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

Average Error: 6.1 → 2.3

Time: 41.7s

Precision: binary64

Cost: 2249

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+30} \lor \neg \left(y \leq 10^{-202}\right):\\ \;\;\;\;\left(\left(\left(\left(z \cdot \left(x \cdot t\right)\right) \cdot \left(18 \cdot y\right) + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right) + a \cdot -4\right) + b \cdot c\right) + \left(x \cdot \left(i \cdot -4\right) + j \cdot \left(k \cdot -27\right)\right)\\ \end{array} \]

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
 (if (or (<= y -2e+30) (not (<= y 1e-202)))
   (+
    (+
     (+ (+ (* (* z (* x t)) (* 18.0 y)) (* t (* a -4.0))) (* b c))
     (* i (* x -4.0)))
    (* k (* j -27.0)))
   (+
    (+ (* t (+ (* (* x 18.0) (* y z)) (* a -4.0))) (* b c))
    (+ (* x (* i -4.0)) (* j (* k -27.0))))))

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 tmp;
	if ((y <= -2e+30) || !(y <= 1e-202)) {
		tmp = (((((z * (x * t)) * (18.0 * y)) + (t * (a * -4.0))) + (b * c)) + (i * (x * -4.0))) + (k * (j * -27.0));
	} else {
		tmp = ((t * (((x * 18.0) * (y * z)) + (a * -4.0))) + (b * c)) + ((x * (i * -4.0)) + (j * (k * -27.0)));
	}
	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) :: tmp
    if ((y <= (-2d+30)) .or. (.not. (y <= 1d-202))) then
        tmp = (((((z * (x * t)) * (18.0d0 * y)) + (t * (a * (-4.0d0)))) + (b * c)) + (i * (x * (-4.0d0)))) + (k * (j * (-27.0d0)))
    else
        tmp = ((t * (((x * 18.0d0) * (y * z)) + (a * (-4.0d0)))) + (b * c)) + ((x * (i * (-4.0d0))) + (j * (k * (-27.0d0))))
    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 tmp;
	if ((y <= -2e+30) || !(y <= 1e-202)) {
		tmp = (((((z * (x * t)) * (18.0 * y)) + (t * (a * -4.0))) + (b * c)) + (i * (x * -4.0))) + (k * (j * -27.0));
	} else {
		tmp = ((t * (((x * 18.0) * (y * z)) + (a * -4.0))) + (b * c)) + ((x * (i * -4.0)) + (j * (k * -27.0)));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t, a, b, c, i, j, k):
	tmp = 0
	if (y <= -2e+30) or not (y <= 1e-202):
		tmp = (((((z * (x * t)) * (18.0 * y)) + (t * (a * -4.0))) + (b * c)) + (i * (x * -4.0))) + (k * (j * -27.0))
	else:
		tmp = ((t * (((x * 18.0) * (y * z)) + (a * -4.0))) + (b * c)) + ((x * (i * -4.0)) + (j * (k * -27.0)))
	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)
	tmp = 0.0
	if ((y <= -2e+30) || !(y <= 1e-202))
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(z * Float64(x * t)) * Float64(18.0 * y)) + Float64(t * Float64(a * -4.0))) + Float64(b * c)) + Float64(i * Float64(x * -4.0))) + Float64(k * Float64(j * -27.0)));
	else
		tmp = Float64(Float64(Float64(t * Float64(Float64(Float64(x * 18.0) * Float64(y * z)) + Float64(a * -4.0))) + Float64(b * c)) + Float64(Float64(x * Float64(i * -4.0)) + Float64(j * Float64(k * -27.0))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z, t, a, b, c, i, j, k)
	tmp = 0.0;
	if ((y <= -2e+30) || ~((y <= 1e-202)))
		tmp = (((((z * (x * t)) * (18.0 * y)) + (t * (a * -4.0))) + (b * c)) + (i * (x * -4.0))) + (k * (j * -27.0));
	else
		tmp = ((t * (((x * 18.0) * (y * z)) + (a * -4.0))) + (b * c)) + ((x * (i * -4.0)) + (j * (k * -27.0)));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := If[Or[LessEqual[y, -2e+30], N[Not[LessEqual[y, 1e-202]], $MachinePrecision]], N[(N[(N[(N[(N[(N[(z * N[(x * t), $MachinePrecision]), $MachinePrecision] * N[(18.0 * y), $MachinePrecision]), $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], N[(N[(N[(t * N[(N[(N[(x * 18.0), $MachinePrecision] * N[(y * z), $MachinePrecision]), $MachinePrecision] + N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision] + N[(N[(x * N[(i * -4.0), $MachinePrecision]), $MachinePrecision] + N[(j * N[(k * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	6.1
Target	1.5
Herbie	2.3

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

Derivation

1. Split input into 2 regimes

2. if y < -2e30 or 1e-202 < y

   1. Initial program 10.5

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

   2. Taylor expanded in x around 0 4.0

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

   3. Simplified 3.2

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

      [Start] 4.0	\[ \left(\left(\left(18 \cdot \left(y \cdot \left(t \cdot \left(z \cdot x\right)\right)\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
      associate-*r* [=>] 4.1	\[ \left(\left(\left(\color{blue}{\left(18 \cdot y\right) \cdot \left(t \cdot \left(z \cdot x\right)\right)} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
      *-commutative [=>] 4.1	\[ \left(\left(\left(\color{blue}{\left(t \cdot \left(z \cdot x\right)\right) \cdot \left(18 \cdot y\right)} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
      associate-*r* [=>] 3.7	\[ \left(\left(\left(\color{blue}{\left(\left(t \cdot z\right) \cdot x\right)} \cdot \left(18 \cdot y\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
      *-commutative [<=] 3.7	\[ \left(\left(\left(\left(\color{blue}{\left(z \cdot t\right)} \cdot x\right) \cdot \left(18 \cdot y\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
      associate-*l* [=>] 3.2	\[ \left(\left(\left(\color{blue}{\left(z \cdot \left(t \cdot x\right)\right)} \cdot \left(18 \cdot y\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]

   if -2e30 < y < 1e-202

   1. Initial program 1.2

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

   2. Simplified 1.2

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

      [Start] 1.2	\[ \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
      associate--l- [=>] 1.2	\[ \color{blue}{\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)} \]
      associate-+l- [=>] 1.2	\[ \color{blue}{\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(\left(a \cdot 4\right) \cdot t - b \cdot c\right)\right)} - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right) \]
      associate-+l- [<=] 1.2	\[ \color{blue}{\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right)} - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right) \]
      distribute-rgt-out-- [=>] 1.2	\[ \left(\color{blue}{t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right)} + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right) \]
      associate-*l* [=>] 1.2	\[ \left(t \cdot \left(\color{blue}{\left(x \cdot 18\right) \cdot \left(y \cdot z\right)} - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right) \]
      associate-*l* [=>] 1.2	\[ \left(t \cdot \left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right) - a \cdot 4\right) + b \cdot c\right) - \left(\color{blue}{x \cdot \left(4 \cdot i\right)} + \left(j \cdot 27\right) \cdot k\right) \]
      associate-*l* [=>] 1.2	\[ \left(t \cdot \left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right) - a \cdot 4\right) + b \cdot c\right) - \left(x \cdot \left(4 \cdot i\right) + \color{blue}{j \cdot \left(27 \cdot k\right)}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+30} \lor \neg \left(y \leq 10^{-202}\right):\\ \;\;\;\;\left(\left(\left(\left(z \cdot \left(x \cdot t\right)\right) \cdot \left(18 \cdot y\right) + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right) + a \cdot -4\right) + b \cdot c\right) + \left(x \cdot \left(i \cdot -4\right) + j \cdot \left(k \cdot -27\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.0
Cost	5321

\[\begin{array}{l} t_1 := t \cdot \left(a \cdot -4\right)\\ t_2 := i \cdot \left(x \cdot -4\right)\\ t_3 := \left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t_1\right) + b \cdot c\right) + t_2\\ t_4 := k \cdot \left(j \cdot -27\right)\\ \mathbf{if}\;t_3 \leq -\infty \lor \neg \left(t_3 \leq 5 \cdot 10^{+304}\right):\\ \;\;\;\;\left(\left(\left(\left(z \cdot \left(x \cdot t\right)\right) \cdot \left(18 \cdot y\right) + t_1\right) + b \cdot c\right) + t_2\right) + t_4\\ \mathbf{else}:\\ \;\;\;\;t_3 + t_4\\ \end{array} \]

Alternative 2
Error	24.0
Cost	3061

\[\begin{array}{l} t_1 := k \cdot \left(j \cdot -27\right)\\ t_2 := b \cdot c + \left(-4 \cdot \left(x \cdot i\right) + -27 \cdot \left(j \cdot k\right)\right)\\ t_3 := -4 \cdot \left(t \cdot a\right)\\ t_4 := t_1 + \left(b \cdot c + t_3\right)\\ t_5 := x \cdot \left(i \cdot -4\right)\\ t_6 := \left(b \cdot c + t \cdot \left(a \cdot -4\right)\right) + t_5\\ t_7 := t_1 + \left(x \cdot \left(\left(y \cdot t\right) \cdot \left(18 \cdot z\right)\right) + t_5\right)\\ \mathbf{if}\;z \leq -8.6 \cdot 10^{-302}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-247}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-215}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-165}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;z \leq 255000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+52}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+110}:\\ \;\;\;\;\left(b \cdot c + t \cdot \left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)\right) + t_5\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+136}:\\ \;\;\;\;t_7\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{+154}:\\ \;\;\;\;\left(b \cdot c + t \cdot \left(\left(18 \cdot y\right) \cdot \left(x \cdot z\right)\right)\right) + t_5\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+168}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{+178}:\\ \;\;\;\;t_1 + \left(b \cdot c + 18 \cdot \left(x \cdot \left(z \cdot \left(y \cdot t\right)\right)\right)\right)\\ \mathbf{elif}\;z \leq 1.26 \cdot 10^{+221} \lor \neg \left(z \leq 3.5 \cdot 10^{+229}\right):\\ \;\;\;\;t_7\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(t_3 + 18 \cdot \left(y \cdot \left(t \cdot \left(x \cdot z\right)\right)\right)\right)\\ \end{array} \]

Alternative 3
Error	37.4
Cost	2812

\[\begin{array}{l} t_1 := k \cdot \left(j \cdot -27\right)\\ t_2 := t_1 + -4 \cdot \left(t \cdot a\right)\\ t_3 := x \cdot \left(i \cdot -4 + 18 \cdot \left(y \cdot \left(z \cdot t\right)\right)\right)\\ t_4 := b \cdot c + x \cdot \left(i \cdot -4\right)\\ t_5 := t \cdot \left(18 \cdot \left(y \cdot \left(x \cdot z\right)\right) + a \cdot -4\right)\\ t_6 := -4 \cdot \left(x \cdot i\right)\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{-250}:\\ \;\;\;\;t_6 + t_1\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{-286}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;z \leq -7.6 \cdot 10^{-303}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-247}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-216}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-167}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 4500000:\\ \;\;\;\;t_6 + -27 \cdot \left(j \cdot k\right)\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{+52}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+72}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+75}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{+117}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+147}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{+206}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{+243}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+307}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(a \cdot -4\right)\\ \end{array} \]

Alternative 4
Error	37.4
Cost	2680

\[\begin{array}{l} t_1 := k \cdot \left(j \cdot -27\right)\\ t_2 := t_1 + -4 \cdot \left(t \cdot a\right)\\ t_3 := b \cdot c + x \cdot \left(i \cdot -4\right)\\ t_4 := -4 \cdot \left(x \cdot i\right)\\ t_5 := x \cdot \left(i \cdot -4 + 18 \cdot \left(y \cdot \left(z \cdot t\right)\right)\right)\\ \mathbf{if}\;z \leq -9.5 \cdot 10^{-245}:\\ \;\;\;\;t_4 + t_1\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{-287}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{-302}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{-247}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{-215}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-168}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 40000:\\ \;\;\;\;t_4 + -27 \cdot \left(j \cdot k\right)\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+53}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{+72}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+75}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 6.9 \cdot 10^{+124}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+133}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 2.75 \cdot 10^{+181}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 10^{+229}:\\ \;\;\;\;18 \cdot \left(\left(x \cdot z\right) \cdot \left(y \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_5\\ \end{array} \]

Alternative 5
Error	25.8
Cost	2668

\[\begin{array}{l} t_1 := k \cdot \left(j \cdot -27\right)\\ t_2 := x \cdot \left(i \cdot -4 + 18 \cdot \left(y \cdot \left(z \cdot t\right)\right)\right)\\ t_3 := t \cdot \left(18 \cdot \left(y \cdot \left(x \cdot z\right)\right) + a \cdot -4\right)\\ t_4 := t_1 + \left(b \cdot c + -4 \cdot \left(t \cdot a\right)\right)\\ t_5 := x \cdot \left(i \cdot -4\right)\\ t_6 := \left(b \cdot c + t \cdot \left(a \cdot -4\right)\right) + t_5\\ t_7 := t_1 + t_2\\ \mathbf{if}\;z \leq -7.5 \cdot 10^{-302}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-245}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;z \leq 1.72 \cdot 10^{-217}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-165}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;z \leq 5500000:\\ \;\;\;\;b \cdot c + \left(-4 \cdot \left(x \cdot i\right) + -27 \cdot \left(j \cdot k\right)\right)\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+52}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{+110}:\\ \;\;\;\;\left(b \cdot c + t \cdot \left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)\right) + t_5\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+141}:\\ \;\;\;\;t_7\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{+196}:\\ \;\;\;\;t_1 + \left(b \cdot c + 18 \cdot \left(x \cdot \left(z \cdot \left(y \cdot t\right)\right)\right)\right)\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+216}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+221}:\\ \;\;\;\;t_7\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+242}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	3.7
Cost	2380

\[\begin{array}{l} t_1 := t \cdot \left(a \cdot -4\right)\\ t_2 := i \cdot \left(x \cdot -4\right)\\ t_3 := k \cdot \left(j \cdot -27\right)\\ t_4 := \left(x \cdot \left(t \cdot \left(18 \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) + \left(x \cdot \left(i \cdot -4\right) + j \cdot \left(k \cdot -27\right)\right)\\ t_5 := 18 \cdot \left(x \cdot z\right)\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{+89}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-95}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(y \cdot t_5\right) + t_1\right) + b \cdot c\right) + t_2\right) + t_3\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+171}:\\ \;\;\;\;\left(\left(\left(t_5 \cdot \left(y \cdot t\right) + t_1\right) + b \cdot c\right) + t_2\right) + t_3\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 7
Error	5.1
Cost	2249

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+106} \lor \neg \left(x \leq 1.4 \cdot 10^{+171}\right):\\ \;\;\;\;\left(x \cdot \left(t \cdot \left(18 \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) + \left(x \cdot \left(i \cdot -4\right) + j \cdot \left(k \cdot -27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(18 \cdot \left(x \cdot z\right)\right) \cdot \left(y \cdot t\right) + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right)\\ \end{array} \]

Alternative 8
Error	45.6
Cost	2168

\[\begin{array}{l} t_1 := t \cdot \left(a \cdot -4\right)\\ t_2 := k \cdot \left(j \cdot -27\right)\\ t_3 := -27 \cdot \left(j \cdot k\right)\\ t_4 := -4 \cdot \left(x \cdot i\right)\\ \mathbf{if}\;a \leq -4.6 \cdot 10^{+261}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;a \leq -2.7 \cdot 10^{+96}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -5.5 \cdot 10^{+61}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -700000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -4.4 \cdot 10^{-98}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;a \leq -1.35 \cdot 10^{-127}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -5.4 \cdot 10^{-219}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;a \leq -2 \cdot 10^{-237}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -1.6 \cdot 10^{-300}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;a \leq 4.85 \cdot 10^{-265}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 3.9 \cdot 10^{-233}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;a \leq 8 \cdot 10^{-146}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 3.8 \cdot 10^{-37}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{+32}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	6.3
Cost	2121

\[\begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{-242} \lor \neg \left(t \leq 1.7 \cdot 10^{-143}\right):\\ \;\;\;\;\left(t \cdot \left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right) + a \cdot -4\right) + b \cdot c\right) + \left(x \cdot \left(i \cdot -4\right) + j \cdot \left(k \cdot -27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot c + \left(-4 \cdot \left(x \cdot i\right) + -27 \cdot \left(j \cdot k\right)\right)\\ \end{array} \]

Alternative 10
Error	24.1
Cost	2018

\[\begin{array}{l} t_1 := b \cdot c + \left(-4 \cdot \left(x \cdot i\right) + -27 \cdot \left(j \cdot k\right)\right)\\ t_2 := t \cdot \left(18 \cdot \left(y \cdot \left(x \cdot z\right)\right) + a \cdot -4\right)\\ \mathbf{if}\;x \leq -8.8 \cdot 10^{+18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-22}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -5.9 \cdot 10^{-235}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-257}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-295} \lor \neg \left(x \leq 1.05 \cdot 10^{-211} \lor \neg \left(x \leq 2.05 \cdot 10^{-123}\right) \land x \leq 2.7 \cdot 10^{-61}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(j \cdot -27\right) + -4 \cdot \left(t \cdot a\right)\\ \end{array} \]

Alternative 11
Error	18.5
Cost	1872

\[\begin{array}{l} t_1 := -4 \cdot \left(t \cdot a\right)\\ t_2 := k \cdot \left(j \cdot -27\right)\\ \mathbf{if}\;x \leq -2.9 \cdot 10^{+30}:\\ \;\;\;\;t_2 + x \cdot \left(i \cdot -4 + 18 \cdot \left(y \cdot \left(z \cdot t\right)\right)\right)\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-175}:\\ \;\;\;\;t_2 + \left(b \cdot c + t_1\right)\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-124}:\\ \;\;\;\;\left(b \cdot c + t \cdot \left(\left(18 \cdot y\right) \cdot \left(x \cdot z\right)\right)\right) + x \cdot \left(i \cdot -4\right)\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{-62}:\\ \;\;\;\;t_2 + \left(t_1 + 18 \cdot \left(y \cdot \left(t \cdot \left(x \cdot z\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot c + \left(-4 \cdot \left(x \cdot i\right) + -27 \cdot \left(j \cdot k\right)\right)\\ \end{array} \]

Alternative 12
Error	9.8
Cost	1865

\[\begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+28} \lor \neg \left(x \leq 2.2 \cdot 10^{-51}\right):\\ \;\;\;\;\left(x \cdot \left(t \cdot \left(18 \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) + \left(x \cdot \left(i \cdot -4\right) + j \cdot \left(k \cdot -27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot c + t \cdot \left(18 \cdot \left(y \cdot \left(x \cdot z\right)\right) + a \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right)\\ \end{array} \]

Alternative 13
Error	35.2
Cost	1764

\[\begin{array}{l} t_1 := b \cdot c + k \cdot \left(j \cdot -27\right)\\ t_2 := t \cdot \left(a \cdot -4\right)\\ t_3 := b \cdot c + x \cdot \left(i \cdot -4\right)\\ \mathbf{if}\;a \leq -4 \cdot 10^{+260}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -9.2 \cdot 10^{+95}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -3 \cdot 10^{-126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1.75 \cdot 10^{-218}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 2.4 \cdot 10^{-162}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 4 \cdot 10^{-34}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{+62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 9 \cdot 10^{+151}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{+199}:\\ \;\;\;\;18 \cdot \left(y \cdot \left(t \cdot \left(x \cdot z\right)\right)\right)\\ \mathbf{elif}\;a \leq 4.6 \cdot 10^{+203}:\\ \;\;\;\;-4 \cdot \left(x \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 14
Error	11.5
Cost	1737

\[\begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{-97} \lor \neg \left(t \leq 6.2 \cdot 10^{-110}\right):\\ \;\;\;\;\left(b \cdot c + t \cdot \left(18 \cdot \left(y \cdot \left(x \cdot z\right)\right) + a \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot c + \left(-4 \cdot \left(x \cdot i\right) + -27 \cdot \left(j \cdot k\right)\right)\\ \end{array} \]

Alternative 15
Error	30.8
Cost	1501

\[\begin{array}{l} t_1 := k \cdot \left(j \cdot -27\right)\\ t_2 := t_1 + -4 \cdot \left(t \cdot a\right)\\ t_3 := -4 \cdot \left(x \cdot i\right) + -27 \cdot \left(j \cdot k\right)\\ t_4 := b \cdot c + t_1\\ \mathbf{if}\;c \leq -8.8 \cdot 10^{-82}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;c \leq 2.8 \cdot 10^{-260}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c \leq 3.8 \cdot 10^{-195}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 7 \cdot 10^{-55}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c \leq 33000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 3.3 \cdot 10^{+103} \lor \neg \left(c \leq 1.4 \cdot 10^{+170}\right):\\ \;\;\;\;b \cdot c + x \cdot \left(i \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 16
Error	32.8
Cost	1499

\[\begin{array}{l} \mathbf{if}\;i \leq -8 \cdot 10^{-12} \lor \neg \left(i \leq 3.2 \cdot 10^{-212}\right) \land \left(i \leq 3.1 \cdot 10^{-179} \lor \neg \left(i \leq 1.85 \cdot 10^{-77} \lor \neg \left(i \leq 6.1 \cdot 10^{+35}\right) \land i \leq 1.2 \cdot 10^{+116}\right)\right):\\ \;\;\;\;b \cdot c + x \cdot \left(i \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(j \cdot -27\right) + -4 \cdot \left(t \cdot a\right)\\ \end{array} \]

Alternative 17
Error	44.3
Cost	1376

\[\begin{array}{l} t_1 := -4 \cdot \left(x \cdot i\right)\\ t_2 := -27 \cdot \left(j \cdot k\right)\\ \mathbf{if}\;c \leq -1.4 \cdot 10^{-82}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;c \leq 1.55 \cdot 10^{-277}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 2.15 \cdot 10^{-110}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 8.2 \cdot 10^{-65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 3.95 \cdot 10^{-26}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 3.1 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 2.5 \cdot 10^{+109}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;c \leq 1.65 \cdot 10^{+149}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;b \cdot c\\ \end{array} \]

Alternative 18
Error	44.3
Cost	1376

\[\begin{array}{l} t_1 := -4 \cdot \left(x \cdot i\right)\\ t_2 := k \cdot \left(j \cdot -27\right)\\ \mathbf{if}\;c \leq -8.8 \cdot 10^{-82}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;c \leq 2.8 \cdot 10^{-278}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 2.9 \cdot 10^{-112}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 5 \cdot 10^{-65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 1.9 \cdot 10^{-26}:\\ \;\;\;\;-27 \cdot \left(j \cdot k\right)\\ \mathbf{elif}\;c \leq 3.2 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 1.55 \cdot 10^{+110}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;c \leq 1.65 \cdot 10^{+149}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;b \cdot c\\ \end{array} \]

Alternative 19
Error	17.3
Cost	1357

\[\begin{array}{l} \mathbf{if}\;t \leq -5.4 \cdot 10^{+113}:\\ \;\;\;\;t \cdot \left(18 \cdot \left(y \cdot \left(x \cdot z\right)\right) + a \cdot -4\right)\\ \mathbf{elif}\;t \leq -7.5 \cdot 10^{-109} \lor \neg \left(t \leq 5.5 \cdot 10^{-47}\right):\\ \;\;\;\;k \cdot \left(j \cdot -27\right) + \left(b \cdot c + -4 \cdot \left(t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot c + \left(-4 \cdot \left(x \cdot i\right) + -27 \cdot \left(j \cdot k\right)\right)\\ \end{array} \]

Alternative 20
Error	35.9
Cost	1236

\[\begin{array}{l} t_1 := t \cdot \left(a \cdot -4\right)\\ t_2 := b \cdot c + k \cdot \left(j \cdot -27\right)\\ t_3 := -4 \cdot \left(x \cdot i\right)\\ \mathbf{if}\;a \leq -4.5 \cdot 10^{+261}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -2.6 \cdot 10^{+95}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 4.4 \cdot 10^{-140}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 8.8 \cdot 10^{-38}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{+62}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 21
Error	43.8
Cost	850

\[\begin{array}{l} \mathbf{if}\;c \leq -9 \cdot 10^{-87} \lor \neg \left(c \leq 38000000000000\right) \land \left(c \leq 3.6 \cdot 10^{+109} \lor \neg \left(c \leq 2.9 \cdot 10^{+149}\right)\right):\\ \;\;\;\;b \cdot c\\ \mathbf{else}:\\ \;\;\;\;-27 \cdot \left(j \cdot k\right)\\ \end{array} \]

Alternative 22
Error	48.4
Cost	192

\[b \cdot c \]

Reproduce

herbie shell --seed 2023060 
(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)))