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

Average Error: 5.6 → 0.7

Time: 41.7s

Precision: binary64

Cost: 6088

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

Math

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

↓

\[\begin{array}{l} t_1 := i \cdot \left(x \cdot -4\right)\\ t_2 := k \cdot \left(j \cdot -27\right)\\ t_3 := t \cdot \left(a \cdot -4\right)\\ t_4 := \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t_3\right) + b \cdot c\right) + t_1\right) + t_2\\ \mathbf{if}\;t_4 \leq -\infty:\\ \;\;\;\;\left(b \cdot c + \left(x \cdot \left(18 \cdot \left(y \cdot \left(z \cdot t\right)\right) + i \cdot -4\right) + -4 \cdot \left(t \cdot a\right)\right)\right) + \left(j \cdot k\right) \cdot -27\\ \mathbf{elif}\;t_4 \leq 10^{+304}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(z \cdot \left(x \cdot t\right)\right) \cdot \left(18 \cdot y\right) + t_3\right)\right) + t_1\right) + t_2\\ \end{array} \]

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 (* i (* x -4.0)))
        (t_2 (* k (* j -27.0)))
        (t_3 (* t (* a -4.0)))
        (t_4 (+ (+ (+ (+ (* (* (* (* x 18.0) y) z) t) t_3) (* b c)) t_1) t_2)))
   (if (<= t_4 (- INFINITY))
     (+
      (+
       (* b c)
       (+ (* x (+ (* 18.0 (* y (* z t))) (* i -4.0))) (* -4.0 (* t a))))
      (* (* j k) -27.0))
     (if (<= t_4 1e+304)
       t_4
       (+ (+ (+ (* b c) (+ (* (* z (* x t)) (* 18.0 y)) t_3)) t_1) t_2)))))

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 = i * (x * -4.0);
	double t_2 = k * (j * -27.0);
	double t_3 = t * (a * -4.0);
	double t_4 = (((((((x * 18.0) * y) * z) * t) + t_3) + (b * c)) + t_1) + t_2;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = ((b * c) + ((x * ((18.0 * (y * (z * t))) + (i * -4.0))) + (-4.0 * (t * a)))) + ((j * k) * -27.0);
	} else if (t_4 <= 1e+304) {
		tmp = t_4;
	} else {
		tmp = (((b * c) + (((z * (x * t)) * (18.0 * y)) + t_3)) + t_1) + t_2;
	}
	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 = i * (x * -4.0);
	double t_2 = k * (j * -27.0);
	double t_3 = t * (a * -4.0);
	double t_4 = (((((((x * 18.0) * y) * z) * t) + t_3) + (b * c)) + t_1) + t_2;
	double tmp;
	if (t_4 <= -Double.POSITIVE_INFINITY) {
		tmp = ((b * c) + ((x * ((18.0 * (y * (z * t))) + (i * -4.0))) + (-4.0 * (t * a)))) + ((j * k) * -27.0);
	} else if (t_4 <= 1e+304) {
		tmp = t_4;
	} else {
		tmp = (((b * c) + (((z * (x * t)) * (18.0 * y)) + t_3)) + t_1) + t_2;
	}
	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 = i * (x * -4.0)
	t_2 = k * (j * -27.0)
	t_3 = t * (a * -4.0)
	t_4 = (((((((x * 18.0) * y) * z) * t) + t_3) + (b * c)) + t_1) + t_2
	tmp = 0
	if t_4 <= -math.inf:
		tmp = ((b * c) + ((x * ((18.0 * (y * (z * t))) + (i * -4.0))) + (-4.0 * (t * a)))) + ((j * k) * -27.0)
	elif t_4 <= 1e+304:
		tmp = t_4
	else:
		tmp = (((b * c) + (((z * (x * t)) * (18.0 * y)) + t_3)) + t_1) + t_2
	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(i * Float64(x * -4.0))
	t_2 = Float64(k * Float64(j * -27.0))
	t_3 = Float64(t * Float64(a * -4.0))
	t_4 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * 18.0) * y) * z) * t) + t_3) + Float64(b * c)) + t_1) + t_2)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(b * c) + Float64(Float64(x * Float64(Float64(18.0 * Float64(y * Float64(z * t))) + Float64(i * -4.0))) + Float64(-4.0 * Float64(t * a)))) + Float64(Float64(j * k) * -27.0));
	elseif (t_4 <= 1e+304)
		tmp = t_4;
	else
		tmp = Float64(Float64(Float64(Float64(b * c) + Float64(Float64(Float64(z * Float64(x * t)) * Float64(18.0 * y)) + t_3)) + t_1) + t_2);
	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 = i * (x * -4.0);
	t_2 = k * (j * -27.0);
	t_3 = t * (a * -4.0);
	t_4 = (((((((x * 18.0) * y) * z) * t) + t_3) + (b * c)) + t_1) + t_2;
	tmp = 0.0;
	if (t_4 <= -Inf)
		tmp = ((b * c) + ((x * ((18.0 * (y * (z * t))) + (i * -4.0))) + (-4.0 * (t * a)))) + ((j * k) * -27.0);
	elseif (t_4 <= 1e+304)
		tmp = t_4;
	else
		tmp = (((b * c) + (((z * (x * t)) * (18.0 * y)) + t_3)) + t_1) + t_2;
	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[(i * N[(x * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(k * N[(j * -27.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(N[(N[(N[(N[(x * 18.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] + t$95$3), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[(N[(b * c), $MachinePrecision] + N[(N[(x * N[(N[(18.0 * N[(y * N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(i * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(j * k), $MachinePrecision] * -27.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 1e+304], t$95$4, N[(N[(N[(N[(b * c), $MachinePrecision] + N[(N[(N[(z * N[(x * t), $MachinePrecision]), $MachinePrecision] * N[(18.0 * y), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision]]]]]]]

Target

Original	5.6
Target	1.6
Herbie	0.7

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

Derivation

1. Split input into 3 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

   1. Initial program 64.0

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

   2. Simplified 41.2

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

      [Start] 64.0	\[ \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
      associate--l- [=>] 64.0	\[ \color{blue}{\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)} \]
      associate-+l- [=>] 64.0	\[ \color{blue}{\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(\left(a \cdot 4\right) \cdot t - b \cdot c\right)\right)} - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right) \]
      associate-+l- [<=] 64.0	\[ \color{blue}{\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right)} - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right) \]
      distribute-rgt-out-- [=>] 64.0	\[ \left(\color{blue}{t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right)} + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right) \]
      associate-*l* [=>] 42.1	\[ \left(t \cdot \left(\color{blue}{\left(x \cdot 18\right) \cdot \left(y \cdot z\right)} - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right) \]
      associate-*l* [=>] 42.1	\[ \left(t \cdot \left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right) - a \cdot 4\right) + b \cdot c\right) - \left(\color{blue}{x \cdot \left(4 \cdot i\right)} + \left(j \cdot 27\right) \cdot k\right) \]
      associate-*l* [=>] 41.2	\[ \left(t \cdot \left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right) - a \cdot 4\right) + b \cdot c\right) - \left(x \cdot \left(4 \cdot i\right) + \color{blue}{j \cdot \left(27 \cdot k\right)}\right) \]

   3. Taylor expanded in x around 0 3.2

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

   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)) < 9.9999999999999994e303

   1. Initial program 0.3

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

   if 9.9999999999999994e303 < (-.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 53.9

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

   2. Taylor expanded in x around 0 13.9

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

   3. Simplified 6.1

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

      [Start] 13.9	\[ \left(\left(\left(18 \cdot \left(y \cdot \left(t \cdot \left(z \cdot x\right)\right)\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
      associate-*r* [=>] 14.3	\[ \left(\left(\left(\color{blue}{\left(18 \cdot y\right) \cdot \left(t \cdot \left(z \cdot x\right)\right)} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
      *-commutative [=>] 14.3	\[ \left(\left(\left(\color{blue}{\left(t \cdot \left(z \cdot x\right)\right) \cdot \left(18 \cdot y\right)} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
      associate-*r* [=>] 6.0	\[ \left(\left(\left(\color{blue}{\left(\left(t \cdot z\right) \cdot x\right)} \cdot \left(18 \cdot y\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
      *-commutative [<=] 6.0	\[ \left(\left(\left(\left(\color{blue}{\left(z \cdot t\right)} \cdot x\right) \cdot \left(18 \cdot y\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
      associate-*l* [=>] 6.1	\[ \left(\left(\left(\color{blue}{\left(z \cdot \left(t \cdot x\right)\right)} \cdot \left(18 \cdot y\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right) \leq -\infty:\\ \;\;\;\;\left(b \cdot c + \left(x \cdot \left(18 \cdot \left(y \cdot \left(z \cdot t\right)\right) + i \cdot -4\right) + -4 \cdot \left(t \cdot a\right)\right)\right) + \left(j \cdot k\right) \cdot -27\\ \mathbf{elif}\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right) \leq 10^{+304}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(z \cdot \left(x \cdot t\right)\right) \cdot \left(18 \cdot y\right) + t \cdot \left(a \cdot -4\right)\right)\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	27.4
Cost	2904

\[\begin{array}{l} t_1 := b \cdot c + k \cdot \left(j \cdot -27\right)\\ t_2 := b \cdot c + -4 \cdot \left(t \cdot a\right)\\ t_3 := \left(j \cdot 27\right) \cdot k\\ t_4 := b \cdot c + -4 \cdot \left(x \cdot i\right)\\ \mathbf{if}\;t_3 \leq -1 \cdot 10^{+97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_3 \leq -5 \cdot 10^{-141}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t_3 \leq -1 \cdot 10^{-322}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_3 \leq 10^{-211}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t_3 \leq 10^{-16}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_3 \leq 50000000000000:\\ \;\;\;\;x \cdot \left(i \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	19.8
Cost	2404

\[\begin{array}{l} t_1 := \left(j \cdot k\right) \cdot -27\\ t_2 := \left(b \cdot c + a \cdot \left(t \cdot -4\right)\right) + -4 \cdot \left(x \cdot i\right)\\ t_3 := b \cdot c + \left(t_1 - 4 \cdot \left(x \cdot i\right)\right)\\ t_4 := k \cdot \left(j \cdot -27\right)\\ t_5 := t \cdot \left(18 \cdot \left(y \cdot \left(x \cdot z\right)\right) + a \cdot -4\right) + t_4\\ t_6 := t_4 + 18 \cdot \left(\left(z \cdot t\right) \cdot \left(x \cdot y\right)\right)\\ t_7 := b \cdot c + -4 \cdot \left(t \cdot a\right)\\ \mathbf{if}\;i \leq -1.9 \cdot 10^{-116}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;i \leq -7 \cdot 10^{-149}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;i \leq -5.5 \cdot 10^{-183}:\\ \;\;\;\;t_7 + t_1\\ \mathbf{elif}\;i \leq -4.5 \cdot 10^{-191}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;i \leq 1.55 \cdot 10^{-38}:\\ \;\;\;\;t_4 + t_7\\ \mathbf{elif}\;i \leq 5.1 \cdot 10^{+14}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;i \leq 7.5 \cdot 10^{+77}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;i \leq 3.65 \cdot 10^{+183}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;i \leq 8 \cdot 10^{+218}:\\ \;\;\;\;t_5\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	12.3
Cost	2133

\[\begin{array}{l} t_1 := -4 \cdot \left(t \cdot a\right)\\ t_2 := x \cdot \left(i \cdot -4\right) + j \cdot \left(k \cdot -27\right)\\ \mathbf{if}\;y \leq -7.7 \cdot 10^{+99}:\\ \;\;\;\;\left(b \cdot c + t \cdot \left(a \cdot -4\right)\right) + t_2\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{+20}:\\ \;\;\;\;\left(b \cdot c + 18 \cdot \left(\left(x \cdot z\right) \cdot \left(y \cdot t\right)\right)\right) + t_2\\ \mathbf{elif}\;y \leq -8.8 \cdot 10^{-26}:\\ \;\;\;\;\left(b \cdot c + \left(t_1 + \frac{x}{\frac{-0.25}{i}}\right)\right) + \left(j \cdot k\right) \cdot -27\\ \mathbf{elif}\;y \leq -7 \cdot 10^{-147} \lor \neg \left(y \leq 2.35 \cdot 10^{-114}\right):\\ \;\;\;\;b \cdot c + \left(x \cdot \left(18 \cdot \left(y \cdot \left(z \cdot t\right)\right) + i \cdot -4\right) + t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot c + -4 \cdot \left(t \cdot a + x \cdot i\right)\right) + k \cdot \left(j \cdot -27\right)\\ \end{array} \]

Alternative 4
Error	4.7
Cost	2121

\[\begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{+37} \lor \neg \left(t \leq 5 \cdot 10^{+179}\right):\\ \;\;\;\;\left(b \cdot c + t \cdot \left(18 \cdot \left(y \cdot \left(x \cdot z\right)\right) + a \cdot -4\right)\right) + -4 \cdot \left(x \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot c + \left(x \cdot \left(18 \cdot \left(y \cdot \left(z \cdot t\right)\right) + i \cdot -4\right) + -4 \cdot \left(t \cdot a\right)\right)\right) + \left(j \cdot k\right) \cdot -27\\ \end{array} \]

Alternative 5
Error	3.4
Cost	2120

\[\begin{array}{l} \mathbf{if}\;t \leq -1.06 \cdot 10^{+37}:\\ \;\;\;\;\left(b \cdot c + t \cdot \left(18 \cdot \left(y \cdot \left(x \cdot z\right)\right) + a \cdot -4\right)\right) + -4 \cdot \left(x \cdot i\right)\\ \mathbf{elif}\;t \leq 10^{-49}:\\ \;\;\;\;\left(b \cdot c + \left(x \cdot \left(18 \cdot \left(y \cdot \left(z \cdot t\right)\right) + i \cdot -4\right) + -4 \cdot \left(t \cdot a\right)\right)\right) + \left(j \cdot k\right) \cdot -27\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot c + t \cdot \left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right) + a \cdot -4\right)\right) + \left(x \cdot \left(i \cdot -4\right) + j \cdot \left(k \cdot -27\right)\right)\\ \end{array} \]

Alternative 6
Error	44.3
Cost	1904

\[\begin{array}{l} t_1 := x \cdot \left(i \cdot -4\right)\\ t_2 := t \cdot \left(a \cdot -4\right)\\ t_3 := \left(j \cdot k\right) \cdot -27\\ \mathbf{if}\;j \leq -9 \cdot 10^{+118}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;j \leq -1.15 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq -1.7 \cdot 10^{-20}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \leq -3.7 \cdot 10^{-62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq -1.45 \cdot 10^{-117}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;j \leq -4.4 \cdot 10^{-128}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \leq -4.9 \cdot 10^{-182}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq -1.65 \cdot 10^{-285}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \leq 9.6 \cdot 10^{-304}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;j \leq 1.55 \cdot 10^{-159}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq 6.5 \cdot 10^{-126}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \leq 5.2 \cdot 10^{-84}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 7
Error	32.3
Cost	1892

\[\begin{array}{l} t_1 := -4 \cdot \left(x \cdot i\right)\\ t_2 := -4 \cdot \left(t \cdot a\right)\\ t_3 := b \cdot c + t_1\\ t_4 := k \cdot \left(j \cdot -27\right)\\ t_5 := t_2 + t_4\\ \mathbf{if}\;x \leq -8.4 \cdot 10^{+42}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-41}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;x \leq -2.5 \cdot 10^{-69}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -1.65 \cdot 10^{-207}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-285}:\\ \;\;\;\;b \cdot c + t_2\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-145}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-30}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 72000000000000:\\ \;\;\;\;t_2 + \left(j \cdot k\right) \cdot -27\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+87}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_4 + t_1\\ \end{array} \]

Alternative 8
Error	32.0
Cost	1892

\[\begin{array}{l} t_1 := k \cdot \left(j \cdot -27\right)\\ t_2 := -4 \cdot \left(t \cdot a\right)\\ t_3 := t_2 + t_1\\ t_4 := -4 \cdot \left(x \cdot i\right)\\ t_5 := b \cdot c + t_4\\ \mathbf{if}\;x \leq -8.8 \cdot 10^{+42}:\\ \;\;\;\;x \cdot \left(18 \cdot \left(y \cdot \left(z \cdot t\right)\right) + i \cdot -4\right)\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-42}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-75}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-207}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-285}:\\ \;\;\;\;b \cdot c + t_2\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-145}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-30}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;x \leq 69000000000000:\\ \;\;\;\;t_2 + \left(j \cdot k\right) \cdot -27\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+83}:\\ \;\;\;\;t_5\\ \mathbf{else}:\\ \;\;\;\;t_1 + t_4\\ \end{array} \]

Alternative 9
Error	32.3
Cost	1892

\[\begin{array}{l} t_1 := -4 \cdot \left(t \cdot a\right)\\ t_2 := k \cdot \left(j \cdot -27\right)\\ t_3 := t_1 + t_2\\ t_4 := -4 \cdot \left(x \cdot i\right)\\ t_5 := b \cdot c + t_4\\ \mathbf{if}\;x \leq -4.6 \cdot 10^{+52}:\\ \;\;\;\;x \cdot \left(18 \cdot \left(y \cdot \left(z \cdot t\right)\right) + i \cdot -4\right)\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-41}:\\ \;\;\;\;t_2 + x \cdot \left(\left(y \cdot z\right) \cdot \left(18 \cdot t\right)\right)\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-68}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-207}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-285}:\\ \;\;\;\;b \cdot c + t_1\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-145}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-32}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;x \leq 50000000000000:\\ \;\;\;\;t_1 + \left(j \cdot k\right) \cdot -27\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{+83}:\\ \;\;\;\;t_5\\ \mathbf{else}:\\ \;\;\;\;t_2 + t_4\\ \end{array} \]

Alternative 10
Error	32.4
Cost	1761

\[\begin{array}{l} t_1 := -4 \cdot \left(t \cdot a\right)\\ t_2 := t_1 + \left(j \cdot k\right) \cdot -27\\ t_3 := b \cdot c + -4 \cdot \left(x \cdot i\right)\\ \mathbf{if}\;x \leq -8.4 \cdot 10^{+42}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-42}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-68}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-207}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-285}:\\ \;\;\;\;b \cdot c + t_1\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-145} \lor \neg \left(x \leq 1.12 \cdot 10^{-30}\right) \land x \leq 75000000000000:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 11
Error	32.4
Cost	1761

\[\begin{array}{l} t_1 := -4 \cdot \left(t \cdot a\right)\\ t_2 := t_1 + k \cdot \left(j \cdot -27\right)\\ t_3 := b \cdot c + -4 \cdot \left(x \cdot i\right)\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{+42}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-42}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -9.2 \cdot 10^{-69}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-207}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-285}:\\ \;\;\;\;b \cdot c + t_1\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-145}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-31} \lor \neg \left(x \leq 3.2 \cdot 10^{+14}\right):\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(j \cdot k\right) \cdot -27\\ \end{array} \]

Alternative 12
Error	9.3
Cost	1737

\[\begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-23} \lor \neg \left(t \leq 4.6 \cdot 10^{+179}\right):\\ \;\;\;\;\left(b \cdot c + t \cdot \left(18 \cdot \left(y \cdot \left(x \cdot z\right)\right) + a \cdot -4\right)\right) + -4 \cdot \left(x \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot c + t \cdot \left(a \cdot -4\right)\right) + \left(x \cdot \left(i \cdot -4\right) + j \cdot \left(k \cdot -27\right)\right)\\ \end{array} \]

Alternative 13
Error	14.5
Cost	1736

\[\begin{array}{l} t_1 := \left(j \cdot k\right) \cdot -27\\ t_2 := \left(j \cdot 27\right) \cdot k\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{+37}:\\ \;\;\;\;b \cdot c + \left(t_1 - 4 \cdot \left(x \cdot i\right)\right)\\ \mathbf{elif}\;t_2 \leq 50000000000000:\\ \;\;\;\;\left(b \cdot c + a \cdot \left(t \cdot -4\right)\right) + -4 \cdot \left(x \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot c + -4 \cdot \left(t \cdot a\right)\right) + t_1\\ \end{array} \]

Alternative 14
Error	11.1
Cost	1736

\[\begin{array}{l} t_1 := k \cdot \left(j \cdot -27\right)\\ \mathbf{if}\;k \leq -1.2 \cdot 10^{-59}:\\ \;\;\;\;y \cdot \left(x \cdot \left(t \cdot \left(18 \cdot z\right)\right)\right) + t_1\\ \mathbf{elif}\;k \leq 4.8 \cdot 10^{-287}:\\ \;\;\;\;b \cdot c + \left(x \cdot \left(18 \cdot \left(y \cdot \left(z \cdot t\right)\right) + i \cdot -4\right) + -4 \cdot \left(t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot c + -4 \cdot \left(t \cdot a + x \cdot i\right)\right) + t_1\\ \end{array} \]

Alternative 15
Error	20.6
Cost	1488

\[\begin{array}{l} t_1 := \left(j \cdot k\right) \cdot -27\\ t_2 := b \cdot c + \left(t_1 - 4 \cdot \left(x \cdot i\right)\right)\\ t_3 := t \cdot \left(18 \cdot \left(y \cdot \left(x \cdot z\right)\right) + a \cdot -4\right)\\ \mathbf{if}\;t \leq -5.5 \cdot 10^{+17}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{+17}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+134}:\\ \;\;\;\;-4 \cdot \left(t \cdot a\right) + t_1\\ \mathbf{elif}\;t \leq 9 \cdot 10^{+182}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 16
Error	9.6
Cost	1476

\[\begin{array}{l} \mathbf{if}\;t \leq 6.4 \cdot 10^{+182}:\\ \;\;\;\;\left(b \cdot c + \left(-4 \cdot \left(t \cdot a\right) + \frac{x}{\frac{-0.25}{i}}\right)\right) + \left(j \cdot k\right) \cdot -27\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(18 \cdot \left(y \cdot \left(x \cdot z\right)\right) + a \cdot -4\right) + -4 \cdot \left(x \cdot i\right)\\ \end{array} \]

Alternative 17
Error	35.1
Cost	1369

\[\begin{array}{l} t_1 := b \cdot c + -4 \cdot \left(t \cdot a\right)\\ t_2 := x \cdot \left(i \cdot -4\right)\\ \mathbf{if}\;i \leq -5.2 \cdot 10^{+160}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;i \leq -7.8 \cdot 10^{+148}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;i \leq -1.7 \cdot 10^{+23}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;i \leq 6.5 \cdot 10^{-274}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;i \leq 3.25 \cdot 10^{+81} \lor \neg \left(i \leq 1.22 \cdot 10^{+166}\right):\\ \;\;\;\;b \cdot c + k \cdot \left(j \cdot -27\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 18
Error	9.6
Cost	1348

\[\begin{array}{l} \mathbf{if}\;t \leq 8.4 \cdot 10^{+184}:\\ \;\;\;\;\left(b \cdot c + -4 \cdot \left(t \cdot a + x \cdot i\right)\right) + k \cdot \left(j \cdot -27\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(18 \cdot \left(y \cdot \left(x \cdot z\right)\right) + a \cdot -4\right) + -4 \cdot \left(x \cdot i\right)\\ \end{array} \]

Alternative 19
Error	17.1
Cost	1225

\[\begin{array}{l} t_1 := \left(j \cdot k\right) \cdot -27\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{+25} \lor \neg \left(x \leq 1.8 \cdot 10^{-109}\right):\\ \;\;\;\;b \cdot c + \left(t_1 - 4 \cdot \left(x \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot c + -4 \cdot \left(t \cdot a\right)\right) + t_1\\ \end{array} \]

Alternative 20
Error	45.5
Cost	1112

\[\begin{array}{l} t_1 := x \cdot \left(i \cdot -4\right)\\ \mathbf{if}\;b \leq -1.35 \cdot 10^{+123}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;b \leq -1.6 \cdot 10^{-35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -3 \cdot 10^{-67}:\\ \;\;\;\;\left(j \cdot k\right) \cdot -27\\ \mathbf{elif}\;b \leq -1.55 \cdot 10^{-95}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -3.2 \cdot 10^{-174}:\\ \;\;\;\;k \cdot \left(j \cdot -27\right)\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-120}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;b \cdot c\\ \end{array} \]

Alternative 21
Error	36.1
Cost	1104

\[\begin{array}{l} t_1 := b \cdot c + -4 \cdot \left(t \cdot a\right)\\ t_2 := x \cdot \left(i \cdot -4\right)\\ \mathbf{if}\;x \leq -8.4 \cdot 10^{+42}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-283}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-257}:\\ \;\;\;\;k \cdot \left(j \cdot -27\right)\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 22
Error	43.7
Cost	585

\[\begin{array}{l} \mathbf{if}\;k \leq -2.05 \cdot 10^{-174} \lor \neg \left(k \leq 1.12 \cdot 10^{-91}\right):\\ \;\;\;\;\left(j \cdot k\right) \cdot -27\\ \mathbf{else}:\\ \;\;\;\;b \cdot c\\ \end{array} \]

Alternative 23
Error	43.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;k \leq -1.7 \cdot 10^{-174}:\\ \;\;\;\;\left(j \cdot k\right) \cdot -27\\ \mathbf{elif}\;k \leq 1.02 \cdot 10^{-91}:\\ \;\;\;\;b \cdot c\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(j \cdot -27\right)\\ \end{array} \]

Alternative 24
Error	47.9
Cost	192

\[b \cdot c \]

Reproduce

herbie shell --seed 2023018 
(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)))