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

Average Error: 5.5 → 4.2

Time: 6.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.2254010690602987 \cdot 10^{95} \lor \neg \left(x \le 1.9490316857655669 \cdot 10^{160}\right):\\ \;\;\;\;t \cdot \left(18 \cdot \left(x \cdot \left(z \cdot y\right)\right) - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(\sqrt[3]{j} \cdot \left(\left(\sqrt[3]{\sqrt[3]{j}} \cdot \sqrt[3]{\sqrt[3]{j}}\right) \cdot \sqrt[3]{\sqrt[3]{j}}\right)\right) \cdot \left(\sqrt[3]{j} \cdot \left(27 \cdot k\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\left(x \cdot \left(18 \cdot y\right)\right) \cdot z - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\\ \end{array}\]

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 (((x <= -1.2254010690602987e+95) || !(x <= 1.949031685765567e+160))) {
		VAR = ((double) (((double) (t * ((double) (((double) (18.0 * ((double) (x * ((double) (z * y)))))) - ((double) (a * 4.0)))))) + ((double) (((double) (b * c)) - ((double) (((double) (((double) (x * 4.0)) * i)) + ((double) (((double) (((double) cbrt(j)) * ((double) (((double) (((double) cbrt(((double) cbrt(j)))) * ((double) cbrt(((double) cbrt(j)))))) * ((double) cbrt(((double) cbrt(j)))))))) * ((double) (((double) cbrt(j)) * ((double) (27.0 * k))))))))))));
	} else {
		VAR = ((double) (((double) (t * ((double) (((double) (((double) (x * ((double) (18.0 * y)))) * z)) - ((double) (a * 4.0)))))) + ((double) (((double) (b * c)) - ((double) (((double) (((double) (x * 4.0)) * i)) + ((double) (((double) (j * 27.0)) * k))))))));
	}
	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	5.5
Target	1.7
Herbie	4.2

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

Derivation

1. Split input into 2 regimes

2. if x < -1.2254010690602987e+95 or 1.949031685765567e+160 < x

   1. Initial program 17.3

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

   2. Simplified 17.3

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

   3. Taylor expanded around inf 9.3

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

   5. Applied associate-*l* 9.3

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

   7. Applied add-cube-cbrt 9.5

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

   8. Applied associate-*l* 9.5

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

   10. Applied add-cube-cbrt 9.5

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

   if -1.2254010690602987e+95 < x < 1.949031685765567e+160

   1. Initial program 3.0

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

   2. Simplified 3.0

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

   4. Applied associate-*l* 3.1

      \[\leadsto t \cdot \left(\color{blue}{\left(x \cdot \left(18 \cdot y\right)\right)} \cdot z - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 4.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.2254010690602987 \cdot 10^{95} \lor \neg \left(x \le 1.9490316857655669 \cdot 10^{160}\right):\\ \;\;\;\;t \cdot \left(18 \cdot \left(x \cdot \left(z \cdot y\right)\right) - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(\sqrt[3]{j} \cdot \left(\left(\sqrt[3]{\sqrt[3]{j}} \cdot \sqrt[3]{\sqrt[3]{j}}\right) \cdot \sqrt[3]{\sqrt[3]{j}}\right)\right) \cdot \left(\sqrt[3]{j} \cdot \left(27 \cdot k\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\left(x \cdot \left(18 \cdot y\right)\right) \cdot z - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020140 
(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)))