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

Average Error: 6.0 → 1.2

Time: 36.3s

Precision: binary64

Cost: 6088

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

Math

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

↓

\[\begin{array}{l} t_1 := k \cdot \left(j \cdot -27\right)\\ t_2 := \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\right) + t_1\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\left(b \cdot c + 18 \cdot \left(y \cdot \left(t \cdot \left(x \cdot z\right)\right)\right)\right) + \left(\left(x \cdot i\right) \cdot -4 + \left(j \cdot k\right) \cdot -27\right)\\ \mathbf{elif}\;t_2 \leq 4 \cdot 10^{+299}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot c + \left(\left(x \cdot \left(\left(y \cdot t\right) \cdot \left(18 \cdot z\right)\right) + x \cdot \left(i \cdot -4\right)\right) + -4 \cdot \left(t \cdot a\right)\right)\right) + t_1\\ \end{array} \]

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 (* k (* j -27.0)))
        (t_2
         (+
          (+
           (+ (+ (* (* (* (* x 18.0) y) z) t) (* t (* a -4.0))) (* b c))
           (* i (* x -4.0)))
          t_1)))
   (if (<= t_2 (- INFINITY))
     (+
      (+ (* b c) (* 18.0 (* y (* t (* x z)))))
      (+ (* (* x i) -4.0) (* (* j k) -27.0)))
     (if (<= t_2 4e+299)
       t_2
       (+
        (+
         (* b c)
         (+
          (+ (* x (* (* y t) (* 18.0 z))) (* x (* i -4.0)))
          (* -4.0 (* t a))))
        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 = k * (j * -27.0);
	double t_2 = (((((((x * 18.0) * y) * z) * t) + (t * (a * -4.0))) + (b * c)) + (i * (x * -4.0))) + t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = ((b * c) + (18.0 * (y * (t * (x * z))))) + (((x * i) * -4.0) + ((j * k) * -27.0));
	} else if (t_2 <= 4e+299) {
		tmp = t_2;
	} else {
		tmp = ((b * c) + (((x * ((y * t) * (18.0 * z))) + (x * (i * -4.0))) + (-4.0 * (t * a)))) + 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 = k * (j * -27.0);
	double t_2 = (((((((x * 18.0) * y) * z) * t) + (t * (a * -4.0))) + (b * c)) + (i * (x * -4.0))) + t_1;
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = ((b * c) + (18.0 * (y * (t * (x * z))))) + (((x * i) * -4.0) + ((j * k) * -27.0));
	} else if (t_2 <= 4e+299) {
		tmp = t_2;
	} else {
		tmp = ((b * c) + (((x * ((y * t) * (18.0 * z))) + (x * (i * -4.0))) + (-4.0 * (t * a)))) + 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 = k * (j * -27.0)
	t_2 = (((((((x * 18.0) * y) * z) * t) + (t * (a * -4.0))) + (b * c)) + (i * (x * -4.0))) + t_1
	tmp = 0
	if t_2 <= -math.inf:
		tmp = ((b * c) + (18.0 * (y * (t * (x * z))))) + (((x * i) * -4.0) + ((j * k) * -27.0))
	elif t_2 <= 4e+299:
		tmp = t_2
	else:
		tmp = ((b * c) + (((x * ((y * t) * (18.0 * z))) + (x * (i * -4.0))) + (-4.0 * (t * a)))) + 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(k * Float64(j * -27.0))
	t_2 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * 18.0) * y) * z) * t) + Float64(t * Float64(a * -4.0))) + Float64(b * c)) + Float64(i * Float64(x * -4.0))) + t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(b * c) + Float64(18.0 * Float64(y * Float64(t * Float64(x * z))))) + Float64(Float64(Float64(x * i) * -4.0) + Float64(Float64(j * k) * -27.0)));
	elseif (t_2 <= 4e+299)
		tmp = t_2;
	else
		tmp = Float64(Float64(Float64(b * c) + Float64(Float64(Float64(x * Float64(Float64(y * t) * Float64(18.0 * z))) + Float64(x * Float64(i * -4.0))) + Float64(-4.0 * Float64(t * a)))) + 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 = k * (j * -27.0);
	t_2 = (((((((x * 18.0) * y) * z) * t) + (t * (a * -4.0))) + (b * c)) + (i * (x * -4.0))) + t_1;
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = ((b * c) + (18.0 * (y * (t * (x * z))))) + (((x * i) * -4.0) + ((j * k) * -27.0));
	elseif (t_2 <= 4e+299)
		tmp = t_2;
	else
		tmp = ((b * c) + (((x * ((y * t) * (18.0 * z))) + (x * (i * -4.0))) + (-4.0 * (t * a)))) + 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[(k * N[(j * -27.0), $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[(t * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision] + N[(i * N[(x * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[(b * c), $MachinePrecision] + N[(18.0 * N[(y * N[(t * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x * i), $MachinePrecision] * -4.0), $MachinePrecision] + N[(N[(j * k), $MachinePrecision] * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+299], t$95$2, N[(N[(N[(b * c), $MachinePrecision] + N[(N[(N[(x * N[(N[(y * t), $MachinePrecision] * N[(18.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(i * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]

Target

Original	6.0
Target	1.7
Herbie	1.2

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

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. Applied egg-rr 7.8

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

   3. Taylor expanded in a around 0 12.2

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

   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)) < 4.0000000000000002e299

   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 4.0000000000000002e299 < (-.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 48.7

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

   2. Simplified 33.2

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

      [Start] 48.7	\[ \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
      cancel-sign-sub-inv [=>] 48.7	\[ \color{blue}{\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) + \left(-j \cdot 27\right) \cdot k} \]
      associate--l+ [=>] 48.7	\[ \color{blue}{\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + \left(b \cdot c - \left(x \cdot 4\right) \cdot i\right)\right)} + \left(-j \cdot 27\right) \cdot k \]

   3. Taylor expanded in x around 0 5.7

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

   4. Applied egg-rr 6.4

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

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right) \leq -\infty:\\ \;\;\;\;\left(b \cdot c + 18 \cdot \left(y \cdot \left(t \cdot \left(x \cdot z\right)\right)\right)\right) + \left(\left(x \cdot i\right) \cdot -4 + \left(j \cdot k\right) \cdot -27\right)\\ \mathbf{elif}\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right) \leq 4 \cdot 10^{+299}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot c + \left(\left(x \cdot \left(\left(y \cdot t\right) \cdot \left(18 \cdot z\right)\right) + x \cdot \left(i \cdot -4\right)\right) + -4 \cdot \left(t \cdot a\right)\right)\right) + k \cdot \left(j \cdot -27\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	36.6
Cost	3212

\[\begin{array}{l} t_1 := b \cdot c + -4 \cdot \left(t \cdot a\right)\\ t_2 := x \cdot \left(i \cdot -4 + 18 \cdot \left(y \cdot \left(z \cdot t\right)\right)\right)\\ t_3 := k \cdot \left(j \cdot -27\right) + t \cdot \left(a \cdot -4\right)\\ t_4 := \left(j \cdot 27\right) \cdot k\\ t_5 := b \cdot c - t_4\\ t_6 := x \cdot \left(i \cdot -4\right) - t_4\\ \mathbf{if}\;y \leq -2.85 \cdot 10^{+249}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -6.2 \cdot 10^{+113}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{+101}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -1.25 \cdot 10^{+79}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;y \leq -8000000:\\ \;\;\;\;t_5\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -9.5 \cdot 10^{-57}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-68}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.95 \cdot 10^{-75}:\\ \;\;\;\;\left(x \cdot i\right) \cdot -4\\ \mathbf{elif}\;y \leq -7.5 \cdot 10^{-104}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;y \leq -1.35 \cdot 10^{-124}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -3.3 \cdot 10^{-149}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.02 \cdot 10^{-181}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;y \leq -4.4 \cdot 10^{-292}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-286}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-229}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-204}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-43}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+94}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	36.6
Cost	3080

\[\begin{array}{l} t_1 := \left(j \cdot 27\right) \cdot k\\ t_2 := x \cdot \left(i \cdot -4\right) - t_1\\ t_3 := b \cdot c + -4 \cdot \left(t \cdot a\right)\\ t_4 := x \cdot \left(i \cdot -4 + 18 \cdot \left(y \cdot \left(z \cdot t\right)\right)\right)\\ t_5 := b \cdot c - t_1\\ t_6 := k \cdot \left(j \cdot -27\right) + t \cdot \left(a \cdot -4\right)\\ \mathbf{if}\;y \leq -1.3 \cdot 10^{+250}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;y \leq -1.08 \cdot 10^{+182}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y \leq -3.4 \cdot 10^{+168}:\\ \;\;\;\;t \cdot \left(18 \cdot \left(y \cdot \left(x \cdot z\right)\right) + a \cdot -4\right)\\ \mathbf{elif}\;y \leq -1.15 \cdot 10^{+89}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{-12}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{-57}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-68}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -6 \cdot 10^{-76}:\\ \;\;\;\;\left(x \cdot i\right) \cdot -4\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-104}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;y \leq -1.9 \cdot 10^{-120}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;y \leq -1.7 \cdot 10^{-150}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -1.15 \cdot 10^{-181}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -3.9 \cdot 10^{-289}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-286}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-229}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;y \leq 2.02 \cdot 10^{-203}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-43}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+71}:\\ \;\;\;\;t_6\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 3
Error	35.7
Cost	2948

\[\begin{array}{l} t_1 := b \cdot c + -4 \cdot \left(t \cdot a\right)\\ t_2 := \left(j \cdot 27\right) \cdot k\\ t_3 := b \cdot c - t_2\\ t_4 := x \cdot \left(i \cdot -4\right) - t_2\\ t_5 := k \cdot \left(j \cdot -27\right)\\ t_6 := t_5 + t \cdot \left(a \cdot -4\right)\\ t_7 := x \cdot \left(i \cdot -4 + 18 \cdot \left(y \cdot \left(z \cdot t\right)\right)\right)\\ \mathbf{if}\;y \leq -1 \cdot 10^{+217}:\\ \;\;\;\;t_5 + 18 \cdot \left(y \cdot \left(z \cdot \left(x \cdot t\right)\right)\right)\\ \mathbf{elif}\;y \leq -6.2 \cdot 10^{+132}:\\ \;\;\;\;t_7\\ \mathbf{elif}\;y \leq -6.8 \cdot 10^{+89}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y \leq -3.8 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{-57}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y \leq -1.25 \cdot 10^{-67}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.95 \cdot 10^{-75}:\\ \;\;\;\;\left(x \cdot i\right) \cdot -4\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{-104}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -1.7 \cdot 10^{-120}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;y \leq -9.5 \cdot 10^{-149}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{-182}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y \leq -2.4 \cdot 10^{-289}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-286}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-229}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-203}:\\ \;\;\;\;t_7\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-43}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 6.7 \cdot 10^{+81}:\\ \;\;\;\;t_6\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	48.6
Cost	2424

\[\begin{array}{l} t_1 := \left(x \cdot i\right) \cdot -4\\ t_2 := -4 \cdot \left(t \cdot a\right)\\ t_3 := j \cdot \left(k \cdot -27\right)\\ \mathbf{if}\;z \leq -1.08 \cdot 10^{-107}:\\ \;\;\;\;18 \cdot \left(y \cdot \left(t \cdot \left(x \cdot z\right)\right)\right)\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-186}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -2.6 \cdot 10^{-301}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{-114}:\\ \;\;\;\;\left(j \cdot k\right) \cdot -27\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+49}:\\ \;\;\;\;k \cdot \left(j \cdot -27\right)\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+70}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{+83}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+102}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;z \leq 6.3 \cdot 10^{+116}:\\ \;\;\;\;t \cdot \left(x \cdot \left(y \cdot \left(18 \cdot z\right)\right)\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+132}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+169}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+177}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+183}:\\ \;\;\;\;18 \cdot \left(\left(z \cdot t\right) \cdot \left(x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot c\\ \end{array} \]

Alternative 5
Error	2.1
Cost	2121

\[\begin{array}{l} \mathbf{if}\;t \leq -2.25 \cdot 10^{+52} \lor \neg \left(t \leq 1.36 \cdot 10^{-11}\right):\\ \;\;\;\;\left(b \cdot c + t \cdot \left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right) + a \cdot -4\right)\right) + \left(x \cdot \left(i \cdot -4\right) + j \cdot \left(k \cdot -27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(j \cdot -27\right) + \left(b \cdot c + \left(-4 \cdot \left(t \cdot a\right) + x \cdot \left(i \cdot -4 + 18 \cdot \left(y \cdot \left(z \cdot t\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 6
Error	5.1
Cost	2120

\[\begin{array}{l} t_1 := x \cdot \left(i \cdot -4\right)\\ \mathbf{if}\;t \leq -3.7 \cdot 10^{+52}:\\ \;\;\;\;\left(b \cdot c + t \cdot \left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right) + a \cdot -4\right)\right) + t_1\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{+186}:\\ \;\;\;\;k \cdot \left(j \cdot -27\right) + \left(b \cdot c + \left(-4 \cdot \left(t \cdot a\right) + x \cdot \left(i \cdot -4 + 18 \cdot \left(y \cdot \left(z \cdot t\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot c + t \cdot \left(a \cdot -4\right)\right) + \left(t_1 + j \cdot \left(k \cdot -27\right)\right)\\ \end{array} \]

Alternative 7
Error	9.2
Cost	1869

\[\begin{array}{l} t_1 := x \cdot \left(i \cdot -4\right)\\ t_2 := t_1 + j \cdot \left(k \cdot -27\right)\\ \mathbf{if}\;z \leq -1.32 \cdot 10^{-102}:\\ \;\;\;\;\left(b \cdot c + 18 \cdot \left(y \cdot \left(t \cdot \left(x \cdot z\right)\right)\right)\right) + t_2\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+251} \lor \neg \left(z \leq 2.9 \cdot 10^{+280}\right):\\ \;\;\;\;\left(b \cdot c + t \cdot \left(a \cdot -4\right)\right) + t_2\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot c + t \cdot \left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right) + a \cdot -4\right)\right) + t_1\\ \end{array} \]

Alternative 8
Error	9.2
Cost	1869

\[\begin{array}{l} t_1 := x \cdot \left(i \cdot -4\right)\\ \mathbf{if}\;z \leq -5.3 \cdot 10^{-105}:\\ \;\;\;\;\left(b \cdot c + 18 \cdot \left(y \cdot \left(t \cdot \left(x \cdot z\right)\right)\right)\right) + \left(\left(x \cdot i\right) \cdot -4 + \left(j \cdot k\right) \cdot -27\right)\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+251} \lor \neg \left(z \leq 3.6 \cdot 10^{+281}\right):\\ \;\;\;\;\left(b \cdot c + t \cdot \left(a \cdot -4\right)\right) + \left(t_1 + j \cdot \left(k \cdot -27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot c + t \cdot \left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right) + a \cdot -4\right)\right) + t_1\\ \end{array} \]

Alternative 9
Error	24.4
Cost	1753

\[\begin{array}{l} t_1 := k \cdot \left(j \cdot -27\right) + \left(b \cdot c + -4 \cdot \left(t \cdot a\right)\right)\\ \mathbf{if}\;t \leq -1.05 \cdot 10^{+127}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3.8 \cdot 10^{-24}:\\ \;\;\;\;t \cdot \left(18 \cdot \left(y \cdot \left(x \cdot z\right)\right) + a \cdot -4\right)\\ \mathbf{elif}\;t \leq -8.8 \cdot 10^{-231}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3.6 \cdot 10^{-276}:\\ \;\;\;\;x \cdot \left(i \cdot -4\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-77} \lor \neg \left(t \leq 1.65 \cdot 10^{-48}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(i \cdot -4 + 18 \cdot \left(y \cdot \left(z \cdot t\right)\right)\right)\\ \end{array} \]

Alternative 10
Error	9.8
Cost	1604

\[\begin{array}{l} t_1 := x \cdot \left(i \cdot -4\right)\\ \mathbf{if}\;t \leq -3.6 \cdot 10^{-7}:\\ \;\;\;\;\left(b \cdot c + t \cdot \left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right) + a \cdot -4\right)\right) + t_1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot c + t \cdot \left(a \cdot -4\right)\right) + \left(t_1 + j \cdot \left(k \cdot -27\right)\right)\\ \end{array} \]

Alternative 11
Error	31.2
Cost	1365

\[\begin{array}{l} t_1 := b \cdot c + -4 \cdot \left(t \cdot a\right)\\ t_2 := \left(j \cdot 27\right) \cdot k\\ t_3 := x \cdot \left(i \cdot -4\right) - t_2\\ \mathbf{if}\;t \leq -1.1 \cdot 10^{-56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3.1 \cdot 10^{-277}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 7.6 \cdot 10^{-88}:\\ \;\;\;\;b \cdot c - t_2\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{+192} \lor \neg \left(t \leq 1.55 \cdot 10^{+259}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 12
Error	9.7
Cost	1344

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

Alternative 13
Error	34.9
Cost	1237

\[\begin{array}{l} t_1 := b \cdot c - \left(j \cdot 27\right) \cdot k\\ t_2 := -4 \cdot \left(t \cdot a\right)\\ \mathbf{if}\;t \leq -8 \cdot 10^{-27}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -7 \cdot 10^{-246}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.75 \cdot 10^{-275}:\\ \;\;\;\;\left(x \cdot i\right) \cdot -4\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{-10} \lor \neg \left(t \leq 7.5 \cdot 10^{+78}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 14
Error	17.2
Cost	1225

\[\begin{array}{l} t_1 := k \cdot \left(j \cdot -27\right)\\ \mathbf{if}\;a \leq -8200000000000 \lor \neg \left(a \leq 0.18\right):\\ \;\;\;\;t_1 + \left(b \cdot c + -4 \cdot \left(t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(b \cdot c + \left(x \cdot i\right) \cdot -4\right)\\ \end{array} \]

Alternative 15
Error	44.6
Cost	1112

\[\begin{array}{l} t_1 := \left(x \cdot i\right) \cdot -4\\ t_2 := -4 \cdot \left(t \cdot a\right)\\ \mathbf{if}\;t \leq -1.15 \cdot 10^{-56}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -7 \cdot 10^{-246}:\\ \;\;\;\;\left(j \cdot k\right) \cdot -27\\ \mathbf{elif}\;t \leq -1.25 \cdot 10^{-275}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-271}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-219}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 10^{-87}:\\ \;\;\;\;b \cdot c\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 16
Error	44.6
Cost	1112

\[\begin{array}{l} t_1 := \left(x \cdot i\right) \cdot -4\\ t_2 := -4 \cdot \left(t \cdot a\right)\\ \mathbf{if}\;t \leq -1.75 \cdot 10^{-56}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -8 \cdot 10^{-246}:\\ \;\;\;\;j \cdot \left(k \cdot -27\right)\\ \mathbf{elif}\;t \leq -1.9 \cdot 10^{-276}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-273}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{-219}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-88}:\\ \;\;\;\;b \cdot c\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 17
Error	44.6
Cost	1112

\[\begin{array}{l} t_1 := \left(x \cdot i\right) \cdot -4\\ t_2 := -4 \cdot \left(t \cdot a\right)\\ \mathbf{if}\;t \leq -1.75 \cdot 10^{-56}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.1 \cdot 10^{-245}:\\ \;\;\;\;k \cdot \left(j \cdot -27\right)\\ \mathbf{elif}\;t \leq -8.7 \cdot 10^{-277}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-269}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-219}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-87}:\\ \;\;\;\;b \cdot c\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 18
Error	30.9
Cost	1104

\[\begin{array}{l} t_1 := b \cdot c - \left(j \cdot 27\right) \cdot k\\ t_2 := b \cdot c + -4 \cdot \left(t \cdot a\right)\\ \mathbf{if}\;t \leq -1.9 \cdot 10^{-54}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -7 \cdot 10^{-246}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.75 \cdot 10^{-275}:\\ \;\;\;\;\left(x \cdot i\right) \cdot -4\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-87}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 19
Error	44.5
Cost	848

\[\begin{array}{l} t_1 := -4 \cdot \left(t \cdot a\right)\\ t_2 := \left(j \cdot k\right) \cdot -27\\ \mathbf{if}\;k \leq -5.5 \cdot 10^{-137}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -1.25 \cdot 10^{-278}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 6 \cdot 10^{-25}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;k \leq 4.4 \cdot 10^{+84}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 20
Error	43.8
Cost	585

\[\begin{array}{l} \mathbf{if}\;k \leq -1.8 \cdot 10^{-113} \lor \neg \left(k \leq 6.4 \cdot 10^{+84}\right):\\ \;\;\;\;\left(j \cdot k\right) \cdot -27\\ \mathbf{else}:\\ \;\;\;\;b \cdot c\\ \end{array} \]

Alternative 21
Error	48.4
Cost	192

\[b \cdot c \]

Reproduce

herbie shell --seed 2023027 
(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)))