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

Average Error: 5.9 → 1.4

Time: 23.2s

Precision: binary64

Cost: 1864

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ (- t z) a) y)) (t_2 (- x (/ (* y (- z t)) a))))
   (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, double a) {
	return x - ((y * (z - t)) / a);
}

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((t - z) / a) * y;
	double t_2 = x - ((y * (z - t)) / a);
	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, 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 = ((t - z) / a) * y;
	double t_2 = x - ((y * (z - t)) / a);
	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, a):
	return x - ((y * (z - t)) / a)

↓

def code(x, y, z, t, a):
	t_1 = ((t - z) / a) * y
	t_2 = x - ((y * (z - t)) / a)
	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, a)
	return Float64(x - Float64(Float64(y * Float64(z - t)) / a))
end

↓

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(t - z) / a) * y)
	t_2 = Float64(x - Float64(Float64(y * Float64(z - t)) / a))
	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, a)
	tmp = x - ((y * (z - t)) / a);
end

↓

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((t - z) / a) * y;
	t_2 = x - ((y * (z - t)) / a);
	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_, 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[(N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(x - N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $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	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 x (/.f64 (*.f64 y (-.f64 z t)) a)) < -inf.0 or 5e307 < (-.f64 x (/.f64 (*.f64 y (-.f64 z t)) a))

   1. Initial program 63.5

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

   2. Taylor expanded in y around inf 12.5

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

   3. Taylor expanded in a around 0 12.5

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

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

   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

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

Alternatives

Alternative 1
Error	13.1
Cost	2060

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

Alternative 2
Error	17.2
Cost	1372

\[\begin{array}{l} t_1 := x - \frac{y \cdot z}{a}\\ t_2 := \frac{t - z}{a} \cdot y\\ t_3 := \frac{y \cdot t}{a} + x\\ \mathbf{if}\;y \leq -2 \cdot 10^{+143}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -3.7 \cdot 10^{+70}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -9.5 \cdot 10^{+31}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -6.1 \cdot 10^{-226}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{-188}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 13000000000000:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{+88}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	29.5
Cost	1044

\[\begin{array}{l} t_1 := \left(-\frac{z}{a}\right) \cdot y\\ \mathbf{if}\;x \leq -1.12 \cdot 10^{-209}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.26 \cdot 10^{-292}:\\ \;\;\;\;\frac{y \cdot t}{a}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-254}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-195}:\\ \;\;\;\;\frac{t}{a} \cdot y\\ \mathbf{elif}\;x \leq 430000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	16.9
Cost	976

\[\begin{array}{l} t_1 := \frac{t - z}{a} \cdot y\\ \mathbf{if}\;y \leq -5.2 \cdot 10^{+141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.26 \cdot 10^{+72}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -9 \cdot 10^{+30}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 10^{+87}:\\ \;\;\;\;\frac{y \cdot t}{a} + x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	29.1
Cost	780

\[\begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-209}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-195}:\\ \;\;\;\;\frac{t}{a} \cdot y\\ \mathbf{elif}\;x \leq 10^{-10}:\\ \;\;\;\;-\frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	19.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00038:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4100000000:\\ \;\;\;\;\frac{t - z}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	28.8
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-209}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-43}:\\ \;\;\;\;\frac{t}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

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, F"
  :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)))