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

Average Error: 5.7 → 0.8

Time: 2.3min

Precision: binary64

Cost: 24008

Math

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

↓

\[\begin{array}{l} t_1 := \left(k \cdot j\right) \cdot -27\\ t_2 := y \cdot \left(t \cdot z\right)\\ t_3 := 18 \cdot \left(t_2 \cdot x\right)\\ t_4 := \left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\\ t_5 := -4 \cdot \left(a \cdot t\right)\\ \mathbf{if}\;t_4 \leq -\infty:\\ \;\;\;\;\left(c \cdot b + \left(\left(18 \cdot t_2 + -4 \cdot i\right) \cdot x + t_5\right)\right) + t_1\\ \mathbf{elif}\;t_4 \leq 10^{+305}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(18 \cdot \left(x \cdot y\right), z, -4 \cdot a\right), \mathsf{fma}\left(b, c, \left(i \cdot x\right) \cdot -4\right)\right) + t_1\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot b + \left(\begin{array}{l} \mathbf{if}\;t_3 \ne 0:\\ \;\;\;\;t_3 \cdot \mathsf{fma}\left(\frac{i}{\frac{t_2 \cdot 18}{-4}}, \frac{x}{x}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(18, t_2, -4 \cdot i\right) \cdot x\\ \end{array} + t_5\right)\right) + t_1\\ \end{array} \]

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 (* y (* t z)))
        (t_3 (* 18.0 (* t_2 x)))
        (t_4
         (-
          (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c))
          (* (* x 4.0) i)))
        (t_5 (* -4.0 (* a t))))
   (if (<= t_4 (- INFINITY))
     (+ (+ (* c b) (+ (* (+ (* 18.0 t_2) (* -4.0 i)) x) t_5)) t_1)
     (if (<= t_4 1e+305)
       (+
        (fma t (fma (* 18.0 (* x y)) z (* -4.0 a)) (fma b c (* (* i x) -4.0)))
        t_1)
       (+
        (+
         (* c b)
         (+
          (if (!= t_3 0.0)
            (* t_3 (fma (/ i (/ (* t_2 18.0) -4.0)) (/ x x) 1.0))
            (* (fma 18.0 t_2 (* -4.0 i)) x))
          t_5))
        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 = y * (t * z);
	double t_3 = 18.0 * (t_2 * x);
	double t_4 = ((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i);
	double t_5 = -4.0 * (a * t);
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = ((c * b) + ((((18.0 * t_2) + (-4.0 * i)) * x) + t_5)) + t_1;
	} else if (t_4 <= 1e+305) {
		tmp = fma(t, fma((18.0 * (x * y)), z, (-4.0 * a)), fma(b, c, ((i * x) * -4.0))) + t_1;
	} else {
		double tmp_1;
		if (t_3 != 0.0) {
			tmp_1 = t_3 * fma((i / ((t_2 * 18.0) / -4.0)), (x / x), 1.0);
		} else {
			tmp_1 = fma(18.0, t_2, (-4.0 * i)) * x;
		}
		tmp = ((c * b) + (tmp_1 + t_5)) + 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(k * j) * -27.0)
	t_2 = Float64(y * Float64(t * z))
	t_3 = Float64(18.0 * Float64(t_2 * x))
	t_4 = 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))
	t_5 = Float64(-4.0 * Float64(a * t))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(c * b) + Float64(Float64(Float64(Float64(18.0 * t_2) + Float64(-4.0 * i)) * x) + t_5)) + t_1);
	elseif (t_4 <= 1e+305)
		tmp = Float64(fma(t, fma(Float64(18.0 * Float64(x * y)), z, Float64(-4.0 * a)), fma(b, c, Float64(Float64(i * x) * -4.0))) + t_1);
	else
		tmp_1 = 0.0
		if (t_3 != 0.0)
			tmp_1 = Float64(t_3 * fma(Float64(i / Float64(Float64(t_2 * 18.0) / -4.0)), Float64(x / x), 1.0));
		else
			tmp_1 = Float64(fma(18.0, t_2, Float64(-4.0 * i)) * x);
		end
		tmp = Float64(Float64(Float64(c * b) + Float64(tmp_1 + t_5)) + t_1);
	end
	return 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[(k * j), $MachinePrecision] * -27.0), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(t * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(18.0 * N[(t$95$2 * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = 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]}, Block[{t$95$5 = N[(-4.0 * N[(a * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[(N[(c * b), $MachinePrecision] + N[(N[(N[(N[(18.0 * t$95$2), $MachinePrecision] + N[(-4.0 * i), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[t$95$4, 1e+305], N[(N[(t * N[(N[(18.0 * N[(x * y), $MachinePrecision]), $MachinePrecision] * z + N[(-4.0 * a), $MachinePrecision]), $MachinePrecision] + N[(b * c + N[(N[(i * x), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(N[(c * b), $MachinePrecision] + N[(If[Unequal[t$95$3, 0.0], N[(t$95$3 * N[(N[(i / N[(N[(t$95$2 * 18.0), $MachinePrecision] / -4.0), $MachinePrecision]), $MachinePrecision] * N[(x / x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(18.0 * t$95$2 + N[(-4.0 * i), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]] + t$95$5), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]]]]

Target

Original	5.7
Target	1.6
Herbie	0.8

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) t) (*.f64 (*.f64 a 4) t)) (*.f64 b c)) (*.f64 (*.f64 x 4) i)) < -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 64.0

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

   3. Taylor expanded in x around 0 5.9

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

   if -inf.0 < (-.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)) < 9.9999999999999994e304

   1. Initial program 0.3

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

   2. Simplified 0.3

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

   if 9.9999999999999994e304 < (-.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))

   1. Initial program 58.3

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

   2. Simplified 57.1

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

   3. Taylor expanded in x around 0 5.5

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

   4. Applied egg-rr 6.0

      \[\leadsto \left(c \cdot b + \left(\color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\left(18 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) \cdot x \ne 0:\\ \;\;\;\;\left(\left(18 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) \cdot x\right) \cdot \left(1 + \frac{\left(-4 \cdot i\right) \cdot x}{\left(18 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(18, y \cdot \left(t \cdot z\right), -4 \cdot i\right) \cdot x\\ } \end{array}} + -4 \cdot \left(a \cdot t\right)\right)\right) + \left(k \cdot j\right) \cdot -27 \]

   5. Simplified 6.1

      \[\leadsto \left(c \cdot b + \left(\color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;18 \cdot \left(\left(y \cdot \left(t \cdot z\right)\right) \cdot x\right) \ne 0:\\ \;\;\;\;\left(18 \cdot \left(\left(y \cdot \left(t \cdot z\right)\right) \cdot x\right)\right) \cdot \mathsf{fma}\left(\frac{i}{\frac{\left(y \cdot \left(t \cdot z\right)\right) \cdot 18}{-4}}, \frac{x}{x}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(18, y \cdot \left(t \cdot z\right), -4 \cdot i\right) \cdot x\\ } \end{array}} + -4 \cdot \left(a \cdot t\right)\right)\right) + \left(k \cdot j\right) \cdot -27 \]
      Proof
3. Recombined 3 regimes into one program.

Alternatives

Alternative 1
Error	1.1
Cost	13772

\[\begin{array}{l} t_1 := y \cdot \left(t \cdot z\right)\\ t_2 := \left(k \cdot j\right) \cdot -27\\ t_3 := \left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\\ t_4 := t_3 - \left(j \cdot 27\right) \cdot k\\ t_5 := -4 \cdot \left(a \cdot t\right)\\ t_6 := 18 \cdot \left(t_1 \cdot x\right)\\ \mathbf{if}\;t_4 \leq -2 \cdot 10^{+304}:\\ \;\;\;\;\left(c \cdot b + \left(\left(18 \cdot t_1 + -4 \cdot i\right) \cdot x + t_5\right)\right) + t_2\\ \mathbf{elif}\;t_4 \leq 10^{+299}:\\ \;\;\;\;t_3 - \left(27 \cdot k\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot b + \left(\begin{array}{l} \mathbf{if}\;t_6 \ne 0:\\ \;\;\;\;t_6 \cdot \mathsf{fma}\left(\frac{i}{\frac{t_1 \cdot 18}{-4}}, \frac{x}{x}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(18, t_1, -4 \cdot i\right) \cdot x\\ \end{array} + t_5\right)\right) + t_2\\ \end{array} \]

Alternative 2
Error	1.0
Cost	6088

\[\begin{array}{l} t_1 := \left(c \cdot b + \left(\left(18 \cdot \left(y \cdot \left(t \cdot z\right)\right) + -4 \cdot i\right) \cdot x + -4 \cdot \left(a \cdot t\right)\right)\right) + \left(k \cdot j\right) \cdot -27\\ t_2 := \left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\\ t_3 := t_2 - \left(j \cdot 27\right) \cdot k\\ \mathbf{if}\;t_3 \leq -2 \cdot 10^{+304}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_3 \leq 10^{+299}:\\ \;\;\;\;t_2 - \left(27 \cdot k\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	28.4
Cost	2676

\[\begin{array}{l} t_1 := -4 \cdot \left(i \cdot x\right)\\ t_2 := -4 \cdot \left(a \cdot t\right)\\ t_3 := c \cdot b + \left(t_1 + t_2\right)\\ t_4 := \left(k \cdot j\right) \cdot -27\\ t_5 := t_1 + t_4\\ t_6 := 18 \cdot \left(y \cdot \left(t \cdot \left(z \cdot x\right)\right)\right) + t_4\\ \mathbf{if}\;a \leq -1.2 \cdot 10^{+50}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -1.7 \cdot 10^{-23}:\\ \;\;\;\;t_2 + t_4\\ \mathbf{elif}\;a \leq -7 \cdot 10^{-194}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -6.2 \cdot 10^{-211}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;a \leq -1.25 \cdot 10^{-276}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -1.12 \cdot 10^{-295}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;a \leq 8.5 \cdot 10^{-271}:\\ \;\;\;\;c \cdot b + \left(-27 \cdot k\right) \cdot j\\ \mathbf{elif}\;a \leq 2.35 \cdot 10^{-122}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;a \leq 1.5 \cdot 10^{-103}:\\ \;\;\;\;c \cdot b + t_1\\ \mathbf{elif}\;a \leq 1.55 \cdot 10^{-56}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;a \leq 4.6 \cdot 10^{-35}:\\ \;\;\;\;c \cdot b + t_4\\ \mathbf{elif}\;a \leq 105000000000:\\ \;\;\;\;t_5\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{+96}:\\ \;\;\;\;t_6\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 4
Error	27.4
Cost	2544

\[\begin{array}{l} t_1 := -4 \cdot \left(i \cdot x\right)\\ t_2 := -4 \cdot \left(a \cdot t\right)\\ t_3 := c \cdot b + \left(t_1 + t_2\right)\\ t_4 := \left(k \cdot j\right) \cdot -27\\ t_5 := t \cdot \left(z \cdot x\right)\\ t_6 := 18 \cdot \left(y \cdot t_5\right) + t_4\\ \mathbf{if}\;a \leq -4 \cdot 10^{+49}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -3.3 \cdot 10^{-24}:\\ \;\;\;\;t_2 + t_4\\ \mathbf{elif}\;a \leq -3.7 \cdot 10^{-194}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -5.1 \cdot 10^{-211}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;a \leq -2.9 \cdot 10^{-274}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -1.46 \cdot 10^{-295}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;a \leq 1.65 \cdot 10^{-270}:\\ \;\;\;\;c \cdot b + \left(-27 \cdot k\right) \cdot j\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{-122}:\\ \;\;\;\;t_1 + t_4\\ \mathbf{elif}\;a \leq 4.4 \cdot 10^{-103}:\\ \;\;\;\;c \cdot b + t_1\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{-56}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;a \leq 1.65 \cdot 10^{+24}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{+96}:\\ \;\;\;\;\left(18 \cdot y\right) \cdot t_5 + t_4\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 5
Error	27.4
Cost	2544

\[\begin{array}{l} t_1 := -4 \cdot \left(i \cdot x\right)\\ t_2 := -4 \cdot \left(a \cdot t\right)\\ t_3 := c \cdot b + \left(t_1 + t_2\right)\\ t_4 := \left(k \cdot j\right) \cdot -27\\ t_5 := t \cdot \left(z \cdot x\right)\\ t_6 := 18 \cdot \left(y \cdot t_5\right) + t_4\\ \mathbf{if}\;a \leq -6.8 \cdot 10^{+47}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -1.55 \cdot 10^{-23}:\\ \;\;\;\;t_2 + t_4\\ \mathbf{elif}\;a \leq -4.8 \cdot 10^{-194}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -4.8 \cdot 10^{-211}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;a \leq -5.2 \cdot 10^{-274}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -1.05 \cdot 10^{-295}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-271}:\\ \;\;\;\;c \cdot b + \left(-27 \cdot k\right) \cdot j\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-122}:\\ \;\;\;\;t_1 + t_4\\ \mathbf{elif}\;a \leq 3.1 \cdot 10^{-103}:\\ \;\;\;\;c \cdot b + t_1\\ \mathbf{elif}\;a \leq 2.9 \cdot 10^{-54}:\\ \;\;\;\;\left(t \cdot z\right) \cdot \left(x \cdot \left(18 \cdot y\right)\right) + t_4\\ \mathbf{elif}\;a \leq 4.7 \cdot 10^{+24}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{+96}:\\ \;\;\;\;\left(18 \cdot y\right) \cdot t_5 + t_4\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 6
Error	1.9
Cost	2248

\[\begin{array}{l} t_1 := \left(\left(\left(\left(\left(18 \cdot x\right) \cdot \left(z \cdot t\right)\right) \cdot y - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{if}\;y \leq -5.5 \cdot 10^{-45}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-112}:\\ \;\;\;\;\left(c \cdot b + \left(\left(\left(\left(18 \cdot y\right) \cdot z\right) \cdot t + -4 \cdot i\right) \cdot x + -4 \cdot \left(a \cdot t\right)\right)\right) + \left(k \cdot j\right) \cdot -27\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	1.7
Cost	2248

\[\begin{array}{l} t_1 := \left(x \cdot 4\right) \cdot i\\ t_2 := \left(j \cdot 27\right) \cdot k\\ t_3 := \left(a \cdot 4\right) \cdot t\\ t_4 := \left(\left(\left(\left(\left(18 \cdot \left(x \cdot y\right)\right) \cdot t\right) \cdot z - t_3\right) + b \cdot c\right) - t_1\right) - t_2\\ \mathbf{if}\;z \leq -1 \cdot 10^{-64}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+22}:\\ \;\;\;\;\left(\left(\left(\left(\left(18 \cdot x\right) \cdot \left(z \cdot t\right)\right) \cdot y - t_3\right) + b \cdot c\right) - t_1\right) - t_2\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 8
Error	10.2
Cost	2128

\[\begin{array}{l} t_1 := \left(k \cdot j\right) \cdot -27\\ t_2 := -4 \cdot \left(i \cdot x\right)\\ t_3 := \left(c \cdot b + \left(t_2 + -4 \cdot \left(a \cdot t\right)\right)\right) + t_1\\ t_4 := \left(c \cdot b + \left(t_2 + 18 \cdot \left(y \cdot \left(t \cdot \left(z \cdot x\right)\right)\right)\right)\right) + t_1\\ \mathbf{if}\;a \leq -2.6 \cdot 10^{-107}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 6.4 \cdot 10^{-160}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;a \leq 3.2 \cdot 10^{-116}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 2700000:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(\left(z \cdot x\right) \cdot \left(18 \cdot y\right) + -4 \cdot a\right) + c \cdot b\right) + t_1\\ \end{array} \]

Alternative 9
Error	9.8
Cost	2128

\[\begin{array}{l} t_1 := -4 \cdot \left(a \cdot t\right)\\ t_2 := \left(k \cdot j\right) \cdot -27\\ t_3 := -4 \cdot \left(i \cdot x\right)\\ t_4 := \left(c \cdot b + \left(t_3 + t_1\right)\right) + t_2\\ t_5 := \left(c \cdot b + \left(t_3 + 18 \cdot \left(y \cdot \left(t \cdot \left(z \cdot x\right)\right)\right)\right)\right) + t_2\\ \mathbf{if}\;a \leq -1.4 \cdot 10^{-107}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{-162}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;a \leq 7 \cdot 10^{-117}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;a \leq 95000000:\\ \;\;\;\;t_5\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot b + \left(\left(18 \cdot y\right) \cdot \left(\left(t \cdot z\right) \cdot x\right) + t_1\right)\right) + t_2\\ \end{array} \]

Alternative 10
Error	4.3
Cost	2120

\[\begin{array}{l} t_1 := \left(k \cdot j\right) \cdot -27\\ t_2 := \left(c \cdot b + \left(\left(18 \cdot \left(y \cdot \left(t \cdot z\right)\right) + -4 \cdot i\right) \cdot x + -4 \cdot \left(a \cdot t\right)\right)\right) + t_1\\ \mathbf{if}\;x \leq -5.3 \cdot 10^{-105}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-39}:\\ \;\;\;\;\left(c \cdot b + \left(18 \cdot \left(y \cdot \left(z \cdot x\right)\right) + -4 \cdot a\right) \cdot t\right) + t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 11
Error	19.0
Cost	2016

\[\begin{array}{l} t_1 := \left(k \cdot j\right) \cdot -27\\ t_2 := -4 \cdot \left(i \cdot x\right)\\ t_3 := -4 \cdot \left(a \cdot t\right)\\ t_4 := \left(c \cdot b + t_2\right) + t_1\\ t_5 := c \cdot b + \left(t_2 + t_3\right)\\ t_6 := \left(c \cdot b + t_3\right) + t_1\\ \mathbf{if}\;a \leq -9 \cdot 10^{+157}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;a \leq -6.5 \cdot 10^{+71}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;a \leq -1.12 \cdot 10^{+48}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;a \leq -700000000000:\\ \;\;\;\;\left(18 \cdot \left(y \cdot \left(z \cdot x\right)\right) + -4 \cdot a\right) \cdot t + \left(j \cdot -27\right) \cdot k\\ \mathbf{elif}\;a \leq -9.2 \cdot 10^{-107}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{-102}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;a \leq 2.55 \cdot 10^{-54}:\\ \;\;\;\;\left(t \cdot z\right) \cdot \left(x \cdot \left(18 \cdot y\right)\right) + t_1\\ \mathbf{elif}\;a \leq 65000000:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_6\\ \end{array} \]

Alternative 12
Error	19.0
Cost	2016

\[\begin{array}{l} t_1 := \left(k \cdot j\right) \cdot -27\\ t_2 := -4 \cdot \left(i \cdot x\right)\\ t_3 := -4 \cdot \left(a \cdot t\right)\\ t_4 := \left(c \cdot b + t_2\right) + t_1\\ t_5 := c \cdot b + \left(t_2 + t_3\right)\\ t_6 := \left(c \cdot b + t_3\right) + t_1\\ \mathbf{if}\;a \leq -1.8 \cdot 10^{+158}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;a \leq -8 \cdot 10^{+71}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;a \leq -1.2 \cdot 10^{+50}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;a \leq -750000000000:\\ \;\;\;\;\left(18 \cdot \left(y \cdot \left(z \cdot x\right)\right) + -4 \cdot a\right) \cdot t + t_1\\ \mathbf{elif}\;a \leq -9.2 \cdot 10^{-107}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{-102}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;a \leq 2.55 \cdot 10^{-54}:\\ \;\;\;\;\left(t \cdot z\right) \cdot \left(x \cdot \left(18 \cdot y\right)\right) + t_1\\ \mathbf{elif}\;a \leq 80000000:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_6\\ \end{array} \]

Alternative 13
Error	31.8
Cost	1892

\[\begin{array}{l} t_1 := \left(k \cdot j\right) \cdot -27\\ t_2 := c \cdot b + \left(-27 \cdot k\right) \cdot j\\ t_3 := -4 \cdot \left(i \cdot x\right)\\ t_4 := t_3 + t_1\\ t_5 := -4 \cdot \left(a \cdot t\right) + t_1\\ \mathbf{if}\;c \leq -11000:\\ \;\;\;\;c \cdot b + t_1\\ \mathbf{elif}\;c \leq -4.9 \cdot 10^{-47}:\\ \;\;\;\;c \cdot b + t_3\\ \mathbf{elif}\;c \leq -6.6 \cdot 10^{-54}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq -4.8 \cdot 10^{-219}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;c \leq 1.35 \cdot 10^{-225}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;c \leq 8.5 \cdot 10^{-90}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;c \leq 9 \cdot 10^{-8}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;c \leq 8 \cdot 10^{+33}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 2.25 \cdot 10^{+187}:\\ \;\;\;\;t_5\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 14
Error	33.0
Cost	1892

\[\begin{array}{l} t_1 := \left(k \cdot j\right) \cdot -27\\ t_2 := c \cdot b + \left(-27 \cdot k\right) \cdot j\\ t_3 := -4 \cdot \left(i \cdot x\right)\\ t_4 := c \cdot b + t_3\\ t_5 := -4 \cdot \left(a \cdot t\right) + t_1\\ \mathbf{if}\;c \leq -5.4:\\ \;\;\;\;c \cdot b + t_1\\ \mathbf{elif}\;c \leq -5.2 \cdot 10^{-47}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;c \leq -6 \cdot 10^{-54}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq -7.5 \cdot 10^{-218}:\\ \;\;\;\;t_3 + t_1\\ \mathbf{elif}\;c \leq 1.8 \cdot 10^{-242}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;c \leq 2.5 \cdot 10^{-137}:\\ \;\;\;\;18 \cdot \left(y \cdot \left(t \cdot \left(z \cdot x\right)\right)\right) + t_1\\ \mathbf{elif}\;c \leq 1.05 \cdot 10^{-6}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;c \leq 2.6 \cdot 10^{+42}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 2.25 \cdot 10^{+187}:\\ \;\;\;\;t_5\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 15
Error	32.3
Cost	1760

\[\begin{array}{l} t_1 := \left(k \cdot j\right) \cdot -27\\ t_2 := c \cdot b + \left(-27 \cdot k\right) \cdot j\\ t_3 := -4 \cdot \left(a \cdot t\right) + t_1\\ t_4 := c \cdot b + -4 \cdot \left(i \cdot x\right)\\ t_5 := c \cdot b + t_1\\ \mathbf{if}\;c \leq -21:\\ \;\;\;\;t_5\\ \mathbf{elif}\;c \leq -6 \cdot 10^{-47}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;c \leq -1.25 \cdot 10^{-79}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;c \leq 1.85 \cdot 10^{-137}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c \leq 1.1 \cdot 10^{-45}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;c \leq 1.65 \cdot 10^{-7}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c \leq 5.4 \cdot 10^{+41}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 3.15 \cdot 10^{+187}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 16
Error	16.9
Cost	1744

\[\begin{array}{l} t_1 := -4 \cdot \left(a \cdot t\right)\\ t_2 := \left(k \cdot j\right) \cdot -27\\ t_3 := \left(18 \cdot \left(y \cdot \left(t \cdot z\right)\right) + -4 \cdot i\right) \cdot x + t_2\\ \mathbf{if}\;x \leq -2.05 \cdot 10^{-25}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-60}:\\ \;\;\;\;c \cdot b + \left(-4 \cdot \left(i \cdot x\right) + t_1\right)\\ \mathbf{elif}\;x \leq -8.6 \cdot 10^{-72}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-36}:\\ \;\;\;\;\left(c \cdot b + t_1\right) + t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 17
Error	8.5
Cost	1736

\[\begin{array}{l} t_1 := \left(k \cdot j\right) \cdot -27\\ t_2 := \left(c \cdot b + \left(18 \cdot \left(y \cdot \left(z \cdot x\right)\right) + -4 \cdot a\right) \cdot t\right) + t_1\\ \mathbf{if}\;t \leq -3.8 \cdot 10^{+59}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-87}:\\ \;\;\;\;\left(c \cdot b + \left(-4 \cdot \left(i \cdot x\right) + -4 \cdot \left(a \cdot t\right)\right)\right) + t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 18
Error	18.6
Cost	1620

\[\begin{array}{l} t_1 := \left(k \cdot j\right) \cdot -27\\ t_2 := -4 \cdot \left(i \cdot x\right)\\ t_3 := -4 \cdot \left(a \cdot t\right)\\ t_4 := \left(c \cdot b + t_3\right) + t_1\\ t_5 := \left(c \cdot b + t_2\right) + t_1\\ \mathbf{if}\;a \leq -9 \cdot 10^{+157}:\\ \;\;\;\;c \cdot b + \left(t_2 + t_3\right)\\ \mathbf{elif}\;a \leq -9.2 \cdot 10^{-107}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{-102}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;a \leq 2.55 \cdot 10^{-54}:\\ \;\;\;\;\left(t \cdot z\right) \cdot \left(x \cdot \left(18 \cdot y\right)\right) + t_1\\ \mathbf{elif}\;a \leq 2000000:\\ \;\;\;\;t_5\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 19
Error	10.7
Cost	1476

\[\begin{array}{l} t_1 := \left(k \cdot j\right) \cdot -27\\ \mathbf{if}\;x \leq -1600000:\\ \;\;\;\;\left(18 \cdot \left(y \cdot \left(t \cdot z\right)\right) + -4 \cdot i\right) \cdot x + t_1\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot b + \left(-4 \cdot \left(i \cdot x\right) + -4 \cdot \left(a \cdot t\right)\right)\right) + t_1\\ \end{array} \]

Alternative 20
Error	17.7
Cost	1224

\[\begin{array}{l} t_1 := -4 \cdot \left(a \cdot t\right)\\ t_2 := \left(c \cdot b + t_1\right) + \left(k \cdot j\right) \cdot -27\\ \mathbf{if}\;k \leq -9.2 \cdot 10^{-69}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 1.7 \cdot 10^{-68}:\\ \;\;\;\;c \cdot b + \left(-4 \cdot \left(i \cdot x\right) + t_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 21
Error	44.1
Cost	1112

\[\begin{array}{l} t_1 := -4 \cdot \left(i \cdot x\right)\\ t_2 := -27 \cdot \left(k \cdot j\right)\\ \mathbf{if}\;k \leq -0.0068:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -1.3 \cdot 10^{-73}:\\ \;\;\;\;c \cdot b\\ \mathbf{elif}\;k \leq -3.1 \cdot 10^{-172}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 9.5 \cdot 10^{-260}:\\ \;\;\;\;c \cdot b\\ \mathbf{elif}\;k \leq 1.7 \cdot 10^{-149}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 2 \cdot 10^{-91}:\\ \;\;\;\;c \cdot b\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 22
Error	44.1
Cost	1112

\[\begin{array}{l} t_1 := -4 \cdot \left(i \cdot x\right)\\ \mathbf{if}\;k \leq -2.8 \cdot 10^{-10}:\\ \;\;\;\;-27 \cdot \left(k \cdot j\right)\\ \mathbf{elif}\;k \leq -5.5 \cdot 10^{-70}:\\ \;\;\;\;c \cdot b\\ \mathbf{elif}\;k \leq -1.15 \cdot 10^{-171}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 5.9 \cdot 10^{-260}:\\ \;\;\;\;c \cdot b\\ \mathbf{elif}\;k \leq 5 \cdot 10^{-149}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 5.4 \cdot 10^{-92}:\\ \;\;\;\;c \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(-27 \cdot j\right) \cdot k\\ \end{array} \]

Alternative 23
Error	34.5
Cost	840

\[\begin{array}{l} t_1 := -27 \cdot \left(k \cdot j\right)\\ \mathbf{if}\;k \leq -3 \cdot 10^{+71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 6.5 \cdot 10^{+198}:\\ \;\;\;\;c \cdot b + -4 \cdot \left(i \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 24
Error	30.3
Cost	840

\[\begin{array}{l} t_1 := c \cdot b + \left(-27 \cdot k\right) \cdot j\\ \mathbf{if}\;k \leq -3.5 \cdot 10^{-70}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 2.4 \cdot 10^{-124}:\\ \;\;\;\;c \cdot b + -4 \cdot \left(i \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 25
Error	30.3
Cost	840

\[\begin{array}{l} \mathbf{if}\;k \leq -7.2 \cdot 10^{-79}:\\ \;\;\;\;c \cdot b + \left(k \cdot j\right) \cdot -27\\ \mathbf{elif}\;k \leq 3.4 \cdot 10^{-125}:\\ \;\;\;\;c \cdot b + -4 \cdot \left(i \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot b + \left(-27 \cdot k\right) \cdot j\\ \end{array} \]

Alternative 26
Error	43.2
Cost	584

\[\begin{array}{l} t_1 := -27 \cdot \left(k \cdot j\right)\\ \mathbf{if}\;k \leq -6.5 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{-91}:\\ \;\;\;\;c \cdot b\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 27
Error	48.1
Cost	192

\[c \cdot b \]

Reproduce

herbie shell --seed 2023033 
(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)))