Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, E

Average Error: 5.7 → 0.9

Time: 1.1min

Precision: binary64

Cost: 6088

\[ \begin{array}{c}[y, z] = \mathsf{sort}([y, z])\\ [j, k] = \mathsf{sort}([j, k])\\ \end{array} \]

Math

\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]

↓

\[\begin{array}{l} t_1 := x \cdot \left(-4 \cdot i - y \cdot \left(z \cdot \left(t \cdot -18\right)\right)\right) + \left(b \cdot c - j \cdot \left(27 \cdot k\right)\right)\\ t_2 := \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+303}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i j k)
 :precision binary64
 (-
  (-
   (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c))
   (* (* x 4.0) i))
  (* (* j 27.0) k)))

↓

(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1
         (+
          (* x (- (* -4.0 i) (* y (* z (* t -18.0)))))
          (- (* b c) (* j (* 27.0 k)))))
        (t_2
         (-
          (-
           (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c))
           (* (* x 4.0) i))
          (* (* j 27.0) k))))
   (if (<= t_2 (- INFINITY)) t_1 (if (<= t_2 5e+303) t_2 t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	return (((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k);
}

↓

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = (x * ((-4.0 * i) - (y * (z * (t * -18.0))))) + ((b * c) - (j * (27.0 * k)));
	double t_2 = (((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 5e+303) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	return (((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k);
}

↓

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = (x * ((-4.0 * i) - (y * (z * (t * -18.0))))) + ((b * c) - (j * (27.0 * k)));
	double t_2 = (((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k);
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= 5e+303) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j, k):
	return (((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k)

↓

def code(x, y, z, t, a, b, c, i, j, k):
	t_1 = (x * ((-4.0 * i) - (y * (z * (t * -18.0))))) + ((b * c) - (j * (27.0 * k)))
	t_2 = (((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k)
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= 5e+303:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j, k)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * 18.0) * y) * z) * t) - Float64(Float64(a * 4.0) * t)) + Float64(b * c)) - Float64(Float64(x * 4.0) * i)) - Float64(Float64(j * 27.0) * k))
end

↓

function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(Float64(x * Float64(Float64(-4.0 * i) - Float64(y * Float64(z * Float64(t * -18.0))))) + Float64(Float64(b * c) - Float64(j * Float64(27.0 * k))))
	t_2 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * 18.0) * y) * z) * t) - Float64(Float64(a * 4.0) * t)) + Float64(b * c)) - Float64(Float64(x * 4.0) * i)) - Float64(Float64(j * 27.0) * k))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 5e+303)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j, k)
	tmp = (((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k);
end

↓

function tmp_2 = code(x, y, z, t, a, b, c, i, j, k)
	t_1 = (x * ((-4.0 * i) - (y * (z * (t * -18.0))))) + ((b * c) - (j * (27.0 * k)));
	t_2 = (((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k);
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= 5e+303)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := N[(N[(N[(N[(N[(N[(N[(N[(x * 18.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision] - N[(N[(x * 4.0), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] - N[(N[(j * 27.0), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(N[(x * N[(N[(-4.0 * i), $MachinePrecision] - N[(y * N[(z * N[(t * -18.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(b * c), $MachinePrecision] - N[(j * N[(27.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(N[(N[(N[(x * 18.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision] - N[(N[(x * 4.0), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] - N[(N[(j * 27.0), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 5e+303], t$95$2, t$95$1]]]]

Target

Original	5.7
Target	1.7
Herbie	0.9

\[\begin{array}{l} \mathbf{if}\;t < -1.6210815397541398 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \mathbf{elif}\;t < 165.68027943805222:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) + \left(c \cdot b - 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) t) (*.f64 (*.f64 a 4) t)) (*.f64 b c)) (*.f64 (*.f64 x 4) i)) (*.f64 (*.f64 j 27) k)) < -inf.0 or 4.9999999999999997e303 < (-.f64 (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) t) (*.f64 (*.f64 a 4) t)) (*.f64 b c)) (*.f64 (*.f64 x 4) i)) (*.f64 (*.f64 j 27) k))

   1. Initial program 58.2

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]

   2. Simplified 36.0

      \[\leadsto \color{blue}{\left(t \cdot \left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right) - a \cdot 4\right) - x \cdot \left(4 \cdot i\right)\right) + \left(b \cdot c - j \cdot \left(27 \cdot k\right)\right)} \]
      Proof

      [Start] 58.2	\[ \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
      rational.json-simplify-48 [=>] 58.2	\[ \color{blue}{\left(b \cdot c + \left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) - \left(x \cdot 4\right) \cdot i\right)\right)} - \left(j \cdot 27\right) \cdot k \]
      rational.json-simplify-48 [=>] 58.2	\[ \color{blue}{\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) - \left(x \cdot 4\right) \cdot i\right) + \left(b \cdot c - \left(j \cdot 27\right) \cdot k\right)} \]
      rational.json-simplify-2 [=>] 58.2	\[ \left(\left(\color{blue}{t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right)} - \left(a \cdot 4\right) \cdot t\right) - \left(x \cdot 4\right) \cdot i\right) + \left(b \cdot c - \left(j \cdot 27\right) \cdot k\right) \]
      rational.json-simplify-52 [=>] 58.2	\[ \left(\color{blue}{t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right)} - \left(x \cdot 4\right) \cdot i\right) + \left(b \cdot c - \left(j \cdot 27\right) \cdot k\right) \]
      rational.json-simplify-2 [=>] 58.2	\[ \left(t \cdot \left(\color{blue}{z \cdot \left(\left(x \cdot 18\right) \cdot y\right)} - a \cdot 4\right) - \left(x \cdot 4\right) \cdot i\right) + \left(b \cdot c - \left(j \cdot 27\right) \cdot k\right) \]
      rational.json-simplify-43 [=>] 36.8	\[ \left(t \cdot \left(\color{blue}{\left(x \cdot 18\right) \cdot \left(y \cdot z\right)} - a \cdot 4\right) - \left(x \cdot 4\right) \cdot i\right) + \left(b \cdot c - \left(j \cdot 27\right) \cdot k\right) \]
      rational.json-simplify-2 [=>] 36.8	\[ \left(t \cdot \left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right) - a \cdot 4\right) - \color{blue}{i \cdot \left(x \cdot 4\right)}\right) + \left(b \cdot c - \left(j \cdot 27\right) \cdot k\right) \]
      rational.json-simplify-43 [=>] 36.8	\[ \left(t \cdot \left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right) - a \cdot 4\right) - \color{blue}{x \cdot \left(4 \cdot i\right)}\right) + \left(b \cdot c - \left(j \cdot 27\right) \cdot k\right) \]
      rational.json-simplify-2 [=>] 36.8	\[ \left(t \cdot \left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right) - a \cdot 4\right) - x \cdot \left(4 \cdot i\right)\right) + \left(b \cdot c - \color{blue}{k \cdot \left(j \cdot 27\right)}\right) \]
      rational.json-simplify-43 [=>] 36.0	\[ \left(t \cdot \left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right) - a \cdot 4\right) - x \cdot \left(4 \cdot i\right)\right) + \left(b \cdot c - \color{blue}{j \cdot \left(27 \cdot k\right)}\right) \]

   3. Taylor expanded in x around -inf 6.8

      \[\leadsto \color{blue}{-1 \cdot \left(\left(-18 \cdot \left(y \cdot \left(t \cdot z\right)\right) - -4 \cdot i\right) \cdot x\right)} + \left(b \cdot c - j \cdot \left(27 \cdot k\right)\right) \]

   4. Simplified 6.8

      \[\leadsto \color{blue}{x \cdot \left(-4 \cdot i - y \cdot \left(z \cdot \left(t \cdot -18\right)\right)\right)} + \left(b \cdot c - j \cdot \left(27 \cdot k\right)\right) \]
      Proof

      [Start] 6.8	\[ -1 \cdot \left(\left(-18 \cdot \left(y \cdot \left(t \cdot z\right)\right) - -4 \cdot i\right) \cdot x\right) + \left(b \cdot c - j \cdot \left(27 \cdot k\right)\right) \]
      rational.json-simplify-2 [=>] 6.8	\[ -1 \cdot \color{blue}{\left(x \cdot \left(-18 \cdot \left(y \cdot \left(t \cdot z\right)\right) - -4 \cdot i\right)\right)} + \left(b \cdot c - j \cdot \left(27 \cdot k\right)\right) \]
      rational.json-simplify-43 [=>] 6.8	\[ \color{blue}{x \cdot \left(\left(-18 \cdot \left(y \cdot \left(t \cdot z\right)\right) - -4 \cdot i\right) \cdot -1\right)} + \left(b \cdot c - j \cdot \left(27 \cdot k\right)\right) \]
      rational.json-simplify-9 [=>] 6.8	\[ x \cdot \color{blue}{\left(-\left(-18 \cdot \left(y \cdot \left(t \cdot z\right)\right) - -4 \cdot i\right)\right)} + \left(b \cdot c - j \cdot \left(27 \cdot k\right)\right) \]
      rational.json-simplify-12 [=>] 6.8	\[ x \cdot \color{blue}{\left(0 - \left(-18 \cdot \left(y \cdot \left(t \cdot z\right)\right) - -4 \cdot i\right)\right)} + \left(b \cdot c - j \cdot \left(27 \cdot k\right)\right) \]
      rational.json-simplify-2 [=>] 6.8	\[ x \cdot \left(0 - \left(-18 \cdot \left(y \cdot \left(t \cdot z\right)\right) - \color{blue}{i \cdot -4}\right)\right) + \left(b \cdot c - j \cdot \left(27 \cdot k\right)\right) \]
      metadata-eval [<=] 6.8	\[ x \cdot \left(0 - \left(-18 \cdot \left(y \cdot \left(t \cdot z\right)\right) - i \cdot \color{blue}{\frac{4}{-1}}\right)\right) + \left(b \cdot c - j \cdot \left(27 \cdot k\right)\right) \]
      rational.json-simplify-49 [<=] 6.8	\[ x \cdot \left(0 - \left(-18 \cdot \left(y \cdot \left(t \cdot z\right)\right) - \color{blue}{\frac{4 \cdot i}{-1}}\right)\right) + \left(b \cdot c - j \cdot \left(27 \cdot k\right)\right) \]
      rational.json-simplify-45 [=>] 6.8	\[ x \cdot \color{blue}{\left(\frac{4 \cdot i}{-1} - \left(-18 \cdot \left(y \cdot \left(t \cdot z\right)\right) - 0\right)\right)} + \left(b \cdot c - j \cdot \left(27 \cdot k\right)\right) \]
      rational.json-simplify-49 [=>] 6.8	\[ x \cdot \left(\color{blue}{i \cdot \frac{4}{-1}} - \left(-18 \cdot \left(y \cdot \left(t \cdot z\right)\right) - 0\right)\right) + \left(b \cdot c - j \cdot \left(27 \cdot k\right)\right) \]
      metadata-eval [=>] 6.8	\[ x \cdot \left(i \cdot \color{blue}{-4} - \left(-18 \cdot \left(y \cdot \left(t \cdot z\right)\right) - 0\right)\right) + \left(b \cdot c - j \cdot \left(27 \cdot k\right)\right) \]
      rational.json-simplify-2 [<=] 6.8	\[ x \cdot \left(\color{blue}{-4 \cdot i} - \left(-18 \cdot \left(y \cdot \left(t \cdot z\right)\right) - 0\right)\right) + \left(b \cdot c - j \cdot \left(27 \cdot k\right)\right) \]
      rational.json-simplify-5 [=>] 6.8	\[ x \cdot \left(-4 \cdot i - \color{blue}{-18 \cdot \left(y \cdot \left(t \cdot z\right)\right)}\right) + \left(b \cdot c - j \cdot \left(27 \cdot k\right)\right) \]
      rational.json-simplify-43 [=>] 6.8	\[ x \cdot \left(-4 \cdot i - \color{blue}{y \cdot \left(\left(t \cdot z\right) \cdot -18\right)}\right) + \left(b \cdot c - j \cdot \left(27 \cdot k\right)\right) \]
      rational.json-simplify-2 [=>] 6.8	\[ x \cdot \left(-4 \cdot i - y \cdot \color{blue}{\left(-18 \cdot \left(t \cdot z\right)\right)}\right) + \left(b \cdot c - j \cdot \left(27 \cdot k\right)\right) \]
      rational.json-simplify-2 [=>] 6.8	\[ x \cdot \left(-4 \cdot i - y \cdot \left(-18 \cdot \color{blue}{\left(z \cdot t\right)}\right)\right) + \left(b \cdot c - j \cdot \left(27 \cdot k\right)\right) \]
      rational.json-simplify-43 [=>] 6.8	\[ x \cdot \left(-4 \cdot i - y \cdot \color{blue}{\left(z \cdot \left(t \cdot -18\right)\right)}\right) + \left(b \cdot c - j \cdot \left(27 \cdot k\right)\right) \]

   if -inf.0 < (-.f64 (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) t) (*.f64 (*.f64 a 4) t)) (*.f64 b c)) (*.f64 (*.f64 x 4) i)) (*.f64 (*.f64 j 27) k)) < 4.9999999999999997e303

   1. Initial program 0.3

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \leq -\infty:\\ \;\;\;\;x \cdot \left(-4 \cdot i - y \cdot \left(z \cdot \left(t \cdot -18\right)\right)\right) + \left(b \cdot c - j \cdot \left(27 \cdot k\right)\right)\\ \mathbf{elif}\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \leq 5 \cdot 10^{+303}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-4 \cdot i - y \cdot \left(z \cdot \left(t \cdot -18\right)\right)\right) + \left(b \cdot c - j \cdot \left(27 \cdot k\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	23.7
Cost	2780

\[\begin{array}{l} t_1 := -4 \cdot \left(a \cdot t\right)\\ t_2 := t_1 - 4 \cdot \left(i \cdot x\right)\\ t_3 := x \cdot \left(-4 \cdot i\right) + \left(b \cdot c - j \cdot \left(27 \cdot k\right)\right)\\ t_4 := c \cdot b + t_1\\ \mathbf{if}\;j \cdot 27 \leq -0.04:\\ \;\;\;\;t_3\\ \mathbf{elif}\;j \cdot 27 \leq -1 \cdot 10^{-151}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;j \cdot 27 \leq -1 \cdot 10^{-202}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;j \cdot 27 \leq -1 \cdot 10^{-222}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;j \cdot 27 \leq -5 \cdot 10^{-290}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \cdot 27 \leq 10^{-274}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;j \cdot 27 \leq 5 \cdot 10^{-211}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 2
Error	19.4
Cost	2776

\[\begin{array}{l} t_1 := c \cdot b + -4 \cdot \left(a \cdot t\right)\\ t_2 := t_1 - 27 \cdot \left(k \cdot j\right)\\ t_3 := b \cdot c - j \cdot \left(27 \cdot k\right)\\ \mathbf{if}\;j \cdot 27 \leq -2 \cdot 10^{+166}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \cdot 27 \leq -1 \cdot 10^{+137}:\\ \;\;\;\;x \cdot \left(-4 \cdot i\right) + t_3\\ \mathbf{elif}\;j \cdot 27 \leq -1 \cdot 10^{+55}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \cdot 27 \leq -2:\\ \;\;\;\;y \cdot \left(t \cdot \left(18 \cdot \left(z \cdot x\right)\right)\right) + t_3\\ \mathbf{elif}\;j \cdot 27 \leq -1 \cdot 10^{-141}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \cdot 27 \leq 10^{-37}:\\ \;\;\;\;t_1 - 4 \cdot \left(i \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;18 \cdot \left(y \cdot \left(x \cdot \left(t \cdot z\right)\right)\right) + t_3\\ \end{array} \]

Alternative 3
Error	18.9
Cost	2256

\[\begin{array}{l} t_1 := c \cdot b + -4 \cdot \left(a \cdot t\right)\\ t_2 := t_1 - 27 \cdot \left(k \cdot j\right)\\ t_3 := b \cdot c - j \cdot \left(27 \cdot k\right)\\ \mathbf{if}\;j \cdot 27 \leq -2 \cdot 10^{+166}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \cdot 27 \leq -1 \cdot 10^{+138}:\\ \;\;\;\;x \cdot \left(-4 \cdot i\right) + t_3\\ \mathbf{elif}\;j \cdot 27 \leq -1 \cdot 10^{-141}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \cdot 27 \leq 10^{-37}:\\ \;\;\;\;t_1 - 4 \cdot \left(i \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;18 \cdot \left(y \cdot \left(x \cdot \left(t \cdot z\right)\right)\right) + t_3\\ \end{array} \]

Alternative 4
Error	4.9
Cost	2120

\[\begin{array}{l} t_1 := b \cdot c - j \cdot \left(27 \cdot k\right)\\ t_2 := t \cdot \left(x \cdot \left(y \cdot \left(18 \cdot z\right)\right) - a \cdot 4\right) + \left(t_1 - x \cdot \left(4 \cdot i\right)\right)\\ \mathbf{if}\;t \leq -3.8 \cdot 10^{-95}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-227}:\\ \;\;\;\;x \cdot \left(-4 \cdot i - y \cdot \left(z \cdot \left(t \cdot -18\right)\right)\right) + t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	4.6
Cost	2120

\[\begin{array}{l} t_1 := j \cdot \left(27 \cdot k\right)\\ t_2 := x \cdot \left(4 \cdot i\right)\\ t_3 := b \cdot c - t_1\\ \mathbf{if}\;t \leq -1.3 \cdot 10^{-94}:\\ \;\;\;\;t \cdot \left(x \cdot \left(y \cdot \left(18 \cdot z\right)\right) - a \cdot 4\right) + \left(t_3 - t_2\right)\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-230}:\\ \;\;\;\;x \cdot \left(-4 \cdot i - y \cdot \left(z \cdot \left(t \cdot -18\right)\right)\right) + t_3\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x \cdot \left(18 \cdot z\right)\right) - a \cdot 4\right) + \left(\left(b \cdot c - t_2\right) - t_1\right)\\ \end{array} \]

Alternative 6
Error	4.6
Cost	2120

\[\begin{array}{l} t_1 := j \cdot \left(27 \cdot k\right)\\ t_2 := x \cdot \left(4 \cdot i\right)\\ \mathbf{if}\;t \leq -2 \cdot 10^{-93}:\\ \;\;\;\;\left(t \cdot \left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right) - a \cdot 4\right) - t_2\right) + \left(b \cdot c - 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-229}:\\ \;\;\;\;x \cdot \left(-4 \cdot i - y \cdot \left(z \cdot \left(t \cdot -18\right)\right)\right) + \left(b \cdot c - t_1\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x \cdot \left(18 \cdot z\right)\right) - a \cdot 4\right) + \left(\left(b \cdot c - t_2\right) - t_1\right)\\ \end{array} \]

Alternative 7
Error	4.6
Cost	2120

\[\begin{array}{l} t_1 := j \cdot \left(27 \cdot k\right)\\ t_2 := x \cdot \left(4 \cdot i\right)\\ t_3 := b \cdot c - t_1\\ \mathbf{if}\;t \leq -5.5 \cdot 10^{-95}:\\ \;\;\;\;\left(t \cdot \left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right) - a \cdot 4\right) - t_2\right) + t_3\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-227}:\\ \;\;\;\;x \cdot \left(-4 \cdot i - y \cdot \left(z \cdot \left(t \cdot -18\right)\right)\right) + t_3\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x \cdot \left(18 \cdot z\right)\right) - a \cdot 4\right) + \left(\left(b \cdot c - t_2\right) - t_1\right)\\ \end{array} \]

Alternative 8
Error	18.3
Cost	2000

\[\begin{array}{l} t_1 := x \cdot \left(-4 \cdot i\right) + \left(b \cdot c - j \cdot \left(27 \cdot k\right)\right)\\ t_2 := \left(c \cdot b + -4 \cdot \left(a \cdot t\right)\right) - 4 \cdot \left(i \cdot x\right)\\ \mathbf{if}\;j \cdot 27 \leq -1 \cdot 10^{+190}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \cdot 27 \leq -5 \cdot 10^{+173}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \cdot 27 \leq -0.04:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \cdot 27 \leq 10^{-37}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;c \cdot b - 27 \cdot \left(k \cdot j\right)\\ \end{array} \]

Alternative 9
Error	18.6
Cost	2000

\[\begin{array}{l} t_1 := c \cdot b + -4 \cdot \left(a \cdot t\right)\\ t_2 := t_1 - 27 \cdot \left(k \cdot j\right)\\ \mathbf{if}\;j \cdot 27 \leq -2 \cdot 10^{+166}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \cdot 27 \leq -1 \cdot 10^{+138}:\\ \;\;\;\;x \cdot \left(-4 \cdot i\right) + \left(b \cdot c - j \cdot \left(27 \cdot k\right)\right)\\ \mathbf{elif}\;j \cdot 27 \leq -1 \cdot 10^{-141}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \cdot 27 \leq 10^{-39}:\\ \;\;\;\;t_1 - 4 \cdot \left(i \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Error	8.3
Cost	1996

\[\begin{array}{l} t_1 := b \cdot c - j \cdot \left(27 \cdot k\right)\\ t_2 := x \cdot \left(4 \cdot i\right)\\ t_3 := \left(y \cdot \left(x \cdot \left(t \cdot \left(18 \cdot z\right)\right)\right) - t_2\right) + t_1\\ t_4 := \left(t \cdot \left(a \cdot -4\right) - t_2\right) + t_1\\ \mathbf{if}\;y \leq -1.05 \cdot 10^{+198}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{+154}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y \leq -4.1 \cdot 10^{+64}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 11
Error	9.8
Cost	1868

\[\begin{array}{l} t_1 := b \cdot c - j \cdot \left(27 \cdot k\right)\\ t_2 := \left(t \cdot \left(a \cdot -4\right) - x \cdot \left(4 \cdot i\right)\right) + t_1\\ \mathbf{if}\;y \leq -5.7 \cdot 10^{+206}:\\ \;\;\;\;y \cdot \left(t \cdot \left(18 \cdot \left(z \cdot x\right)\right)\right) + t_1\\ \mathbf{elif}\;y \leq -2.9 \cdot 10^{+156}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{+64}:\\ \;\;\;\;x \cdot \left(-4 \cdot i - y \cdot \left(z \cdot \left(t \cdot -18\right)\right)\right) + t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 12
Error	9.8
Cost	1868

\[\begin{array}{l} t_1 := b \cdot c - j \cdot \left(27 \cdot k\right)\\ t_2 := \left(t \cdot \left(a \cdot -4\right) - x \cdot \left(4 \cdot i\right)\right) + t_1\\ \mathbf{if}\;y \leq -2.8 \cdot 10^{+202}:\\ \;\;\;\;y \cdot \left(t \cdot \left(18 \cdot \left(z \cdot x\right)\right)\right) + t_1\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{+155}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -8 \cdot 10^{+63}:\\ \;\;\;\;\left(18 \cdot \left(y \cdot \left(t \cdot z\right)\right) - 4 \cdot i\right) \cdot x + t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 13
Error	44.3
Cost	1508

\[\begin{array}{l} t_1 := -27 \cdot \left(k \cdot j\right)\\ t_2 := -4 \cdot \left(a \cdot t\right)\\ \mathbf{if}\;j \leq -3 \cdot 10^{+69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq -4 \cdot 10^{+34}:\\ \;\;\;\;c \cdot b\\ \mathbf{elif}\;j \leq -0.00056:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq -1.36 \cdot 10^{-155}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \leq -2.35 \cdot 10^{-207}:\\ \;\;\;\;-4 \cdot \left(i \cdot x\right)\\ \mathbf{elif}\;j \leq -2.1 \cdot 10^{-291}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \leq 1.16 \cdot 10^{-272}:\\ \;\;\;\;c \cdot b\\ \mathbf{elif}\;j \leq 1.92 \cdot 10^{-208}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \leq 57000:\\ \;\;\;\;c \cdot b\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	44.2
Cost	1508

\[\begin{array}{l} t_1 := -4 \cdot \left(a \cdot t\right)\\ t_2 := -27 \cdot \left(k \cdot j\right)\\ \mathbf{if}\;j \leq -2.3 \cdot 10^{+70}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \leq -1.05 \cdot 10^{+35}:\\ \;\;\;\;c \cdot b\\ \mathbf{elif}\;j \leq -9 \cdot 10^{-5}:\\ \;\;\;\;k \cdot \left(j \cdot -27\right)\\ \mathbf{elif}\;j \leq -2.35 \cdot 10^{-154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq -6.2 \cdot 10^{-209}:\\ \;\;\;\;-4 \cdot \left(i \cdot x\right)\\ \mathbf{elif}\;j \leq -2.6 \cdot 10^{-291}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq 8.5 \cdot 10^{-276}:\\ \;\;\;\;c \cdot b\\ \mathbf{elif}\;j \leq 9.5 \cdot 10^{-208}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq 0.042:\\ \;\;\;\;c \cdot b\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 15
Error	33.7
Cost	1500

\[\begin{array}{l} t_1 := x \cdot \left(-4 \cdot i\right) + c \cdot b\\ t_2 := c \cdot b + -4 \cdot \left(a \cdot t\right)\\ t_3 := -27 \cdot \left(k \cdot j\right)\\ \mathbf{if}\;j \leq -2.55 \cdot 10^{+188}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;j \leq -3.6 \cdot 10^{+162}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \leq -8.8 \cdot 10^{+135}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq -1.3 \cdot 10^{-155}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \leq 1.28 \cdot 10^{-285}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq 2.4 \cdot 10^{-207}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \leq 340000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 16
Error	9.5
Cost	1476

\[\begin{array}{l} t_1 := b \cdot c - j \cdot \left(27 \cdot k\right)\\ \mathbf{if}\;y \leq -4.6 \cdot 10^{+207}:\\ \;\;\;\;y \cdot \left(t \cdot \left(18 \cdot \left(z \cdot x\right)\right)\right) + t_1\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(a \cdot -4\right) - x \cdot \left(4 \cdot i\right)\right) + t_1\\ \end{array} \]

Alternative 17
Error	45.0
Cost	1376

\[\begin{array}{l} t_1 := -27 \cdot \left(k \cdot j\right)\\ t_2 := -4 \cdot \left(a \cdot t\right)\\ \mathbf{if}\;t \leq -8.8 \cdot 10^{+62}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.05 \cdot 10^{-82}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -5.5 \cdot 10^{-179}:\\ \;\;\;\;c \cdot b\\ \mathbf{elif}\;t \leq -1.35 \cdot 10^{-262}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-226}:\\ \;\;\;\;c \cdot b\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.12 \cdot 10^{-10}:\\ \;\;\;\;c \cdot b\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 18
Error	30.3
Cost	1368

\[\begin{array}{l} t_1 := 27 \cdot \left(k \cdot j\right)\\ t_2 := -4 \cdot \left(a \cdot t\right)\\ t_3 := t_2 - t_1\\ t_4 := t_2 - 4 \cdot \left(i \cdot x\right)\\ t_5 := c \cdot b - t_1\\ \mathbf{if}\;c \leq -5.2 \cdot 10^{-154}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;c \leq -2.1 \cdot 10^{-200}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;c \leq 2.2 \cdot 10^{-256}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c \leq 1.45 \cdot 10^{-212}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;c \leq 5.2 \cdot 10^{-24}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c \leq 6.2 \cdot 10^{+77}:\\ \;\;\;\;c \cdot b + t_2\\ \mathbf{else}:\\ \;\;\;\;t_5\\ \end{array} \]

Alternative 19
Error	30.2
Cost	1368

\[\begin{array}{l} t_1 := 27 \cdot \left(k \cdot j\right)\\ t_2 := -4 \cdot \left(a \cdot t\right)\\ t_3 := t_2 - t_1\\ t_4 := c \cdot b - t_1\\ \mathbf{if}\;c \leq -3.95 \cdot 10^{-156}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;c \leq -2.4 \cdot 10^{-200}:\\ \;\;\;\;t_2 - 4 \cdot \left(i \cdot x\right)\\ \mathbf{elif}\;c \leq 5 \cdot 10^{-265}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c \leq 1.12 \cdot 10^{-212}:\\ \;\;\;\;-4 \cdot \left(i \cdot x\right) - k \cdot \left(j \cdot 27\right)\\ \mathbf{elif}\;c \leq 6.5 \cdot 10^{-25}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c \leq 1.16 \cdot 10^{+78}:\\ \;\;\;\;c \cdot b + t_2\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 20
Error	29.5
Cost	1236

\[\begin{array}{l} t_1 := x \cdot \left(-4 \cdot i\right) + c \cdot b\\ t_2 := c \cdot b + -4 \cdot \left(a \cdot t\right)\\ t_3 := c \cdot b - 27 \cdot \left(k \cdot j\right)\\ \mathbf{if}\;j \leq -5.8 \cdot 10^{-5}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;j \leq -4.7 \cdot 10^{-154}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \leq 3.2 \cdot 10^{-283}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq 2.7 \cdot 10^{-208}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \leq 1.6 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 21
Error	29.8
Cost	1236

\[\begin{array}{l} t_1 := x \cdot \left(-4 \cdot i\right) + c \cdot b\\ t_2 := -4 \cdot \left(a \cdot t\right)\\ t_3 := c \cdot b - 27 \cdot \left(k \cdot j\right)\\ \mathbf{if}\;j \leq -8.6 \cdot 10^{-6}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;j \leq -3.4 \cdot 10^{-155}:\\ \;\;\;\;c \cdot b + t_2\\ \mathbf{elif}\;j \leq 2.3 \cdot 10^{-273}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq 1.2 \cdot 10^{-212}:\\ \;\;\;\;t_2 - 4 \cdot \left(i \cdot x\right)\\ \mathbf{elif}\;j \leq 2.9 \cdot 10^{-36}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 22
Error	33.7
Cost	840

\[\begin{array}{l} t_1 := -27 \cdot \left(k \cdot j\right)\\ \mathbf{if}\;j \leq -2.6 \cdot 10^{+188}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq 120000:\\ \;\;\;\;c \cdot b + -4 \cdot \left(a \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 23
Error	43.1
Cost	584

\[\begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{+78}:\\ \;\;\;\;c \cdot b\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-111}:\\ \;\;\;\;-27 \cdot \left(k \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot b\\ \end{array} \]

Alternative 24
Error	48.1
Cost	192

\[c \cdot b \]

Reproduce

herbie shell --seed 2023075 
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, E"
  :precision binary64

  :herbie-target
  (if (< t -1.6210815397541398e-69) (- (- (* (* 18.0 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4.0)) (- (* (* k j) 27.0) (* c b))) (if (< t 165.68027943805222) (+ (- (* (* 18.0 y) (* x (* z t))) (* (+ (* a t) (* i x)) 4.0)) (- (* c b) (* 27.0 (* k j)))) (- (- (* (* 18.0 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4.0)) (- (* (* k j) 27.0) (* c b)))))

  (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)))