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

Average Error: 5.8 → 1.8

Time: 29.0s

Precision: binary64

Cost: 6088

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

Math

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

↓

\[\begin{array}{l} t_1 := \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(t \cdot \left(x \cdot z\right)\right) + b \cdot c\right) + \left(j \cdot \left(k \cdot -27\right) - x \cdot \left(4 \cdot i\right)\right)\\ \mathbf{elif}\;t_1 \leq 10^{+308}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(18 \cdot \left(y \cdot \left(z \cdot t\right)\right) + i \cdot -4\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1
         (+
          (+
           (+ (+ (* (* (* (* x 18.0) y) z) t) (* t (* a -4.0))) (* b c))
           (* i (* x -4.0)))
          (* k (* j -27.0)))))
   (if (<= t_1 (- INFINITY))
     (+
      (+ (* (* 18.0 y) (* t (* x z))) (* b c))
      (- (* j (* k -27.0)) (* x (* 4.0 i))))
     (if (<= t_1 1e+308) t_1 (* x (+ (* 18.0 (* y (* z t))) (* i -4.0)))))))

C

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

↓

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = (((((((x * 18.0) * y) * z) * t) + (t * (a * -4.0))) + (b * c)) + (i * (x * -4.0))) + (k * (j * -27.0));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (((18.0 * y) * (t * (x * z))) + (b * c)) + ((j * (k * -27.0)) - (x * (4.0 * i)));
	} else if (t_1 <= 1e+308) {
		tmp = t_1;
	} else {
		tmp = x * ((18.0 * (y * (z * t))) + (i * -4.0));
	}
	return tmp;
}

Java

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

↓

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = (((((((x * 18.0) * y) * z) * t) + (t * (a * -4.0))) + (b * c)) + (i * (x * -4.0))) + (k * (j * -27.0));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (((18.0 * y) * (t * (x * z))) + (b * c)) + ((j * (k * -27.0)) - (x * (4.0 * i)));
	} else if (t_1 <= 1e+308) {
		tmp = t_1;
	} else {
		tmp = x * ((18.0 * (y * (z * t))) + (i * -4.0));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t, a, b, c, i, j, k):
	t_1 = (((((((x * 18.0) * y) * z) * t) + (t * (a * -4.0))) + (b * c)) + (i * (x * -4.0))) + (k * (j * -27.0))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (((18.0 * y) * (t * (x * z))) + (b * c)) + ((j * (k * -27.0)) - (x * (4.0 * i)))
	elif t_1 <= 1e+308:
		tmp = t_1
	else:
		tmp = x * ((18.0 * (y * (z * t))) + (i * -4.0))
	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(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))) + Float64(k * Float64(j * -27.0)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(18.0 * y) * Float64(t * Float64(x * z))) + Float64(b * c)) + Float64(Float64(j * Float64(k * -27.0)) - Float64(x * Float64(4.0 * i))));
	elseif (t_1 <= 1e+308)
		tmp = t_1;
	else
		tmp = Float64(x * Float64(Float64(18.0 * Float64(y * Float64(z * t))) + Float64(i * -4.0)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z, t, a, b, c, i, j, k)
	t_1 = (((((((x * 18.0) * y) * z) * t) + (t * (a * -4.0))) + (b * c)) + (i * (x * -4.0))) + (k * (j * -27.0));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (((18.0 * y) * (t * (x * z))) + (b * c)) + ((j * (k * -27.0)) - (x * (4.0 * i)));
	elseif (t_1 <= 1e+308)
		tmp = t_1;
	else
		tmp = x * ((18.0 * (y * (z * t))) + (i * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(N[(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] + N[(k * N[(j * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(18.0 * y), $MachinePrecision] * N[(t * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision] + N[(N[(j * N[(k * -27.0), $MachinePrecision]), $MachinePrecision] - N[(x * N[(4.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+308], t$95$1, N[(x * N[(N[(18.0 * N[(y * N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(i * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	5.8
Target	1.8
Herbie	1.8

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.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 38.9

      \[\leadsto \color{blue}{\left(t \cdot \left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right) - a \cdot 4\right) + b \cdot c\right) - \left(x \cdot \left(4 \cdot i\right) + j \cdot \left(27 \cdot k\right)\right)} \]
      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* [=>] 40.0	\[ \left(t \cdot \left(\color{blue}{\left(x \cdot 18\right) \cdot \left(y \cdot z\right)} - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right) \]
      associate-*l* [=>] 40.0	\[ \left(t \cdot \left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right) - a \cdot 4\right) + b \cdot c\right) - \left(\color{blue}{x \cdot \left(4 \cdot i\right)} + \left(j \cdot 27\right) \cdot k\right) \]
      associate-*l* [=>] 38.9	\[ \left(t \cdot \left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right) - a \cdot 4\right) + b \cdot c\right) - \left(x \cdot \left(4 \cdot i\right) + \color{blue}{j \cdot \left(27 \cdot k\right)}\right) \]

   3. Taylor expanded in x around inf 13.1

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

   4. Simplified 13.4

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

      [Start] 13.1	\[ \left(18 \cdot \left(y \cdot \left(t \cdot \left(z \cdot x\right)\right)\right) + b \cdot c\right) - \left(x \cdot \left(4 \cdot i\right) + j \cdot \left(27 \cdot k\right)\right) \]
      associate-*r* [=>] 13.4	\[ \left(\color{blue}{\left(18 \cdot y\right) \cdot \left(t \cdot \left(z \cdot x\right)\right)} + b \cdot c\right) - \left(x \cdot \left(4 \cdot i\right) + j \cdot \left(27 \cdot k\right)\right) \]

   if -inf.0 < (-.f64 (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) t) (*.f64 (*.f64 a 4) t)) (*.f64 b c)) (*.f64 (*.f64 x 4) i)) (*.f64 (*.f64 j 27) k)) < 1e308

   1. Initial program 0.2

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

   if 1e308 < (-.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 63.5

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

   2. Simplified 38.8

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

      [Start] 63.5	\[ \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
      sub-neg [=>] 63.5	\[ \color{blue}{\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) + \left(-\left(j \cdot 27\right) \cdot k\right)} \]
      +-commutative [=>] 63.5	\[ \color{blue}{\left(-\left(j \cdot 27\right) \cdot k\right) + \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right)} \]
      associate-*l* [=>] 62.5	\[ \left(-\color{blue}{j \cdot \left(27 \cdot k\right)}\right) + \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) \]
      distribute-rgt-neg-in [=>] 62.5	\[ \color{blue}{j \cdot \left(-27 \cdot k\right)} + \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) \]
      fma-def [=>] 62.5	\[ \color{blue}{\mathsf{fma}\left(j, -27 \cdot k, \left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right)} \]
      *-commutative [=>] 62.5	\[ \mathsf{fma}\left(j, -\color{blue}{k \cdot 27}, \left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) \]
      distribute-rgt-neg-in [=>] 62.5	\[ \mathsf{fma}\left(j, \color{blue}{k \cdot \left(-27\right)}, \left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) \]
      metadata-eval [=>] 62.5	\[ \mathsf{fma}\left(j, k \cdot \color{blue}{-27}, \left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) \]
      sub-neg [=>] 62.5	\[ \mathsf{fma}\left(j, k \cdot -27, \color{blue}{\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) + \left(-\left(x \cdot 4\right) \cdot i\right)}\right) \]
      +-commutative [=>] 62.5	\[ \mathsf{fma}\left(j, k \cdot -27, \color{blue}{\left(-\left(x \cdot 4\right) \cdot i\right) + \left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right)}\right) \]
      associate-*l* [=>] 62.5	\[ \mathsf{fma}\left(j, k \cdot -27, \left(-\color{blue}{x \cdot \left(4 \cdot i\right)}\right) + \left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right)\right) \]
      distribute-rgt-neg-in [=>] 62.5	\[ \mathsf{fma}\left(j, k \cdot -27, \color{blue}{x \cdot \left(-4 \cdot i\right)} + \left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right)\right) \]
      fma-def [=>] 62.5	\[ \mathsf{fma}\left(j, k \cdot -27, \color{blue}{\mathsf{fma}\left(x, -4 \cdot i, \left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right)}\right) \]
      *-commutative [=>] 62.5	\[ \mathsf{fma}\left(j, k \cdot -27, \mathsf{fma}\left(x, -\color{blue}{i \cdot 4}, \left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right)\right) \]
      distribute-rgt-neg-in [=>] 62.5	\[ \mathsf{fma}\left(j, k \cdot -27, \mathsf{fma}\left(x, \color{blue}{i \cdot \left(-4\right)}, \left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right)\right) \]
      metadata-eval [=>] 62.5	\[ \mathsf{fma}\left(j, k \cdot -27, \mathsf{fma}\left(x, i \cdot \color{blue}{-4}, \left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right)\right) \]
      distribute-rgt-out-- [=>] 62.5	\[ \mathsf{fma}\left(j, k \cdot -27, \mathsf{fma}\left(x, i \cdot -4, \color{blue}{t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right)} + b \cdot c\right)\right) \]
      fma-def [=>] 62.5	\[ \mathsf{fma}\left(j, k \cdot -27, \mathsf{fma}\left(x, i \cdot -4, \color{blue}{\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c\right)}\right)\right) \]
      associate-*l* [=>] 61.6	\[ \mathsf{fma}\left(j, k \cdot -27, \mathsf{fma}\left(x, i \cdot -4, \mathsf{fma}\left(t, \color{blue}{\left(x \cdot \left(18 \cdot y\right)\right)} \cdot z - a \cdot 4, b \cdot c\right)\right)\right) \]
      associate-*l* [=>] 38.8	\[ \mathsf{fma}\left(j, k \cdot -27, \mathsf{fma}\left(x, i \cdot -4, \mathsf{fma}\left(t, \color{blue}{x \cdot \left(\left(18 \cdot y\right) \cdot z\right)} - a \cdot 4, b \cdot c\right)\right)\right) \]
      fma-neg [=>] 38.8	\[ \mathsf{fma}\left(j, k \cdot -27, \mathsf{fma}\left(x, i \cdot -4, \mathsf{fma}\left(t, \color{blue}{\mathsf{fma}\left(x, \left(18 \cdot y\right) \cdot z, -a \cdot 4\right)}, b \cdot c\right)\right)\right) \]
      *-commutative [=>] 38.8	\[ \mathsf{fma}\left(j, k \cdot -27, \mathsf{fma}\left(x, i \cdot -4, \mathsf{fma}\left(t, \mathsf{fma}\left(x, \color{blue}{z \cdot \left(18 \cdot y\right)}, -a \cdot 4\right), b \cdot c\right)\right)\right) \]
      distribute-rgt-neg-in [=>] 38.8	\[ \mathsf{fma}\left(j, k \cdot -27, \mathsf{fma}\left(x, i \cdot -4, \mathsf{fma}\left(t, \mathsf{fma}\left(x, z \cdot \left(18 \cdot y\right), \color{blue}{a \cdot \left(-4\right)}\right), b \cdot c\right)\right)\right) \]
      metadata-eval [=>] 38.8	\[ \mathsf{fma}\left(j, k \cdot -27, \mathsf{fma}\left(x, i \cdot -4, \mathsf{fma}\left(t, \mathsf{fma}\left(x, z \cdot \left(18 \cdot y\right), a \cdot \color{blue}{-4}\right), b \cdot c\right)\right)\right) \]

   3. Taylor expanded in x around inf 21.9

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

4. Final simplification 1.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right) \leq -\infty:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(t \cdot \left(x \cdot z\right)\right) + b \cdot c\right) + \left(j \cdot \left(k \cdot -27\right) - x \cdot \left(4 \cdot i\right)\right)\\ \mathbf{elif}\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right) \leq 10^{+308}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(18 \cdot \left(y \cdot \left(z \cdot t\right)\right) + i \cdot -4\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	5.6
Cost	2120

\[\begin{array}{l} t_1 := j \cdot \left(k \cdot -27\right) - x \cdot \left(4 \cdot i\right)\\ \mathbf{if}\;y \leq -4.6 \cdot 10^{+202}:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(t \cdot \left(x \cdot z\right)\right) + b \cdot c\right) + t_1\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+22}:\\ \;\;\;\;\left(t \cdot \left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right) + a \cdot -4\right) + b \cdot c\right) + t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(18 \cdot \left(y \cdot \left(z \cdot t\right)\right) + i \cdot -4\right)\\ \end{array} \]

Alternative 2
Error	34.0
Cost	2024

\[\begin{array}{l} t_1 := x \cdot \left(18 \cdot \left(y \cdot \left(z \cdot t\right)\right) + i \cdot -4\right)\\ t_2 := b \cdot c + -4 \cdot \left(t \cdot a\right)\\ t_3 := k \cdot \left(j \cdot -27\right)\\ t_4 := t_3 + i \cdot \left(x \cdot -4\right)\\ \mathbf{if}\;j \leq -1.6 \cdot 10^{+163}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;j \leq -9.2 \cdot 10^{+98}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \leq -1.4 \cdot 10^{+83}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;j \leq -2.6 \cdot 10^{+72}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \leq -9.5 \cdot 10^{-78}:\\ \;\;\;\;b \cdot c + t_3\\ \mathbf{elif}\;j \leq -4.5 \cdot 10^{-144}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \leq -3.6 \cdot 10^{-185}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq -8.5 \cdot 10^{-277}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \leq 3.4 \cdot 10^{-244}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq 8.5 \cdot 10^{-145}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 3
Error	9.6
Cost	2001

\[\begin{array}{l} t_1 := j \cdot \left(k \cdot -27\right) - x \cdot \left(4 \cdot i\right)\\ t_2 := \left(b \cdot c - t \cdot \left(a \cdot 4\right)\right) + t_1\\ \mathbf{if}\;j \leq -2.5 \cdot 10^{+41}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \leq -3.8 \cdot 10^{-19}:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(t \cdot \left(x \cdot z\right)\right) + b \cdot c\right) + t_1\\ \mathbf{elif}\;j \leq -6.8 \cdot 10^{-91} \lor \neg \left(j \leq 5.9 \cdot 10^{-148}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;b \cdot c + \left(t \cdot \left(18 \cdot \left(y \cdot \left(x \cdot z\right)\right) + a \cdot -4\right) + -4 \cdot \left(x \cdot i\right)\right)\\ \end{array} \]

Alternative 4
Error	20.4
Cost	1884

\[\begin{array}{l} t_1 := -4 \cdot \left(x \cdot i\right)\\ t_2 := k \cdot \left(j \cdot -27\right)\\ t_3 := t_2 + \left(b \cdot c + -4 \cdot \left(t \cdot a\right)\right)\\ t_4 := b \cdot c + \left(t_1 + -27 \cdot \left(j \cdot k\right)\right)\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+78}:\\ \;\;\;\;x \cdot \left(18 \cdot \left(y \cdot \left(z \cdot t\right)\right) + i \cdot -4\right)\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{+19}:\\ \;\;\;\;t \cdot \left(18 \cdot \left(y \cdot \left(x \cdot z\right)\right) + a \cdot -4\right) + t_2\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-187}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-127}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-12}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{+66}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+84}:\\ \;\;\;\;t_2 + y \cdot \left(t \cdot \left(x \cdot \left(18 \cdot z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot c - t \cdot \left(a \cdot 4\right)\right) + t_1\\ \end{array} \]

Alternative 5
Error	34.6
Cost	1764

\[\begin{array}{l} t_1 := k \cdot \left(j \cdot -27\right)\\ t_2 := b \cdot c + t_1\\ t_3 := -4 \cdot \left(t \cdot a\right)\\ t_4 := b \cdot c + t_3\\ t_5 := x \cdot \left(i \cdot -4\right)\\ t_6 := t_1 + t_3\\ \mathbf{if}\;i \leq -2.75 \cdot 10^{+173}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;i \leq -9.6 \cdot 10^{+107}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;i \leq -4.2 \cdot 10^{+17}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;i \leq -3.6 \cdot 10^{-174}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;i \leq -1.46 \cdot 10^{-213}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;i \leq -9.2 \cdot 10^{-307}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;i \leq 2.8 \cdot 10^{-230}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;i \leq 6.6 \cdot 10^{-149}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;i \leq 8.2 \cdot 10^{+186}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_5\\ \end{array} \]

Alternative 6
Error	25.1
Cost	1752

\[\begin{array}{l} t_1 := x \cdot \left(18 \cdot \left(y \cdot \left(z \cdot t\right)\right) + i \cdot -4\right)\\ t_2 := b \cdot c + \left(-4 \cdot \left(x \cdot i\right) + -27 \cdot \left(j \cdot k\right)\right)\\ t_3 := b \cdot c + -4 \cdot \left(t \cdot a\right)\\ \mathbf{if}\;j \leq -6 \cdot 10^{-91}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \leq -6.8 \cdot 10^{-143}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;j \leq -3.45 \cdot 10^{-180}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq -1.9 \cdot 10^{-277}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;j \leq 4.6 \cdot 10^{-244}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq 1.95 \cdot 10^{-162}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	9.1
Cost	1737

\[\begin{array}{l} \mathbf{if}\;j \leq -2.1 \cdot 10^{-90} \lor \neg \left(j \leq 1.3 \cdot 10^{-146}\right):\\ \;\;\;\;\left(b \cdot c - t \cdot \left(a \cdot 4\right)\right) + \left(j \cdot \left(k \cdot -27\right) - x \cdot \left(4 \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot c + \left(t \cdot \left(18 \cdot \left(y \cdot \left(x \cdot z\right)\right) + a \cdot -4\right) + -4 \cdot \left(x \cdot i\right)\right)\\ \end{array} \]

Alternative 8
Error	19.4
Cost	1620

\[\begin{array}{l} t_1 := k \cdot \left(j \cdot -27\right)\\ t_2 := t_1 + \left(b \cdot c + -4 \cdot \left(t \cdot a\right)\right)\\ t_3 := b \cdot c + \left(-4 \cdot \left(x \cdot i\right) + -27 \cdot \left(j \cdot k\right)\right)\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{+77}:\\ \;\;\;\;x \cdot \left(18 \cdot \left(y \cdot \left(z \cdot t\right)\right) + i \cdot -4\right)\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-127}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-13}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{+66}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+84}:\\ \;\;\;\;t_1 + y \cdot \left(t \cdot \left(x \cdot \left(18 \cdot z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 9
Error	32.5
Cost	1496

\[\begin{array}{l} t_1 := b \cdot c + -4 \cdot \left(t \cdot a\right)\\ t_2 := k \cdot \left(j \cdot -27\right)\\ t_3 := t_2 + i \cdot \left(x \cdot -4\right)\\ \mathbf{if}\;j \leq -5 \cdot 10^{+162}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;j \leq -9.2 \cdot 10^{+98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq -1.95 \cdot 10^{+83}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;j \leq -1.6 \cdot 10^{+66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq -4.8 \cdot 10^{-79}:\\ \;\;\;\;b \cdot c + t_2\\ \mathbf{elif}\;j \leq 1.35 \cdot 10^{-143}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 10
Error	18.3
Cost	1489

\[\begin{array}{l} t_1 := -4 \cdot \left(x \cdot i\right)\\ \mathbf{if}\;a \leq -3 \cdot 10^{+66}:\\ \;\;\;\;\left(b \cdot c - t \cdot \left(a \cdot 4\right)\right) + t_1\\ \mathbf{elif}\;a \leq 1.15 \cdot 10^{-17} \lor \neg \left(a \leq 2.3 \cdot 10^{+76}\right) \land a \leq 6.5 \cdot 10^{+154}:\\ \;\;\;\;b \cdot c + \left(t_1 + -27 \cdot \left(j \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(j \cdot -27\right) + \left(b \cdot c + -4 \cdot \left(t \cdot a\right)\right)\\ \end{array} \]

Alternative 11
Error	10.1
Cost	1476

\[\begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{+99}:\\ \;\;\;\;\left(b \cdot c - t \cdot \left(a \cdot 4\right)\right) + \left(j \cdot \left(k \cdot -27\right) - x \cdot \left(4 \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(18 \cdot \left(y \cdot \left(x \cdot z\right)\right) + a \cdot -4\right) + k \cdot \left(j \cdot -27\right)\\ \end{array} \]

Alternative 12
Error	45.6
Cost	1376

\[\begin{array}{l} t_1 := a \cdot \left(t \cdot -4\right)\\ t_2 := x \cdot \left(i \cdot -4\right)\\ t_3 := -27 \cdot \left(j \cdot k\right)\\ \mathbf{if}\;j \leq -9 \cdot 10^{+149}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;j \leq -9.5 \cdot 10^{+98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq -8.2 \cdot 10^{+83}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \leq -4.7 \cdot 10^{-144}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;j \leq -6.2 \cdot 10^{-195}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \leq 2.1 \cdot 10^{-293}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;j \leq 5.5 \cdot 10^{-178}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq 6.5 \cdot 10^{-144}:\\ \;\;\;\;b \cdot c\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 13
Error	31.2
Cost	1106

\[\begin{array}{l} \mathbf{if}\;j \leq -9 \cdot 10^{+149} \lor \neg \left(j \leq -4.8 \cdot 10^{+72}\right) \land \left(j \leq -3.2 \cdot 10^{-79} \lor \neg \left(j \leq 3.9 \cdot 10^{-144}\right)\right):\\ \;\;\;\;b \cdot c + k \cdot \left(j \cdot -27\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot c + -4 \cdot \left(t \cdot a\right)\\ \end{array} \]

Alternative 14
Error	45.5
Cost	848

\[\begin{array}{l} t_1 := -27 \cdot \left(j \cdot k\right)\\ \mathbf{if}\;j \leq -6.8 \cdot 10^{+159}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq 4.4 \cdot 10^{-294}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;j \leq 1.65 \cdot 10^{-177}:\\ \;\;\;\;a \cdot \left(t \cdot -4\right)\\ \mathbf{elif}\;j \leq 4.8 \cdot 10^{-147}:\\ \;\;\;\;b \cdot c\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	37.4
Cost	841

\[\begin{array}{l} \mathbf{if}\;j \leq -2.6 \cdot 10^{+166} \lor \neg \left(j \leq 1.8 \cdot 10^{-143}\right):\\ \;\;\;\;-27 \cdot \left(j \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot c + -4 \cdot \left(t \cdot a\right)\\ \end{array} \]

Alternative 16
Error	45.3
Cost	585

\[\begin{array}{l} \mathbf{if}\;j \leq -6.8 \cdot 10^{+159} \lor \neg \left(j \leq 1.8 \cdot 10^{-143}\right):\\ \;\;\;\;-27 \cdot \left(j \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot c\\ \end{array} \]

Alternative 17
Error	48.5
Cost	192

\[b \cdot c \]

Reproduce

herbie shell --seed 2022354 
(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)))