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

Average Error: 5.2 → 1.3

Time: 43.6s

Precision: binary64

Cost: 5192

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

Math

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

↓

\[\begin{array}{l} t_1 := \left(x \cdot 4\right) \cdot i\\ t_2 := \left(b \cdot c - t_1\right) - j \cdot \left(27 \cdot k\right)\\ t_3 := \left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - t_1\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;\left(y \cdot \left(\left(z \cdot x\right) \cdot \left(18 \cdot t\right)\right) - x \cdot \left(i \cdot 4\right)\right) + \left(b \cdot c - j \cdot \left(k \cdot 27\right)\right)\\ \mathbf{elif}\;t_3 \leq 10^{+303}:\\ \;\;\;\;t \cdot \left(18 \cdot \left(z \cdot \left(x \cdot y\right)\right) - a \cdot 4\right) + t_2\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(18 \cdot \left(x \cdot t\right)\right)\right) + t_2\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i j k)
 :precision binary64
 (-
  (-
   (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c))
   (* (* x 4.0) i))
  (* (* j 27.0) k)))

↓

(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (* (* x 4.0) i))
        (t_2 (- (- (* b c) t_1) (* j (* 27.0 k))))
        (t_3
         (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) t_1)))
   (if (<= t_3 (- INFINITY))
     (+
      (- (* y (* (* z x) (* 18.0 t))) (* x (* i 4.0)))
      (- (* b c) (* j (* k 27.0))))
     (if (<= t_3 1e+303)
       (+ (* t (- (* 18.0 (* z (* x y))) (* a 4.0))) t_2)
       (+ (* y (* z (* 18.0 (* x t)))) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	return (((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k);
}

↓

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = (x * 4.0) * i;
	double t_2 = ((b * c) - t_1) - (j * (27.0 * k));
	double t_3 = ((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - t_1;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = ((y * ((z * x) * (18.0 * t))) - (x * (i * 4.0))) + ((b * c) - (j * (k * 27.0)));
	} else if (t_3 <= 1e+303) {
		tmp = (t * ((18.0 * (z * (x * y))) - (a * 4.0))) + t_2;
	} else {
		tmp = (y * (z * (18.0 * (x * t)))) + t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	return (((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k);
}

↓

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = (x * 4.0) * i;
	double t_2 = ((b * c) - t_1) - (j * (27.0 * k));
	double t_3 = ((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - t_1;
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = ((y * ((z * x) * (18.0 * t))) - (x * (i * 4.0))) + ((b * c) - (j * (k * 27.0)));
	} else if (t_3 <= 1e+303) {
		tmp = (t * ((18.0 * (z * (x * y))) - (a * 4.0))) + t_2;
	} else {
		tmp = (y * (z * (18.0 * (x * t)))) + t_2;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t, a, b, c, i, j, k):
	t_1 = (x * 4.0) * i
	t_2 = ((b * c) - t_1) - (j * (27.0 * k))
	t_3 = ((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - t_1
	tmp = 0
	if t_3 <= -math.inf:
		tmp = ((y * ((z * x) * (18.0 * t))) - (x * (i * 4.0))) + ((b * c) - (j * (k * 27.0)))
	elif t_3 <= 1e+303:
		tmp = (t * ((18.0 * (z * (x * y))) - (a * 4.0))) + t_2
	else:
		tmp = (y * (z * (18.0 * (x * t)))) + t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j, k)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * 18.0) * y) * z) * t) - Float64(Float64(a * 4.0) * t)) + Float64(b * c)) - Float64(Float64(x * 4.0) * i)) - Float64(Float64(j * 27.0) * k))
end

↓

function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(Float64(x * 4.0) * i)
	t_2 = Float64(Float64(Float64(b * c) - t_1) - Float64(j * Float64(27.0 * k)))
	t_3 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * 18.0) * y) * z) * t) - Float64(Float64(a * 4.0) * t)) + Float64(b * c)) - t_1)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(y * Float64(Float64(z * x) * Float64(18.0 * t))) - Float64(x * Float64(i * 4.0))) + Float64(Float64(b * c) - Float64(j * Float64(k * 27.0))));
	elseif (t_3 <= 1e+303)
		tmp = Float64(Float64(t * Float64(Float64(18.0 * Float64(z * Float64(x * y))) - Float64(a * 4.0))) + t_2);
	else
		tmp = Float64(Float64(y * Float64(z * Float64(18.0 * Float64(x * t)))) + t_2);
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z, t, a, b, c, i, j, k)
	t_1 = (x * 4.0) * i;
	t_2 = ((b * c) - t_1) - (j * (27.0 * k));
	t_3 = ((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - t_1;
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = ((y * ((z * x) * (18.0 * t))) - (x * (i * 4.0))) + ((b * c) - (j * (k * 27.0)));
	elseif (t_3 <= 1e+303)
		tmp = (t * ((18.0 * (z * (x * y))) - (a * 4.0))) + t_2;
	else
		tmp = (y * (z * (18.0 * (x * t)))) + t_2;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(N[(x * 4.0), $MachinePrecision] * i), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(b * c), $MachinePrecision] - t$95$1), $MachinePrecision] - N[(j * N[(27.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = 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] - t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[(y * N[(N[(z * x), $MachinePrecision] * N[(18.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(b * c), $MachinePrecision] - N[(j * N[(k * 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+303], N[(N[(t * N[(N[(18.0 * N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision], N[(N[(y * N[(z * N[(18.0 * N[(x * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]]]]]]

Target

Original	5.2
Target	1.5
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 41.5

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

      [Start] 64.0	\[ \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
      rational_best_oopsla_all_46_json_45_simplify-107 [=>] 64.0	\[ \color{blue}{\left(b \cdot c + \left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) - \left(x \cdot 4\right) \cdot i\right)\right)} - \left(j \cdot 27\right) \cdot k \]
      rational_best_oopsla_all_46_json_45_simplify-107 [=>] 64.0	\[ \color{blue}{\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) - \left(x \cdot 4\right) \cdot i\right) + \left(b \cdot c - \left(j \cdot 27\right) \cdot k\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 64.0	\[ \left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \color{blue}{t \cdot \left(a \cdot 4\right)}\right) - \left(x \cdot 4\right) \cdot i\right) + \left(b \cdot c - \left(j \cdot 27\right) \cdot k\right) \]
      rational_best_oopsla_all_46_json_45_simplify-102 [=>] 64.0	\[ \left(\color{blue}{t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right)} - \left(x \cdot 4\right) \cdot i\right) + \left(b \cdot c - \left(j \cdot 27\right) \cdot k\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 64.0	\[ \left(t \cdot \left(\color{blue}{z \cdot \left(\left(x \cdot 18\right) \cdot y\right)} - a \cdot 4\right) - \left(x \cdot 4\right) \cdot i\right) + \left(b \cdot c - \left(j \cdot 27\right) \cdot k\right) \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 41.5	\[ \left(t \cdot \left(\color{blue}{\left(x \cdot 18\right) \cdot \left(z \cdot y\right)} - a \cdot 4\right) - \left(x \cdot 4\right) \cdot i\right) + \left(b \cdot c - \left(j \cdot 27\right) \cdot k\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 41.5	\[ \left(t \cdot \left(\left(x \cdot 18\right) \cdot \left(z \cdot y\right) - a \cdot 4\right) - \color{blue}{i \cdot \left(x \cdot 4\right)}\right) + \left(b \cdot c - \left(j \cdot 27\right) \cdot k\right) \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 41.5	\[ \left(t \cdot \left(\left(x \cdot 18\right) \cdot \left(z \cdot y\right) - a \cdot 4\right) - \color{blue}{x \cdot \left(i \cdot 4\right)}\right) + \left(b \cdot c - \left(j \cdot 27\right) \cdot k\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 41.5	\[ \left(t \cdot \left(\left(x \cdot 18\right) \cdot \left(z \cdot y\right) - a \cdot 4\right) - x \cdot \left(i \cdot 4\right)\right) + \left(b \cdot c - \color{blue}{k \cdot \left(j \cdot 27\right)}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 41.5	\[ \left(t \cdot \left(\left(x \cdot 18\right) \cdot \left(z \cdot y\right) - a \cdot 4\right) - x \cdot \left(i \cdot 4\right)\right) + \left(b \cdot c - \color{blue}{j \cdot \left(k \cdot 27\right)}\right) \]

   3. Taylor expanded in x around inf 11.7

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

   4. Simplified 11.7

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

      [Start] 11.7	\[ \left(18 \cdot \left(y \cdot \left(t \cdot \left(z \cdot x\right)\right)\right) - x \cdot \left(i \cdot 4\right)\right) + \left(b \cdot c - j \cdot \left(k \cdot 27\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 11.7	\[ \left(\color{blue}{y \cdot \left(18 \cdot \left(t \cdot \left(z \cdot x\right)\right)\right)} - x \cdot \left(i \cdot 4\right)\right) + \left(b \cdot c - j \cdot \left(k \cdot 27\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 11.7	\[ \left(y \cdot \left(18 \cdot \color{blue}{\left(\left(z \cdot x\right) \cdot t\right)}\right) - x \cdot \left(i \cdot 4\right)\right) + \left(b \cdot c - j \cdot \left(k \cdot 27\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 11.7	\[ \left(y \cdot \color{blue}{\left(\left(z \cdot x\right) \cdot \left(18 \cdot t\right)\right)} - x \cdot \left(i \cdot 4\right)\right) + \left(b \cdot c - j \cdot \left(k \cdot 27\right)\right) \]

   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)) < 1e303

   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. Simplified 0.4

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

      [Start] 0.3	\[ \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.3	\[ \left(\color{blue}{\left(b \cdot c + \left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right)\right)} - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
      rational_best_oopsla_all_46_json_45_simplify-107 [=>] 0.3	\[ \color{blue}{\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + \left(b \cdot c - \left(x \cdot 4\right) \cdot i\right)\right)} - \left(j \cdot 27\right) \cdot k \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.3	\[ \color{blue}{\left(\left(b \cdot c - \left(x \cdot 4\right) \cdot i\right) + \left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right)\right)} - \left(j \cdot 27\right) \cdot k \]
      rational_best_oopsla_all_46_json_45_simplify-107 [=>] 0.3	\[ \color{blue}{\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + \left(\left(b \cdot c - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\right)} \]

   if 1e303 < (-.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 54.3

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

   2. Simplified 53.6

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

      [Start] 54.3	\[ \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 54.3	\[ \left(\color{blue}{\left(b \cdot c + \left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right)\right)} - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
      rational_best_oopsla_all_46_json_45_simplify-107 [=>] 54.3	\[ \color{blue}{\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + \left(b \cdot c - \left(x \cdot 4\right) \cdot i\right)\right)} - \left(j \cdot 27\right) \cdot k \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 54.3	\[ \color{blue}{\left(\left(b \cdot c - \left(x \cdot 4\right) \cdot i\right) + \left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right)\right)} - \left(j \cdot 27\right) \cdot k \]
      rational_best_oopsla_all_46_json_45_simplify-107 [=>] 54.3	\[ \color{blue}{\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + \left(\left(b \cdot c - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\right)} \]

   3. Taylor expanded in z around inf 15.9

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

   4. Simplified 10.6

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

      [Start] 15.9	\[ 18 \cdot \left(y \cdot \left(t \cdot \left(z \cdot x\right)\right)\right) + \left(\left(b \cdot c - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 16.0	\[ \color{blue}{y \cdot \left(18 \cdot \left(t \cdot \left(z \cdot x\right)\right)\right)} + \left(\left(b \cdot c - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 10.6	\[ y \cdot \left(18 \cdot \color{blue}{\left(z \cdot \left(t \cdot x\right)\right)}\right) + \left(\left(b \cdot c - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 10.6	\[ y \cdot \color{blue}{\left(z \cdot \left(18 \cdot \left(t \cdot x\right)\right)\right)} + \left(\left(b \cdot c - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 10.6	\[ y \cdot \left(z \cdot \left(18 \cdot \color{blue}{\left(x \cdot t\right)}\right)\right) + \left(\left(b \cdot c - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i \leq -\infty:\\ \;\;\;\;\left(y \cdot \left(\left(z \cdot x\right) \cdot \left(18 \cdot t\right)\right) - x \cdot \left(i \cdot 4\right)\right) + \left(b \cdot c - j \cdot \left(k \cdot 27\right)\right)\\ \mathbf{elif}\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i \leq 10^{+303}:\\ \;\;\;\;t \cdot \left(18 \cdot \left(z \cdot \left(x \cdot y\right)\right) - a \cdot 4\right) + \left(\left(b \cdot c - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(18 \cdot \left(x \cdot t\right)\right)\right) + \left(\left(b \cdot c - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	13.4
Cost	2896

\[\begin{array}{l} t_1 := c \cdot b + x \cdot \left(t \cdot \left(18 \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right)\\ t_2 := \left(j \cdot 27\right) \cdot k\\ t_3 := b \cdot c + \left(\left(t \cdot \left(a \cdot -4\right) - \left(x \cdot 4\right) \cdot i\right) - t_2\right)\\ \mathbf{if}\;t_2 \leq -4 \cdot 10^{-224}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{-254}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{-150}:\\ \;\;\;\;c \cdot b + t \cdot \left(y \cdot \left(18 \cdot \left(x \cdot z\right)\right) - 4 \cdot a\right)\\ \mathbf{elif}\;t_2 \leq 10^{-25}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 2
Error	13.4
Cost	2896

\[\begin{array}{l} t_1 := t \cdot \left(a \cdot -4\right) + \left(\left(b \cdot c - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\right)\\ t_2 := c \cdot b + x \cdot \left(t \cdot \left(18 \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right)\\ t_3 := \left(j \cdot 27\right) \cdot k\\ \mathbf{if}\;t_3 \leq -4 \cdot 10^{-224}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_3 \leq 5 \cdot 10^{-254}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_3 \leq 2 \cdot 10^{-150}:\\ \;\;\;\;c \cdot b + t \cdot \left(y \cdot \left(18 \cdot \left(x \cdot z\right)\right) - 4 \cdot a\right)\\ \mathbf{elif}\;t_3 \leq 10^{-25}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	8.4
Cost	2636

\[\begin{array}{l} t_1 := \left(x \cdot 4\right) \cdot i\\ t_2 := t \cdot \left(a \cdot -4\right) + \left(\left(b \cdot c - t_1\right) - j \cdot \left(27 \cdot k\right)\right)\\ t_3 := \left(j \cdot 27\right) \cdot k\\ \mathbf{if}\;t_3 \leq -1 \cdot 10^{+180}:\\ \;\;\;\;b \cdot c + \left(\left(18 \cdot \left(y \cdot \left(z \cdot \left(t \cdot x\right)\right)\right) - t_1\right) - t_3\right)\\ \mathbf{elif}\;t_3 \leq -5 \cdot 10^{-201}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_3 \leq 5 \cdot 10^{-18}:\\ \;\;\;\;\left(t \cdot \left(\left(x \cdot 18\right) \cdot \left(z \cdot y\right) - a \cdot 4\right) - x \cdot \left(i \cdot 4\right)\right) + c \cdot b\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	27.5
Cost	2264

\[\begin{array}{l} t_1 := -27 \cdot \left(k \cdot j\right)\\ t_2 := t_1 - 4 \cdot \left(i \cdot x\right)\\ \mathbf{if}\;b \cdot c \leq -1 \cdot 10^{-55}:\\ \;\;\;\;b \cdot c + j \cdot \left(k \cdot -27\right)\\ \mathbf{elif}\;b \cdot c \leq -2 \cdot 10^{-145}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \cdot c \leq -1.2 \cdot 10^{-257}:\\ \;\;\;\;-4 \cdot \left(a \cdot t\right) + t_1\\ \mathbf{elif}\;b \cdot c \leq 10^{+41}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \cdot c \leq 10^{+83}:\\ \;\;\;\;c \cdot b + t \cdot \left(a \cdot -4\right)\\ \mathbf{elif}\;b \cdot c \leq 5 \cdot 10^{+131}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;c \cdot b - i \cdot \left(4 \cdot x\right)\\ \end{array} \]

Alternative 5
Error	28.4
Cost	2264

\[\begin{array}{l} t_1 := -27 \cdot \left(k \cdot j\right) - 4 \cdot \left(i \cdot x\right)\\ \mathbf{if}\;b \cdot c \leq -1 \cdot 10^{-55}:\\ \;\;\;\;b \cdot c + j \cdot \left(k \cdot -27\right)\\ \mathbf{elif}\;b \cdot c \leq -5 \cdot 10^{-159}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \cdot c \leq -1 \cdot 10^{-264}:\\ \;\;\;\;t \cdot \left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a\right)\\ \mathbf{elif}\;b \cdot c \leq 10^{+41}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \cdot c \leq 10^{+83}:\\ \;\;\;\;c \cdot b + t \cdot \left(a \cdot -4\right)\\ \mathbf{elif}\;b \cdot c \leq 5 \cdot 10^{+131}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;c \cdot b - i \cdot \left(4 \cdot x\right)\\ \end{array} \]

Alternative 6
Error	8.1
Cost	2248

\[\begin{array}{l} t_1 := t \cdot \left(a \cdot -4\right) + \left(\left(b \cdot c - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\right)\\ t_2 := \left(j \cdot 27\right) \cdot k\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{-201}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{-18}:\\ \;\;\;\;\left(t \cdot \left(\left(x \cdot 18\right) \cdot \left(z \cdot y\right) - a \cdot 4\right) - x \cdot \left(i \cdot 4\right)\right) + c \cdot b\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	4.2
Cost	1988

\[\begin{array}{l} \mathbf{if}\;y \leq -2.15 \cdot 10^{+136}:\\ \;\;\;\;\left(18 \cdot \left(y \cdot \left(\left(z \cdot x\right) \cdot t\right)\right) - x \cdot \left(i \cdot 4\right)\right) + \left(b \cdot c - j \cdot \left(k \cdot 27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot c + \left(\left(t \cdot \left(18 \cdot \left(z \cdot \left(x \cdot y\right)\right) - a \cdot 4\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\right)\\ \end{array} \]

Alternative 8
Error	4.2
Cost	1988

\[\begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+136}:\\ \;\;\;\;\left(18 \cdot \left(y \cdot \left(\left(z \cdot x\right) \cdot t\right)\right) - x \cdot \left(i \cdot 4\right)\right) + \left(b \cdot c - j \cdot \left(k \cdot 27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(18 \cdot \left(z \cdot \left(x \cdot y\right)\right) - a \cdot 4\right) + \left(\left(b \cdot c - \left(x \cdot 4\right) \cdot i\right) - 27 \cdot \left(k \cdot j\right)\right)\\ \end{array} \]

Alternative 9
Error	20.8
Cost	1880

\[\begin{array}{l} t_1 := c \cdot b - \left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(k \cdot j\right)\right)\\ t_2 := c \cdot b + t \cdot \left(y \cdot \left(18 \cdot \left(x \cdot z\right)\right) - 4 \cdot a\right)\\ \mathbf{if}\;j \leq -1.25 \cdot 10^{-86}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq -2.7 \cdot 10^{-143}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \leq -9.6 \cdot 10^{-203}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq -6.2 \cdot 10^{-271}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \leq -7 \cdot 10^{-288}:\\ \;\;\;\;c \cdot b - i \cdot \left(4 \cdot x\right)\\ \mathbf{elif}\;j \leq 1.75 \cdot 10^{-181}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	20.7
Cost	1880

\[\begin{array}{l} t_1 := c \cdot b - \left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(k \cdot j\right)\right)\\ t_2 := c \cdot b + t \cdot \left(y \cdot \left(18 \cdot \left(x \cdot z\right)\right) - 4 \cdot a\right)\\ \mathbf{if}\;j \leq -2.6 \cdot 10^{-87}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq -8.8 \cdot 10^{-141}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \leq -1.72 \cdot 10^{-205}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq -5.9 \cdot 10^{-271}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \leq -1.16 \cdot 10^{-287}:\\ \;\;\;\;c \cdot b + x \cdot \left(t \cdot \left(18 \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right)\\ \mathbf{elif}\;j \leq 1.22 \cdot 10^{-185}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	23.3
Cost	1752

\[\begin{array}{l} t_1 := c \cdot b - \left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{if}\;i \leq -1.7 \cdot 10^{-228}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;i \leq 4.5 \cdot 10^{-283}:\\ \;\;\;\;t \cdot \left(18 \cdot \left(x \cdot \left(y \cdot z\right)\right) - 4 \cdot a\right)\\ \mathbf{elif}\;i \leq 1.4 \cdot 10^{-235}:\\ \;\;\;\;b \cdot c + j \cdot \left(k \cdot -27\right)\\ \mathbf{elif}\;i \leq 10^{-200}:\\ \;\;\;\;-4 \cdot \left(a \cdot t\right) + -27 \cdot \left(k \cdot j\right)\\ \mathbf{elif}\;i \leq 2.8 \cdot 10^{-156}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;i \leq 5.1 \cdot 10^{-113}:\\ \;\;\;\;c \cdot b + 18 \cdot \left(z \cdot \left(y \cdot \left(x \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	30.5
Cost	1632

\[\begin{array}{l} t_1 := -4 \cdot \left(a \cdot t\right) + -27 \cdot \left(k \cdot j\right)\\ t_2 := b \cdot c + j \cdot \left(k \cdot -27\right)\\ t_3 := c \cdot b - i \cdot \left(4 \cdot x\right)\\ \mathbf{if}\;i \leq -8.2 \cdot 10^{+54}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;i \leq -5 \cdot 10^{-253}:\\ \;\;\;\;b \cdot c + k \cdot \left(j \cdot -27\right)\\ \mathbf{elif}\;i \leq 4.2 \cdot 10^{-207}:\\ \;\;\;\;c \cdot b + t \cdot \left(a \cdot -4\right)\\ \mathbf{elif}\;i \leq 1.35 \cdot 10^{-68}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;i \leq 0.17:\\ \;\;\;\;t_1\\ \mathbf{elif}\;i \leq 1.42 \cdot 10^{+63}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;i \leq 5.5 \cdot 10^{+74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;i \leq 6.8 \cdot 10^{+148}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 13
Error	29.1
Cost	1368

\[\begin{array}{l} t_1 := c \cdot b + t \cdot \left(a \cdot -4\right)\\ t_2 := b \cdot c + j \cdot \left(k \cdot -27\right)\\ t_3 := c \cdot b - i \cdot \left(4 \cdot x\right)\\ \mathbf{if}\;j \leq -2.4 \cdot 10^{-13}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \leq -1.15 \cdot 10^{-137}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq -3.6 \cdot 10^{-138}:\\ \;\;\;\;18 \cdot \left(y \cdot \left(x \cdot \left(t \cdot z\right)\right)\right)\\ \mathbf{elif}\;j \leq -9.5 \cdot 10^{-288}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;j \leq 9.5 \cdot 10^{-167}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq 2.05 \cdot 10^{-112}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 14
Error	42.8
Cost	1244

\[\begin{array}{l} t_1 := \left(i \cdot x\right) \cdot -4\\ t_2 := -27 \cdot \left(k \cdot j\right)\\ \mathbf{if}\;j \leq -4.5 \cdot 10^{-13}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \leq -1.4 \cdot 10^{-164}:\\ \;\;\;\;c \cdot b\\ \mathbf{elif}\;j \leq -2.85 \cdot 10^{-205}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq -4.9 \cdot 10^{-272}:\\ \;\;\;\;c \cdot b\\ \mathbf{elif}\;j \leq -7.5 \cdot 10^{-288}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq 5.2 \cdot 10^{-167}:\\ \;\;\;\;t \cdot \left(a \cdot -4\right)\\ \mathbf{elif}\;j \leq 6 \cdot 10^{-135}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 15
Error	30.0
Cost	1236

\[\begin{array}{l} t_1 := \left(i \cdot x\right) \cdot -4\\ t_2 := c \cdot b + t \cdot \left(a \cdot -4\right)\\ t_3 := b \cdot c + j \cdot \left(k \cdot -27\right)\\ \mathbf{if}\;j \leq -2.3 \cdot 10^{-13}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;j \leq -2 \cdot 10^{-162}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \leq -6.9 \cdot 10^{-199}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq 7.8 \cdot 10^{-163}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \leq 8 \cdot 10^{-139}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 16
Error	34.9
Cost	1104

\[\begin{array}{l} t_1 := \left(i \cdot x\right) \cdot -4\\ t_2 := b \cdot c + j \cdot \left(k \cdot -27\right)\\ \mathbf{if}\;j \leq -1.05 \cdot 10^{-163}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \leq -2.55 \cdot 10^{-210}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq -1.6 \cdot 10^{-272}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \leq 8 \cdot 10^{-139}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 17
Error	42.9
Cost	716

\[\begin{array}{l} t_1 := -27 \cdot \left(k \cdot j\right)\\ \mathbf{if}\;j \leq -4.8 \cdot 10^{-13}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq -2.4 \cdot 10^{-285}:\\ \;\;\;\;c \cdot b\\ \mathbf{elif}\;j \leq 1.3 \cdot 10^{-146}:\\ \;\;\;\;t \cdot \left(a \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 18
Error	42.9
Cost	584

\[\begin{array}{l} t_1 := -27 \cdot \left(k \cdot j\right)\\ \mathbf{if}\;k \leq -2 \cdot 10^{-52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 6.5 \cdot 10^{-60}:\\ \;\;\;\;c \cdot b\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 19
Error	48.0
Cost	192

\[c \cdot b \]

Reproduce

herbie shell --seed 2023090 
(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)))