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

Average Error: 5.2 → 2.4

Time: 27.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -2.33418308625431402 \cdot 10^{121}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right)\right) - t \cdot \left(a \cdot 4\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{elif}\;t \le 3849175216108120570000:\\ \;\;\;\;\left(\left(b \cdot c + \left(x \cdot \left(\left(18 \cdot y\right) \cdot \left(t \cdot z\right)\right) - t \cdot \left(a \cdot 4\right)\right)\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(t \cdot \left(z \cdot \left(\left(x \cdot 18\right) \cdot y\right)\right) - t \cdot \left(a \cdot 4\right)\right)\right) - \left(x \cdot 4\right) \cdot i\right) - \sqrt[3]{k} \cdot \left(j \cdot \left(27 \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)\right)\right)\\ \end{array}\]

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	return ((double) (((double) (((double) (((double) (((double) (((double) (((double) (((double) (x * 18.0)) * y)) * z)) * t)) - ((double) (((double) (a * 4.0)) * t)))) + ((double) (b * c)))) - ((double) (((double) (x * 4.0)) * i)))) - ((double) (((double) (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 VAR;
	if ((t <= -2.334183086254314e+121)) {
		VAR = ((double) (((double) (((double) (((double) (((double) (t * ((double) (((double) (x * 18.0)) * ((double) (y * z)))))) - ((double) (t * ((double) (a * 4.0)))))) + ((double) (b * c)))) - ((double) (((double) (x * 4.0)) * i)))) - ((double) (((double) (j * 27.0)) * k))));
	} else {
		double VAR_1;
		if ((t <= 3.8491752161081206e+21)) {
			VAR_1 = ((double) (((double) (((double) (((double) (b * c)) + ((double) (((double) (x * ((double) (((double) (18.0 * y)) * ((double) (t * z)))))) - ((double) (t * ((double) (a * 4.0)))))))) - ((double) (((double) (x * 4.0)) * i)))) - ((double) (((double) (j * 27.0)) * k))));
		} else {
			VAR_1 = ((double) (((double) (((double) (((double) (b * c)) + ((double) (((double) (t * ((double) (z * ((double) (((double) (x * 18.0)) * y)))))) - ((double) (t * ((double) (a * 4.0)))))))) - ((double) (((double) (x * 4.0)) * i)))) - ((double) (((double) cbrt(k)) * ((double) (j * ((double) (27.0 * ((double) (((double) cbrt(k)) * ((double) cbrt(k))))))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus i

Bits error versus j

Bits error versus k

Target

Original	5.2
Target	1.6
Herbie	2.4

\[\begin{array}{l} \mathbf{if}\;t \lt -1.6210815397541398 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \mathbf{elif}\;t \lt 165.680279438052224:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) + \left(c \cdot b - 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if t < -2.33418308625431402e121

   1. Initial program 1.3

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

   3. Applied associate-*l* 1.2

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

   if -2.33418308625431402e121 < t < 3849175216108120570000

   1. Initial program 6.4

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

   2. Taylor expanded around inf 7.4

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

   3. Simplified 2.6

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

   if 3849175216108120570000 < t

   1. Initial program 1.9

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

   3. Applied add-cube-cbrt 2.0

      \[\leadsto \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot \color{blue}{\left(\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right) \cdot \sqrt[3]{k}\right)}\]

   4. Applied associate-*r* 2.0

      \[\leadsto \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \color{blue}{\left(\left(j \cdot 27\right) \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)\right) \cdot \sqrt[3]{k}}\]

   5. Simplified 1.9

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

4. Final simplification 2.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.33418308625431402 \cdot 10^{121}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right)\right) - t \cdot \left(a \cdot 4\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{elif}\;t \le 3849175216108120570000:\\ \;\;\;\;\left(\left(b \cdot c + \left(x \cdot \left(\left(18 \cdot y\right) \cdot \left(t \cdot z\right)\right) - t \cdot \left(a \cdot 4\right)\right)\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(t \cdot \left(z \cdot \left(\left(x \cdot 18\right) \cdot y\right)\right) - t \cdot \left(a \cdot 4\right)\right)\right) - \left(x \cdot 4\right) \cdot i\right) - \sqrt[3]{k} \cdot \left(j \cdot \left(27 \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(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)))