Average Error: 5.0 → 5.1
Time: 44.7s
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
↓
\[\mathsf{fma}\left(z \cdot \left(\left(\sqrt{18.0} \cdot \left(x \cdot y\right)\right) \cdot \sqrt{18.0}\right) - a \cdot 4.0, t, b \cdot c - \mathsf{fma}\left(k, 27.0 \cdot j, \left(x \cdot 4.0\right) \cdot i\right)\right)\]
\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k
↓
\mathsf{fma}\left(z \cdot \left(\left(\sqrt{18.0} \cdot \left(x \cdot y\right)\right) \cdot \sqrt{18.0}\right) - a \cdot 4.0, t, b \cdot c - \mathsf{fma}\left(k, 27.0 \cdot j, \left(x \cdot 4.0\right) \cdot i\right)\right)
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r3818425 = x;
        double r3818426 = 18.0;
        double r3818427 = r3818425 * r3818426;
        double r3818428 = y;
        double r3818429 = r3818427 * r3818428;
        double r3818430 = z;
        double r3818431 = r3818429 * r3818430;
        double r3818432 = t;
        double r3818433 = r3818431 * r3818432;
        double r3818434 = a;
        double r3818435 = 4.0;
        double r3818436 = r3818434 * r3818435;
        double r3818437 = r3818436 * r3818432;
        double r3818438 = r3818433 - r3818437;
        double r3818439 = b;
        double r3818440 = c;
        double r3818441 = r3818439 * r3818440;
        double r3818442 = r3818438 + r3818441;
        double r3818443 = r3818425 * r3818435;
        double r3818444 = i;
        double r3818445 = r3818443 * r3818444;
        double r3818446 = r3818442 - r3818445;
        double r3818447 = j;
        double r3818448 = 27.0;
        double r3818449 = r3818447 * r3818448;
        double r3818450 = k;
        double r3818451 = r3818449 * r3818450;
        double r3818452 = r3818446 - r3818451;
        return r3818452;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r3818453 = z;
        double r3818454 = 18.0;
        double r3818455 = sqrt(r3818454);
        double r3818456 = x;
        double r3818457 = y;
        double r3818458 = r3818456 * r3818457;
        double r3818459 = r3818455 * r3818458;
        double r3818460 = r3818459 * r3818455;
        double r3818461 = r3818453 * r3818460;
        double r3818462 = a;
        double r3818463 = 4.0;
        double r3818464 = r3818462 * r3818463;
        double r3818465 = r3818461 - r3818464;
        double r3818466 = t;
        double r3818467 = b;
        double r3818468 = c;
        double r3818469 = r3818467 * r3818468;
        double r3818470 = k;
        double r3818471 = 27.0;
        double r3818472 = j;
        double r3818473 = r3818471 * r3818472;
        double r3818474 = r3818456 * r3818463;
        double r3818475 = i;
        double r3818476 = r3818474 * r3818475;
        double r3818477 = fma(r3818470, r3818473, r3818476);
        double r3818478 = r3818469 - r3818477;
        double r3818479 = fma(r3818465, r3818466, r3818478);
        return r3818479;
}

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

Derivation

  1. Initial program 5.0

    \[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

  2. Simplified5.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z - a \cdot 4.0, t, c \cdot b - \mathsf{fma}\left(k, 27.0 \cdot j, \left(x \cdot 4.0\right) \cdot i\right)\right)}\]

  3. Taylor expanded around 0 5.0

    \[\leadsto \mathsf{fma}\left(\color{blue}{\left(18.0 \cdot \left(x \cdot y\right)\right)} \cdot z - a \cdot 4.0, t, c \cdot b - \mathsf{fma}\left(k, 27.0 \cdot j, \left(x \cdot 4.0\right) \cdot i\right)\right)\]
  4. Using strategy rm

  5. Applied add-sqr-sqrt5.1

    \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\sqrt{18.0} \cdot \sqrt{18.0}\right)} \cdot \left(x \cdot y\right)\right) \cdot z - a \cdot 4.0, t, c \cdot b - \mathsf{fma}\left(k, 27.0 \cdot j, \left(x \cdot 4.0\right) \cdot i\right)\right)\]

  6. Applied associate-*l*5.1

    \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\sqrt{18.0} \cdot \left(\sqrt{18.0} \cdot \left(x \cdot y\right)\right)\right)} \cdot z - a \cdot 4.0, t, c \cdot b - \mathsf{fma}\left(k, 27.0 \cdot j, \left(x \cdot 4.0\right) \cdot i\right)\right)\]

  7. Final simplification5.1

    \[\leadsto \mathsf{fma}\left(z \cdot \left(\left(\sqrt{18.0} \cdot \left(x \cdot y\right)\right) \cdot \sqrt{18.0}\right) - a \cdot 4.0, t, b \cdot c - \mathsf{fma}\left(k, 27.0 \cdot j, \left(x \cdot 4.0\right) \cdot i\right)\right)\]

Reproduce

herbie shell --seed 2019135 +o rules:numerics
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1"
  (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)))