x \leq -3.733380478835523 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -6.106572808413942 \cdot 10^{-81}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.0661811553637774 \cdot 10^{-99}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.836244289550675 \cdot 10^{-187}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -4.2913794126638345 \cdot 10^{-230}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.7544209452101333 \cdot 10^{-133}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	27.8
Cost	1112

\[\begin{array}{l} t_1 := \frac{z}{\frac{t}{y}}\\ \mathbf{if}\;x \leq -3.733380478835523 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -6.106572808413942 \cdot 10^{-81}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.0661811553637774 \cdot 10^{-99}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.836244289550675 \cdot 10^{-187}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -4.2913794126638345 \cdot 10^{-230}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.7544209452101333 \cdot 10^{-133}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	27.9
Cost	1112

\[\begin{array}{l} t_1 := \frac{z}{\frac{t}{y}}\\ \mathbf{if}\;x \leq -3.733380478835523 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -6.106572808413942 \cdot 10^{-81}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.0661811553637774 \cdot 10^{-99}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.836244289550675 \cdot 10^{-187}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -4.2913794126638345 \cdot 10^{-230}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.7544209452101333 \cdot 10^{-133}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	11.9
Cost	976

\[\begin{array}{l} t_1 := \frac{z - x}{\frac{t}{y}}\\ \mathbf{if}\;y \leq -3.5 \cdot 10^{+205}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -6 \cdot 10^{+78}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;y \leq -70000000000:\\ \;\;\;\;x - \frac{x}{\frac{t}{y}}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+106}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	11.9
Cost	976

\[\begin{array}{l} t_1 := \frac{z - x}{\frac{t}{y}}\\ \mathbf{if}\;y \leq -3.5 \cdot 10^{+205}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -4.6 \cdot 10^{+86}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;y \leq -70000000000:\\ \;\;\;\;x + \frac{y}{-\frac{t}{x}}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+106}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	5.1
Cost	972

\[\begin{array}{l} t_1 := x + \frac{y}{\frac{t}{z - x}}\\ \mathbf{if}\;t \leq -2.4 \cdot 10^{-100}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -4.8 \cdot 10^{-240}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \mathbf{elif}\;t \leq 10^{-129}:\\ \;\;\;\;\frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	11.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -10840709.461642133:\\ \;\;\;\;x - x \cdot \frac{y}{t}\\ \mathbf{elif}\;x \leq 3.2292834975240435 \cdot 10^{-112}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{\frac{t}{y}}\\ \end{array} \]

Alternative 10
Error	31.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022308 
(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)))