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

Average Error: 5.9 → 1.4

Time: 24.3s

Precision: binary64

Cost: 1352

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ t_2 := \frac{z - t}{a} \cdot y\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;x + \frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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))) (t_2 (* (/ (- z t) a) y)))
   (if (<= t_1 (- INFINITY)) t_2 (if (<= t_1 2e+307) (+ x (/ t_1 a)) t_2))))

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 t_2 = ((z - t) / a) * y;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= 2e+307) {
		tmp = x + (t_1 / a);
	} else {
		tmp = t_2;
	}
	return tmp;
}

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

Target

Original	5.9
Target	0.6
Herbie	1.4

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (-.f64 z t)) < -inf.0 or 1.99999999999999997e307 < (*.f64 y (-.f64 z t))

   1. Initial program 63.6

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

   2. Taylor expanded in y around inf 12.2

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

   3. Taylor expanded in a around 0 12.2

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

   if -inf.0 < (*.f64 y (-.f64 z t)) < 1.99999999999999997e307

   1. Initial program 0.4

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

4. Final simplification 1.4

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

Alternatives

Alternative 1
Error	13.1
Cost	1996

\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ t_2 := \frac{z - t}{a} \cdot y\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+222}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+67}:\\ \;\;\;\;x - \frac{y \cdot t}{a}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+298}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	29.0
Cost	1112

\[\begin{array}{l} t_1 := \frac{z}{a} \cdot y\\ t_2 := -\frac{y \cdot t}{a}\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{-209}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-292}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-200}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 10^{-192}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-134}:\\ \;\;\;\;\frac{z \cdot y}{a}\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-10}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	16.3
Cost	1108

\[\begin{array}{l} t_1 := x - \frac{y \cdot t}{a}\\ t_2 := \frac{z - t}{a} \cdot y\\ t_3 := \frac{y \cdot z}{a} + x\\ \mathbf{if}\;y \leq -7.8 \cdot 10^{+135}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -5.4 \cdot 10^{-226}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.85 \cdot 10^{-187}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 1400000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+87}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	29.1
Cost	980

\[\begin{array}{l} t_1 := \frac{z}{a} \cdot y\\ \mathbf{if}\;x \leq -1 \cdot 10^{-209}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-291}:\\ \;\;\;\;-\frac{y \cdot t}{a}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-254}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{-195}:\\ \;\;\;\;\left(-\frac{t}{a}\right) \cdot y\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-10}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	17.3
Cost	976

\[\begin{array}{l} t_1 := \frac{z - t}{a} \cdot y\\ \mathbf{if}\;y \leq -3 \cdot 10^{+159}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -5 \cdot 10^{+69}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.3 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{+88}:\\ \;\;\;\;\frac{y \cdot z}{a} + x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	19.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -920000:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1750000000:\\ \;\;\;\;\frac{z - t}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	28.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-166}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 10^{-10}:\\ \;\;\;\;\frac{z}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	31.1
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023077 
(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)))