Average Error: 5.7 → 4.8
Time: 18.6s
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.816104989340000263533513256286904752193 \cdot 10^{-100}:\\ \;\;\;\;\mathsf{fma}\left(t, \left(x \cdot 18\right) \cdot \left(y \cdot z\right) - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{elif}\;t \le 1.945396783892281696257471720471681450496 \cdot 10^{-140}:\\ \;\;\;\;\mathsf{fma}\left(t, 0 - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \left(x \cdot \left(18 \cdot y\right)\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \end{array}\]
\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 -1.816104989340000263533513256286904752193 \cdot 10^{-100}:\\
\;\;\;\;\mathsf{fma}\left(t, \left(x \cdot 18\right) \cdot \left(y \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)\\

\mathbf{elif}\;t \le 1.945396783892281696257471720471681450496 \cdot 10^{-140}:\\
\;\;\;\;\mathsf{fma}\left(t, 0 - 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, \left(x \cdot \left(18 \cdot 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)\\

\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 r736368 = x;
        double r736369 = 18.0;
        double r736370 = r736368 * r736369;
        double r736371 = y;
        double r736372 = r736370 * r736371;
        double r736373 = z;
        double r736374 = r736372 * r736373;
        double r736375 = t;
        double r736376 = r736374 * r736375;
        double r736377 = a;
        double r736378 = 4.0;
        double r736379 = r736377 * r736378;
        double r736380 = r736379 * r736375;
        double r736381 = r736376 - r736380;
        double r736382 = b;
        double r736383 = c;
        double r736384 = r736382 * r736383;
        double r736385 = r736381 + r736384;
        double r736386 = r736368 * r736378;
        double r736387 = i;
        double r736388 = r736386 * r736387;
        double r736389 = r736385 - r736388;
        double r736390 = j;
        double r736391 = 27.0;
        double r736392 = r736390 * r736391;
        double r736393 = k;
        double r736394 = r736392 * r736393;
        double r736395 = r736389 - r736394;
        return r736395;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r736396 = t;
        double r736397 = -1.8161049893400003e-100;
        bool r736398 = r736396 <= r736397;
        double r736399 = x;
        double r736400 = 18.0;
        double r736401 = r736399 * r736400;
        double r736402 = y;
        double r736403 = z;
        double r736404 = r736402 * r736403;
        double r736405 = r736401 * r736404;
        double r736406 = a;
        double r736407 = 4.0;
        double r736408 = r736406 * r736407;
        double r736409 = r736405 - r736408;
        double r736410 = b;
        double r736411 = c;
        double r736412 = r736410 * r736411;
        double r736413 = i;
        double r736414 = r736407 * r736413;
        double r736415 = j;
        double r736416 = 27.0;
        double r736417 = r736415 * r736416;
        double r736418 = k;
        double r736419 = r736417 * r736418;
        double r736420 = fma(r736399, r736414, r736419);
        double r736421 = r736412 - r736420;
        double r736422 = fma(r736396, r736409, r736421);
        double r736423 = 1.9453967838922817e-140;
        bool r736424 = r736396 <= r736423;
        double r736425 = 0.0;
        double r736426 = r736425 - r736408;
        double r736427 = fma(r736396, r736426, r736421);
        double r736428 = r736400 * r736402;
        double r736429 = r736399 * r736428;
        double r736430 = r736429 * r736403;
        double r736431 = r736430 - r736408;
        double r736432 = fma(r736396, r736431, r736421);
        double r736433 = r736424 ? r736427 : r736432;
        double r736434 = r736398 ? r736422 : r736433;
        return r736434;
}

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

Original5.7
Target2.0
Herbie4.8
\[\begin{array}{l} \mathbf{if}\;t \lt -1.62108153975413982700795070153457058168 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \mathbf{elif}\;t \lt 165.6802794380522243500308832153677940369:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) + \left(c \cdot b - 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if t < -1.8161049893400003e-100

    1. Initial program 3.3

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

    2. Simplified3.3

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

    4. Applied associate-*l*3.7

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

    if -1.8161049893400003e-100 < t < 1.9453967838922817e-140

    1. Initial program 8.9

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

    2. Simplified8.9

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

    3. Taylor expanded around 0 6.2

      \[\leadsto \mathsf{fma}\left(t, \color{blue}{0} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\]

    if 1.9453967838922817e-140 < t

    1. Initial program 3.8

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

    2. Simplified3.8

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

    4. Applied associate-*l*3.9

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

  4. Final simplification4.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.816104989340000263533513256286904752193 \cdot 10^{-100}:\\ \;\;\;\;\mathsf{fma}\left(t, \left(x \cdot 18\right) \cdot \left(y \cdot z\right) - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{elif}\;t \le 1.945396783892281696257471720471681450496 \cdot 10^{-140}:\\ \;\;\;\;\mathsf{fma}\left(t, 0 - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \left(x \cdot \left(18 \cdot y\right)\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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)))