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

Average Error: 6.1 → 0.4

Time: 19.0s

Precision: binary64

Cost: 1608

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;x - \frac{\frac{y}{a}}{\frac{1}{z - t}}\\ \mathbf{elif}\;t_1 \leq 10^{+262}:\\ \;\;\;\;x - \frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\left(z - t\right) \cdot \frac{2}{a}}{\frac{2}{y}}\\ \end{array} \]

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) (/ 1.0 (- z t))))
     (if (<= t_1 1e+262)
       (- x (/ t_1 a))
       (- x (/ (* (- z t) (/ 2.0 a)) (/ 2.0 y)))))))

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) / (1.0 / (z - t)));
	} else if (t_1 <= 1e+262) {
		tmp = x - (t_1 / a);
	} else {
		tmp = x - (((z - t) * (2.0 / a)) / (2.0 / y));
	}
	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) / (1.0 / (z - t)));
	} else if (t_1 <= 1e+262) {
		tmp = x - (t_1 / a);
	} else {
		tmp = x - (((z - t) * (2.0 / a)) / (2.0 / y));
	}
	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) / (1.0 / (z - t)))
	elif t_1 <= 1e+262:
		tmp = x - (t_1 / a)
	else:
		tmp = x - (((z - t) * (2.0 / a)) / (2.0 / y))
	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(y / a) / Float64(1.0 / Float64(z - t))));
	elseif (t_1 <= 1e+262)
		tmp = Float64(x - Float64(t_1 / a));
	else
		tmp = Float64(x - Float64(Float64(Float64(z - t) * Float64(2.0 / a)) / Float64(2.0 / y)));
	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) / (1.0 / (z - t)));
	elseif (t_1 <= 1e+262)
		tmp = x - (t_1 / a);
	else
		tmp = x - (((z - t) * (2.0 / a)) / (2.0 / y));
	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[(y / a), $MachinePrecision] / N[(1.0 / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+262], N[(x - N[(t$95$1 / a), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(z - t), $MachinePrecision] * N[(2.0 / a), $MachinePrecision]), $MachinePrecision] / N[(2.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	6.1
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

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

      [Start] 64.0	\[ x - \frac{y \cdot \left(z - t\right)}{a} \]
      rational.json-simplify-2 [=>] 64.0	\[ x - \frac{\color{blue}{\left(z - t\right) \cdot y}}{a} \]
      rational.json-simplify-49 [=>] 0.2	\[ x - \color{blue}{y \cdot \frac{z - t}{a}} \]

   3. Applied egg-rr 0.3

      \[\leadsto x - \color{blue}{\frac{\frac{y}{a}}{\frac{1}{z - t}}} \]

   if -inf.0 < (*.f64 y (-.f64 z t)) < 1e262

   1. Initial program 0.4

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

   if 1e262 < (*.f64 y (-.f64 z t))

   1. Initial program 43.0

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

   2. Simplified 0.3

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

      [Start] 43.0	\[ x - \frac{y \cdot \left(z - t\right)}{a} \]
      rational.json-simplify-2 [=>] 43.0	\[ x - \frac{\color{blue}{\left(z - t\right) \cdot y}}{a} \]
      rational.json-simplify-49 [=>] 0.3	\[ x - \color{blue}{y \cdot \frac{z - t}{a}} \]

   3. Applied egg-rr 0.4

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -\infty:\\ \;\;\;\;x - \frac{\frac{y}{a}}{\frac{1}{z - t}}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq 10^{+262}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\left(z - t\right) \cdot \frac{2}{a}}{\frac{2}{y}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	1352

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -4 \cdot 10^{+260}:\\ \;\;\;\;x - y \cdot \frac{z - t}{a}\\ \mathbf{elif}\;t_1 \leq 10^{+217}:\\ \;\;\;\;x - \frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Alternative 2
Error	0.4
Cost	1352

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;x - \frac{\frac{y}{a}}{\frac{1}{z - t}}\\ \mathbf{elif}\;t_1 \leq 10^{+217}:\\ \;\;\;\;x - \frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Alternative 3
Error	29.0
Cost	980

\[\begin{array}{l} t_1 := \frac{t}{\frac{a}{y}}\\ \mathbf{if}\;x \leq -1.9 \cdot 10^{-13}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-172}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-225}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -9.2 \cdot 10^{-299}:\\ \;\;\;\;\frac{y \cdot t}{a}\\ \mathbf{elif}\;x \leq 1.46 \cdot 10^{-66}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	28.7
Cost	980

\[\begin{array}{l} t_1 := \frac{t}{\frac{a}{y}}\\ \mathbf{if}\;x \leq -5 \cdot 10^{-14}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-182}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-227}:\\ \;\;\;\;-y \cdot \frac{z}{a}\\ \mathbf{elif}\;x \leq -1.56 \cdot 10^{-299}:\\ \;\;\;\;\frac{y \cdot t}{a}\\ \mathbf{elif}\;x \leq 1.04 \cdot 10^{-64}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	28.7
Cost	980

\[\begin{array}{l} t_1 := \frac{t}{\frac{a}{y}}\\ \mathbf{if}\;x \leq -7 \cdot 10^{-14}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-182}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-228}:\\ \;\;\;\;-\frac{y}{\frac{a}{z}}\\ \mathbf{elif}\;x \leq -1.02 \cdot 10^{-298}:\\ \;\;\;\;\frac{y \cdot t}{a}\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-67}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	28.5
Cost	980

\[\begin{array}{l} t_1 := \frac{t}{\frac{a}{y}}\\ \mathbf{if}\;x \leq -5 \cdot 10^{-13}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -5.6 \cdot 10^{-179}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-225}:\\ \;\;\;\;z \cdot \frac{y}{-a}\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-299}:\\ \;\;\;\;\frac{y \cdot t}{a}\\ \mathbf{elif}\;x \leq 7.1 \cdot 10^{-65}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	6.4
Cost	840

\[\begin{array}{l} t_1 := x - y \cdot \frac{z - t}{a}\\ \mathbf{if}\;a \leq -2.1 \cdot 10^{-82}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.75 \cdot 10^{-269}:\\ \;\;\;\;x - \frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	20.7
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-95}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.18 \cdot 10^{+21}:\\ \;\;\;\;y \cdot \frac{t - z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	15.8
Cost	712

\[\begin{array}{l} t_1 := y \cdot \frac{t}{a} + x\\ \mathbf{if}\;x \leq -3.3 \cdot 10^{-181}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{-136}:\\ \;\;\;\;y \cdot \frac{t - z}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	12.0
Cost	712

\[\begin{array}{l} t_1 := y \cdot \frac{t}{a} + x\\ \mathbf{if}\;t \leq -5 \cdot 10^{-13}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-56}:\\ \;\;\;\;x - z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	28.6
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-13}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.14 \cdot 10^{-63}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12
Error	2.5
Cost	576

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

Alternative 13
Error	31.3
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023074 
(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)))