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

Average Error: 6.5 → 0.4

Time: 10.4s

Precision: binary64

Cost: 1352

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;t_1 \leq 10^{+223}:\\ \;\;\;\;x + \frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \end{array} \]

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 (/ y (/ a (- z t))))
     (if (<= t_1 1e+223) (+ x (/ t_1 a)) (+ x (* (- z t) (/ y 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 + (y / (a / (z - t)));
	} else if (t_1 <= 1e+223) {
		tmp = x + (t_1 / a);
	} else {
		tmp = x + ((z - t) * (y / 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 + (y / (a / (z - t)));
	} else if (t_1 <= 1e+223) {
		tmp = x + (t_1 / a);
	} else {
		tmp = x + ((z - t) * (y / 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 + (y / (a / (z - t)))
	elif t_1 <= 1e+223:
		tmp = x + (t_1 / a)
	else:
		tmp = x + ((z - t) * (y / 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(y / Float64(a / Float64(z - t))));
	elseif (t_1 <= 1e+223)
		tmp = Float64(x + Float64(t_1 / a));
	else
		tmp = Float64(x + Float64(Float64(z - t) * Float64(y / 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 + (y / (a / (z - t)));
	elseif (t_1 <= 1e+223)
		tmp = x + (t_1 / a);
	else
		tmp = x + ((z - t) * (y / 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[(y / N[(a / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+223], N[(x + N[(t$95$1 / a), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(z - t), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	6.5
Target	0.6
Herbie	0.4

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

Derivation

1. Split input into 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 0.2

      \[\leadsto \color{blue}{x + \frac{y}{\frac{a}{z - t}}} \]
      Proof
      (+.f64 x (/.f64 y (/.f64 a (-.f64 z t)))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y (-.f64 z t)) a))): 0 points increase in error, 0 points decrease in error

   if -inf.0 < (*.f64 y (-.f64 z t)) < 1.00000000000000005e223

   1. Initial program 0.3

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

   if 1.00000000000000005e223 < (*.f64 y (-.f64 z t))

   1. Initial program 34.5

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

   2. Simplified 0.7

      \[\leadsto \color{blue}{x + \frac{y}{a} \cdot \left(z - t\right)} \]
      Proof
      (+.f64 x (/.f64 y (/.f64 a (-.f64 z t)))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y (-.f64 z t)) a))): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -\infty:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq 10^{+223}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	32.9
Cost	1704

\[\begin{array}{l} t_1 := \frac{y}{a} \cdot \left(-t\right)\\ \mathbf{if}\;a \leq -2.65 \cdot 10^{-17}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq -1.75 \cdot 10^{-53}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -4.5 \cdot 10^{-262}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 9.8 \cdot 10^{-227}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{elif}\;a \leq 1.04 \cdot 10^{-172}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{-96}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{-25}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{-12}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{+114}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 2 \cdot 10^{+168}:\\ \;\;\;\;y \cdot \left(-\frac{t}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	29.4
Cost	849

\[\begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-132}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-208} \lor \neg \left(x \leq 1.85 \cdot 10^{+15}\right) \land x \leq 4.5 \cdot 10^{+66}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	29.1
Cost	848

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-131}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-197}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+15}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+66}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	29.1
Cost	848

\[\begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-132}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-192}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+15}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+66}:\\ \;\;\;\;\frac{y}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	29.2
Cost	848

\[\begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-93}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{-123}:\\ \;\;\;\;\frac{-y}{\frac{a}{t}}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+15}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+66}:\\ \;\;\;\;\frac{y}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	29.2
Cost	848

\[\begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{-93}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-126}:\\ \;\;\;\;y \cdot \left(-\frac{t}{a}\right)\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+15}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+66}:\\ \;\;\;\;\frac{y}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	14.7
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-88} \lor \neg \left(x \leq 1.58 \cdot 10^{-204}\right):\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - t}{a}\\ \end{array} \]

Alternative 8
Error	9.5
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{-40} \lor \neg \left(t \leq 1.8 \cdot 10^{-58}\right):\\ \;\;\;\;x - t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \end{array} \]

Alternative 9
Error	20.7
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-86}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+66}:\\ \;\;\;\;y \cdot \frac{z - t}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	10.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{-40}:\\ \;\;\;\;x - t \cdot \frac{y}{a}\\ \mathbf{elif}\;t \leq 1.46 \cdot 10^{-62}:\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{t}}\\ \end{array} \]

Alternative 11
Error	2.7
Cost	576

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

Alternative 12
Error	31.5
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022343 
(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)))