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

Average Error: 5.8 → 1.0

Time: 40.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i \le -1.591976256934889034187455180953636735584 \cdot 10^{302} \lor \neg \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i \le 2.618961756211518564383950228017988163092 \cdot 10^{294}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(\left(z \cdot t\right) \cdot y\right) \cdot x, 18, \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(j \cdot 27\right) \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r456439 = x;
        double r456440 = 18.0;
        double r456441 = r456439 * r456440;
        double r456442 = y;
        double r456443 = r456441 * r456442;
        double r456444 = z;
        double r456445 = r456443 * r456444;
        double r456446 = t;
        double r456447 = r456445 * r456446;
        double r456448 = a;
        double r456449 = 4.0;
        double r456450 = r456448 * r456449;
        double r456451 = r456450 * r456446;
        double r456452 = r456447 - r456451;
        double r456453 = b;
        double r456454 = c;
        double r456455 = r456453 * r456454;
        double r456456 = r456452 + r456455;
        double r456457 = r456439 * r456449;
        double r456458 = i;
        double r456459 = r456457 * r456458;
        double r456460 = r456456 - r456459;
        double r456461 = j;
        double r456462 = 27.0;
        double r456463 = r456461 * r456462;
        double r456464 = k;
        double r456465 = r456463 * r456464;
        double r456466 = r456460 - r456465;
        return r456466;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r456467 = x;
        double r456468 = 18.0;
        double r456469 = r456467 * r456468;
        double r456470 = y;
        double r456471 = r456469 * r456470;
        double r456472 = z;
        double r456473 = r456471 * r456472;
        double r456474 = t;
        double r456475 = r456473 * r456474;
        double r456476 = a;
        double r456477 = 4.0;
        double r456478 = r456476 * r456477;
        double r456479 = r456478 * r456474;
        double r456480 = r456475 - r456479;
        double r456481 = b;
        double r456482 = c;
        double r456483 = r456481 * r456482;
        double r456484 = r456480 + r456483;
        double r456485 = r456467 * r456477;
        double r456486 = i;
        double r456487 = r456485 * r456486;
        double r456488 = r456484 - r456487;
        double r456489 = -1.591976256934889e+302;
        bool r456490 = r456488 <= r456489;
        double r456491 = 2.6189617562115186e+294;
        bool r456492 = r456488 <= r456491;
        double r456493 = !r456492;
        bool r456494 = r456490 || r456493;
        double r456495 = r456472 * r456474;
        double r456496 = r456495 * r456470;
        double r456497 = r456496 * r456467;
        double r456498 = r456467 * r456486;
        double r456499 = fma(r456474, r456476, r456498);
        double r456500 = j;
        double r456501 = 27.0;
        double r456502 = r456500 * r456501;
        double r456503 = k;
        double r456504 = r456502 * r456503;
        double r456505 = fma(r456477, r456499, r456504);
        double r456506 = -r456505;
        double r456507 = fma(r456482, r456481, r456506);
        double r456508 = fma(r456497, r456468, r456507);
        double r456509 = r456488 - r456504;
        double r456510 = r456494 ? r456508 : r456509;
        return r456510;
}

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	5.8
Target	1.6
Herbie	1.0

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

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)) < -1.591976256934889e+302 or 2.6189617562115186e+294 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i))

   1. Initial program 49.0

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

   2. Simplified 13.5

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

   4. Applied associate-*r* 7.4

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

   6. Applied associate-*l* 16.5

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

   7. Simplified 16.5

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

   9. Applied associate-*r* 6.0

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

   10. Simplified 6.0

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

   if -1.591976256934889e+302 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < 2.6189617562115186e+294

   1. Initial program 0.3

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

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i \le -1.591976256934889034187455180953636735584 \cdot 10^{302} \lor \neg \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i \le 2.618961756211518564383950228017988163092 \cdot 10^{294}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(\left(z \cdot t\right) \cdot y\right) \cdot x, 18, \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(j \cdot 27\right) \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +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 165.68027943805222) (+ (- (* (* 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)))