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

Average Error: 6.0 → 1.9

Time: 36.4s

Precision: binary64

Cost: 6088

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

Math

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

↓

\[\begin{array}{l} t_1 := i \cdot \left(x \cdot -4\right)\\ t_2 := -27 \cdot \left(j \cdot k\right) + -4 \cdot \left(t \cdot a\right)\\ t_3 := k \cdot \left(j \cdot -27\right)\\ t_4 := t \cdot \left(a \cdot -4\right)\\ t_5 := \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t_4\right) + b \cdot c\right) + t_1\right) + t_3\\ \mathbf{if}\;t_5 \leq -2 \cdot 10^{+301}:\\ \;\;\;\;b \cdot c + \left(x \cdot \left(18 \cdot \left(y \cdot \left(z \cdot t\right)\right) + i \cdot -4\right) + t_2\right)\\ \mathbf{elif}\;t_5 \leq 4 \cdot 10^{+297}:\\ \;\;\;\;\left(\left(b \cdot c + \left(t \cdot \left(z \cdot \left(18 \cdot \left(x \cdot y\right)\right)\right) + t_4\right)\right) + t_1\right) + t_3\\ \mathbf{else}:\\ \;\;\;\;b \cdot c + \left(t_2 + x \cdot \left(\left(y \cdot t\right) \cdot \left(18 \cdot z\right)\right)\right)\\ \end{array} \]

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 (* i (* x -4.0)))
        (t_2 (+ (* -27.0 (* j k)) (* -4.0 (* t a))))
        (t_3 (* k (* j -27.0)))
        (t_4 (* t (* a -4.0)))
        (t_5 (+ (+ (+ (+ (* (* (* (* x 18.0) y) z) t) t_4) (* b c)) t_1) t_3)))
   (if (<= t_5 -2e+301)
     (+ (* b c) (+ (* x (+ (* 18.0 (* y (* z t))) (* i -4.0))) t_2))
     (if (<= t_5 4e+297)
       (+ (+ (+ (* b c) (+ (* t (* z (* 18.0 (* x y)))) t_4)) t_1) t_3)
       (+ (* b c) (+ t_2 (* x (* (* y t) (* 18.0 z)))))))))

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 = i * (x * -4.0);
	double t_2 = (-27.0 * (j * k)) + (-4.0 * (t * a));
	double t_3 = k * (j * -27.0);
	double t_4 = t * (a * -4.0);
	double t_5 = (((((((x * 18.0) * y) * z) * t) + t_4) + (b * c)) + t_1) + t_3;
	double tmp;
	if (t_5 <= -2e+301) {
		tmp = (b * c) + ((x * ((18.0 * (y * (z * t))) + (i * -4.0))) + t_2);
	} else if (t_5 <= 4e+297) {
		tmp = (((b * c) + ((t * (z * (18.0 * (x * y)))) + t_4)) + t_1) + t_3;
	} else {
		tmp = (b * c) + (t_2 + (x * ((y * t) * (18.0 * z))));
	}
	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) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_1 = i * (x * (-4.0d0))
    t_2 = ((-27.0d0) * (j * k)) + ((-4.0d0) * (t * a))
    t_3 = k * (j * (-27.0d0))
    t_4 = t * (a * (-4.0d0))
    t_5 = (((((((x * 18.0d0) * y) * z) * t) + t_4) + (b * c)) + t_1) + t_3
    if (t_5 <= (-2d+301)) then
        tmp = (b * c) + ((x * ((18.0d0 * (y * (z * t))) + (i * (-4.0d0)))) + t_2)
    else if (t_5 <= 4d+297) then
        tmp = (((b * c) + ((t * (z * (18.0d0 * (x * y)))) + t_4)) + t_1) + t_3
    else
        tmp = (b * c) + (t_2 + (x * ((y * t) * (18.0d0 * z))))
    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 = i * (x * -4.0);
	double t_2 = (-27.0 * (j * k)) + (-4.0 * (t * a));
	double t_3 = k * (j * -27.0);
	double t_4 = t * (a * -4.0);
	double t_5 = (((((((x * 18.0) * y) * z) * t) + t_4) + (b * c)) + t_1) + t_3;
	double tmp;
	if (t_5 <= -2e+301) {
		tmp = (b * c) + ((x * ((18.0 * (y * (z * t))) + (i * -4.0))) + t_2);
	} else if (t_5 <= 4e+297) {
		tmp = (((b * c) + ((t * (z * (18.0 * (x * y)))) + t_4)) + t_1) + t_3;
	} else {
		tmp = (b * c) + (t_2 + (x * ((y * t) * (18.0 * z))));
	}
	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 = i * (x * -4.0)
	t_2 = (-27.0 * (j * k)) + (-4.0 * (t * a))
	t_3 = k * (j * -27.0)
	t_4 = t * (a * -4.0)
	t_5 = (((((((x * 18.0) * y) * z) * t) + t_4) + (b * c)) + t_1) + t_3
	tmp = 0
	if t_5 <= -2e+301:
		tmp = (b * c) + ((x * ((18.0 * (y * (z * t))) + (i * -4.0))) + t_2)
	elif t_5 <= 4e+297:
		tmp = (((b * c) + ((t * (z * (18.0 * (x * y)))) + t_4)) + t_1) + t_3
	else:
		tmp = (b * c) + (t_2 + (x * ((y * t) * (18.0 * z))))
	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(i * Float64(x * -4.0))
	t_2 = Float64(Float64(-27.0 * Float64(j * k)) + Float64(-4.0 * Float64(t * a)))
	t_3 = Float64(k * Float64(j * -27.0))
	t_4 = Float64(t * Float64(a * -4.0))
	t_5 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * 18.0) * y) * z) * t) + t_4) + Float64(b * c)) + t_1) + t_3)
	tmp = 0.0
	if (t_5 <= -2e+301)
		tmp = Float64(Float64(b * c) + Float64(Float64(x * Float64(Float64(18.0 * Float64(y * Float64(z * t))) + Float64(i * -4.0))) + t_2));
	elseif (t_5 <= 4e+297)
		tmp = Float64(Float64(Float64(Float64(b * c) + Float64(Float64(t * Float64(z * Float64(18.0 * Float64(x * y)))) + t_4)) + t_1) + t_3);
	else
		tmp = Float64(Float64(b * c) + Float64(t_2 + Float64(x * Float64(Float64(y * t) * Float64(18.0 * z)))));
	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 = i * (x * -4.0);
	t_2 = (-27.0 * (j * k)) + (-4.0 * (t * a));
	t_3 = k * (j * -27.0);
	t_4 = t * (a * -4.0);
	t_5 = (((((((x * 18.0) * y) * z) * t) + t_4) + (b * c)) + t_1) + t_3;
	tmp = 0.0;
	if (t_5 <= -2e+301)
		tmp = (b * c) + ((x * ((18.0 * (y * (z * t))) + (i * -4.0))) + t_2);
	elseif (t_5 <= 4e+297)
		tmp = (((b * c) + ((t * (z * (18.0 * (x * y)))) + t_4)) + t_1) + t_3;
	else
		tmp = (b * c) + (t_2 + (x * ((y * t) * (18.0 * z))));
	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[(i * N[(x * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-27.0 * N[(j * k), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(k * N[(j * -27.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[(N[(N[(N[(N[(x * 18.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] + t$95$4), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision]}, If[LessEqual[t$95$5, -2e+301], 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] + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 4e+297], N[(N[(N[(N[(b * c), $MachinePrecision] + N[(N[(t * N[(z * N[(18.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision], N[(N[(b * c), $MachinePrecision] + N[(t$95$2 + N[(x * N[(N[(y * t), $MachinePrecision] * N[(18.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Target

Original	6.0
Target	1.8
Herbie	1.9

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

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)) < -2.00000000000000011e301

   1. Initial program 50.6

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

   2. Simplified 30.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(j, k \cdot -27, \mathsf{fma}\left(t, \mathsf{fma}\left(x, y \cdot \left(18 \cdot z\right), a \cdot -4\right), \mathsf{fma}\left(b, c, i \cdot \left(x \cdot -4\right)\right)\right)\right)} \]
      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))))): 2 points increase in error, 3 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))))): 7 points increase in error, 10 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))))): 4 points increase in error, 2 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))))): 0 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, 1 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)))): 3 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))): 11 points increase in error, 14 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 5.4

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

   if -2.00000000000000011e301 < (-.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)) < 4.0000000000000001e297

   1. Initial program 0.3

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

   2. Taylor expanded in x around 0 0.3

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

   if 4.0000000000000001e297 < (-.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 45.3

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

   2. Simplified 31.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(j, k \cdot -27, \mathsf{fma}\left(t, \mathsf{fma}\left(x, y \cdot \left(18 \cdot z\right), a \cdot -4\right), \mathsf{fma}\left(b, c, i \cdot \left(x \cdot -4\right)\right)\right)\right)} \]
      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))))): 2 points increase in error, 3 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))))): 7 points increase in error, 10 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))))): 4 points increase in error, 2 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))))): 0 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, 1 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)))): 3 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))): 11 points increase in error, 14 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 6.3

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

   4. Taylor expanded in y around inf 24.1

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

   5. Simplified 21.3

      \[\leadsto c \cdot b + \left(\color{blue}{x \cdot \left(\left(y \cdot t\right) \cdot \left(18 \cdot z\right)\right)} + \left(-27 \cdot \left(k \cdot j\right) + -4 \cdot \left(a \cdot t\right)\right)\right) \]
      Proof
      (*.f64 x (*.f64 (*.f64 y t) (*.f64 18 z))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 y t) 18) z))): 23 points increase in error, 25 points decrease in error
      (*.f64 x (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 18 (*.f64 y t))) z)): 0 points increase in error, 0 points decrease in error
      (*.f64 x (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 18 y) t)) z)): 15 points increase in error, 20 points decrease in error
      (*.f64 x (Rewrite<= associate-*r*_binary64 (*.f64 (*.f64 18 y) (*.f64 t z)))): 40 points increase in error, 38 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (*.f64 18 y) (*.f64 t z)) x)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 (*.f64 18 y) (*.f64 (*.f64 t z) x))): 43 points increase in error, 37 points decrease in error
      (*.f64 (*.f64 18 y) (Rewrite<= associate-*r*_binary64 (*.f64 t (*.f64 z x)))): 34 points increase in error, 42 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 18 (*.f64 y (*.f64 t (*.f64 z x))))): 28 points increase in error, 22 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 1.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right) \leq -2 \cdot 10^{+301}:\\ \;\;\;\;b \cdot c + \left(x \cdot \left(18 \cdot \left(y \cdot \left(z \cdot t\right)\right) + i \cdot -4\right) + \left(-27 \cdot \left(j \cdot k\right) + -4 \cdot \left(t \cdot a\right)\right)\right)\\ \mathbf{elif}\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right) \leq 4 \cdot 10^{+297}:\\ \;\;\;\;\left(\left(b \cdot c + \left(t \cdot \left(z \cdot \left(18 \cdot \left(x \cdot y\right)\right)\right) + t \cdot \left(a \cdot -4\right)\right)\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot c + \left(\left(-27 \cdot \left(j \cdot k\right) + -4 \cdot \left(t \cdot a\right)\right) + x \cdot \left(\left(y \cdot t\right) \cdot \left(18 \cdot z\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	21.3
Cost	3296

\[\begin{array}{l} t_1 := i \cdot \left(x \cdot -4\right)\\ t_2 := -4 \cdot \left(t \cdot a\right)\\ t_3 := -27 \cdot \left(j \cdot k\right)\\ t_4 := k \cdot \left(j \cdot -27\right)\\ t_5 := -4 \cdot \left(x \cdot i\right)\\ t_6 := b \cdot c + \left(t_3 + t_5\right)\\ t_7 := b \cdot c + \left(t_2 + 18 \cdot \left(y \cdot \left(t \cdot \left(x \cdot z\right)\right)\right)\right)\\ \mathbf{if}\;j \cdot 27 \leq -2 \cdot 10^{+140}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;j \cdot 27 \leq -1 \cdot 10^{-33}:\\ \;\;\;\;b \cdot c + \left(t_3 + t_2\right)\\ \mathbf{elif}\;j \cdot 27 \leq -2 \cdot 10^{-86}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;j \cdot 27 \leq -5 \cdot 10^{-221}:\\ \;\;\;\;t_7\\ \mathbf{elif}\;j \cdot 27 \leq -1 \cdot 10^{-241}:\\ \;\;\;\;b \cdot c + t_5\\ \mathbf{elif}\;j \cdot 27 \leq 5 \cdot 10^{-252}:\\ \;\;\;\;t_7\\ \mathbf{elif}\;j \cdot 27 \leq 2 \cdot 10^{-223}:\\ \;\;\;\;\left(t_1 + t \cdot \left(a \cdot -4\right)\right) + t_4\\ \mathbf{elif}\;j \cdot 27 \leq 5 \cdot 10^{-199}:\\ \;\;\;\;t_7\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot c + t_1\right) + t_4\\ \end{array} \]

Alternative 2
Error	4.0
Cost	2248

\[\begin{array}{l} t_1 := -27 \cdot \left(j \cdot k\right)\\ t_2 := t_1 + -4 \cdot \left(t \cdot a\right)\\ \mathbf{if}\;y \leq -1 \cdot 10^{+76}:\\ \;\;\;\;b \cdot c + \left(x \cdot \left(18 \cdot \left(y \cdot \left(z \cdot t\right)\right) + i \cdot -4\right) + t_2\right)\\ \mathbf{elif}\;y \leq 10^{+100}:\\ \;\;\;\;b \cdot c + \left(t_2 + \left(x \cdot \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right) + x \cdot \left(i \cdot -4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot c + \left(t_1 + 18 \cdot \left(y \cdot \left(t \cdot \left(x \cdot z\right)\right)\right)\right)\\ \end{array} \]

Alternative 3
Error	4.0
Cost	2248

\[\begin{array}{l} t_1 := -27 \cdot \left(j \cdot k\right)\\ \mathbf{if}\;y \leq -2.9442684637353802 \cdot 10^{-95}:\\ \;\;\;\;\left(\left(\left(b \cdot c + x \cdot \left(18 \cdot \left(y \cdot \left(z \cdot t\right)\right)\right)\right) + t \cdot \left(a \cdot -4\right)\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right)\\ \mathbf{elif}\;y \leq 10^{+100}:\\ \;\;\;\;b \cdot c + \left(\left(t_1 + -4 \cdot \left(t \cdot a\right)\right) + \left(x \cdot \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right) + x \cdot \left(i \cdot -4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot c + \left(t_1 + 18 \cdot \left(y \cdot \left(t \cdot \left(x \cdot z\right)\right)\right)\right)\\ \end{array} \]

Alternative 4
Error	9.4
Cost	1996

\[\begin{array}{l} t_1 := -27 \cdot \left(j \cdot k\right)\\ t_2 := t_1 + -4 \cdot \left(t \cdot a\right)\\ t_3 := b \cdot c + \left(t_2 + -4 \cdot \left(x \cdot i\right)\right)\\ \mathbf{if}\;z \leq -1.1275845448844931 \cdot 10^{-24}:\\ \;\;\;\;b \cdot c + \left(t_1 + 18 \cdot \left(y \cdot \left(t \cdot \left(x \cdot z\right)\right)\right)\right)\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+216}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 10^{+253}:\\ \;\;\;\;b \cdot c + \left(t_2 + x \cdot \left(t \cdot \left(z \cdot \left(18 \cdot y\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 5
Error	4.5
Cost	1988

\[\begin{array}{l} t_1 := -27 \cdot \left(j \cdot k\right)\\ \mathbf{if}\;t \leq 10^{+85}:\\ \;\;\;\;b \cdot c + \left(x \cdot \left(18 \cdot \left(y \cdot \left(z \cdot t\right)\right) + i \cdot -4\right) + \left(t_1 + -4 \cdot \left(t \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot c + t_1\right) + t \cdot \left(18 \cdot \left(y \cdot \left(x \cdot z\right)\right) + a \cdot -4\right)\\ \end{array} \]

Alternative 6
Error	32.4
Cost	1892

\[\begin{array}{l} t_1 := -4 \cdot \left(x \cdot i\right)\\ t_2 := -4 \cdot \left(t \cdot a\right)\\ t_3 := t_2 + t_1\\ t_4 := -27 \cdot \left(j \cdot k\right)\\ t_5 := b \cdot c + t_4\\ \mathbf{if}\;i \leq -3.8 \cdot 10^{+209}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;i \leq -1.3669725379649949 \cdot 10^{-115}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;i \leq -3.985409677101012 \cdot 10^{-246}:\\ \;\;\;\;t_4 + t_2\\ \mathbf{elif}\;i \leq 7.119618615129933 \cdot 10^{-61}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;i \leq 3.02720100853735 \cdot 10^{-29}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;i \leq 1.0214612855972758 \cdot 10^{-19}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;i \leq 10^{+89}:\\ \;\;\;\;b \cdot c + t_1\\ \mathbf{elif}\;i \leq 1.2 \cdot 10^{+126}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;i \leq 7.5 \cdot 10^{+214}:\\ \;\;\;\;t_5\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 7
Error	9.3
Cost	1864

\[\begin{array}{l} t_1 := -27 \cdot \left(j \cdot k\right) + -4 \cdot \left(t \cdot a\right)\\ t_2 := b \cdot c + \left(t_1 + -4 \cdot \left(x \cdot i\right)\right)\\ \mathbf{if}\;i \leq -2.8 \cdot 10^{+76}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;i \leq 3.0861559053375176 \cdot 10^{-134}:\\ \;\;\;\;b \cdot c + \left(t_1 + x \cdot \left(18 \cdot \left(y \cdot \left(z \cdot t\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	9.2
Cost	1864

\[\begin{array}{l} t_1 := -27 \cdot \left(j \cdot k\right)\\ t_2 := t_1 + -4 \cdot \left(t \cdot a\right)\\ \mathbf{if}\;z \leq -1.1275845448844931 \cdot 10^{-24}:\\ \;\;\;\;b \cdot c + \left(t_1 + 18 \cdot \left(y \cdot \left(t \cdot \left(x \cdot z\right)\right)\right)\right)\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+207}:\\ \;\;\;\;b \cdot c + \left(t_2 + -4 \cdot \left(x \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot c + \left(t_2 + x \cdot \left(\left(y \cdot t\right) \cdot \left(18 \cdot z\right)\right)\right)\\ \end{array} \]

Alternative 9
Error	9.0
Cost	1864

\[\begin{array}{l} t_1 := -4 \cdot \left(x \cdot i\right)\\ t_2 := -27 \cdot \left(j \cdot k\right)\\ t_3 := t_2 + -4 \cdot \left(t \cdot a\right)\\ \mathbf{if}\;z \leq -1.9335245506119153 \cdot 10^{-155}:\\ \;\;\;\;b \cdot c + \left(18 \cdot \left(y \cdot \left(t \cdot \left(x \cdot z\right)\right)\right) + \left(t_2 + t_1\right)\right)\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+207}:\\ \;\;\;\;b \cdot c + \left(t_3 + t_1\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot c + \left(t_3 + x \cdot \left(\left(y \cdot t\right) \cdot \left(18 \cdot z\right)\right)\right)\\ \end{array} \]

Alternative 10
Error	8.8
Cost	1604

\[\begin{array}{l} t_1 := -27 \cdot \left(j \cdot k\right)\\ \mathbf{if}\;t \leq -1.65 \cdot 10^{+20}:\\ \;\;\;\;\left(b \cdot c + t_1\right) + t \cdot \left(18 \cdot \left(y \cdot \left(x \cdot z\right)\right) + a \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot c + \left(\left(t_1 + -4 \cdot \left(t \cdot a\right)\right) + -4 \cdot \left(x \cdot i\right)\right)\\ \end{array} \]

Alternative 11
Error	9.2
Cost	1476

\[\begin{array}{l} t_1 := -27 \cdot \left(j \cdot k\right)\\ \mathbf{if}\;z \leq -1.1275845448844931 \cdot 10^{-24}:\\ \;\;\;\;b \cdot c + \left(t_1 + 18 \cdot \left(y \cdot \left(t \cdot \left(x \cdot z\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot c + \left(\left(t_1 + -4 \cdot \left(t \cdot a\right)\right) + -4 \cdot \left(x \cdot i\right)\right)\\ \end{array} \]

Alternative 12
Error	32.3
Cost	1364

\[\begin{array}{l} t_1 := -27 \cdot \left(j \cdot k\right)\\ t_2 := t_1 + -4 \cdot \left(t \cdot a\right)\\ t_3 := b \cdot c + t_1\\ t_4 := b \cdot c + -4 \cdot \left(x \cdot i\right)\\ \mathbf{if}\;i \leq -6.4 \cdot 10^{+232}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;i \leq -1.3669725379649949 \cdot 10^{-115}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;i \leq -3.985409677101012 \cdot 10^{-246}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;i \leq 3.0861559053375176 \cdot 10^{-134}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;i \leq 4.727670455937416 \cdot 10^{-34}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 13
Error	20.5
Cost	1224

\[\begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{+44}:\\ \;\;\;\;t \cdot \left(18 \cdot \left(y \cdot \left(x \cdot z\right)\right) + a \cdot -4\right)\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+143}:\\ \;\;\;\;\left(b \cdot c + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right)\\ \mathbf{else}:\\ \;\;\;\;-27 \cdot \left(j \cdot k\right) + -4 \cdot \left(t \cdot a\right)\\ \end{array} \]

Alternative 14
Error	20.4
Cost	1224

\[\begin{array}{l} t_1 := -27 \cdot \left(j \cdot k\right)\\ \mathbf{if}\;t \leq -7.2 \cdot 10^{+44}:\\ \;\;\;\;t \cdot \left(18 \cdot \left(y \cdot \left(x \cdot z\right)\right) + a \cdot -4\right)\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+143}:\\ \;\;\;\;b \cdot c + \left(t_1 + -4 \cdot \left(x \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + -4 \cdot \left(t \cdot a\right)\\ \end{array} \]

Alternative 15
Error	17.3
Cost	1224

\[\begin{array}{l} t_1 := -27 \cdot \left(j \cdot k\right)\\ \mathbf{if}\;i \leq -1.747758207024607 \cdot 10^{-52}:\\ \;\;\;\;b \cdot c + \left(t_1 + -4 \cdot \left(x \cdot i\right)\right)\\ \mathbf{elif}\;i \leq 4.727670455937416 \cdot 10^{-34}:\\ \;\;\;\;b \cdot c + \left(t_1 + -4 \cdot \left(t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot c + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right)\\ \end{array} \]

Alternative 16
Error	44.0
Cost	1112

\[\begin{array}{l} t_1 := t \cdot \left(a \cdot -4\right)\\ t_2 := k \cdot \left(j \cdot -27\right)\\ \mathbf{if}\;j \leq -3.0037768621546007 \cdot 10^{-46}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \leq -1.4005285539616526 \cdot 10^{-112}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;j \leq -1.9898012388619925 \cdot 10^{-220}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq -3.7557276113319725 \cdot 10^{-243}:\\ \;\;\;\;i \cdot \left(x \cdot -4\right)\\ \mathbf{elif}\;j \leq -2.3775614218637233 \cdot 10^{-294}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq 6.875195805254046 \cdot 10^{-81}:\\ \;\;\;\;b \cdot c\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 17
Error	43.9
Cost	1112

\[\begin{array}{l} t_1 := t \cdot \left(a \cdot -4\right)\\ \mathbf{if}\;j \leq -3.0037768621546007 \cdot 10^{-46}:\\ \;\;\;\;-27 \cdot \left(j \cdot k\right)\\ \mathbf{elif}\;j \leq -1.4005285539616526 \cdot 10^{-112}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;j \leq -1.9898012388619925 \cdot 10^{-220}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq -3.7557276113319725 \cdot 10^{-243}:\\ \;\;\;\;i \cdot \left(x \cdot -4\right)\\ \mathbf{elif}\;j \leq -2.3775614218637233 \cdot 10^{-294}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq 6.875195805254046 \cdot 10^{-81}:\\ \;\;\;\;b \cdot c\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(j \cdot -27\right)\\ \end{array} \]

Alternative 18
Error	43.5
Cost	848

\[\begin{array}{l} t_1 := k \cdot \left(j \cdot -27\right)\\ \mathbf{if}\;b \leq -6.558178820824638 \cdot 10^{+50}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;b \leq -1.5742549839408044 \cdot 10^{+26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -3.0788137146454116 \cdot 10^{-47}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;b \leq 1.9910013802631265 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;b \cdot c\\ \end{array} \]

Alternative 19
Error	43.9
Cost	848

\[\begin{array}{l} t_1 := k \cdot \left(j \cdot -27\right)\\ \mathbf{if}\;j \leq -3.0037768621546007 \cdot 10^{-46}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq -1.4005285539616526 \cdot 10^{-112}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;j \leq -2.3775614218637233 \cdot 10^{-294}:\\ \;\;\;\;t \cdot \left(a \cdot -4\right)\\ \mathbf{elif}\;j \leq 6.875195805254046 \cdot 10^{-81}:\\ \;\;\;\;b \cdot c\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 20
Error	33.6
Cost	840

\[\begin{array}{l} \mathbf{if}\;j \leq -2.15 \cdot 10^{+153}:\\ \;\;\;\;-27 \cdot \left(j \cdot k\right)\\ \mathbf{elif}\;j \leq 1.1341173910228675 \cdot 10^{-79}:\\ \;\;\;\;b \cdot c + -4 \cdot \left(x \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(j \cdot -27\right)\\ \end{array} \]

Alternative 21
Error	30.6
Cost	840

\[\begin{array}{l} \mathbf{if}\;j \leq -1.9468450531376673 \cdot 10^{-42}:\\ \;\;\;\;b \cdot c + -27 \cdot \left(j \cdot k\right)\\ \mathbf{elif}\;j \leq 1.1341173910228675 \cdot 10^{-79}:\\ \;\;\;\;b \cdot c + -4 \cdot \left(x \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(j \cdot -27\right)\\ \end{array} \]

Alternative 22
Error	48.1
Cost	192

\[b \cdot c \]

Reproduce

herbie shell --seed 2022297 
(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)))