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

Average Error: 5.6 → 1.8

Time: 30.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i = -\infty \lor \neg \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i \le 1.191425524707375816183030560675713891259 \cdot 10^{284}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(t \cdot y\right) \cdot \left(z \cdot x\right), 18, \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(j \cdot 27\right) \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(i \cdot x, -4, \mathsf{fma}\left(b, c, t \cdot \left(\left(18 \cdot \left(x \cdot y\right)\right) \cdot z - a \cdot 4\right)\right)\right) + \left(x \cdot 4\right) \cdot \left(\left(-i\right) + i\right)\right) - j \cdot \left(27 \cdot k\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r431472 = x;
        double r431473 = 18.0;
        double r431474 = r431472 * r431473;
        double r431475 = y;
        double r431476 = r431474 * r431475;
        double r431477 = z;
        double r431478 = r431476 * r431477;
        double r431479 = t;
        double r431480 = r431478 * r431479;
        double r431481 = a;
        double r431482 = 4.0;
        double r431483 = r431481 * r431482;
        double r431484 = r431483 * r431479;
        double r431485 = r431480 - r431484;
        double r431486 = b;
        double r431487 = c;
        double r431488 = r431486 * r431487;
        double r431489 = r431485 + r431488;
        double r431490 = r431472 * r431482;
        double r431491 = i;
        double r431492 = r431490 * r431491;
        double r431493 = r431489 - r431492;
        double r431494 = j;
        double r431495 = 27.0;
        double r431496 = r431494 * r431495;
        double r431497 = k;
        double r431498 = r431496 * r431497;
        double r431499 = r431493 - r431498;
        return r431499;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r431500 = x;
        double r431501 = 18.0;
        double r431502 = r431500 * r431501;
        double r431503 = y;
        double r431504 = r431502 * r431503;
        double r431505 = z;
        double r431506 = r431504 * r431505;
        double r431507 = t;
        double r431508 = r431506 * r431507;
        double r431509 = a;
        double r431510 = 4.0;
        double r431511 = r431509 * r431510;
        double r431512 = r431511 * r431507;
        double r431513 = r431508 - r431512;
        double r431514 = b;
        double r431515 = c;
        double r431516 = r431514 * r431515;
        double r431517 = r431513 + r431516;
        double r431518 = r431500 * r431510;
        double r431519 = i;
        double r431520 = r431518 * r431519;
        double r431521 = r431517 - r431520;
        double r431522 = -inf.0;
        bool r431523 = r431521 <= r431522;
        double r431524 = 1.1914255247073758e+284;
        bool r431525 = r431521 <= r431524;
        double r431526 = !r431525;
        bool r431527 = r431523 || r431526;
        double r431528 = r431507 * r431503;
        double r431529 = r431505 * r431500;
        double r431530 = r431528 * r431529;
        double r431531 = r431500 * r431519;
        double r431532 = fma(r431507, r431509, r431531);
        double r431533 = j;
        double r431534 = 27.0;
        double r431535 = r431533 * r431534;
        double r431536 = k;
        double r431537 = r431535 * r431536;
        double r431538 = fma(r431510, r431532, r431537);
        double r431539 = -r431538;
        double r431540 = fma(r431515, r431514, r431539);
        double r431541 = fma(r431530, r431501, r431540);
        double r431542 = r431519 * r431500;
        double r431543 = -r431510;
        double r431544 = r431500 * r431503;
        double r431545 = r431501 * r431544;
        double r431546 = r431545 * r431505;
        double r431547 = r431546 - r431511;
        double r431548 = r431507 * r431547;
        double r431549 = fma(r431514, r431515, r431548);
        double r431550 = fma(r431542, r431543, r431549);
        double r431551 = -r431519;
        double r431552 = r431551 + r431519;
        double r431553 = r431518 * r431552;
        double r431554 = r431550 + r431553;
        double r431555 = r431534 * r431536;
        double r431556 = r431533 * r431555;
        double r431557 = r431554 - r431556;
        double r431558 = r431527 ? r431541 : r431557;
        return r431558;
}

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	5.6
Target	1.5
Herbie	1.8

\[\begin{array}{l} \mathbf{if}\;t \lt -1.62108153975413982700795070153457058168 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \mathbf{elif}\;t \lt 165.6802794380522243500308832153677940369:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) + \left(c \cdot b - 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < -inf.0 or 1.1914255247073758e+284 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i))

   1. Initial program 46.8

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

   2. Simplified 13.0

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

   if -inf.0 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < 1.1914255247073758e+284

   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. Using strategy rm

   3. Applied associate-*l* 0.4

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

   5. Applied associate-*l* 3.3

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

   6. Simplified 3.3

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

   8. Applied add-sqr-sqrt 34.3

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

   9. Applied prod-diff 34.3

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

   10. Simplified 0.4

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

   11. Simplified 0.4

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

4. Final simplification 1.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i = -\infty \lor \neg \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i \le 1.191425524707375816183030560675713891259 \cdot 10^{284}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(t \cdot y\right) \cdot \left(z \cdot x\right), 18, \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(j \cdot 27\right) \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(i \cdot x, -4, \mathsf{fma}\left(b, c, t \cdot \left(\left(18 \cdot \left(x \cdot y\right)\right) \cdot z - a \cdot 4\right)\right)\right) + \left(x \cdot 4\right) \cdot \left(\left(-i\right) + i\right)\right) - j \cdot \left(27 \cdot k\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(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 165.680279438052224) (+ (- (* (* 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)))