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

Average Error: 5.5 → 1.8

Time: 22.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -2.47214238657750342 \cdot 10^{-31}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - a \cdot \left(4 \cdot t\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{elif}\;t \le 1.2407054045064306 \cdot 10^{-25}:\\ \;\;\;\;\left(\left(\left(\left(x \cdot 18\right) \cdot \left(y \cdot \left(z \cdot t\right)\right) - a \cdot \left(4 \cdot t\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - a \cdot \left(4 \cdot t\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(\left(j \cdot 27\right) \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)\right) \cdot \sqrt[3]{k}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r4571 = x;
        double r4572 = 18.0;
        double r4573 = r4571 * r4572;
        double r4574 = y;
        double r4575 = r4573 * r4574;
        double r4576 = z;
        double r4577 = r4575 * r4576;
        double r4578 = t;
        double r4579 = r4577 * r4578;
        double r4580 = a;
        double r4581 = 4.0;
        double r4582 = r4580 * r4581;
        double r4583 = r4582 * r4578;
        double r4584 = r4579 - r4583;
        double r4585 = b;
        double r4586 = c;
        double r4587 = r4585 * r4586;
        double r4588 = r4584 + r4587;
        double r4589 = r4571 * r4581;
        double r4590 = i;
        double r4591 = r4589 * r4590;
        double r4592 = r4588 - r4591;
        double r4593 = j;
        double r4594 = 27.0;
        double r4595 = r4593 * r4594;
        double r4596 = k;
        double r4597 = r4595 * r4596;
        double r4598 = r4592 - r4597;
        return r4598;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r4599 = t;
        double r4600 = -2.4721423865775034e-31;
        bool r4601 = r4599 <= r4600;
        double r4602 = x;
        double r4603 = 18.0;
        double r4604 = r4602 * r4603;
        double r4605 = y;
        double r4606 = r4604 * r4605;
        double r4607 = z;
        double r4608 = r4606 * r4607;
        double r4609 = r4608 * r4599;
        double r4610 = a;
        double r4611 = 4.0;
        double r4612 = r4611 * r4599;
        double r4613 = r4610 * r4612;
        double r4614 = r4609 - r4613;
        double r4615 = b;
        double r4616 = c;
        double r4617 = r4615 * r4616;
        double r4618 = r4614 + r4617;
        double r4619 = r4602 * r4611;
        double r4620 = i;
        double r4621 = r4619 * r4620;
        double r4622 = r4618 - r4621;
        double r4623 = j;
        double r4624 = 27.0;
        double r4625 = k;
        double r4626 = r4624 * r4625;
        double r4627 = r4623 * r4626;
        double r4628 = r4622 - r4627;
        double r4629 = 1.2407054045064306e-25;
        bool r4630 = r4599 <= r4629;
        double r4631 = r4607 * r4599;
        double r4632 = r4605 * r4631;
        double r4633 = r4604 * r4632;
        double r4634 = r4633 - r4613;
        double r4635 = r4634 + r4617;
        double r4636 = r4635 - r4621;
        double r4637 = r4623 * r4624;
        double r4638 = r4637 * r4625;
        double r4639 = r4636 - r4638;
        double r4640 = cbrt(r4625);
        double r4641 = r4640 * r4640;
        double r4642 = r4637 * r4641;
        double r4643 = r4642 * r4640;
        double r4644 = r4622 - r4643;
        double r4645 = r4630 ? r4639 : r4644;
        double r4646 = r4601 ? r4628 : r4645;
        return r4646;
}

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	1.3
Herbie	1.8

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

Derivation

1. Split input into 3 regimes

2. if t < -2.4721423865775034e-31

   1. Initial program 1.8

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

   3. Applied associate-*l* 1.9

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

   5. Applied associate-*l* 1.8

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

   if -2.4721423865775034e-31 < t < 1.2407054045064306e-25

   1. Initial program 8.3

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

   3. Applied associate-*l* 8.3

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

   5. Applied associate-*l* 4.4

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

   7. Applied associate-*l* 1.7

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

   if 1.2407054045064306e-25 < t

   1. Initial program 1.7

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

   3. Applied associate-*l* 1.8

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

   5. Applied add-cube-cbrt 2.0

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

   6. Applied associate-*r* 2.0

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

4. Final simplification 1.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.47214238657750342 \cdot 10^{-31}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - a \cdot \left(4 \cdot t\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{elif}\;t \le 1.2407054045064306 \cdot 10^{-25}:\\ \;\;\;\;\left(\left(\left(\left(x \cdot 18\right) \cdot \left(y \cdot \left(z \cdot t\right)\right) - a \cdot \left(4 \cdot t\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - a \cdot \left(4 \cdot t\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(\left(j \cdot 27\right) \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)\right) \cdot \sqrt[3]{k}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(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 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4)) (- (* (* k j) 27) (* c b))) (if (< t 165.68027943805222) (+ (- (* (* 18 y) (* x (* z t))) (* (+ (* a t) (* i x)) 4)) (- (* c b) (* 27 (* k j)))) (- (- (* (* 18 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4)) (- (* (* k j) 27) (* c b)))))

  (- (- (+ (- (* (* (* (* x 18) y) z) t) (* (* a 4) t)) (* b c)) (* (* x 4) i)) (* (* j 27) k)))