Average Error: 5.5 → 1.0
Time: 38.9s
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot j\right) \cdot k = -\infty:\\ \;\;\;\;\mathsf{fma}\left(b, c, 18 \cdot \left(y \cdot \left(z \cdot \left(t \cdot x\right)\right)\right) - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \sqrt[3]{j} \cdot \left(\left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(k \cdot 27\right)\right)\right)\right)\\ \mathbf{elif}\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot j\right) \cdot k \le 5.292429442197061463553396675246931672241 \cdot 10^{278}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot j\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, c, 18 \cdot \left(y \cdot \left(x \cdot \left(t \cdot z\right)\right)\right) - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(k \cdot 27\right) \cdot j\right)\right)\\ \end{array}\]
\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}\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot j\right) \cdot k = -\infty:\\
\;\;\;\;\mathsf{fma}\left(b, c, 18 \cdot \left(y \cdot \left(z \cdot \left(t \cdot x\right)\right)\right) - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \sqrt[3]{j} \cdot \left(\left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(k \cdot 27\right)\right)\right)\right)\\

\mathbf{elif}\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot j\right) \cdot k \le 5.292429442197061463553396675246931672241 \cdot 10^{278}:\\
\;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot j\right) \cdot k\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(b, c, 18 \cdot \left(y \cdot \left(x \cdot \left(t \cdot z\right)\right)\right) - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(k \cdot 27\right) \cdot j\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 r32192433 = x;
        double r32192434 = 18.0;
        double r32192435 = r32192433 * r32192434;
        double r32192436 = y;
        double r32192437 = r32192435 * r32192436;
        double r32192438 = z;
        double r32192439 = r32192437 * r32192438;
        double r32192440 = t;
        double r32192441 = r32192439 * r32192440;
        double r32192442 = a;
        double r32192443 = 4.0;
        double r32192444 = r32192442 * r32192443;
        double r32192445 = r32192444 * r32192440;
        double r32192446 = r32192441 - r32192445;
        double r32192447 = b;
        double r32192448 = c;
        double r32192449 = r32192447 * r32192448;
        double r32192450 = r32192446 + r32192449;
        double r32192451 = r32192433 * r32192443;
        double r32192452 = i;
        double r32192453 = r32192451 * r32192452;
        double r32192454 = r32192450 - r32192453;
        double r32192455 = j;
        double r32192456 = 27.0;
        double r32192457 = r32192455 * r32192456;
        double r32192458 = k;
        double r32192459 = r32192457 * r32192458;
        double r32192460 = r32192454 - r32192459;
        return r32192460;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r32192461 = t;
        double r32192462 = x;
        double r32192463 = 18.0;
        double r32192464 = r32192462 * r32192463;
        double r32192465 = y;
        double r32192466 = r32192464 * r32192465;
        double r32192467 = z;
        double r32192468 = r32192466 * r32192467;
        double r32192469 = r32192461 * r32192468;
        double r32192470 = a;
        double r32192471 = 4.0;
        double r32192472 = r32192470 * r32192471;
        double r32192473 = r32192472 * r32192461;
        double r32192474 = r32192469 - r32192473;
        double r32192475 = c;
        double r32192476 = b;
        double r32192477 = r32192475 * r32192476;
        double r32192478 = r32192474 + r32192477;
        double r32192479 = r32192462 * r32192471;
        double r32192480 = i;
        double r32192481 = r32192479 * r32192480;
        double r32192482 = r32192478 - r32192481;
        double r32192483 = 27.0;
        double r32192484 = j;
        double r32192485 = r32192483 * r32192484;
        double r32192486 = k;
        double r32192487 = r32192485 * r32192486;
        double r32192488 = r32192482 - r32192487;
        double r32192489 = -inf.0;
        bool r32192490 = r32192488 <= r32192489;
        double r32192491 = r32192461 * r32192462;
        double r32192492 = r32192467 * r32192491;
        double r32192493 = r32192465 * r32192492;
        double r32192494 = r32192463 * r32192493;
        double r32192495 = r32192462 * r32192480;
        double r32192496 = fma(r32192461, r32192470, r32192495);
        double r32192497 = cbrt(r32192484);
        double r32192498 = r32192497 * r32192497;
        double r32192499 = r32192486 * r32192483;
        double r32192500 = r32192498 * r32192499;
        double r32192501 = r32192497 * r32192500;
        double r32192502 = fma(r32192471, r32192496, r32192501);
        double r32192503 = r32192494 - r32192502;
        double r32192504 = fma(r32192476, r32192475, r32192503);
        double r32192505 = 5.2924294421970615e+278;
        bool r32192506 = r32192488 <= r32192505;
        double r32192507 = r32192461 * r32192467;
        double r32192508 = r32192462 * r32192507;
        double r32192509 = r32192465 * r32192508;
        double r32192510 = r32192463 * r32192509;
        double r32192511 = r32192499 * r32192484;
        double r32192512 = fma(r32192471, r32192496, r32192511);
        double r32192513 = r32192510 - r32192512;
        double r32192514 = fma(r32192476, r32192475, r32192513);
        double r32192515 = r32192506 ? r32192488 : r32192514;
        double r32192516 = r32192490 ? r32192504 : r32192515;
        return r32192516;
}

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

Original5.5
Target1.4
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;t \lt -1.62108153975413982700795070153457058168 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \mathbf{elif}\;t \lt 165.6802794380522243500308832153677940369:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) + \left(c \cdot b - 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)) < -inf.0

    1. Initial program 64.0

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

    2. Simplified13.1

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

    4. Applied associate-*r*5.0

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

    6. Applied associate-*r*4.6

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

    8. Applied associate-*r*4.6

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

    10. Applied add-cube-cbrt4.7

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

    11. Applied associate-*r*4.7

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

    if -inf.0 < (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)) < 5.2924294421970615e+278

    1. Initial program 0.3

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

    if 5.2924294421970615e+278 < (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k))

    1. Initial program 28.5

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

    2. Simplified9.8

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

    4. Applied associate-*r*6.5

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

    6. Applied associate-*r*6.2

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

    8. Applied associate-*r*6.3

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

    10. Applied associate-*r*6.4

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot j\right) \cdot k = -\infty:\\ \;\;\;\;\mathsf{fma}\left(b, c, 18 \cdot \left(y \cdot \left(z \cdot \left(t \cdot x\right)\right)\right) - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \sqrt[3]{j} \cdot \left(\left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(k \cdot 27\right)\right)\right)\right)\\ \mathbf{elif}\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot j\right) \cdot k \le 5.292429442197061463553396675246931672241 \cdot 10^{278}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot j\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, c, 18 \cdot \left(y \cdot \left(x \cdot \left(t \cdot z\right)\right)\right) - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(k \cdot 27\right) \cdot j\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +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"

  :herbie-target
  (if (< t -1.6210815397541398e-69) (- (- (* (* 18.0 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4.0)) (- (* (* k j) 27.0) (* c b))) (if (< t 165.68027943805222) (+ (- (* (* 18.0 y) (* x (* z t))) (* (+ (* a t) (* i x)) 4.0)) (- (* c b) (* 27.0 (* k j)))) (- (- (* (* 18.0 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4.0)) (- (* (* k j) 27.0) (* c b)))))

  (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)))