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

Average Error: 5.7 → 1.0

Time: 39.2s

Precision: binary64

Cost: 6088

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

Math

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

↓

\[\begin{array}{l} t_1 := i \cdot \left(x \cdot -4\right)\\ t_2 := k \cdot \left(j \cdot -27\right)\\ t_3 := t \cdot \left(a \cdot -4\right)\\ t_4 := \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t_3\right) + b \cdot c\right) + t_1\right) + t_2\\ t_5 := y \cdot \left(z \cdot t\right)\\ \mathbf{if}\;t_4 \leq -\infty:\\ \;\;\;\;\left(b \cdot c + x \cdot \left(18 \cdot t_5 + i \cdot -4\right)\right) + \left(\left(t \cdot a\right) \cdot -4 - 27 \cdot \left(j \cdot k\right)\right)\\ \mathbf{elif}\;t_4 \leq 1.62 \cdot 10^{+297}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(x \cdot 18\right) \cdot t_5 + t_3\right)\right) + t_1\right) + t_2\\ \end{array} \]

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))
        (t_5 (* y (* z t))))
   (if (<= t_4 (- INFINITY))
     (+
      (+ (* b c) (* x (+ (* 18.0 t_5) (* i -4.0))))
      (- (* (* t a) -4.0) (* 27.0 (* j k))))
     (if (<= t_4 1.62e+297)
       t_4
       (+ (+ (+ (* b c) (+ (* (* x 18.0) t_5) 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 t_5 = y * (z * t);
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = ((b * c) + (x * ((18.0 * t_5) + (i * -4.0)))) + (((t * a) * -4.0) - (27.0 * (j * k)));
	} else if (t_4 <= 1.62e+297) {
		tmp = t_4;
	} else {
		tmp = (((b * c) + (((x * 18.0) * t_5) + 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 t_5 = y * (z * t);
	double tmp;
	if (t_4 <= -Double.POSITIVE_INFINITY) {
		tmp = ((b * c) + (x * ((18.0 * t_5) + (i * -4.0)))) + (((t * a) * -4.0) - (27.0 * (j * k)));
	} else if (t_4 <= 1.62e+297) {
		tmp = t_4;
	} else {
		tmp = (((b * c) + (((x * 18.0) * t_5) + 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
	t_5 = y * (z * t)
	tmp = 0
	if t_4 <= -math.inf:
		tmp = ((b * c) + (x * ((18.0 * t_5) + (i * -4.0)))) + (((t * a) * -4.0) - (27.0 * (j * k)))
	elif t_4 <= 1.62e+297:
		tmp = t_4
	else:
		tmp = (((b * c) + (((x * 18.0) * t_5) + 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)
	t_5 = Float64(y * Float64(z * t))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(b * c) + Float64(x * Float64(Float64(18.0 * t_5) + Float64(i * -4.0)))) + Float64(Float64(Float64(t * a) * -4.0) - Float64(27.0 * Float64(j * k))));
	elseif (t_4 <= 1.62e+297)
		tmp = t_4;
	else
		tmp = Float64(Float64(Float64(Float64(b * c) + Float64(Float64(Float64(x * 18.0) * t_5) + 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;
	t_5 = y * (z * t);
	tmp = 0.0;
	if (t_4 <= -Inf)
		tmp = ((b * c) + (x * ((18.0 * t_5) + (i * -4.0)))) + (((t * a) * -4.0) - (27.0 * (j * k)));
	elseif (t_4 <= 1.62e+297)
		tmp = t_4;
	else
		tmp = (((b * c) + (((x * 18.0) * t_5) + 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]}, Block[{t$95$5 = N[(y * N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[(N[(b * c), $MachinePrecision] + N[(x * N[(N[(18.0 * t$95$5), $MachinePrecision] + N[(i * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t * a), $MachinePrecision] * -4.0), $MachinePrecision] - N[(27.0 * N[(j * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 1.62e+297], t$95$4, N[(N[(N[(N[(b * c), $MachinePrecision] + N[(N[(N[(x * 18.0), $MachinePrecision] * t$95$5), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision]]]]]]]]

Target

Original	5.7
Target	1.6
Herbie	1.0

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

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 5.5

      \[\leadsto \left(\left(\left(\color{blue}{\left(0 + \left(x \cdot 18\right) \cdot \left(y \cdot \left(z \cdot t\right)\right)\right)} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]

   3. Taylor expanded in x around 0 4.1

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

   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)) < 1.61999999999999999e297

   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 1.61999999999999999e297 < (-.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 46.0

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

   2. Applied egg-rr 8.8

      \[\leadsto \left(\left(\left(\color{blue}{\left(0 + \left(x \cdot 18\right) \cdot \left(y \cdot \left(z \cdot t\right)\right)\right)} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
3. Recombined 3 regimes into one program.

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right) \leq -\infty:\\ \;\;\;\;\left(b \cdot c + x \cdot \left(18 \cdot \left(y \cdot \left(z \cdot t\right)\right) + i \cdot -4\right)\right) + \left(\left(t \cdot a\right) \cdot -4 - 27 \cdot \left(j \cdot k\right)\right)\\ \mathbf{elif}\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right) \leq 1.62 \cdot 10^{+297}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(x \cdot 18\right) \cdot \left(y \cdot \left(z \cdot t\right)\right) + t \cdot \left(a \cdot -4\right)\right)\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.9
Cost	6088

\[\begin{array}{l} t_1 := \left(b \cdot c + x \cdot \left(18 \cdot \left(y \cdot \left(z \cdot t\right)\right) + i \cdot -4\right)\right) + \left(\left(t \cdot a\right) \cdot -4 - 27 \cdot \left(j \cdot k\right)\right)\\ t_2 := \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right)\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 1.62 \cdot 10^{+297}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	33.2
Cost	2416

\[\begin{array}{l} t_1 := \left(t \cdot a\right) \cdot -4 - 27 \cdot \left(j \cdot k\right)\\ t_2 := t \cdot \left(18 \cdot \left(y \cdot \left(x \cdot z\right)\right) + a \cdot -4\right)\\ t_3 := b \cdot c - 4 \cdot \left(x \cdot i\right)\\ t_4 := \left(j \cdot k\right) \cdot -27\\ t_5 := t_4 + -4 \cdot \left(x \cdot i\right)\\ t_6 := b \cdot c + t_4\\ \mathbf{if}\;t \leq -1 \cdot 10^{+120}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.8 \cdot 10^{+57}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -1.4182308078943914 \cdot 10^{-5}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -2.0767034434138977 \cdot 10^{-95}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.1795578848883674 \cdot 10^{-133}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -4.1574391057457075 \cdot 10^{-214}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.0120573589287873 \cdot 10^{-252}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;t \leq 4.5206646605173066 \cdot 10^{-273}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;t \leq 7.972191750369296 \cdot 10^{-204}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 5.420980006714453 \cdot 10^{-152}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;t \leq 249.2737578366813:\\ \;\;\;\;t_6\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{+203}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_6\\ \end{array} \]

Alternative 3
Error	33.4
Cost	2156

\[\begin{array}{l} t_1 := \left(j \cdot k\right) \cdot -27\\ t_2 := b \cdot c + t_1\\ t_3 := b \cdot c - 4 \cdot \left(x \cdot i\right)\\ t_4 := \left(t \cdot a\right) \cdot -4\\ t_5 := b \cdot c + t_4\\ t_6 := t_4 - 27 \cdot \left(j \cdot k\right)\\ \mathbf{if}\;k \leq -6.854494792984612 \cdot 10^{-35}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;k \leq -1.7743175610010773 \cdot 10^{-101}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq -3.3810979634221506 \cdot 10^{-163}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;k \leq -3.2103538446130872 \cdot 10^{-255}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;k \leq 9.826577919179858 \cdot 10^{-263}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq 2.6034481745893545 \cdot 10^{-233}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;k \leq 1.700312636834686 \cdot 10^{-162}:\\ \;\;\;\;18 \cdot \left(\left(z \cdot t\right) \cdot \left(x \cdot y\right)\right)\\ \mathbf{elif}\;k \leq 1.314584988576366 \cdot 10^{-106}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;k \leq 78034045611846.67:\\ \;\;\;\;t_6\\ \mathbf{elif}\;k \leq 10^{+147}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 10^{+180}:\\ \;\;\;\;t_1 + -4 \cdot \left(x \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	32.6
Cost	2156

\[\begin{array}{l} t_1 := \left(j \cdot k\right) \cdot -27\\ t_2 := b \cdot c + t_1\\ t_3 := b \cdot c - 4 \cdot \left(x \cdot i\right)\\ t_4 := \left(t \cdot a\right) \cdot -4\\ t_5 := b \cdot c + t_4\\ t_6 := t_4 - 27 \cdot \left(j \cdot k\right)\\ \mathbf{if}\;k \leq -6.854494792984612 \cdot 10^{-35}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;k \leq -1.7743175610010773 \cdot 10^{-101}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq -3.3810979634221506 \cdot 10^{-163}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;k \leq -3.2103538446130872 \cdot 10^{-255}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;k \leq 9.826577919179858 \cdot 10^{-263}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq 1.824687048610874 \cdot 10^{-240}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;k \leq 1.700312636834686 \cdot 10^{-162}:\\ \;\;\;\;x \cdot \left(18 \cdot \left(y \cdot \left(z \cdot t\right)\right) + i \cdot -4\right)\\ \mathbf{elif}\;k \leq 1.314584988576366 \cdot 10^{-106}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;k \leq 78034045611846.67:\\ \;\;\;\;t_6\\ \mathbf{elif}\;k \leq 10^{+147}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 10^{+180}:\\ \;\;\;\;t_1 + -4 \cdot \left(x \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	4.1
Cost	2120

\[\begin{array}{l} t_1 := \left(b \cdot c + x \cdot \left(18 \cdot \left(y \cdot \left(z \cdot t\right)\right) + i \cdot -4\right)\right) + \left(\left(t \cdot a\right) \cdot -4 - 27 \cdot \left(j \cdot k\right)\right)\\ \mathbf{if}\;x \leq -3.157937627403369 \cdot 10^{-177}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4.060431915396524 \cdot 10^{-125}:\\ \;\;\;\;b \cdot c + \left(\left(j \cdot k\right) \cdot -27 + t \cdot \left(18 \cdot \left(y \cdot \left(x \cdot z\right)\right) + a \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	3.1
Cost	2116

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+30}:\\ \;\;\;\;\left(b \cdot c + x \cdot \left(18 \cdot \left(y \cdot \left(z \cdot t\right)\right) + i \cdot -4\right)\right) + \left(\left(t \cdot a\right) \cdot -4 - 27 \cdot \left(j \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(18 \cdot y\right) \cdot \left(t \cdot \left(x \cdot z\right)\right) + t \cdot \left(a \cdot -4\right)\right)\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right)\\ \end{array} \]

Alternative 7
Error	9.5
Cost	2000

\[\begin{array}{l} t_1 := b \cdot c + \left(\left(j \cdot k\right) \cdot -27 + t \cdot \left(18 \cdot \left(y \cdot \left(x \cdot z\right)\right) + a \cdot -4\right)\right)\\ t_2 := \left(t \cdot a\right) \cdot -4 - 27 \cdot \left(j \cdot k\right)\\ t_3 := b \cdot c + \left(-4 \cdot \left(x \cdot i\right) + t_2\right)\\ \mathbf{if}\;i \leq -6.920103541444589 \cdot 10^{+87}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;i \leq -4.543085541508697 \cdot 10^{-285}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;i \leq 1.3820384148815683 \cdot 10^{-192}:\\ \;\;\;\;\left(b \cdot c + 18 \cdot \left(x \cdot \left(z \cdot \left(y \cdot t\right)\right)\right)\right) + t_2\\ \mathbf{elif}\;i \leq 1.0438660273824667 \cdot 10^{-109}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 8
Error	32.3
Cost	1760

\[\begin{array}{l} t_1 := \left(j \cdot k\right) \cdot -27\\ t_2 := b \cdot c + \left(t \cdot a\right) \cdot -4\\ t_3 := b \cdot c + t_1\\ \mathbf{if}\;k \leq -2.6507720112970817 \cdot 10^{-125}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq -3.2103538446130872 \cdot 10^{-255}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 9.826577919179858 \cdot 10^{-263}:\\ \;\;\;\;b \cdot c - 4 \cdot \left(x \cdot i\right)\\ \mathbf{elif}\;k \leq 2.6034481745893545 \cdot 10^{-233}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 1.700312636834686 \cdot 10^{-162}:\\ \;\;\;\;18 \cdot \left(\left(z \cdot t\right) \cdot \left(x \cdot y\right)\right)\\ \mathbf{elif}\;k \leq 8680293.68987565:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 10^{+147}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq 10^{+180}:\\ \;\;\;\;t_1 + -4 \cdot \left(x \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 9
Error	23.8
Cost	1752

\[\begin{array}{l} t_1 := b \cdot c + \left(\left(t \cdot a\right) \cdot -4 - 27 \cdot \left(j \cdot k\right)\right)\\ \mathbf{if}\;k \leq -3.2103538446130872 \cdot 10^{-255}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 9.826577919179858 \cdot 10^{-263}:\\ \;\;\;\;b \cdot c - 4 \cdot \left(x \cdot i\right)\\ \mathbf{elif}\;k \leq 5.977014559598515 \cdot 10^{-171}:\\ \;\;\;\;t \cdot \left(18 \cdot \left(y \cdot \left(x \cdot z\right)\right) + a \cdot -4\right)\\ \mathbf{elif}\;k \leq 1.700312636834686 \cdot 10^{-162}:\\ \;\;\;\;x \cdot \left(18 \cdot \left(y \cdot \left(z \cdot t\right)\right) + i \cdot -4\right)\\ \mathbf{elif}\;k \leq 5 \cdot 10^{+149}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 10^{+180}:\\ \;\;\;\;\left(j \cdot k\right) \cdot -27 + -4 \cdot \left(x \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	9.6
Cost	1736

\[\begin{array}{l} t_1 := b \cdot c + \left(-4 \cdot \left(x \cdot i\right) + \left(\left(t \cdot a\right) \cdot -4 - 27 \cdot \left(j \cdot k\right)\right)\right)\\ \mathbf{if}\;i \leq -6.920103541444589 \cdot 10^{+87}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;i \leq 1.0438660273824667 \cdot 10^{-109}:\\ \;\;\;\;b \cdot c + \left(\left(j \cdot k\right) \cdot -27 + t \cdot \left(18 \cdot \left(y \cdot \left(x \cdot z\right)\right) + a \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	9.8
Cost	1608

\[\begin{array}{l} t_1 := b \cdot c + \left(-4 \cdot \left(x \cdot i\right) + \left(\left(t \cdot a\right) \cdot -4 - 27 \cdot \left(j \cdot k\right)\right)\right)\\ \mathbf{if}\;b \leq -4.707814081205205 \cdot 10^{-145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -3.4476386091500946 \cdot 10^{-159}:\\ \;\;\;\;x \cdot \left(18 \cdot \left(y \cdot \left(z \cdot t\right)\right) + i \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	45.9
Cost	1508

\[\begin{array}{l} t_1 := \left(t \cdot a\right) \cdot -4\\ t_2 := k \cdot \left(j \cdot -27\right)\\ \mathbf{if}\;t \leq -4 \cdot 10^{+103}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -6.7 \cdot 10^{+50}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;t \leq -5.030361573776496 \cdot 10^{-158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.5206646605173066 \cdot 10^{-273}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 7.972191750369296 \cdot 10^{-204}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;t \leq 9.798485222824493 \cdot 10^{-195}:\\ \;\;\;\;\left(j \cdot k\right) \cdot -27\\ \mathbf{elif}\;t \leq 5.420980006714453 \cdot 10^{-152}:\\ \;\;\;\;-4 \cdot \left(x \cdot i\right)\\ \mathbf{elif}\;t \leq 1.5237648582652415 \cdot 10^{-129}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;t \leq 3.8419251003645414 \cdot 10^{-23}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	45.9
Cost	1508

\[\begin{array}{l} t_1 := \left(t \cdot a\right) \cdot -4\\ \mathbf{if}\;t \leq -4 \cdot 10^{+103}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -6.7 \cdot 10^{+50}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;t \leq -5.030361573776496 \cdot 10^{-158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.5206646605173066 \cdot 10^{-273}:\\ \;\;\;\;k \cdot \left(j \cdot -27\right)\\ \mathbf{elif}\;t \leq 7.972191750369296 \cdot 10^{-204}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;t \leq 9.798485222824493 \cdot 10^{-195}:\\ \;\;\;\;\left(j \cdot k\right) \cdot -27\\ \mathbf{elif}\;t \leq 5.420980006714453 \cdot 10^{-152}:\\ \;\;\;\;-4 \cdot \left(x \cdot i\right)\\ \mathbf{elif}\;t \leq 1.5237648582652415 \cdot 10^{-129}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;t \leq 3.8419251003645414 \cdot 10^{-23}:\\ \;\;\;\;j \cdot \left(k \cdot -27\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	19.2
Cost	1488

\[\begin{array}{l} t_1 := 27 \cdot \left(j \cdot k\right)\\ t_2 := b \cdot c - \left(t_1 + 4 \cdot \left(x \cdot i\right)\right)\\ t_3 := b \cdot c + \left(\left(t \cdot a\right) \cdot -4 - t_1\right)\\ \mathbf{if}\;t \leq -1 \cdot 10^{+120}:\\ \;\;\;\;t \cdot \left(18 \cdot \left(y \cdot \left(x \cdot z\right)\right) + a \cdot -4\right)\\ \mathbf{elif}\;t \leq -2.15 \cdot 10^{+66}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -4.1574391057457075 \cdot 10^{-214}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 1.39435008475039 \cdot 10^{-137}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 15
Error	19.2
Cost	1488

\[\begin{array}{l} t_1 := 27 \cdot \left(j \cdot k\right)\\ t_2 := b \cdot c + \left(\left(t \cdot a\right) \cdot -4 - t_1\right)\\ \mathbf{if}\;t \leq -1 \cdot 10^{+120}:\\ \;\;\;\;t \cdot \left(18 \cdot \left(y \cdot \left(x \cdot z\right)\right) + a \cdot -4\right)\\ \mathbf{elif}\;t \leq -2.6 \cdot 10^{+65}:\\ \;\;\;\;b \cdot c + \left(-4 \cdot \left(x \cdot i\right) + k \cdot \left(j \cdot -27\right)\right)\\ \mathbf{elif}\;t \leq -4.1574391057457075 \cdot 10^{-214}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.39435008475039 \cdot 10^{-137}:\\ \;\;\;\;b \cdot c - \left(t_1 + 4 \cdot \left(x \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 16
Error	37.8
Cost	1368

\[\begin{array}{l} t_1 := k \cdot \left(j \cdot -27\right)\\ t_2 := b \cdot c + \left(t \cdot a\right) \cdot -4\\ \mathbf{if}\;t \leq -6.799772488926225 \cdot 10^{-185}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 4.5206646605173066 \cdot 10^{-273}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.4053112057294966 \cdot 10^{-197}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 5.420980006714453 \cdot 10^{-152}:\\ \;\;\;\;-4 \cdot \left(x \cdot i\right)\\ \mathbf{elif}\;t \leq 2.7501339688677334 \cdot 10^{-87}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 8.451982303300754 \cdot 10^{-28}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 17
Error	32.3
Cost	1368

\[\begin{array}{l} t_1 := b \cdot c + \left(t \cdot a\right) \cdot -4\\ t_2 := b \cdot c + \left(j \cdot k\right) \cdot -27\\ \mathbf{if}\;k \leq -2.6507720112970817 \cdot 10^{-125}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -3.2103538446130872 \cdot 10^{-255}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 9.826577919179858 \cdot 10^{-263}:\\ \;\;\;\;b \cdot c - 4 \cdot \left(x \cdot i\right)\\ \mathbf{elif}\;k \leq 2.6034481745893545 \cdot 10^{-233}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 1.700312636834686 \cdot 10^{-162}:\\ \;\;\;\;18 \cdot \left(\left(z \cdot t\right) \cdot \left(x \cdot y\right)\right)\\ \mathbf{elif}\;k \leq 8680293.68987565:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 18
Error	44.8
Cost	1112

\[\begin{array}{l} t_1 := -4 \cdot \left(x \cdot i\right)\\ \mathbf{if}\;j \leq -4.3 \cdot 10^{+68}:\\ \;\;\;\;\left(j \cdot k\right) \cdot -27\\ \mathbf{elif}\;j \leq -1.377093232029791 \cdot 10^{-20}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;j \leq -3.877650154002631 \cdot 10^{-146}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq -1.059598516641876 \cdot 10^{-237}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;j \leq 3.435709863455254 \cdot 10^{-287}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq 2.201494454230296 \cdot 10^{-169}:\\ \;\;\;\;b \cdot c\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(j \cdot -27\right)\\ \end{array} \]

Alternative 19
Error	30.6
Cost	1104

\[\begin{array}{l} t_1 := b \cdot c + \left(t \cdot a\right) \cdot -4\\ t_2 := b \cdot c + \left(j \cdot k\right) \cdot -27\\ \mathbf{if}\;k \leq -2.6507720112970817 \cdot 10^{-125}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -3.2103538446130872 \cdot 10^{-255}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 9.826577919179858 \cdot 10^{-263}:\\ \;\;\;\;b \cdot c - 4 \cdot \left(x \cdot i\right)\\ \mathbf{elif}\;k \leq 8680293.68987565:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 20
Error	30.3
Cost	840

\[\begin{array}{l} t_1 := b \cdot c + \left(j \cdot k\right) \cdot -27\\ \mathbf{if}\;k \leq -2.6507720112970817 \cdot 10^{-125}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 8680293.68987565:\\ \;\;\;\;b \cdot c + \left(t \cdot a\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 21
Error	44.4
Cost	584

\[\begin{array}{l} t_1 := k \cdot \left(j \cdot -27\right)\\ \mathbf{if}\;j \leq -4.3 \cdot 10^{+68}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq 2.201494454230296 \cdot 10^{-169}:\\ \;\;\;\;b \cdot c\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 22
Error	44.4
Cost	584

\[\begin{array}{l} \mathbf{if}\;j \leq -4.3 \cdot 10^{+68}:\\ \;\;\;\;\left(j \cdot k\right) \cdot -27\\ \mathbf{elif}\;j \leq 2.201494454230296 \cdot 10^{-169}:\\ \;\;\;\;b \cdot c\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(j \cdot -27\right)\\ \end{array} \]

Alternative 23
Error	48.2
Cost	192

\[b \cdot c \]

Reproduce

herbie shell --seed 2022217 
(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)))