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

Average Error: 5.7 → 1.1

Time: 46.0s

Precision: binary64

Cost: 6088

Math

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

↓

\[\begin{array}{l} t_1 := \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;b \cdot c + \left(x \cdot \left(18 \cdot \left(y \cdot \left(z \cdot t\right)\right) + i \cdot -4\right) + \left(-27 \cdot \left(j \cdot k\right) + -4 \cdot \left(t \cdot a\right)\right)\right)\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;18 \cdot \left(y \cdot \left(z \cdot \left(x \cdot t\right)\right)\right) + \left(b \cdot c + -4 \cdot \left(x \cdot i\right)\right)\\ \end{array} \]

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))
     (+
      (* b c)
      (+
       (* x (+ (* 18.0 (* y (* z t))) (* i -4.0)))
       (+ (* -27.0 (* j k)) (* -4.0 (* t a)))))
     (if (<= t_1 5e+306)
       t_1
       (+ (* 18.0 (* y (* z (* x t)))) (+ (* b c) (* -4.0 (* x i))))))))

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 = (b * c) + ((x * ((18.0 * (y * (z * t))) + (i * -4.0))) + ((-27.0 * (j * k)) + (-4.0 * (t * a))));
	} else if (t_1 <= 5e+306) {
		tmp = t_1;
	} else {
		tmp = (18.0 * (y * (z * (x * t)))) + ((b * c) + (-4.0 * (x * i)));
	}
	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 = (b * c) + ((x * ((18.0 * (y * (z * t))) + (i * -4.0))) + ((-27.0 * (j * k)) + (-4.0 * (t * a))));
	} else if (t_1 <= 5e+306) {
		tmp = t_1;
	} else {
		tmp = (18.0 * (y * (z * (x * t)))) + ((b * c) + (-4.0 * (x * i)));
	}
	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 = (b * c) + ((x * ((18.0 * (y * (z * t))) + (i * -4.0))) + ((-27.0 * (j * k)) + (-4.0 * (t * a))))
	elif t_1 <= 5e+306:
		tmp = t_1
	else:
		tmp = (18.0 * (y * (z * (x * t)))) + ((b * c) + (-4.0 * (x * i)))
	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(b * c) + Float64(Float64(x * Float64(Float64(18.0 * Float64(y * Float64(z * t))) + Float64(i * -4.0))) + Float64(Float64(-27.0 * Float64(j * k)) + Float64(-4.0 * Float64(t * a)))));
	elseif (t_1 <= 5e+306)
		tmp = t_1;
	else
		tmp = Float64(Float64(18.0 * Float64(y * Float64(z * Float64(x * t)))) + Float64(Float64(b * c) + Float64(-4.0 * Float64(x * i))));
	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 = (b * c) + ((x * ((18.0 * (y * (z * t))) + (i * -4.0))) + ((-27.0 * (j * k)) + (-4.0 * (t * a))));
	elseif (t_1 <= 5e+306)
		tmp = t_1;
	else
		tmp = (18.0 * (y * (z * (x * t)))) + ((b * c) + (-4.0 * (x * i)));
	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[(b * c), $MachinePrecision] + N[(N[(x * N[(N[(18.0 * N[(y * N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(i * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-27.0 * N[(j * k), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+306], t$95$1, N[(N[(18.0 * N[(y * N[(z * N[(x * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(b * c), $MachinePrecision] + N[(-4.0 * N[(x * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	5.7
Target	1.5
Herbie	1.1

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.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 41.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(j, k \cdot -27, \mathsf{fma}\left(t, \mathsf{fma}\left(x, y \cdot \left(18 \cdot z\right), a \cdot -4\right), \mathsf{fma}\left(b, c, i \cdot \left(x \cdot -4\right)\right)\right)\right)} \]
      Proof
      (fma.f64 j (*.f64 k -27) (fma.f64 t (fma.f64 x (*.f64 y (*.f64 18 z)) (*.f64 a -4)) (fma.f64 b c (*.f64 i (*.f64 x -4))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 j (*.f64 k (Rewrite<= metadata-eval (neg.f64 27))) (fma.f64 t (fma.f64 x (*.f64 y (*.f64 18 z)) (*.f64 a -4)) (fma.f64 b c (*.f64 i (*.f64 x -4))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 j (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 k 27))) (fma.f64 t (fma.f64 x (*.f64 y (*.f64 18 z)) (*.f64 a -4)) (fma.f64 b c (*.f64 i (*.f64 x -4))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 j (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 27 k))) (fma.f64 t (fma.f64 x (*.f64 y (*.f64 18 z)) (*.f64 a -4)) (fma.f64 b c (*.f64 i (*.f64 x -4))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (fma.f64 t (fma.f64 x (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 y 18) z)) (*.f64 a -4)) (fma.f64 b c (*.f64 i (*.f64 x -4))))): 5 points increase in error, 2 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (fma.f64 t (fma.f64 x (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 18 y)) z) (*.f64 a -4)) (fma.f64 b c (*.f64 i (*.f64 x -4))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (fma.f64 t (fma.f64 x (*.f64 (*.f64 18 y) z) (*.f64 a (Rewrite<= metadata-eval (neg.f64 4)))) (fma.f64 b c (*.f64 i (*.f64 x -4))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (fma.f64 t (fma.f64 x (*.f64 (*.f64 18 y) z) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 a 4)))) (fma.f64 b c (*.f64 i (*.f64 x -4))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (fma.f64 t (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 x (*.f64 (*.f64 18 y) z)) (*.f64 a 4))) (fma.f64 b c (*.f64 i (*.f64 x -4))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (fma.f64 t (-.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x (*.f64 18 y)) z)) (*.f64 a 4)) (fma.f64 b c (*.f64 i (*.f64 x -4))))): 8 points increase in error, 21 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (fma.f64 t (-.f64 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x 18) y)) z) (*.f64 a 4)) (fma.f64 b c (*.f64 i (*.f64 x -4))))): 3 points increase in error, 1 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (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))))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (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))))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (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))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (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)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (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))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (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))))): 2 points increase in error, 0 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (+.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)))): 0 points increase in error, 2 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (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)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 j (neg.f64 (*.f64 27 k))) (-.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 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 j (*.f64 27 k)))) (-.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))): 0 points increase in error, 0 points decrease in error
      (+.f64 (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 j 27) k))) (-.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 points increase in error, 9 points decrease in error
      (+.f64 (Rewrite<= distribute-lft-neg-out_binary64 (*.f64 (neg.f64 (*.f64 j 27)) k)) (-.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))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_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 0 5.7

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

   if -inf.0 < (-.f64 (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) t) (*.f64 (*.f64 a 4) t)) (*.f64 b c)) (*.f64 (*.f64 x 4) i)) (*.f64 (*.f64 j 27) k)) < 4.99999999999999993e306

   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 4.99999999999999993e306 < (-.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 60.3

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

   2. Simplified 39.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(j, k \cdot -27, \mathsf{fma}\left(t, \mathsf{fma}\left(x, y \cdot \left(18 \cdot z\right), a \cdot -4\right), \mathsf{fma}\left(b, c, i \cdot \left(x \cdot -4\right)\right)\right)\right)} \]
      Proof
      (fma.f64 j (*.f64 k -27) (fma.f64 t (fma.f64 x (*.f64 y (*.f64 18 z)) (*.f64 a -4)) (fma.f64 b c (*.f64 i (*.f64 x -4))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 j (*.f64 k (Rewrite<= metadata-eval (neg.f64 27))) (fma.f64 t (fma.f64 x (*.f64 y (*.f64 18 z)) (*.f64 a -4)) (fma.f64 b c (*.f64 i (*.f64 x -4))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 j (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 k 27))) (fma.f64 t (fma.f64 x (*.f64 y (*.f64 18 z)) (*.f64 a -4)) (fma.f64 b c (*.f64 i (*.f64 x -4))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 j (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 27 k))) (fma.f64 t (fma.f64 x (*.f64 y (*.f64 18 z)) (*.f64 a -4)) (fma.f64 b c (*.f64 i (*.f64 x -4))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (fma.f64 t (fma.f64 x (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 y 18) z)) (*.f64 a -4)) (fma.f64 b c (*.f64 i (*.f64 x -4))))): 5 points increase in error, 2 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (fma.f64 t (fma.f64 x (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 18 y)) z) (*.f64 a -4)) (fma.f64 b c (*.f64 i (*.f64 x -4))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (fma.f64 t (fma.f64 x (*.f64 (*.f64 18 y) z) (*.f64 a (Rewrite<= metadata-eval (neg.f64 4)))) (fma.f64 b c (*.f64 i (*.f64 x -4))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (fma.f64 t (fma.f64 x (*.f64 (*.f64 18 y) z) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 a 4)))) (fma.f64 b c (*.f64 i (*.f64 x -4))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (fma.f64 t (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 x (*.f64 (*.f64 18 y) z)) (*.f64 a 4))) (fma.f64 b c (*.f64 i (*.f64 x -4))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (fma.f64 t (-.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x (*.f64 18 y)) z)) (*.f64 a 4)) (fma.f64 b c (*.f64 i (*.f64 x -4))))): 8 points increase in error, 21 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (fma.f64 t (-.f64 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x 18) y)) z) (*.f64 a 4)) (fma.f64 b c (*.f64 i (*.f64 x -4))))): 3 points increase in error, 1 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (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))))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (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))))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (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))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (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)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (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))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (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))))): 2 points increase in error, 0 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (+.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)))): 0 points increase in error, 2 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (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)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 j (neg.f64 (*.f64 27 k))) (-.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 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 j (*.f64 27 k)))) (-.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))): 0 points increase in error, 0 points decrease in error
      (+.f64 (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 j 27) k))) (-.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 points increase in error, 9 points decrease in error
      (+.f64 (Rewrite<= distribute-lft-neg-out_binary64 (*.f64 (neg.f64 (*.f64 j 27)) k)) (-.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))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_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 j around 0 43.5

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

   4. Taylor expanded in y around inf 20.0

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

   5. Simplified 14.6

      \[\leadsto \color{blue}{18 \cdot \left(y \cdot \left(z \cdot \left(x \cdot t\right)\right)\right)} + \left(c \cdot b + -4 \cdot \left(i \cdot x\right)\right) \]
      Proof
      (*.f64 18 (*.f64 y (*.f64 z (*.f64 x t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 18 (*.f64 y (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 z x) t)))): 34 points increase in error, 40 points decrease in error
      (*.f64 18 (*.f64 y (Rewrite<= *-commutative_binary64 (*.f64 t (*.f64 z x))))): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right) \leq -\infty:\\ \;\;\;\;b \cdot c + \left(x \cdot \left(18 \cdot \left(y \cdot \left(z \cdot t\right)\right) + i \cdot -4\right) + \left(-27 \cdot \left(j \cdot k\right) + -4 \cdot \left(t \cdot a\right)\right)\right)\\ \mathbf{elif}\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right) \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right)\\ \mathbf{else}:\\ \;\;\;\;18 \cdot \left(y \cdot \left(z \cdot \left(x \cdot t\right)\right)\right) + \left(b \cdot c + -4 \cdot \left(x \cdot i\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	32.3
Cost	2420

\[\begin{array}{l} t_1 := -4 \cdot \left(x \cdot i\right)\\ t_2 := -4 \cdot \left(t \cdot a\right) + t_1\\ t_3 := b \cdot c + t \cdot \left(a \cdot -4\right)\\ t_4 := b \cdot c + j \cdot \left(k \cdot -27\right)\\ \mathbf{if}\;b \leq -1 \cdot 10^{+172}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;b \leq -2 \cdot 10^{+81}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq -1.7693268580124588 \cdot 10^{+50}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;b \leq -2.4501728892961686 \cdot 10^{+21}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -1949185137545.3115:\\ \;\;\;\;t_4\\ \mathbf{elif}\;b \leq -1480723.7694898874:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq -217693.2077571571:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -1.134729701796594 \cdot 10^{-34}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;b \leq -1.4657321651704276 \cdot 10^{-97}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -4.956503523313797 \cdot 10^{-133}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;b \leq -7.218454735387566 \cdot 10^{-274}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -1.6275284268398332 \cdot 10^{-292}:\\ \;\;\;\;k \cdot \left(j \cdot -27\right)\\ \mathbf{elif}\;b \leq 2.7946399442015396 \cdot 10^{-165}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 2
Error	2.6
Cost	2248

\[\begin{array}{l} t_1 := b \cdot c + \left(x \cdot \left(18 \cdot \left(y \cdot \left(z \cdot t\right)\right) + i \cdot -4\right) + \left(-27 \cdot \left(j \cdot k\right) + -4 \cdot \left(t \cdot a\right)\right)\right)\\ \mathbf{if}\;x \leq -1 \cdot 10^{+15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.7409508007207653 \cdot 10^{-203}:\\ \;\;\;\;\left(\left(b \cdot c + \left(t \cdot \left(18 \cdot \left(y \cdot \left(x \cdot z\right)\right)\right) + t \cdot \left(a \cdot -4\right)\right)\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	33.2
Cost	2156

\[\begin{array}{l} t_1 := j \cdot \left(k \cdot -27\right)\\ t_2 := -4 \cdot \left(x \cdot i\right)\\ t_3 := b \cdot c + t \cdot \left(a \cdot -4\right)\\ t_4 := b \cdot c + t_1\\ t_5 := -4 \cdot \left(t \cdot a\right)\\ t_6 := t_5 + t_1\\ t_7 := t_5 + t_2\\ t_8 := b \cdot c + t_2\\ \mathbf{if}\;b \leq -1 \cdot 10^{+172}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;b \leq -2 \cdot 10^{+81}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq -1.7693268580124588 \cdot 10^{+50}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;b \leq -2.4501728892961686 \cdot 10^{+21}:\\ \;\;\;\;t_7\\ \mathbf{elif}\;b \leq -209692056.41619214:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq -1.2242203182343165 \cdot 10^{-55}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;b \leq -1.7030443287942985 \cdot 10^{-84}:\\ \;\;\;\;t_8\\ \mathbf{elif}\;b \leq -6.456745236750323 \cdot 10^{-221}:\\ \;\;\;\;t_7\\ \mathbf{elif}\;b \leq -1.6275284268398332 \cdot 10^{-292}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;b \leq 8.226554320390236 \cdot 10^{-234}:\\ \;\;\;\;t_8\\ \mathbf{elif}\;b \leq 1.6499421565109947 \cdot 10^{-138}:\\ \;\;\;\;t_6\\ \mathbf{else}:\\ \;\;\;\;t_8\\ \end{array} \]

Alternative 4
Error	4.0
Cost	2120

\[\begin{array}{l} t_1 := -27 \cdot \left(j \cdot k\right) + -4 \cdot \left(t \cdot a\right)\\ t_2 := b \cdot c + \left(x \cdot \left(18 \cdot \left(y \cdot \left(z \cdot t\right)\right) + i \cdot -4\right) + t_1\right)\\ \mathbf{if}\;x \leq -5.955743381124295 \cdot 10^{-77}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.7409508007207653 \cdot 10^{-203}:\\ \;\;\;\;b \cdot c + \left(t_1 + 18 \cdot \left(y \cdot \left(t \cdot \left(x \cdot z\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	4.0
Cost	2120

\[\begin{array}{l} t_1 := -27 \cdot \left(j \cdot k\right) + -4 \cdot \left(t \cdot a\right)\\ \mathbf{if}\;x \leq -1.3223608542122258 \cdot 10^{-101}:\\ \;\;\;\;b \cdot c + \left(t_1 + \left(x \cdot \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right) + x \cdot \left(i \cdot -4\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.7409508007207653 \cdot 10^{-203}:\\ \;\;\;\;b \cdot c + \left(t_1 + 18 \cdot \left(y \cdot \left(t \cdot \left(x \cdot z\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot c + \left(x \cdot \left(18 \cdot \left(y \cdot \left(z \cdot t\right)\right) + i \cdot -4\right) + t_1\right)\\ \end{array} \]

Alternative 6
Error	33.7
Cost	1896

\[\begin{array}{l} t_1 := b \cdot c + t \cdot \left(a \cdot -4\right)\\ t_2 := b \cdot c + j \cdot \left(k \cdot -27\right)\\ t_3 := -4 \cdot \left(x \cdot i\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{+167}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -1.95 \cdot 10^{+77}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{+61}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{+58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.5341313772304282 \cdot 10^{-106}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -2.5621834633735013 \cdot 10^{-202}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.051015994925748 \cdot 10^{-170}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 5.05372033707518 \cdot 10^{-139}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9.569437403422831 \cdot 10^{-123}:\\ \;\;\;\;-27 \cdot \left(j \cdot k\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+91}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 7
Error	25.1
Cost	1884

\[\begin{array}{l} t_1 := -4 \cdot \left(t \cdot a\right)\\ t_2 := j \cdot \left(k \cdot -27\right)\\ t_3 := b \cdot c + \left(t_1 + -4 \cdot \left(x \cdot i\right)\right)\\ t_4 := t_1 + t_2\\ \mathbf{if}\;c \leq -7.030207950188279 \cdot 10^{-177}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c \leq -7.893711437436827 \cdot 10^{-264}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;c \leq 3.1300839777318157 \cdot 10^{-237}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c \leq 2.4259210401252363 \cdot 10^{-191}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;c \leq 2672936120857637000:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c \leq 5.9 \cdot 10^{+53}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;c \leq 10^{+240}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;b \cdot c + t_2\\ \end{array} \]

Alternative 8
Error	13.4
Cost	1864

\[\begin{array}{l} t_1 := -4 \cdot \left(x \cdot i\right)\\ t_2 := -27 \cdot \left(j \cdot k\right) + -4 \cdot \left(t \cdot a\right)\\ \mathbf{if}\;z \leq -3.8623287741706324 \cdot 10^{-23}:\\ \;\;\;\;\left(b \cdot c + t_1\right) + 18 \cdot \left(y \cdot \left(t \cdot \left(x \cdot z\right)\right)\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+253}:\\ \;\;\;\;b \cdot c + \left(t_2 + t_1\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot c + \left(t_2 + x \cdot \left(\left(y \cdot t\right) \cdot \left(18 \cdot z\right)\right)\right)\\ \end{array} \]

Alternative 9
Error	7.6
Cost	1864

\[\begin{array}{l} t_1 := -27 \cdot \left(j \cdot k\right) + -4 \cdot \left(t \cdot a\right)\\ \mathbf{if}\;i \leq -4.77186366681625 \cdot 10^{-116}:\\ \;\;\;\;b \cdot c + \left(t_1 + -4 \cdot \left(x \cdot i\right)\right)\\ \mathbf{elif}\;i \leq 1.1378808904644298 \cdot 10^{-158}:\\ \;\;\;\;b \cdot c + \left(t_1 + 18 \cdot \left(y \cdot \left(t \cdot \left(x \cdot z\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + t \cdot \left(a \cdot -4\right)\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right)\\ \end{array} \]

Alternative 10
Error	17.6
Cost	1744

\[\begin{array}{l} t_1 := -27 \cdot \left(j \cdot k\right)\\ t_2 := -4 \cdot \left(x \cdot i\right)\\ t_3 := b \cdot c + \left(t_1 + t_2\right)\\ \mathbf{if}\;x \leq -1.3223608542122258 \cdot 10^{-101}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 5.05372033707518 \cdot 10^{-139}:\\ \;\;\;\;b \cdot c + \left(t_1 + -4 \cdot \left(t \cdot a\right)\right)\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{+46}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+270}:\\ \;\;\;\;18 \cdot \left(y \cdot \left(z \cdot \left(x \cdot t\right)\right)\right) + \left(b \cdot c + t_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 11
Error	18.7
Cost	1620

\[\begin{array}{l} t_1 := -4 \cdot \left(x \cdot i\right)\\ t_2 := -27 \cdot \left(j \cdot k\right)\\ t_3 := b \cdot c + \left(t_2 + t_1\right)\\ t_4 := -4 \cdot \left(t \cdot a\right)\\ t_5 := b \cdot c + \left(t_2 + t_4\right)\\ \mathbf{if}\;k \leq -6.1013102691733786 \cdot 10^{-142}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;k \leq -7.535644471942728 \cdot 10^{-270}:\\ \;\;\;\;\left(b \cdot c + t_1\right) + 18 \cdot \left(y \cdot \left(t \cdot \left(x \cdot z\right)\right)\right)\\ \mathbf{elif}\;k \leq 1159302.420475264:\\ \;\;\;\;b \cdot c + \left(t_4 + t_1\right)\\ \mathbf{elif}\;k \leq 10^{+119}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq 2 \cdot 10^{+202}:\\ \;\;\;\;t_5\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 12
Error	11.2
Cost	1608

\[\begin{array}{l} t_1 := \left(\left(b \cdot c + t \cdot \left(a \cdot -4\right)\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right)\\ \mathbf{if}\;t \leq -2.2 \cdot 10^{+121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.3616782127305156 \cdot 10^{-38}:\\ \;\;\;\;b \cdot c + \left(-4 \cdot \left(t \cdot a\right) + 18 \cdot \left(y \cdot \left(t \cdot \left(x \cdot z\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	11.1
Cost	1608

\[\begin{array}{l} t_1 := -4 \cdot \left(t \cdot a\right)\\ \mathbf{if}\;t \leq -2.2 \cdot 10^{+121}:\\ \;\;\;\;\left(\left(b \cdot c + t \cdot \left(a \cdot -4\right)\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right)\\ \mathbf{elif}\;t \leq -1.3616782127305156 \cdot 10^{-38}:\\ \;\;\;\;b \cdot c + \left(t_1 + 18 \cdot \left(y \cdot \left(t \cdot \left(x \cdot z\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot c + \left(\left(-27 \cdot \left(j \cdot k\right) + t_1\right) + -4 \cdot \left(x \cdot i\right)\right)\\ \end{array} \]

Alternative 14
Error	30.6
Cost	1500

\[\begin{array}{l} t_1 := b \cdot c + t \cdot \left(a \cdot -4\right)\\ t_2 := b \cdot c + j \cdot \left(k \cdot -27\right)\\ t_3 := b \cdot c + -4 \cdot \left(x \cdot i\right)\\ \mathbf{if}\;x \leq -1.26 \cdot 10^{+43}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -1.5341313772304282 \cdot 10^{-106}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -2.5621834633735013 \cdot 10^{-202}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.051015994925748 \cdot 10^{-170}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 5.05372033707518 \cdot 10^{-139}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9.569437403422831 \cdot 10^{-123}:\\ \;\;\;\;-27 \cdot \left(j \cdot k\right)\\ \mathbf{elif}\;x \leq 2.1660779834117468 \cdot 10^{-84}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 15
Error	19.9
Cost	1488

\[\begin{array}{l} t_1 := -27 \cdot \left(j \cdot k\right)\\ t_2 := -4 \cdot \left(t \cdot a\right)\\ t_3 := t_2 + j \cdot \left(k \cdot -27\right)\\ \mathbf{if}\;t \leq -2.2 \cdot 10^{+121}:\\ \;\;\;\;b \cdot c + \left(t_1 + t_2\right)\\ \mathbf{elif}\;t \leq -2.6426294061935176 \cdot 10^{-52}:\\ \;\;\;\;b \cdot c + t \cdot \left(18 \cdot \left(y \cdot \left(x \cdot z\right)\right) + a \cdot -4\right)\\ \mathbf{elif}\;t \leq -1.8726023254563807 \cdot 10^{-109}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{+163}:\\ \;\;\;\;b \cdot c + \left(t_1 + -4 \cdot \left(x \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 16
Error	45.7
Cost	1376

\[\begin{array}{l} t_1 := j \cdot \left(k \cdot -27\right)\\ t_2 := -4 \cdot \left(x \cdot i\right)\\ \mathbf{if}\;c \leq -7.030207950188279 \cdot 10^{-177}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;c \leq 2.4196513079853168 \cdot 10^{-278}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 9.705949036431725 \cdot 10^{-256}:\\ \;\;\;\;t \cdot \left(a \cdot -4\right)\\ \mathbf{elif}\;c \leq 6.589019238327619 \cdot 10^{-188}:\\ \;\;\;\;k \cdot \left(j \cdot -27\right)\\ \mathbf{elif}\;c \leq 1.3826079386658574 \cdot 10^{-44}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 5.9 \cdot 10^{+53}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 2.7 \cdot 10^{+82}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;c \leq 8.1 \cdot 10^{+158}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;b \cdot c\\ \end{array} \]

Alternative 17
Error	45.7
Cost	1376

\[\begin{array}{l} t_1 := j \cdot \left(k \cdot -27\right)\\ t_2 := -4 \cdot \left(x \cdot i\right)\\ \mathbf{if}\;c \leq -7.030207950188279 \cdot 10^{-177}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;c \leq 2.4196513079853168 \cdot 10^{-278}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 9.705949036431725 \cdot 10^{-256}:\\ \;\;\;\;t \cdot \left(a \cdot -4\right)\\ \mathbf{elif}\;c \leq 6.589019238327619 \cdot 10^{-188}:\\ \;\;\;\;-27 \cdot \left(j \cdot k\right)\\ \mathbf{elif}\;c \leq 1.3826079386658574 \cdot 10^{-44}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 5.9 \cdot 10^{+53}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 2.7 \cdot 10^{+82}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;c \leq 8.1 \cdot 10^{+158}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;b \cdot c\\ \end{array} \]

Alternative 18
Error	17.9
Cost	1224

\[\begin{array}{l} t_1 := -4 \cdot \left(t \cdot a\right)\\ t_2 := b \cdot c + \left(t_1 + -4 \cdot \left(x \cdot i\right)\right)\\ \mathbf{if}\;x \leq -1.26 \cdot 10^{+43}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2.1660779834117468 \cdot 10^{-84}:\\ \;\;\;\;b \cdot c + \left(-27 \cdot \left(j \cdot k\right) + t_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 19
Error	16.8
Cost	1224

\[\begin{array}{l} t_1 := -27 \cdot \left(j \cdot k\right)\\ t_2 := b \cdot c + \left(t_1 + -4 \cdot \left(x \cdot i\right)\right)\\ \mathbf{if}\;x \leq -1.3223608542122258 \cdot 10^{-101}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 5.05372033707518 \cdot 10^{-139}:\\ \;\;\;\;b \cdot c + \left(t_1 + -4 \cdot \left(t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 20
Error	45.6
Cost	1112

\[\begin{array}{l} t_1 := -4 \cdot \left(x \cdot i\right)\\ \mathbf{if}\;c \leq -7.030207950188279 \cdot 10^{-177}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;c \leq 6.589019238327619 \cdot 10^{-188}:\\ \;\;\;\;k \cdot \left(j \cdot -27\right)\\ \mathbf{elif}\;c \leq 1.3826079386658574 \cdot 10^{-44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 5.9 \cdot 10^{+53}:\\ \;\;\;\;j \cdot \left(k \cdot -27\right)\\ \mathbf{elif}\;c \leq 2.7 \cdot 10^{+82}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;c \leq 8.1 \cdot 10^{+158}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;b \cdot c\\ \end{array} \]

Alternative 21
Error	33.0
Cost	840

\[\begin{array}{l} t_1 := -4 \cdot \left(x \cdot i\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{+167}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+91}:\\ \;\;\;\;b \cdot c + j \cdot \left(k \cdot -27\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 22
Error	42.9
Cost	584

\[\begin{array}{l} t_1 := k \cdot \left(j \cdot -27\right)\\ \mathbf{if}\;j \leq -10216137.05408264:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq 3.0188512521784534 \cdot 10^{-66}:\\ \;\;\;\;b \cdot c\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 23
Error	42.8
Cost	584

\[\begin{array}{l} t_1 := j \cdot \left(k \cdot -27\right)\\ \mathbf{if}\;j \leq -10216137.05408264:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq 3.0188512521784534 \cdot 10^{-66}:\\ \;\;\;\;b \cdot c\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 24
Error	48.3
Cost	192

\[b \cdot c \]

Reproduce

herbie shell --seed 2022308 
(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)))