Average Error: 5.3 → 4.3
Time: 18.0s
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -7.802449997962363245079599405702539273998 \cdot 10^{-129} \lor \neg \left(t \le 7.690769560303890058907581900892530132644 \cdot 10^{-134}\right):\\ \;\;\;\;\mathsf{fma}\left(t, \left(\left(y \cdot 18\right) \cdot x\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, 0 - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\ \end{array}\]
\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 -7.802449997962363245079599405702539273998 \cdot 10^{-129} \lor \neg \left(t \le 7.690769560303890058907581900892530132644 \cdot 10^{-134}\right):\\
\;\;\;\;\mathsf{fma}\left(t, \left(\left(y \cdot 18\right) \cdot x\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(t, 0 - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r953333 = x;
        double r953334 = 18.0;
        double r953335 = r953333 * r953334;
        double r953336 = y;
        double r953337 = r953335 * r953336;
        double r953338 = z;
        double r953339 = r953337 * r953338;
        double r953340 = t;
        double r953341 = r953339 * r953340;
        double r953342 = a;
        double r953343 = 4.0;
        double r953344 = r953342 * r953343;
        double r953345 = r953344 * r953340;
        double r953346 = r953341 - r953345;
        double r953347 = b;
        double r953348 = c;
        double r953349 = r953347 * r953348;
        double r953350 = r953346 + r953349;
        double r953351 = r953333 * r953343;
        double r953352 = i;
        double r953353 = r953351 * r953352;
        double r953354 = r953350 - r953353;
        double r953355 = j;
        double r953356 = 27.0;
        double r953357 = r953355 * r953356;
        double r953358 = k;
        double r953359 = r953357 * r953358;
        double r953360 = r953354 - r953359;
        return r953360;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r953361 = t;
        double r953362 = -7.802449997962363e-129;
        bool r953363 = r953361 <= r953362;
        double r953364 = 7.69076956030389e-134;
        bool r953365 = r953361 <= r953364;
        double r953366 = !r953365;
        bool r953367 = r953363 || r953366;
        double r953368 = y;
        double r953369 = 18.0;
        double r953370 = r953368 * r953369;
        double r953371 = x;
        double r953372 = r953370 * r953371;
        double r953373 = z;
        double r953374 = r953372 * r953373;
        double r953375 = a;
        double r953376 = 4.0;
        double r953377 = r953375 * r953376;
        double r953378 = r953374 - r953377;
        double r953379 = b;
        double r953380 = c;
        double r953381 = r953379 * r953380;
        double r953382 = i;
        double r953383 = r953376 * r953382;
        double r953384 = j;
        double r953385 = 27.0;
        double r953386 = r953384 * r953385;
        double r953387 = k;
        double r953388 = r953386 * r953387;
        double r953389 = fma(r953371, r953383, r953388);
        double r953390 = r953381 - r953389;
        double r953391 = fma(r953361, r953378, r953390);
        double r953392 = 0.0;
        double r953393 = r953392 - r953377;
        double r953394 = r953385 * r953387;
        double r953395 = r953384 * r953394;
        double r953396 = fma(r953371, r953383, r953395);
        double r953397 = r953381 - r953396;
        double r953398 = fma(r953361, r953393, r953397);
        double r953399 = r953367 ? r953391 : r953398;
        return r953399;
}

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

Original5.3
Target1.3
Herbie4.3
\[\begin{array}{l} \mathbf{if}\;t \lt -1.62108153975413982700795070153457058168 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \mathbf{elif}\;t \lt 165.6802794380522243500308832153677940369:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) + \left(c \cdot b - 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -7.802449997962363e-129 or 7.69076956030389e-134 < t

    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. Simplified3.1

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

    4. Applied add-cube-cbrt3.2

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

    5. Applied associate-*r*3.2

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

    7. Applied *-un-lft-identity3.2

      \[\leadsto \mathsf{fma}\left(t, \left(\left(\left(x \cdot 18\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot \sqrt[3]{y}\right) \cdot \color{blue}{\left(1 \cdot z\right)} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\]

    8. Applied associate-*r*3.2

      \[\leadsto \mathsf{fma}\left(t, \color{blue}{\left(\left(\left(\left(x \cdot 18\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot \sqrt[3]{y}\right) \cdot 1\right) \cdot z} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\]

    9. Simplified3.1

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

    if -7.802449997962363e-129 < t < 7.69076956030389e-134

    1. Initial program 9.1

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

    2. Simplified9.1

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

    4. Applied associate-*l*9.1

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

    5. Taylor expanded around 0 6.3

      \[\leadsto \mathsf{fma}\left(t, \color{blue}{0} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification4.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -7.802449997962363245079599405702539273998 \cdot 10^{-129} \lor \neg \left(t \le 7.690769560303890058907581900892530132644 \cdot 10^{-134}\right):\\ \;\;\;\;\mathsf{fma}\left(t, \left(\left(y \cdot 18\right) \cdot x\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, 0 - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 +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)))