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

Average Error: 5.7 → 1.5

Time: 48.5s

Precision: binary64

Cost: 6088

\[ \begin{array}{c}[j, k] = \mathsf{sort}([j, k])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_1 := \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+302}:\\ \;\;\;\;\left(\left(18 \cdot \left(y \cdot \left(\left(z \cdot x\right) \cdot t\right)\right) + b \cdot c\right) - x \cdot \left(i \cdot 4\right)\right) - j \cdot \left(k \cdot 27\right)\\ \mathbf{elif}\;t_1 \leq 10^{+308}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot \left(18 \cdot \left(z \cdot \left(x \cdot t\right)\right)\right) + b \cdot c\right) - x \cdot \left(4 \cdot i\right)\right) - j \cdot \left(27 \cdot k\right)\\ \end{array} \]

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) (* (* a 4.0) t)) (* b c))
           (* (* x 4.0) i))
          (* (* j 27.0) k))))
   (if (<= t_1 -2e+302)
     (-
      (- (+ (* 18.0 (* y (* (* z x) t))) (* b c)) (* x (* i 4.0)))
      (* j (* k 27.0)))
     (if (<= t_1 1e+308)
       t_1
       (-
        (- (+ (* y (* 18.0 (* z (* x t)))) (* b c)) (* x (* 4.0 i)))
        (* j (* 27.0 k)))))))

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) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k);
	double tmp;
	if (t_1 <= -2e+302) {
		tmp = (((18.0 * (y * ((z * x) * t))) + (b * c)) - (x * (i * 4.0))) - (j * (k * 27.0));
	} else if (t_1 <= 1e+308) {
		tmp = t_1;
	} else {
		tmp = (((y * (18.0 * (z * (x * t)))) + (b * c)) - (x * (4.0 * i))) - (j * (27.0 * k));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j, k)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    code = (((((((x * 18.0d0) * y) * z) * t) - ((a * 4.0d0) * t)) + (b * c)) - ((x * 4.0d0) * i)) - ((j * 27.0d0) * k)
end function

↓

real(8) function code(x, y, z, t, a, b, c, i, j, k)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (((((((x * 18.0d0) * y) * z) * t) - ((a * 4.0d0) * t)) + (b * c)) - ((x * 4.0d0) * i)) - ((j * 27.0d0) * k)
    if (t_1 <= (-2d+302)) then
        tmp = (((18.0d0 * (y * ((z * x) * t))) + (b * c)) - (x * (i * 4.0d0))) - (j * (k * 27.0d0))
    else if (t_1 <= 1d+308) then
        tmp = t_1
    else
        tmp = (((y * (18.0d0 * (z * (x * t)))) + (b * c)) - (x * (4.0d0 * i))) - (j * (27.0d0 * k))
    end if
    code = tmp
end function

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) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k);
	double tmp;
	if (t_1 <= -2e+302) {
		tmp = (((18.0 * (y * ((z * x) * t))) + (b * c)) - (x * (i * 4.0))) - (j * (k * 27.0));
	} else if (t_1 <= 1e+308) {
		tmp = t_1;
	} else {
		tmp = (((y * (18.0 * (z * (x * t)))) + (b * c)) - (x * (4.0 * i))) - (j * (27.0 * k));
	}
	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) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k)
	tmp = 0
	if t_1 <= -2e+302:
		tmp = (((18.0 * (y * ((z * x) * t))) + (b * c)) - (x * (i * 4.0))) - (j * (k * 27.0))
	elif t_1 <= 1e+308:
		tmp = t_1
	else:
		tmp = (((y * (18.0 * (z * (x * t)))) + (b * c)) - (x * (4.0 * i))) - (j * (27.0 * k))
	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(Float64(a * 4.0) * t)) + Float64(b * c)) - Float64(Float64(x * 4.0) * i)) - Float64(Float64(j * 27.0) * k))
	tmp = 0.0
	if (t_1 <= -2e+302)
		tmp = Float64(Float64(Float64(Float64(18.0 * Float64(y * Float64(Float64(z * x) * t))) + Float64(b * c)) - Float64(x * Float64(i * 4.0))) - Float64(j * Float64(k * 27.0)));
	elseif (t_1 <= 1e+308)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(Float64(y * Float64(18.0 * Float64(z * Float64(x * t)))) + Float64(b * c)) - Float64(x * Float64(4.0 * i))) - Float64(j * Float64(27.0 * k)));
	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) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k);
	tmp = 0.0;
	if (t_1 <= -2e+302)
		tmp = (((18.0 * (y * ((z * x) * t))) + (b * c)) - (x * (i * 4.0))) - (j * (k * 27.0));
	elseif (t_1 <= 1e+308)
		tmp = t_1;
	else
		tmp = (((y * (18.0 * (z * (x * t)))) + (b * c)) - (x * (4.0 * i))) - (j * (27.0 * k));
	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[(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]}, If[LessEqual[t$95$1, -2e+302], N[(N[(N[(N[(18.0 * N[(y * N[(N[(z * x), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision] - N[(x * N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(j * N[(k * 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+308], t$95$1, N[(N[(N[(N[(y * N[(18.0 * N[(z * N[(x * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision] - N[(x * N[(4.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(j * N[(27.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	5.7
Target	1.8
Herbie	1.5

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

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)) < -2.0000000000000002e302

   1. Initial program 50.8

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

   2. Simplified 32.8

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

      [Start] 50.8	\[ \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
      rational_best-simplify-2 [=>] 50.8	\[ \left(\left(\left(\color{blue}{t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right)} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
      rational_best-simplify-2 [=>] 50.8	\[ \left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \color{blue}{t \cdot \left(a \cdot 4\right)}\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
      rational_best-simplify-50 [=>] 50.8	\[ \left(\left(\color{blue}{t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right)} + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
      rational_best-simplify-2 [=>] 50.8	\[ \left(\left(t \cdot \left(\color{blue}{z \cdot \left(\left(x \cdot 18\right) \cdot y\right)} - a \cdot 4\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
      rational_best-simplify-44 [=>] 33.1	\[ \left(\left(t \cdot \left(\color{blue}{\left(x \cdot 18\right) \cdot \left(z \cdot y\right)} - a \cdot 4\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
      rational_best-simplify-2 [=>] 33.1	\[ \left(\left(t \cdot \left(\left(x \cdot 18\right) \cdot \left(z \cdot y\right) - a \cdot 4\right) + b \cdot c\right) - \color{blue}{i \cdot \left(x \cdot 4\right)}\right) - \left(j \cdot 27\right) \cdot k \]
      rational_best-simplify-44 [=>] 33.1	\[ \left(\left(t \cdot \left(\left(x \cdot 18\right) \cdot \left(z \cdot y\right) - a \cdot 4\right) + b \cdot c\right) - \color{blue}{x \cdot \left(i \cdot 4\right)}\right) - \left(j \cdot 27\right) \cdot k \]
      rational_best-simplify-2 [=>] 33.1	\[ \left(\left(t \cdot \left(\left(x \cdot 18\right) \cdot \left(z \cdot y\right) - a \cdot 4\right) + b \cdot c\right) - x \cdot \left(i \cdot 4\right)\right) - \color{blue}{k \cdot \left(j \cdot 27\right)} \]
      rational_best-simplify-44 [=>] 32.8	\[ \left(\left(t \cdot \left(\left(x \cdot 18\right) \cdot \left(z \cdot y\right) - a \cdot 4\right) + b \cdot c\right) - x \cdot \left(i \cdot 4\right)\right) - \color{blue}{j \cdot \left(k \cdot 27\right)} \]

   3. Taylor expanded in x around inf 16.8

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

   4. Simplified 16.8

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

      [Start] 16.8	\[ \left(\left(18 \cdot \left(y \cdot \left(t \cdot \left(z \cdot x\right)\right)\right) + b \cdot c\right) - x \cdot \left(i \cdot 4\right)\right) - j \cdot \left(k \cdot 27\right) \]
      rational_best-simplify-2 [=>] 16.8	\[ \left(\left(18 \cdot \left(y \cdot \color{blue}{\left(\left(z \cdot x\right) \cdot t\right)}\right) + b \cdot c\right) - x \cdot \left(i \cdot 4\right)\right) - j \cdot \left(k \cdot 27\right) \]

   if -2.0000000000000002e302 < (-.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 62.9

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

   2. Simplified 38.2

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

      [Start] 62.9	\[ \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
      rational_best-simplify-2 [=>] 62.9	\[ \left(\left(\left(\color{blue}{t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right)} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
      rational_best-simplify-2 [=>] 62.9	\[ \left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \color{blue}{t \cdot \left(a \cdot 4\right)}\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
      rational_best-simplify-50 [=>] 62.9	\[ \left(\left(\color{blue}{t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right)} + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
      rational_best-simplify-2 [=>] 62.9	\[ \left(\left(t \cdot \left(\color{blue}{z \cdot \left(\left(x \cdot 18\right) \cdot y\right)} - a \cdot 4\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
      rational_best-simplify-2 [=>] 62.9	\[ \left(\left(t \cdot \left(z \cdot \color{blue}{\left(y \cdot \left(x \cdot 18\right)\right)} - a \cdot 4\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
      rational_best-simplify-44 [=>] 39.9	\[ \left(\left(t \cdot \left(\color{blue}{y \cdot \left(z \cdot \left(x \cdot 18\right)\right)} - a \cdot 4\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
      rational_best-simplify-2 [=>] 39.9	\[ \left(\left(t \cdot \left(y \cdot \left(z \cdot \color{blue}{\left(18 \cdot x\right)}\right) - a \cdot 4\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
      rational_best-simplify-44 [=>] 39.7	\[ \left(\left(t \cdot \left(y \cdot \color{blue}{\left(18 \cdot \left(z \cdot x\right)\right)} - a \cdot 4\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
      rational_best-simplify-2 [=>] 39.7	\[ \left(\left(t \cdot \left(y \cdot \left(18 \cdot \color{blue}{\left(x \cdot z\right)}\right) - a \cdot 4\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
      rational_best-simplify-2 [=>] 39.7	\[ \left(\left(t \cdot \left(y \cdot \left(18 \cdot \left(x \cdot z\right)\right) - a \cdot 4\right) + b \cdot c\right) - \color{blue}{i \cdot \left(x \cdot 4\right)}\right) - \left(j \cdot 27\right) \cdot k \]
      rational_best-simplify-44 [=>] 39.7	\[ \left(\left(t \cdot \left(y \cdot \left(18 \cdot \left(x \cdot z\right)\right) - a \cdot 4\right) + b \cdot c\right) - \color{blue}{x \cdot \left(i \cdot 4\right)}\right) - \left(j \cdot 27\right) \cdot k \]
      rational_best-simplify-2 [=>] 39.7	\[ \left(\left(t \cdot \left(y \cdot \left(18 \cdot \left(x \cdot z\right)\right) - a \cdot 4\right) + b \cdot c\right) - x \cdot \color{blue}{\left(4 \cdot i\right)}\right) - \left(j \cdot 27\right) \cdot k \]
      rational_best-simplify-2 [=>] 39.7	\[ \left(\left(t \cdot \left(y \cdot \left(18 \cdot \left(x \cdot z\right)\right) - a \cdot 4\right) + b \cdot c\right) - x \cdot \left(4 \cdot i\right)\right) - \color{blue}{k \cdot \left(j \cdot 27\right)} \]

   3. Taylor expanded in y around inf 13.6

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

   4. Simplified 7.8

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

      [Start] 13.6	\[ \left(\left(18 \cdot \left(y \cdot \left(t \cdot \left(z \cdot x\right)\right)\right) + b \cdot c\right) - x \cdot \left(4 \cdot i\right)\right) - j \cdot \left(27 \cdot k\right) \]
      rational_best-simplify-44 [=>] 13.5	\[ \left(\left(\color{blue}{y \cdot \left(18 \cdot \left(t \cdot \left(z \cdot x\right)\right)\right)} + b \cdot c\right) - x \cdot \left(4 \cdot i\right)\right) - j \cdot \left(27 \cdot k\right) \]
      rational_best-simplify-44 [=>] 7.8	\[ \left(\left(y \cdot \left(18 \cdot \color{blue}{\left(z \cdot \left(t \cdot x\right)\right)}\right) + b \cdot c\right) - x \cdot \left(4 \cdot i\right)\right) - j \cdot \left(27 \cdot k\right) \]
      rational_best-simplify-2 [=>] 7.8	\[ \left(\left(y \cdot \left(18 \cdot \left(z \cdot \color{blue}{\left(x \cdot t\right)}\right)\right) + b \cdot c\right) - x \cdot \left(4 \cdot i\right)\right) - j \cdot \left(27 \cdot k\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \leq -2 \cdot 10^{+302}:\\ \;\;\;\;\left(\left(18 \cdot \left(y \cdot \left(\left(z \cdot x\right) \cdot t\right)\right) + b \cdot c\right) - x \cdot \left(i \cdot 4\right)\right) - j \cdot \left(k \cdot 27\right)\\ \mathbf{elif}\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \leq 10^{+308}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot \left(18 \cdot \left(z \cdot \left(x \cdot t\right)\right)\right) + b \cdot c\right) - x \cdot \left(4 \cdot i\right)\right) - j \cdot \left(27 \cdot k\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	4.1
Cost	2120

\[\begin{array}{l} t_1 := \left(\left(t \cdot \left(a \cdot -4 - z \cdot \left(y \cdot \left(x \cdot -18\right)\right)\right) + b \cdot c\right) - x \cdot \left(i \cdot 4\right)\right) - j \cdot \left(k \cdot 27\right)\\ \mathbf{if}\;t \leq -4 \cdot 10^{-167}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-169}:\\ \;\;\;\;\left(\left(y \cdot \left(18 \cdot \left(z \cdot \left(x \cdot t\right)\right)\right) + b \cdot c\right) - x \cdot \left(4 \cdot i\right)\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	4.5
Cost	2120

\[\begin{array}{l} t_1 := x \cdot \left(i \cdot 4\right)\\ \mathbf{if}\;t \leq -2.15 \cdot 10^{-94}:\\ \;\;\;\;\left(\left(t \cdot \left(\left(x \cdot 18\right) \cdot \left(z \cdot y\right) - a \cdot 4\right) + b \cdot c\right) - t_1\right) - 27 \cdot \left(k \cdot j\right)\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-169}:\\ \;\;\;\;\left(\left(y \cdot \left(18 \cdot \left(z \cdot \left(x \cdot t\right)\right)\right) + b \cdot c\right) - x \cdot \left(4 \cdot i\right)\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t \cdot \left(a \cdot -4 - z \cdot \left(y \cdot \left(x \cdot -18\right)\right)\right) + b \cdot c\right) - t_1\right) - j \cdot \left(k \cdot 27\right)\\ \end{array} \]

Alternative 3
Error	22.9
Cost	2012

\[\begin{array}{l} t_1 := -4 \cdot \left(a \cdot t\right)\\ t_2 := j \cdot \left(k \cdot 27\right)\\ t_3 := \left(t_1 - i \cdot \left(4 \cdot x\right)\right) - t_2\\ t_4 := x \cdot \left(4 \cdot i\right)\\ t_5 := \left(c \cdot b + t_1\right) - t_2\\ t_6 := \left(c \cdot b + \left(18 \cdot \left(y \cdot z\right)\right) \cdot \left(t \cdot x\right)\right) - t_4\\ \mathbf{if}\;k \leq -6.7 \cdot 10^{-140}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;k \leq 3.15 \cdot 10^{-176}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;k \leq 2.5 \cdot 10^{-78}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq 1.1 \cdot 10^{-10}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;k \leq 3.4 \cdot 10^{+76}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;k \leq 1.35 \cdot 10^{+97}:\\ \;\;\;\;c \cdot b - \left(k \cdot \left(27 \cdot j\right) + t_4\right)\\ \mathbf{elif}\;k \leq 4.4 \cdot 10^{+108}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_5\\ \end{array} \]

Alternative 4
Error	9.7
Cost	1996

\[\begin{array}{l} t_1 := x \cdot \left(4 \cdot i\right)\\ t_2 := j \cdot \left(27 \cdot k\right)\\ t_3 := j \cdot \left(k \cdot 27\right)\\ \mathbf{if}\;z \leq 7.2 \cdot 10^{+71}:\\ \;\;\;\;\left(\left(b \cdot c + -4 \cdot \left(a \cdot t + x \cdot i\right)\right) + t_3\right) + j \cdot \left(k \cdot -54\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+141}:\\ \;\;\;\;\left(\left(18 \cdot \left(\left(y \cdot x\right) \cdot \left(z \cdot t\right)\right) + b \cdot c\right) - t_1\right) - t_2\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+241}:\\ \;\;\;\;\left(\left(a \cdot \left(-4 \cdot t\right) + b \cdot c\right) - x \cdot \left(i \cdot 4\right)\right) - t_3\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot \left(18 \cdot \left(z \cdot \left(x \cdot t\right)\right)\right) + b \cdot c\right) - t_1\right) - t_2\\ \end{array} \]

Alternative 5
Error	9.6
Cost	1996

\[\begin{array}{l} t_1 := j \cdot \left(k \cdot 27\right)\\ t_2 := x \cdot \left(4 \cdot i\right)\\ t_3 := j \cdot \left(27 \cdot k\right)\\ \mathbf{if}\;z \leq 7 \cdot 10^{+71}:\\ \;\;\;\;\left(\left(c \cdot b + a \cdot \left(t \cdot -8\right)\right) + 4 \cdot \left(a \cdot t + i \cdot \left(-x\right)\right)\right) - t_1\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+144}:\\ \;\;\;\;\left(\left(18 \cdot \left(\left(y \cdot x\right) \cdot \left(z \cdot t\right)\right) + b \cdot c\right) - t_2\right) - t_3\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+241}:\\ \;\;\;\;\left(\left(a \cdot \left(-4 \cdot t\right) + b \cdot c\right) - x \cdot \left(i \cdot 4\right)\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot \left(18 \cdot \left(z \cdot \left(x \cdot t\right)\right)\right) + b \cdot c\right) - t_2\right) - t_3\\ \end{array} \]

Alternative 6
Error	31.4
Cost	1896

\[\begin{array}{l} t_1 := -4 \cdot \left(a \cdot t\right)\\ t_2 := c \cdot b - k \cdot \left(27 \cdot j\right)\\ t_3 := j \cdot \left(k \cdot 27\right)\\ t_4 := i \cdot \left(x \cdot -4\right) - t_3\\ t_5 := t_1 - t_3\\ \mathbf{if}\;c \leq -7 \cdot 10^{-111}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq -9.8 \cdot 10^{-301}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;c \leq 5.7 \cdot 10^{-179}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;c \leq 6.1 \cdot 10^{-165}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;c \leq 4.5 \cdot 10^{-124}:\\ \;\;\;\;x \cdot \left(y \cdot \left(z \cdot \left(18 \cdot t\right)\right) - 4 \cdot i\right)\\ \mathbf{elif}\;c \leq 1.14 \cdot 10^{-100}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;c \leq 9 \cdot 10^{-54}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;c \leq 4.9 \cdot 10^{+17}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;c \leq 9 \cdot 10^{+216}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 5.5 \cdot 10^{+272}:\\ \;\;\;\;c \cdot b + t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	8.8
Cost	1864

\[\begin{array}{l} t_1 := x \cdot \left(i \cdot 4\right)\\ t_2 := j \cdot \left(k \cdot 27\right)\\ t_3 := \left(\left(a \cdot \left(-4 \cdot t\right) + b \cdot c\right) - t_1\right) - t_2\\ \mathbf{if}\;a \leq -1.6 \cdot 10^{-123}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{+30}:\\ \;\;\;\;\left(\left(18 \cdot \left(y \cdot \left(x \cdot \left(z \cdot t\right)\right)\right) + b \cdot c\right) - t_1\right) - t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 8
Error	9.2
Cost	1864

\[\begin{array}{l} t_1 := \left(\left(a \cdot \left(-4 \cdot t\right) + b \cdot c\right) - x \cdot \left(i \cdot 4\right)\right) - j \cdot \left(k \cdot 27\right)\\ \mathbf{if}\;a \leq -3 \cdot 10^{-138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 3400000000000:\\ \;\;\;\;\left(\left(18 \cdot \left(\left(y \cdot x\right) \cdot \left(z \cdot t\right)\right) + b \cdot c\right) - x \cdot \left(4 \cdot i\right)\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	9.0
Cost	1736

\[\begin{array}{l} t_1 := \left(\left(a \cdot \left(-4 \cdot t\right) + b \cdot c\right) - x \cdot \left(i \cdot 4\right)\right) - j \cdot \left(k \cdot 27\right)\\ \mathbf{if}\;a \leq -1.2 \cdot 10^{-113}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 3.2 \cdot 10^{-176}:\\ \;\;\;\;\left(b \cdot c + x \cdot \left(t \cdot \left(z \cdot \left(18 \cdot y\right)\right) + i \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	45.4
Cost	1640

\[\begin{array}{l} t_1 := -27 \cdot \left(k \cdot j\right)\\ t_2 := -4 \cdot \left(i \cdot x\right)\\ t_3 := -4 \cdot \left(a \cdot t\right)\\ \mathbf{if}\;a \leq -3.55 \cdot 10^{+80}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -3.2 \cdot 10^{-113}:\\ \;\;\;\;c \cdot b\\ \mathbf{elif}\;a \leq -4.6 \cdot 10^{-126}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -5.2 \cdot 10^{-287}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{-257}:\\ \;\;\;\;c \cdot b\\ \mathbf{elif}\;a \leq 7.8 \cdot 10^{-218}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 3.4 \cdot 10^{-164}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 105000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.75 \cdot 10^{+69}:\\ \;\;\;\;c \cdot b\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{+174}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 11
Error	45.4
Cost	1640

\[\begin{array}{l} t_1 := j \cdot \left(k \cdot -27\right)\\ t_2 := -4 \cdot \left(i \cdot x\right)\\ t_3 := -4 \cdot \left(a \cdot t\right)\\ \mathbf{if}\;a \leq -1.3 \cdot 10^{+80}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -2.25 \cdot 10^{-113}:\\ \;\;\;\;c \cdot b\\ \mathbf{elif}\;a \leq -3.7 \cdot 10^{-126}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -8 \cdot 10^{-290}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2 \cdot 10^{-259}:\\ \;\;\;\;c \cdot b\\ \mathbf{elif}\;a \leq 3.2 \cdot 10^{-218}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.25 \cdot 10^{-164}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 51000000:\\ \;\;\;\;-27 \cdot \left(k \cdot j\right)\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{+68}:\\ \;\;\;\;c \cdot b\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{+174}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 12
Error	45.6
Cost	1640

\[\begin{array}{l} t_1 := j \cdot \left(k \cdot -27\right)\\ t_2 := -4 \cdot \left(i \cdot x\right)\\ t_3 := -4 \cdot \left(a \cdot t\right)\\ \mathbf{if}\;a \leq -1 \cdot 10^{+79}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -5 \cdot 10^{-112}:\\ \;\;\;\;c \cdot b\\ \mathbf{elif}\;a \leq -8 \cdot 10^{-126}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -6 \cdot 10^{-291}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 9 \cdot 10^{-259}:\\ \;\;\;\;c \cdot b\\ \mathbf{elif}\;a \leq 1.5 \cdot 10^{-218}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{-165}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 0.038:\\ \;\;\;\;k \cdot \left(j \cdot -27\right)\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{+65}:\\ \;\;\;\;c \cdot b\\ \mathbf{elif}\;a \leq 2.9 \cdot 10^{+174}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 13
Error	31.0
Cost	1632

\[\begin{array}{l} t_1 := -4 \cdot \left(a \cdot t\right)\\ t_2 := c \cdot b - k \cdot \left(27 \cdot j\right)\\ t_3 := j \cdot \left(k \cdot 27\right)\\ t_4 := i \cdot \left(x \cdot -4\right) - t_3\\ t_5 := t_1 - t_3\\ \mathbf{if}\;c \leq -1.05 \cdot 10^{-110}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq -2.8 \cdot 10^{-301}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;c \leq 1.16 \cdot 10^{-182}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;c \leq 1.5 \cdot 10^{-165}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;c \leq 9 \cdot 10^{-54}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;c \leq 1.15 \cdot 10^{+19}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;c \leq 3.5 \cdot 10^{+217}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 3.2 \cdot 10^{+272}:\\ \;\;\;\;c \cdot b + t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 14
Error	19.0
Cost	1616

\[\begin{array}{l} t_1 := c \cdot b - \left(k \cdot \left(27 \cdot j\right) + x \cdot \left(4 \cdot i\right)\right)\\ t_2 := \left(-4 \cdot \left(a \cdot t\right) - i \cdot \left(4 \cdot x\right)\right) - j \cdot \left(k \cdot 27\right)\\ \mathbf{if}\;c \leq -9.5 \cdot 10^{-111}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 6.2 \cdot 10^{-67}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 1.4 \cdot 10^{-58}:\\ \;\;\;\;\left(c \cdot b + 18 \cdot \left(y \cdot \left(t \cdot \left(z \cdot x\right)\right)\right)\right) - 27 \cdot \left(k \cdot j\right)\\ \mathbf{elif}\;c \leq 5.2 \cdot 10^{+120}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	32.2
Cost	1500

\[\begin{array}{l} t_1 := c \cdot b - k \cdot \left(27 \cdot j\right)\\ t_2 := -4 \cdot \left(a \cdot t\right)\\ t_3 := -4 \cdot \left(i \cdot x\right)\\ t_4 := c \cdot b + t_2\\ \mathbf{if}\;t \leq -1 \cdot 10^{-66}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq -1.8 \cdot 10^{-117}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -1.65 \cdot 10^{-268}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -7.5 \cdot 10^{-292}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-37}:\\ \;\;\;\;t_2 - j \cdot \left(k \cdot 27\right)\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 16
Error	22.1
Cost	1488

\[\begin{array}{l} t_1 := c \cdot b - \left(k \cdot \left(27 \cdot j\right) + x \cdot \left(4 \cdot i\right)\right)\\ t_2 := -4 \cdot \left(a \cdot t\right)\\ t_3 := t_2 - j \cdot \left(k \cdot 27\right)\\ \mathbf{if}\;a \leq -1.5 \cdot 10^{+161}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -420000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -2.6 \cdot 10^{-86}:\\ \;\;\;\;c \cdot b + t_2\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{+175}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 17
Error	34.9
Cost	1368

\[\begin{array}{l} t_1 := c \cdot b + -4 \cdot \left(a \cdot t\right)\\ t_2 := k \cdot \left(j \cdot -27\right)\\ \mathbf{if}\;j \leq -2.05 \cdot 10^{+251}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \leq -7 \cdot 10^{+201}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq -2.8 \cdot 10^{+155}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \leq -6.3 \cdot 10^{+15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq -4 \cdot 10^{-10}:\\ \;\;\;\;-4 \cdot \left(i \cdot x\right)\\ \mathbf{elif}\;j \leq 1.45 \cdot 10^{-50}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-27 \cdot \left(k \cdot j\right)\\ \end{array} \]

Alternative 18
Error	17.6
Cost	1352

\[\begin{array}{l} t_1 := -4 \cdot \left(a \cdot t\right)\\ t_2 := j \cdot \left(k \cdot 27\right)\\ \mathbf{if}\;a \leq -1.75 \cdot 10^{-113}:\\ \;\;\;\;\left(c \cdot b + t_1\right) - t_2\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{+73}:\\ \;\;\;\;c \cdot b - \left(k \cdot \left(27 \cdot j\right) + x \cdot \left(4 \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 - i \cdot \left(4 \cdot x\right)\right) - t_2\\ \end{array} \]

Alternative 19
Error	9.8
Cost	1344

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

Alternative 20
Error	45.0
Cost	1244

\[\begin{array}{l} t_1 := -4 \cdot \left(a \cdot t\right)\\ t_2 := -27 \cdot \left(k \cdot j\right)\\ \mathbf{if}\;a \leq -4.5 \cdot 10^{+79}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -2 \cdot 10^{-24}:\\ \;\;\;\;c \cdot b\\ \mathbf{elif}\;a \leq -1.35 \cdot 10^{-87}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -6.2 \cdot 10^{-287}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 8.5 \cdot 10^{-79}:\\ \;\;\;\;c \cdot b\\ \mathbf{elif}\;a \leq 720000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{+74}:\\ \;\;\;\;c \cdot b\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 21
Error	32.0
Cost	1236

\[\begin{array}{l} t_1 := c \cdot b - k \cdot \left(27 \cdot j\right)\\ t_2 := -4 \cdot \left(i \cdot x\right)\\ t_3 := c \cdot b + -4 \cdot \left(a \cdot t\right)\\ \mathbf{if}\;t \leq -5.8 \cdot 10^{-67}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -2.2 \cdot 10^{-117}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -2.4 \cdot 10^{-268}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -7.3 \cdot 10^{-292}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 22
Error	18.1
Cost	1224

\[\begin{array}{l} t_1 := \left(c \cdot b + -4 \cdot \left(a \cdot t\right)\right) - j \cdot \left(k \cdot 27\right)\\ \mathbf{if}\;a \leq -1.7 \cdot 10^{-113}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.9 \cdot 10^{+174}:\\ \;\;\;\;c \cdot b - \left(k \cdot \left(27 \cdot j\right) + x \cdot \left(4 \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 23
Error	43.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;c \leq -3.2 \cdot 10^{-79}:\\ \;\;\;\;c \cdot b\\ \mathbf{elif}\;c \leq 1.8 \cdot 10^{+122}:\\ \;\;\;\;-27 \cdot \left(k \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot b\\ \end{array} \]

Alternative 24
Error	48.3
Cost	192

\[c \cdot b \]

Reproduce

herbie shell --seed 2023096 
(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)))