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

Average Error: 5.7 → 3.1

Time: 33.2s

Precision: binary64

Cost: 15556

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

Math

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

↓

\[\begin{array}{l} t_1 := k \cdot \left(j \cdot -27\right)\\ t_2 := i \cdot \left(x \cdot -4\right)\\ t_3 := \left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + t_2\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(t, a \cdot -4, \mathsf{fma}\left(b, c, t_2\right)\right) + t_1\\ \mathbf{elif}\;t_3 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;t_3 + t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(18 \cdot \left(y \cdot \left(z \cdot t\right)\right) + i \cdot -4\right)\\ \end{array} \]

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 (* k (* j -27.0)))
        (t_2 (* i (* x -4.0)))
        (t_3
         (+
          (+ (+ (* (* (* (* x 18.0) y) z) t) (* t (* a -4.0))) (* b c))
          t_2)))
   (if (<= t_3 (- INFINITY))
     (+ (fma t (* a -4.0) (fma b c t_2)) t_1)
     (if (<= t_3 2e+302)
       (+ t_3 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 = k * (j * -27.0);
	double t_2 = i * (x * -4.0);
	double t_3 = ((((((x * 18.0) * y) * z) * t) + (t * (a * -4.0))) + (b * c)) + t_2;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = fma(t, (a * -4.0), fma(b, c, t_2)) + t_1;
	} else if (t_3 <= 2e+302) {
		tmp = t_3 + 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(k * Float64(j * -27.0))
	t_2 = Float64(i * Float64(x * -4.0))
	t_3 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * 18.0) * y) * z) * t) + Float64(t * Float64(a * -4.0))) + Float64(b * c)) + t_2)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(fma(t, Float64(a * -4.0), fma(b, c, t_2)) + t_1);
	elseif (t_3 <= 2e+302)
		tmp = Float64(t_3 + t_1);
	else
		tmp = Float64(x * Float64(Float64(18.0 * Float64(y * Float64(z * t))) + Float64(i * -4.0)));
	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[(k * N[(j * -27.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(x * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = 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] + t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(t * N[(a * -4.0), $MachinePrecision] + N[(b * c + t$95$2), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[t$95$3, 2e+302], N[(t$95$3 + t$95$1), $MachinePrecision], N[(x * N[(N[(18.0 * N[(y * N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(i * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	5.7
Target	1.7
Herbie	3.1

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

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 40.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(x \cdot 18, y \cdot z, a \cdot -4\right), \mathsf{fma}\left(b, c, i \cdot \left(x \cdot -4\right)\right)\right) + k \cdot \left(j \cdot -27\right)} \]
      Proof
      (+.f64 (fma.f64 t (fma.f64 (*.f64 x 18) (*.f64 y z) (*.f64 a -4)) (fma.f64 b c (*.f64 i (*.f64 x -4)))) (*.f64 k (*.f64 j -27))): 0 points increase in error, 0 points decrease in error
      (+.f64 (fma.f64 t (fma.f64 (*.f64 x 18) (*.f64 y z) (*.f64 a (Rewrite<= metadata-eval (neg.f64 4)))) (fma.f64 b c (*.f64 i (*.f64 x -4)))) (*.f64 k (*.f64 j -27))): 0 points increase in error, 0 points decrease in error
      (+.f64 (fma.f64 t (fma.f64 (*.f64 x 18) (*.f64 y z) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 a 4)))) (fma.f64 b c (*.f64 i (*.f64 x -4)))) (*.f64 k (*.f64 j -27))): 0 points increase in error, 0 points decrease in error
      (+.f64 (fma.f64 t (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (*.f64 x 18) (*.f64 y z)) (*.f64 a 4))) (fma.f64 b c (*.f64 i (*.f64 x -4)))) (*.f64 k (*.f64 j -27))): 0 points increase in error, 0 points decrease in error
      (+.f64 (fma.f64 t (-.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 x 18) y) z)) (*.f64 a 4)) (fma.f64 b c (*.f64 i (*.f64 x -4)))) (*.f64 k (*.f64 j -27))): 14 points increase in error, 23 points decrease in error
      (+.f64 (fma.f64 t (-.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) (*.f64 a 4)) (fma.f64 b c (*.f64 i (*.f64 x (Rewrite<= metadata-eval (neg.f64 4)))))) (*.f64 k (*.f64 j -27))): 0 points increase in error, 0 points decrease in error
      (+.f64 (fma.f64 t (-.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) (*.f64 a 4)) (fma.f64 b c (*.f64 i (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 x 4)))))) (*.f64 k (*.f64 j -27))): 0 points increase in error, 0 points decrease in error
      (+.f64 (fma.f64 t (-.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) (*.f64 a 4)) (fma.f64 b c (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 (*.f64 x 4)) i)))) (*.f64 k (*.f64 j -27))): 0 points increase in error, 0 points decrease in error
      (+.f64 (fma.f64 t (-.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) (*.f64 a 4)) (fma.f64 b c (Rewrite=> distribute-lft-neg-out_binary64 (neg.f64 (*.f64 (*.f64 x 4) i))))) (*.f64 k (*.f64 j -27))): 0 points increase in error, 0 points decrease in error
      (+.f64 (fma.f64 t (-.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) (*.f64 a 4)) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 b c) (*.f64 (*.f64 x 4) i)))) (*.f64 k (*.f64 j -27))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 t (-.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) (*.f64 a 4))) (-.f64 (*.f64 b c) (*.f64 (*.f64 x 4) i)))) (*.f64 k (*.f64 j -27))): 2 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) t) (*.f64 (*.f64 a 4) t))) (-.f64 (*.f64 b c) (*.f64 (*.f64 x 4) i))) (*.f64 k (*.f64 j -27))): 3 points increase in error, 1 points decrease in error
      (+.f64 (Rewrite<= associate--l+_binary64 (-.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 k (*.f64 j -27))): 0 points increase in error, 0 points decrease in error
      (+.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 k (*.f64 j (Rewrite<= metadata-eval (neg.f64 27))))): 0 points increase in error, 0 points decrease in error
      (+.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 k (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 j 27))))): 0 points increase in error, 0 points decrease in error
      (+.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)) (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 (*.f64 j 27)) k))): 0 points increase in error, 0 points decrease in error
      (+.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)) (Rewrite=> distribute-lft-neg-out_binary64 (neg.f64 (*.f64 (*.f64 j 27) k)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= sub-neg_binary64 (-.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))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in x around 0 36.1

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

   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)) < 2.0000000000000002e302

   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 2.0000000000000002e302 < (-.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 53.6

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

   2. Simplified 36.2

      \[\leadsto \color{blue}{\left(t \cdot \left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right) - a \cdot 4\right) + b \cdot c\right) - \left(x \cdot \left(4 \cdot i\right) + j \cdot \left(27 \cdot k\right)\right)} \]
      Proof
      (-.f64 (+.f64 (*.f64 t (-.f64 (*.f64 (*.f64 x 18) (*.f64 y z)) (*.f64 a 4))) (*.f64 b c)) (+.f64 (*.f64 x (*.f64 4 i)) (*.f64 j (*.f64 27 k)))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (*.f64 t (-.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 x 18) y) z)) (*.f64 a 4))) (*.f64 b c)) (+.f64 (*.f64 x (*.f64 4 i)) (*.f64 j (*.f64 27 k)))): 14 points increase in error, 23 points decrease in error
      (-.f64 (+.f64 (*.f64 t (Rewrite=> sub-neg_binary64 (+.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) (neg.f64 (*.f64 a 4))))) (*.f64 b c)) (+.f64 (*.f64 x (*.f64 4 i)) (*.f64 j (*.f64 27 k)))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (*.f64 t (Rewrite=> +-commutative_binary64 (+.f64 (neg.f64 (*.f64 a 4)) (*.f64 (*.f64 (*.f64 x 18) y) z)))) (*.f64 b c)) (+.f64 (*.f64 x (*.f64 4 i)) (*.f64 j (*.f64 27 k)))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (neg.f64 (*.f64 a 4)) t) (*.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) t))) (*.f64 b c)) (+.f64 (*.f64 x (*.f64 4 i)) (*.f64 j (*.f64 27 k)))): 3 points increase in error, 1 points decrease in error
      (-.f64 (+.f64 (+.f64 (Rewrite=> distribute-lft-neg-out_binary64 (neg.f64 (*.f64 (*.f64 a 4) t))) (*.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) t)) (*.f64 b c)) (+.f64 (*.f64 x (*.f64 4 i)) (*.f64 j (*.f64 27 k)))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (+.f64 (Rewrite=> neg-sub0_binary64 (-.f64 0 (*.f64 (*.f64 a 4) t))) (*.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) t)) (*.f64 b c)) (+.f64 (*.f64 x (*.f64 4 i)) (*.f64 j (*.f64 27 k)))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (Rewrite=> associate-+l-_binary64 (-.f64 0 (-.f64 (*.f64 (*.f64 a 4) t) (*.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) t)))) (*.f64 b c)) (+.f64 (*.f64 x (*.f64 4 i)) (*.f64 j (*.f64 27 k)))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 (*.f64 (*.f64 a 4) t)) (*.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) t))) (*.f64 b c)) (+.f64 (*.f64 x (*.f64 4 i)) (*.f64 j (*.f64 27 k)))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (*.f64 (*.f64 a 4) t))) (*.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) t)) (*.f64 b c)) (+.f64 (*.f64 x (*.f64 4 i)) (*.f64 j (*.f64 27 k)))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (+.f64 (Rewrite<= distribute-lft-neg-out_binary64 (*.f64 (neg.f64 (*.f64 a 4)) t)) (*.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) t)) (*.f64 b c)) (+.f64 (*.f64 x (*.f64 4 i)) (*.f64 j (*.f64 27 k)))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) t) (*.f64 (neg.f64 (*.f64 a 4)) t))) (*.f64 b c)) (+.f64 (*.f64 x (*.f64 4 i)) (*.f64 j (*.f64 27 k)))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) t) (*.f64 (*.f64 a 4) t))) (*.f64 b c)) (+.f64 (*.f64 x (*.f64 4 i)) (*.f64 j (*.f64 27 k)))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) t) (*.f64 (*.f64 a 4) t)) (*.f64 b c)) (+.f64 (*.f64 x (*.f64 4 i)) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 j 27) k)))): 7 points increase in error, 10 points decrease in error
      (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) t) (*.f64 (*.f64 a 4) t)) (*.f64 b c)) (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x 4) i)) (*.f64 (*.f64 j 27) k))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) t) (*.f64 (*.f64 a 4) t)) (*.f64 b c)) (Rewrite<= cancel-sign-sub_binary64 (-.f64 (*.f64 (*.f64 x 4) i) (*.f64 (neg.f64 (*.f64 j 27)) k)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+l-_binary64 (+.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 (neg.f64 (*.f64 j 27)) k))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= cancel-sign-sub-inv_binary64 (-.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))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in x around inf 24.8

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

4. Final simplification 3.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(t, a \cdot -4, \mathsf{fma}\left(b, c, i \cdot \left(x \cdot -4\right)\right)\right) + k \cdot \left(j \cdot -27\right)\\ \mathbf{elif}\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right) \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(18 \cdot \left(y \cdot \left(z \cdot t\right)\right) + i \cdot -4\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	3.1
Cost	5320

\[\begin{array}{l} t_1 := t \cdot \left(a \cdot -4\right)\\ t_2 := \left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t_1\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\left(b \cdot c + t_1\right) + \left(x \cdot \left(i \cdot -4\right) - j \cdot \left(k \cdot 27\right)\right)\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;t_2 + k \cdot \left(j \cdot -27\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(18 \cdot \left(y \cdot \left(z \cdot t\right)\right) + i \cdot -4\right)\\ \end{array} \]

Alternative 2
Error	11.0
Cost	2896

\[\begin{array}{l} t_1 := \left(b \cdot c + t \cdot \left(a \cdot -4\right)\right) + \left(x \cdot \left(i \cdot -4\right) - j \cdot \left(k \cdot 27\right)\right)\\ t_2 := k \cdot \left(j \cdot 27\right)\\ \mathbf{if}\;t_2 \leq -2 \cdot 10^{-221}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{-265}:\\ \;\;\;\;t \cdot \left(a \cdot -4 + 18 \cdot \left(y \cdot \left(x \cdot z\right)\right)\right)\\ \mathbf{elif}\;t_2 \leq 10^{-27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 0.0004:\\ \;\;\;\;18 \cdot \left(y \cdot \left(t \cdot \left(x \cdot z\right)\right)\right) + -4 \cdot \left(t \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	11.0
Cost	2896

\[\begin{array}{l} t_1 := \left(b \cdot c + t \cdot \left(a \cdot -4\right)\right) + \left(x \cdot \left(i \cdot -4\right) - j \cdot \left(k \cdot 27\right)\right)\\ t_2 := -4 \cdot \left(t \cdot a\right)\\ t_3 := k \cdot \left(j \cdot 27\right)\\ \mathbf{if}\;t_3 \leq -2 \cdot 10^{-221}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_3 \leq -1 \cdot 10^{-265}:\\ \;\;\;\;t \cdot \left(a \cdot -4 + 18 \cdot \left(y \cdot \left(x \cdot z\right)\right)\right)\\ \mathbf{elif}\;t_3 \leq 10^{-27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_3 \leq 0.0004:\\ \;\;\;\;18 \cdot \left(y \cdot \left(t \cdot \left(x \cdot z\right)\right)\right) + t_2\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(j \cdot -27\right) + \left(\left(b \cdot c + t_2\right) + i \cdot \left(x \cdot -4\right)\right)\\ \end{array} \]

Alternative 4
Error	30.5
Cost	2152

\[\begin{array}{l} t_1 := x \cdot \left(18 \cdot \left(z \cdot \left(y \cdot t\right)\right) + i \cdot -4\right)\\ t_2 := k \cdot \left(j \cdot -27\right) + b \cdot c\\ t_3 := t \cdot \left(a \cdot -4 + 18 \cdot \left(y \cdot \left(x \cdot z\right)\right)\right)\\ \mathbf{if}\;t \leq -3.1 \cdot 10^{-6}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -5.5 \cdot 10^{-101}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -7.3 \cdot 10^{-124}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-303}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-292}:\\ \;\;\;\;x \cdot \left(i \cdot -4\right) - j \cdot \left(k \cdot 27\right)\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{-222}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-205}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 0.00048:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 9.6 \cdot 10^{+112}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+147}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 5
Error	5.6
Cost	1984

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

Alternative 6
Error	5.5
Cost	1856

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

Alternative 7
Error	5.6
Cost	1856

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

Alternative 8
Error	30.6
Cost	1760

\[\begin{array}{l} t_1 := x \cdot \left(18 \cdot \left(z \cdot \left(y \cdot t\right)\right) + i \cdot -4\right)\\ t_2 := k \cdot \left(j \cdot -27\right)\\ t_3 := t_2 + -4 \cdot \left(t \cdot a\right)\\ t_4 := t_2 + b \cdot c\\ \mathbf{if}\;c \leq -4.9 \cdot 10^{-49}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;c \leq -2.4 \cdot 10^{-220}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c \leq 1.15 \cdot 10^{-297}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 2.9 \cdot 10^{-186}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c \leq 3.1 \cdot 10^{-176}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 1.25 \cdot 10^{-141}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c \leq 4.6 \cdot 10^{-92}:\\ \;\;\;\;x \cdot \left(i \cdot -4\right) - j \cdot \left(k \cdot 27\right)\\ \mathbf{elif}\;c \leq 5.8 \cdot 10^{-67}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 9
Error	8.6
Cost	1736

\[\begin{array}{l} t_1 := \left(b \cdot c + t \cdot \left(a \cdot -4 + 18 \cdot \left(y \cdot \left(x \cdot z\right)\right)\right)\right) + -27 \cdot \left(k \cdot j\right)\\ \mathbf{if}\;t \leq -1.9 \cdot 10^{+50}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-49}:\\ \;\;\;\;k \cdot \left(j \cdot -27\right) + \left(\left(b \cdot c + -4 \cdot \left(t \cdot a\right)\right) + i \cdot \left(x \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	35.3
Cost	1632

\[\begin{array}{l} t_1 := k \cdot \left(j \cdot -27\right) + b \cdot c\\ t_2 := x \cdot \left(i \cdot -4\right)\\ t_3 := -4 \cdot \left(t \cdot a\right)\\ \mathbf{if}\;t \leq -1.7 \cdot 10^{+202}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{+77}:\\ \;\;\;\;t \cdot \left(18 \cdot \left(y \cdot \left(x \cdot z\right)\right)\right)\\ \mathbf{elif}\;t \leq -5 \cdot 10^{-101}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -7.3 \cdot 10^{-124}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-303}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-292}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 0.052:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+81}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	32.6
Cost	1628

\[\begin{array}{l} t_1 := x \cdot \left(i \cdot -4\right)\\ t_2 := k \cdot \left(j \cdot -27\right)\\ t_3 := t_2 + b \cdot c\\ t_4 := t_2 + -4 \cdot \left(t \cdot a\right)\\ \mathbf{if}\;t \leq -4.7 \cdot 10^{+200}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq -2.9 \cdot 10^{+77}:\\ \;\;\;\;t \cdot \left(18 \cdot \left(y \cdot \left(x \cdot z\right)\right)\right)\\ \mathbf{elif}\;t \leq -8.5 \cdot 10^{-101}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -7.3 \cdot 10^{-124}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-303}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-292}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{-38}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 12
Error	18.2
Cost	1620

\[\begin{array}{l} t_1 := -4 \cdot \left(t \cdot a\right)\\ t_2 := -27 \cdot \left(k \cdot j\right)\\ t_3 := \left(b \cdot c + t_1\right) + t_2\\ t_4 := k \cdot \left(j \cdot -27\right)\\ \mathbf{if}\;t \leq -1.6 \cdot 10^{+200}:\\ \;\;\;\;t_4 + \left(t_1 + i \cdot \left(x \cdot -4\right)\right)\\ \mathbf{elif}\;t \leq -2.55 \cdot 10^{+125}:\\ \;\;\;\;t_4 + 18 \cdot \left(y \cdot \left(t \cdot \left(x \cdot z\right)\right)\right)\\ \mathbf{elif}\;t \leq -4.1 \cdot 10^{+77}:\\ \;\;\;\;t \cdot \left(a \cdot -4 + 18 \cdot \left(y \cdot \left(x \cdot z\right)\right)\right)\\ \mathbf{elif}\;t \leq -4.1 \cdot 10^{-53}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-56}:\\ \;\;\;\;b \cdot c + \left(t_2 + -4 \cdot \left(x \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 13
Error	44.2
Cost	1508

\[\begin{array}{l} t_1 := -27 \cdot \left(k \cdot j\right)\\ t_2 := -4 \cdot \left(t \cdot a\right)\\ t_3 := x \cdot \left(i \cdot -4\right)\\ \mathbf{if}\;c \leq -4.7 \cdot 10^{-48}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;c \leq -1.5 \cdot 10^{-179}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq -5 \cdot 10^{-184}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -3.6 \cdot 10^{-224}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c \leq -1.1 \cdot 10^{-248}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -7 \cdot 10^{-299}:\\ \;\;\;\;18 \cdot \left(y \cdot \left(z \cdot \left(x \cdot t\right)\right)\right)\\ \mathbf{elif}\;c \leq 3.9 \cdot 10^{-287}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c \leq 3.8 \cdot 10^{-239}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 2.2 \cdot 10^{-59}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;b \cdot c\\ \end{array} \]

Alternative 14
Error	44.4
Cost	1508

\[\begin{array}{l} t_1 := -27 \cdot \left(k \cdot j\right)\\ t_2 := -4 \cdot \left(t \cdot a\right)\\ t_3 := x \cdot \left(i \cdot -4\right)\\ \mathbf{if}\;c \leq -2.2 \cdot 10^{-47}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;c \leq -2.6 \cdot 10^{-179}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq -1.04 \cdot 10^{-187}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -3.3 \cdot 10^{-227}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c \leq -4 \cdot 10^{-248}:\\ \;\;\;\;x \cdot \left(z \cdot \left(y \cdot \left(18 \cdot t\right)\right)\right)\\ \mathbf{elif}\;c \leq -6.8 \cdot 10^{-299}:\\ \;\;\;\;18 \cdot \left(y \cdot \left(z \cdot \left(x \cdot t\right)\right)\right)\\ \mathbf{elif}\;c \leq 3.8 \cdot 10^{-287}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c \leq 8.2 \cdot 10^{-239}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 2.7 \cdot 10^{-61}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;b \cdot c\\ \end{array} \]

Alternative 15
Error	20.3
Cost	1488

\[\begin{array}{l} t_1 := b \cdot c + \left(-27 \cdot \left(k \cdot j\right) + -4 \cdot \left(x \cdot i\right)\right)\\ t_2 := t \cdot \left(a \cdot -4 + 18 \cdot \left(y \cdot \left(x \cdot z\right)\right)\right)\\ \mathbf{if}\;t \leq -1.02 \cdot 10^{+52}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 0.6:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+109}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{+149}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 16
Error	17.8
Cost	1356

\[\begin{array}{l} t_1 := -27 \cdot \left(k \cdot j\right)\\ t_2 := \left(b \cdot c + -4 \cdot \left(t \cdot a\right)\right) + t_1\\ \mathbf{if}\;t \leq -4.5 \cdot 10^{+77}:\\ \;\;\;\;t \cdot \left(a \cdot -4 + 18 \cdot \left(y \cdot \left(x \cdot z\right)\right)\right)\\ \mathbf{elif}\;t \leq -2.75 \cdot 10^{-55}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-55}:\\ \;\;\;\;b \cdot c + \left(t_1 + -4 \cdot \left(x \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 17
Error	30.7
Cost	1100

\[\begin{array}{l} t_1 := k \cdot \left(j \cdot -27\right)\\ t_2 := t_1 + b \cdot c\\ \mathbf{if}\;b \leq -1.9 \cdot 10^{+60}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -2.6 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \left(i \cdot -4\right) - j \cdot \left(k \cdot 27\right)\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-114}:\\ \;\;\;\;t_1 + -4 \cdot \left(t \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 18
Error	30.7
Cost	1100

\[\begin{array}{l} t_1 := k \cdot \left(j \cdot -27\right)\\ t_2 := t_1 + b \cdot c\\ \mathbf{if}\;b \leq -4 \cdot 10^{+60}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -8.2 \cdot 10^{-15}:\\ \;\;\;\;t_1 + x \cdot \left(i \cdot -4\right)\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-113}:\\ \;\;\;\;t_1 + -4 \cdot \left(t \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 19
Error	44.3
Cost	980

\[\begin{array}{l} t_1 := -4 \cdot \left(t \cdot a\right)\\ t_2 := -27 \cdot \left(k \cdot j\right)\\ \mathbf{if}\;b \leq -2.8 \cdot 10^{-6}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;b \leq -4.7 \cdot 10^{-81}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -1.7 \cdot 10^{-137}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -3.7 \cdot 10^{-285}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 2.32 \cdot 10^{-144}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;b \cdot c\\ \end{array} \]

Alternative 20
Error	44.3
Cost	980

\[\begin{array}{l} t_1 := -4 \cdot \left(t \cdot a\right)\\ \mathbf{if}\;b \leq -2.8 \cdot 10^{-6}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;b \leq -2.9 \cdot 10^{-82}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -3 \cdot 10^{-139}:\\ \;\;\;\;k \cdot \left(j \cdot -27\right)\\ \mathbf{elif}\;b \leq -1.6 \cdot 10^{-281}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-144}:\\ \;\;\;\;-27 \cdot \left(k \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot c\\ \end{array} \]

Alternative 21
Error	44.8
Cost	848

\[\begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{+60}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;b \leq -1.7 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \left(i \cdot -4\right)\\ \mathbf{elif}\;b \leq -3.8 \cdot 10^{-282}:\\ \;\;\;\;-4 \cdot \left(t \cdot a\right)\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{-144}:\\ \;\;\;\;-27 \cdot \left(k \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot c\\ \end{array} \]

Alternative 22
Error	43.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{-21}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-144}:\\ \;\;\;\;-27 \cdot \left(k \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot c\\ \end{array} \]

Alternative 23
Error	48.7
Cost	192

\[b \cdot c \]

Reproduce

herbie shell --seed 2022325 
(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)))