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

Average Error: 5.5 → 0.5

Time: 20.6s

Precision: binary64

Cost: 1352

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+166}:\\ \;\;\;\;x + \frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \end{array} \]

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))))
   (if (<= t_1 (- INFINITY))
     (+ x (* (- z t) (/ y a)))
     (if (<= t_1 2e+166) (+ x (/ t_1 a)) (+ x (* y (/ (- z t) a)))))))

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 tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = x + ((z - t) * (y / a));
	} else if (t_1 <= 2e+166) {
		tmp = x + (t_1 / a);
	} else {
		tmp = x + (y * ((z - t) / a));
	}
	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 tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = x + ((z - t) * (y / a));
	} else if (t_1 <= 2e+166) {
		tmp = x + (t_1 / a);
	} else {
		tmp = x + (y * ((z - t) / a));
	}
	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)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = x + ((z - t) * (y / a))
	elif t_1 <= 2e+166:
		tmp = x + (t_1 / a)
	else:
		tmp = x + (y * ((z - t) / a))
	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))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(x + Float64(Float64(z - t) * Float64(y / a)));
	elseif (t_1 <= 2e+166)
		tmp = Float64(x + Float64(t_1 / a));
	else
		tmp = Float64(x + Float64(y * Float64(Float64(z - t) / a)));
	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);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = x + ((z - t) * (y / a));
	elseif (t_1 <= 2e+166)
		tmp = x + (t_1 / a);
	else
		tmp = x + (y * ((z - t) / a));
	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]}, If[LessEqual[t$95$1, (-Infinity)], N[(x + N[(N[(z - t), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+166], N[(x + N[(t$95$1 / a), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	5.5
Target	0.7
Herbie	0.5

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 64.0

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

   2. Simplified 0.3

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

      [Start] 64.0	\[ x + \frac{y \cdot \left(z - t\right)}{a} \]
      rational.json-simplify-49 [=>] 0.3	\[ x + \color{blue}{\left(z - t\right) \cdot \frac{y}{a}} \]

   if -inf.0 < (*.f64 y (-.f64 z t)) < 1.99999999999999988e166

   1. Initial program 0.3

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

   if 1.99999999999999988e166 < (*.f64 y (-.f64 z t))

   1. Initial program 21.4

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

   2. Simplified 1.8

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

      [Start] 21.4	\[ x + \frac{y \cdot \left(z - t\right)}{a} \]
      rational.json-simplify-2 [=>] 21.4	\[ x + \frac{\color{blue}{\left(z - t\right) \cdot y}}{a} \]
      rational.json-simplify-49 [=>] 1.8	\[ x + \color{blue}{y \cdot \frac{z - t}{a}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -\infty:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq 2 \cdot 10^{+166}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	12.5
Cost	1996

\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+35}:\\ \;\;\;\;\frac{y}{a} \cdot \left(z - t\right)\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+40}:\\ \;\;\;\;\frac{y \cdot z}{a} + x\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+303}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{a} + x\\ \end{array} \]

Alternative 2
Error	28.8
Cost	1440

\[\begin{array}{l} t_1 := \frac{y}{a} \cdot \left(-t\right)\\ \mathbf{if}\;x \leq -9.2 \cdot 10^{-42}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-157}:\\ \;\;\;\;\frac{t}{a} \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq -6.4 \cdot 10^{-190}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-287}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{elif}\;x \leq 5.9 \cdot 10^{-263}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-225}:\\ \;\;\;\;\frac{y}{\frac{a}{z}}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-138}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-87}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	28.8
Cost	1440

\[\begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-43}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-157}:\\ \;\;\;\;\frac{t}{a} \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq -2.45 \cdot 10^{-189}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.05 \cdot 10^{-298}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{-263}:\\ \;\;\;\;\frac{y}{a} \cdot \left(-t\right)\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-225}:\\ \;\;\;\;\frac{y}{\frac{a}{z}}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-139}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-87}:\\ \;\;\;\;\frac{-t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	28.7
Cost	1440

\[\begin{array}{l} \mathbf{if}\;x \leq -3.55 \cdot 10^{-45}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-157}:\\ \;\;\;\;\frac{y \cdot \left(-t\right)}{a}\\ \mathbf{elif}\;x \leq -1.65 \cdot 10^{-189}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -9.8 \cdot 10^{-288}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-263}:\\ \;\;\;\;\frac{y}{a} \cdot \left(-t\right)\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-225}:\\ \;\;\;\;\frac{y}{\frac{a}{z}}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-138}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-87}:\\ \;\;\;\;\frac{-t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	28.8
Cost	1244

\[\begin{array}{l} t_1 := z \cdot \frac{y}{a}\\ t_2 := \frac{t}{a} \cdot \left(-y\right)\\ \mathbf{if}\;x \leq -1.38 \cdot 10^{-39}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-157}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{-189}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.6 \cdot 10^{-206}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.7 \cdot 10^{-219}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-225}:\\ \;\;\;\;\frac{y}{\frac{a}{z}}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-136}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	20.9
Cost	1240

\[\begin{array}{l} t_1 := \frac{y}{a} \cdot \left(z - t\right)\\ \mathbf{if}\;x \leq -7.8 \cdot 10^{+117}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -6.4 \cdot 10^{+56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-38}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-87}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+20}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 10^{+84}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	2.6
Cost	1096

\[\begin{array}{l} t_1 := x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{if}\;z - t \leq -5 \cdot 10^{+191}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z - t \leq 10^{+86}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	28.4
Cost	980

\[\begin{array}{l} t_1 := z \cdot \frac{y}{a}\\ \mathbf{if}\;x \leq -9.1 \cdot 10^{-41}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{-149}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-189}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-225}:\\ \;\;\;\;\frac{y}{\frac{a}{z}}\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-138}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	15.8
Cost	976

\[\begin{array}{l} t_1 := z \cdot \frac{y}{a} + x\\ t_2 := \frac{y}{a} \cdot \left(z - t\right)\\ \mathbf{if}\;x \leq -7.8 \cdot 10^{+117}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{+79}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -4.7 \cdot 10^{-42}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-87}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	16.3
Cost	976

\[\begin{array}{l} t_1 := \frac{y}{a} \cdot \left(z - t\right)\\ \mathbf{if}\;x \leq -7.8 \cdot 10^{+117}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -8.6 \cdot 10^{+78}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{-43}:\\ \;\;\;\;z \cdot \frac{y}{a} + x\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-87}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot z}{a} + x\\ \end{array} \]

Alternative 11
Error	6.1
Cost	972

\[\begin{array}{l} t_1 := x + y \cdot \frac{z - t}{a}\\ \mathbf{if}\;t \leq 2 \cdot 10^{-259}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{-56}:\\ \;\;\;\;z \cdot \frac{y}{a} + x\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+198}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y}{-a} + x\\ \end{array} \]

Alternative 12
Error	9.4
Cost	776

\[\begin{array}{l} t_1 := z \cdot \frac{y}{a} + x\\ \mathbf{if}\;z \leq -8.5 \cdot 10^{+49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 62000000:\\ \;\;\;\;\left(-\frac{t}{\frac{a}{y}}\right) + x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	20.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-39}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-87}:\\ \;\;\;\;y \cdot \frac{z - t}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 14
Error	28.1
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-189}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-147}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 15
Error	28.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-41}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-139}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 16
Error	30.5
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023075 
(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)))