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

Average Error: 5.5 → 1.3

Time: 36.0s

Precision: binary64

Cost: 15684

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

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot \left(a \cdot -4\right)\\ t_2 := i \cdot \left(x \cdot -4\right)\\ t_3 := \left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t_1\right) + b \cdot c\right) + t_2\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(z, y \cdot \left(18 \cdot t\right), i \cdot -4\right), b \cdot c\right) + j \cdot \left(k \cdot -27\right)\\ \mathbf{elif}\;t_3 \leq 2 \cdot 10^{+277}:\\ \;\;\;\;t_3 + -27 \cdot \left(j \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(x \cdot 18\right) \cdot \left(y \cdot \left(z \cdot t\right)\right) + t_1\right)\right) + t_2\right) + k \cdot \left(j \cdot -27\right)\\ \end{array} \]

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 (* t (* a -4.0)))
        (t_2 (* i (* x -4.0)))
        (t_3 (+ (+ (+ (* (* (* (* x 18.0) y) z) t) t_1) (* b c)) t_2)))
   (if (<= t_3 (- INFINITY))
     (+ (fma x (fma z (* y (* 18.0 t)) (* i -4.0)) (* b c)) (* j (* k -27.0)))
     (if (<= t_3 2e+277)
       (+ t_3 (* -27.0 (* j k)))
       (+
        (+ (+ (* b c) (+ (* (* x 18.0) (* y (* z t))) t_1)) t_2)
        (* k (* j -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 t_1 = t * (a * -4.0);
	double t_2 = i * (x * -4.0);
	double t_3 = ((((((x * 18.0) * y) * z) * t) + t_1) + (b * c)) + t_2;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = fma(x, fma(z, (y * (18.0 * t)), (i * -4.0)), (b * c)) + (j * (k * -27.0));
	} else if (t_3 <= 2e+277) {
		tmp = t_3 + (-27.0 * (j * k));
	} else {
		tmp = (((b * c) + (((x * 18.0) * (y * (z * t))) + t_1)) + t_2) + (k * (j * -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)
	t_1 = Float64(t * Float64(a * -4.0))
	t_2 = Float64(i * Float64(x * -4.0))
	t_3 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * 18.0) * y) * z) * t) + t_1) + Float64(b * c)) + t_2)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(fma(x, fma(z, Float64(y * Float64(18.0 * t)), Float64(i * -4.0)), Float64(b * c)) + Float64(j * Float64(k * -27.0)));
	elseif (t_3 <= 2e+277)
		tmp = Float64(t_3 + Float64(-27.0 * Float64(j * k)));
	else
		tmp = Float64(Float64(Float64(Float64(b * c) + Float64(Float64(Float64(x * 18.0) * Float64(y * Float64(z * t))) + t_1)) + t_2) + Float64(k * Float64(j * -27.0)));
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(t * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(x * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(N[(N[(x * 18.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(x * N[(z * N[(y * N[(18.0 * t), $MachinePrecision]), $MachinePrecision] + N[(i * -4.0), $MachinePrecision]), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision] + N[(j * N[(k * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+277], N[(t$95$3 + N[(-27.0 * N[(j * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(b * c), $MachinePrecision] + N[(N[(N[(x * 18.0), $MachinePrecision] * N[(y * N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + N[(k * N[(j * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	5.5
Target	1.4
Herbie	1.3

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) t) (*.f64 (*.f64 a 4) t)) (*.f64 b c)) (*.f64 (*.f64 x 4) i)) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 43.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(j, k \cdot -27, \mathsf{fma}\left(t, \mathsf{fma}\left(x, y \cdot \left(18 \cdot z\right), a \cdot -4\right), \mathsf{fma}\left(b, c, i \cdot \left(x \cdot -4\right)\right)\right)\right)} \]
      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))))): 4 points increase in error, 2 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (fma.f64 t (fma.f64 x (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 18 y)) z) (*.f64 a -4)) (fma.f64 b c (*.f64 i (*.f64 x -4))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (fma.f64 t (fma.f64 x (*.f64 (*.f64 18 y) z) (*.f64 a (Rewrite<= metadata-eval (neg.f64 4)))) (fma.f64 b c (*.f64 i (*.f64 x -4))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (fma.f64 t (fma.f64 x (*.f64 (*.f64 18 y) z) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 a 4)))) (fma.f64 b c (*.f64 i (*.f64 x -4))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (fma.f64 t (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 x (*.f64 (*.f64 18 y) z)) (*.f64 a 4))) (fma.f64 b c (*.f64 i (*.f64 x -4))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (fma.f64 t (-.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x (*.f64 18 y)) z)) (*.f64 a 4)) (fma.f64 b c (*.f64 i (*.f64 x -4))))): 7 points increase in error, 20 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))))): 7 points increase in error, 4 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))))): 2 points increase in error, 0 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 t (-.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) (*.f64 a 4))) (-.f64 (*.f64 b c) (*.f64 (*.f64 x 4) i))))): 2 points increase in error, 0 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (+.f64 (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) t) (*.f64 (*.f64 a 4) t))) (-.f64 (*.f64 b c) (*.f64 (*.f64 x 4) i)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 j (neg.f64 (*.f64 27 k)) (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) t) (*.f64 (*.f64 a 4) t)) (*.f64 b c)) (*.f64 (*.f64 x 4) i)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 j (neg.f64 (*.f64 27 k))) (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) t) (*.f64 (*.f64 a 4) t)) (*.f64 b c)) (*.f64 (*.f64 x 4) i)))): 1 points increase in error, 1 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))): 10 points increase in error, 10 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 a around 0 16.9

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

   4. Simplified 7.2

      \[\leadsto \mathsf{fma}\left(j, k \cdot -27, \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(-4, i, z \cdot \left(y \cdot \left(18 \cdot t\right)\right)\right), c \cdot b\right)}\right) \]
      Proof
      (fma.f64 x (fma.f64 -4 i (*.f64 z (*.f64 y (*.f64 18 t)))) (*.f64 c b)): 0 points increase in error, 0 points decrease in error
      (fma.f64 x (fma.f64 -4 i (*.f64 z (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 y 18) t)))) (*.f64 c b)): 4 points increase in error, 9 points decrease in error
      (fma.f64 x (fma.f64 -4 i (*.f64 z (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 18 y)) t))) (*.f64 c b)): 0 points increase in error, 0 points decrease in error
      (fma.f64 x (fma.f64 -4 i (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (*.f64 18 y) t) z))) (*.f64 c b)): 0 points increase in error, 0 points decrease in error
      (fma.f64 x (fma.f64 -4 i (Rewrite<= associate-*r*_binary64 (*.f64 (*.f64 18 y) (*.f64 t z)))) (*.f64 c b)): 24 points increase in error, 13 points decrease in error
      (fma.f64 x (fma.f64 -4 i (Rewrite<= associate-*r*_binary64 (*.f64 18 (*.f64 y (*.f64 t z))))) (*.f64 c b)): 9 points increase in error, 9 points decrease in error
      (fma.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -4 i) (*.f64 18 (*.f64 y (*.f64 t z))))) (*.f64 c b)): 0 points increase in error, 0 points decrease in error
      (fma.f64 x (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 18 (*.f64 y (*.f64 t z))) (*.f64 -4 i))) (*.f64 c b)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (+.f64 (*.f64 18 (*.f64 y (*.f64 t z))) (*.f64 -4 i))) (*.f64 c b))): 3 points increase in error, 1 points decrease in error
      (+.f64 (*.f64 x (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 -4 i) (*.f64 18 (*.f64 y (*.f64 t z)))))) (*.f64 c b)): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> distribute-rgt-in_binary64 (+.f64 (*.f64 (*.f64 -4 i) x) (*.f64 (*.f64 18 (*.f64 y (*.f64 t z))) x))) (*.f64 c b)): 1 points increase in error, 1 points decrease in error
      (+.f64 (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 -4 (*.f64 i x))) (*.f64 (*.f64 18 (*.f64 y (*.f64 t z))) x)) (*.f64 c b)): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 -4 (*.f64 i x)) (Rewrite=> associate-*l*_binary64 (*.f64 18 (*.f64 (*.f64 y (*.f64 t z)) x)))) (*.f64 c b)): 11 points increase in error, 12 points decrease in error
      (+.f64 (+.f64 (*.f64 -4 (*.f64 i x)) (*.f64 18 (Rewrite<= associate-*r*_binary64 (*.f64 y (*.f64 (*.f64 t z) x))))) (*.f64 c b)): 19 points increase in error, 18 points decrease in error
      (+.f64 (+.f64 (*.f64 -4 (*.f64 i x)) (*.f64 18 (*.f64 y (Rewrite<= associate-*r*_binary64 (*.f64 t (*.f64 z x)))))) (*.f64 c b)): 16 points increase in error, 15 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 c b) (+.f64 (*.f64 -4 (*.f64 i x)) (*.f64 18 (*.f64 y (*.f64 t (*.f64 z x))))))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 7.2

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

   if -inf.0 < (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) t) (*.f64 (*.f64 a 4) t)) (*.f64 b c)) (*.f64 (*.f64 x 4) i)) < 2.00000000000000001e277

   1. Initial program 0.4

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

   2. Taylor expanded in j around 0 0.2

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

   if 2.00000000000000001e277 < (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) t) (*.f64 (*.f64 a 4) t)) (*.f64 b c)) (*.f64 (*.f64 x 4) i))

   1. Initial program 29.9

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

   2. Applied egg-rr 9.7

      \[\leadsto \left(\left(\left(\color{blue}{{\left(\left(x \cdot 18\right) \cdot \left(y \cdot \left(z \cdot t\right)\right)\right)}^{1}} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
3. Recombined 3 regimes into one program.

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(z, y \cdot \left(18 \cdot t\right), i \cdot -4\right), b \cdot c\right) + j \cdot \left(k \cdot -27\right)\\ \mathbf{elif}\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right) \leq 2 \cdot 10^{+277}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\right) + -27 \cdot \left(j \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(x \cdot 18\right) \cdot \left(y \cdot \left(z \cdot t\right)\right) + t \cdot \left(a \cdot -4\right)\right)\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	2.0
Cost	5320

\[\begin{array}{l} t_1 := -27 \cdot \left(j \cdot k\right)\\ t_2 := t_1 + x \cdot \left(i \cdot -4 + 18 \cdot \left(y \cdot \left(z \cdot t\right)\right)\right)\\ t_3 := \left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_3 \leq 2 \cdot 10^{+298}:\\ \;\;\;\;t_3 + t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	1.2
Cost	5320

\[\begin{array}{l} t_1 := t \cdot \left(a \cdot -4\right)\\ t_2 := k \cdot \left(j \cdot -27\right)\\ t_3 := i \cdot \left(x \cdot -4\right)\\ t_4 := \left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t_1\right) + b \cdot c\right) + t_3\\ \mathbf{if}\;t_4 \leq -\infty:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(x \cdot 18\right) \cdot \left(z \cdot \left(y \cdot t\right)\right) + t_1\right)\right) + t_3\right) + t_2\\ \mathbf{elif}\;t_4 \leq 2 \cdot 10^{+292}:\\ \;\;\;\;t_4 + -27 \cdot \left(j \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(z \cdot \left(18 \cdot \left(x \cdot \left(y \cdot t\right)\right)\right) + t_1\right)\right) + t_3\right) + t_2\\ \end{array} \]

Alternative 3
Error	1.2
Cost	5320

\[\begin{array}{l} t_1 := t \cdot \left(a \cdot -4\right)\\ t_2 := k \cdot \left(j \cdot -27\right)\\ t_3 := i \cdot \left(x \cdot -4\right)\\ t_4 := \left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t_1\right) + b \cdot c\right) + t_3\\ \mathbf{if}\;t_4 \leq -\infty:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(x \cdot 18\right) \cdot \left(z \cdot \left(y \cdot t\right)\right) + t_1\right)\right) + t_3\right) + t_2\\ \mathbf{elif}\;t_4 \leq 2 \cdot 10^{+277}:\\ \;\;\;\;t_4 + -27 \cdot \left(j \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(x \cdot 18\right) \cdot \left(y \cdot \left(z \cdot t\right)\right) + t_1\right)\right) + t_3\right) + t_2\\ \end{array} \]

Alternative 4
Error	34.6
Cost	2280

\[\begin{array}{l} t_1 := j \cdot \left(k \cdot -27\right)\\ t_2 := t_1 + \left(t \cdot \left(x \cdot z\right)\right) \cdot \left(18 \cdot y\right)\\ t_3 := b \cdot c + t_1\\ t_4 := t_1 + x \cdot \left(i \cdot -4\right)\\ \mathbf{if}\;i \leq -7.6 \cdot 10^{+127}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;i \leq -1.324391717834849 \cdot 10^{+65}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;i \leq -3.5504891846598274 \cdot 10^{+48}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;i \leq -3.234762927889179 \cdot 10^{-177}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;i \leq -7.258968238765895 \cdot 10^{-239}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;i \leq 1.251052339216281 \cdot 10^{-264}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;i \leq 1.3204895779196463 \cdot 10^{-177}:\\ \;\;\;\;b \cdot c - 27 \cdot \left(j \cdot k\right)\\ \mathbf{elif}\;i \leq 1.0954237086127396 \cdot 10^{-146}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;i \leq 1.728071561851088 \cdot 10^{-26}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;i \leq 674167788.5833592:\\ \;\;\;\;t_2\\ \mathbf{elif}\;i \leq 2.431378639110073 \cdot 10^{+59}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 5
Error	34.6
Cost	2280

\[\begin{array}{l} t_1 := t \cdot \left(x \cdot z\right)\\ t_2 := j \cdot \left(k \cdot -27\right)\\ t_3 := t_2 + 18 \cdot \left(y \cdot t_1\right)\\ t_4 := b \cdot c + t_2\\ t_5 := t_2 + x \cdot \left(i \cdot -4\right)\\ \mathbf{if}\;i \leq -7.6 \cdot 10^{+127}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;i \leq -1.324391717834849 \cdot 10^{+65}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;i \leq -3.5504891846598274 \cdot 10^{+48}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;i \leq -3.234762927889179 \cdot 10^{-177}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;i \leq -7.258968238765895 \cdot 10^{-239}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;i \leq 1.251052339216281 \cdot 10^{-264}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;i \leq 1.3204895779196463 \cdot 10^{-177}:\\ \;\;\;\;b \cdot c - 27 \cdot \left(j \cdot k\right)\\ \mathbf{elif}\;i \leq 1.0954237086127396 \cdot 10^{-146}:\\ \;\;\;\;t_2 + t_1 \cdot \left(18 \cdot y\right)\\ \mathbf{elif}\;i \leq 1.728071561851088 \cdot 10^{-26}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;i \leq 674167788.5833592:\\ \;\;\;\;t_3\\ \mathbf{elif}\;i \leq 2.431378639110073 \cdot 10^{+59}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_5\\ \end{array} \]

Alternative 6
Error	1.5
Cost	2248

\[\begin{array}{l} t_1 := i \cdot \left(x \cdot -4\right)\\ t_2 := k \cdot \left(j \cdot -27\right)\\ t_3 := t \cdot \left(a \cdot -4\right)\\ \mathbf{if}\;t \leq -1 \cdot 10^{+32}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t_3\right) + b \cdot c\right) + t_1\right) + -27 \cdot \left(j \cdot k\right)\\ \mathbf{elif}\;t \leq 10^{+20}:\\ \;\;\;\;\left(\left(b \cdot c + \left(z \cdot \left(18 \cdot \left(x \cdot \left(y \cdot t\right)\right)\right) + t_3\right)\right) + t_1\right) + t_2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(t \cdot \left(18 \cdot \left(y \cdot \left(x \cdot z\right)\right)\right) + t_3\right)\right) + t_1\right) + t_2\\ \end{array} \]

Alternative 7
Error	21.9
Cost	1744

\[\begin{array}{l} t_1 := j \cdot \left(k \cdot -27\right)\\ t_2 := t_1 + \left(b \cdot c + -4 \cdot \left(x \cdot i\right)\right)\\ \mathbf{if}\;i \leq -3.5504891846598274 \cdot 10^{+48}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;i \leq -3.234762927889179 \cdot 10^{-177}:\\ \;\;\;\;t_1 + \left(b \cdot c + 18 \cdot \left(y \cdot \left(t \cdot \left(x \cdot z\right)\right)\right)\right)\\ \mathbf{elif}\;i \leq -4.7004647834414655 \cdot 10^{-231}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;i \leq 1.0954237086127396 \cdot 10^{-146}:\\ \;\;\;\;t_1 + \left(b \cdot c + \left(y \cdot t\right) \cdot \left(x \cdot \left(18 \cdot z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	17.2
Cost	1732

\[\begin{array}{l} t_1 := -27 \cdot \left(j \cdot k\right)\\ \mathbf{if}\;x \leq 10^{+183}:\\ \;\;\;\;b \cdot c + \left(18 \cdot \left(y \cdot \left(t \cdot \left(x \cdot z\right)\right)\right) + \left(t_1 + -4 \cdot \left(x \cdot i\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + x \cdot \left(i \cdot -4 + 18 \cdot \left(y \cdot \left(z \cdot t\right)\right)\right)\\ \end{array} \]

Alternative 9
Error	22.5
Cost	1612

\[\begin{array}{l} t_1 := j \cdot \left(k \cdot -27\right)\\ t_2 := t_1 + \left(b \cdot c + -4 \cdot \left(x \cdot i\right)\right)\\ \mathbf{if}\;i \leq -953244808720.1964:\\ \;\;\;\;t_2\\ \mathbf{elif}\;i \leq -7.535897168032784 \cdot 10^{-186}:\\ \;\;\;\;-27 \cdot \left(j \cdot k\right) + x \cdot \left(i \cdot -4 + 18 \cdot \left(y \cdot \left(z \cdot t\right)\right)\right)\\ \mathbf{elif}\;i \leq 1.0954237086127396 \cdot 10^{-146}:\\ \;\;\;\;t_1 + \left(b \cdot c + \left(y \cdot t\right) \cdot \left(x \cdot \left(18 \cdot z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Error	24.0
Cost	1488

\[\begin{array}{l} t_1 := j \cdot \left(k \cdot -27\right)\\ t_2 := t_1 + \left(b \cdot c + -4 \cdot \left(x \cdot i\right)\right)\\ \mathbf{if}\;i \leq -7.258968238765895 \cdot 10^{-239}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;i \leq 1.251052339216281 \cdot 10^{-264}:\\ \;\;\;\;t_1 + 18 \cdot \left(y \cdot \left(t \cdot \left(x \cdot z\right)\right)\right)\\ \mathbf{elif}\;i \leq 1.3204895779196463 \cdot 10^{-177}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;i \leq 1.0954237086127396 \cdot 10^{-146}:\\ \;\;\;\;-27 \cdot \left(j \cdot k\right) + x \cdot \left(\left(y \cdot t\right) \cdot \left(18 \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 11
Error	24.0
Cost	1488

\[\begin{array}{l} t_1 := -4 \cdot \left(x \cdot i\right)\\ t_2 := -27 \cdot \left(j \cdot k\right)\\ t_3 := j \cdot \left(k \cdot -27\right)\\ t_4 := t_3 + \left(b \cdot c + t_1\right)\\ \mathbf{if}\;i \leq -7.258968238765895 \cdot 10^{-239}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;i \leq 1.251052339216281 \cdot 10^{-264}:\\ \;\;\;\;t_3 + 18 \cdot \left(y \cdot \left(t \cdot \left(x \cdot z\right)\right)\right)\\ \mathbf{elif}\;i \leq 1.3204895779196463 \cdot 10^{-177}:\\ \;\;\;\;b \cdot c + \left(t_2 + t_1\right)\\ \mathbf{elif}\;i \leq 1.0954237086127396 \cdot 10^{-146}:\\ \;\;\;\;t_2 + x \cdot \left(\left(y \cdot t\right) \cdot \left(18 \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 12
Error	21.6
Cost	1480

\[\begin{array}{l} t_1 := j \cdot \left(k \cdot -27\right)\\ t_2 := t_1 + \left(b \cdot c + -4 \cdot \left(x \cdot i\right)\right)\\ \mathbf{if}\;i \leq -7.683608765675223 \cdot 10^{+38}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;i \leq 1.0954237086127396 \cdot 10^{-146}:\\ \;\;\;\;t_1 + \left(b \cdot c + \left(y \cdot t\right) \cdot \left(x \cdot \left(18 \cdot z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 13
Error	36.1
Cost	1368

\[\begin{array}{l} t_1 := -4 \cdot \left(x \cdot i\right)\\ t_2 := b \cdot c + j \cdot \left(k \cdot -27\right)\\ \mathbf{if}\;c \leq -3.822369009102349 \cdot 10^{-100}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq -6.557449595564455 \cdot 10^{-149}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -2.422260213109983 \cdot 10^{-210}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq -7.750217602983084 \cdot 10^{-267}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 1.3013955668821897 \cdot 10^{-238}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 1.9789739609960656 \cdot 10^{-184}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 14
Error	36.1
Cost	1368

\[\begin{array}{l} t_1 := -4 \cdot \left(x \cdot i\right)\\ t_2 := b \cdot c + j \cdot \left(k \cdot -27\right)\\ \mathbf{if}\;c \leq -3.822369009102349 \cdot 10^{-100}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq -6.557449595564455 \cdot 10^{-149}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -2.422260213109983 \cdot 10^{-210}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq -7.750217602983084 \cdot 10^{-267}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 1.3013955668821897 \cdot 10^{-238}:\\ \;\;\;\;b \cdot c - 27 \cdot \left(j \cdot k\right)\\ \mathbf{elif}\;c \leq 1.9789739609960656 \cdot 10^{-184}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 15
Error	45.3
Cost	1244

\[\begin{array}{l} t_1 := -4 \cdot \left(x \cdot i\right)\\ t_2 := j \cdot \left(k \cdot -27\right)\\ \mathbf{if}\;c \leq -216182.65284283462:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;c \leq -7.750217602983084 \cdot 10^{-267}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 1.3013955668821897 \cdot 10^{-238}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 4.331987772929981 \cdot 10^{-102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 3.503586629425069 \cdot 10^{-69}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 3.2857186398915778 \cdot 10^{-24}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;c \leq 2.1 \cdot 10^{+105}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;b \cdot c\\ \end{array} \]

Alternative 16
Error	45.3
Cost	1244

\[\begin{array}{l} t_1 := -4 \cdot \left(x \cdot i\right)\\ t_2 := j \cdot \left(k \cdot -27\right)\\ \mathbf{if}\;c \leq -216182.65284283462:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;c \leq -7.750217602983084 \cdot 10^{-267}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 1.3013955668821897 \cdot 10^{-238}:\\ \;\;\;\;-27 \cdot \left(j \cdot k\right)\\ \mathbf{elif}\;c \leq 4.331987772929981 \cdot 10^{-102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 3.503586629425069 \cdot 10^{-69}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 3.2857186398915778 \cdot 10^{-24}:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;c \leq 2.1 \cdot 10^{+105}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;b \cdot c\\ \end{array} \]

Alternative 17
Error	23.7
Cost	1224

\[\begin{array}{l} t_1 := j \cdot \left(k \cdot -27\right)\\ t_2 := t_1 + \left(b \cdot c + -4 \cdot \left(x \cdot i\right)\right)\\ \mathbf{if}\;i \leq -7.258968238765895 \cdot 10^{-239}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;i \leq 1.251052339216281 \cdot 10^{-264}:\\ \;\;\;\;t_1 + 18 \cdot \left(y \cdot \left(t \cdot \left(x \cdot z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 18
Error	31.1
Cost	968

\[\begin{array}{l} t_1 := j \cdot \left(k \cdot -27\right)\\ t_2 := b \cdot c + t_1\\ \mathbf{if}\;c \leq -1.1006119887051711 \cdot 10^{-82}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 4.331987772929981 \cdot 10^{-102}:\\ \;\;\;\;t_1 + x \cdot \left(i \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 19
Error	45.1
Cost	584

\[\begin{array}{l} \mathbf{if}\;c \leq -216182.65284283462:\\ \;\;\;\;b \cdot c\\ \mathbf{elif}\;c \leq 4.0481654046340834 \cdot 10^{-70}:\\ \;\;\;\;-4 \cdot \left(x \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot c\\ \end{array} \]

Alternative 20
Error	48.7
Cost	192

\[b \cdot c \]

Reproduce

herbie shell --seed 2022316 
(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)))