Average Error: 3.8 → 1.2
Time: 17.5s
Precision: 64
\[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -5.710514206256463262912944743704297947195 \cdot 10^{-143} \lor \neg \left(t \le 3.266284109707386469243141298434361497441 \cdot 10^{-36}\right):\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(\left(\left(27 \cdot b\right) \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a} - t \cdot \left(\left(9 \cdot y\right) \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(27 \cdot b\right) \cdot a - y \cdot \left(t \cdot \left(9 \cdot z\right)\right)\right)\\ \end{array}\]
\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b
↓
\begin{array}{l}
\mathbf{if}\;t \le -5.710514206256463262912944743704297947195 \cdot 10^{-143} \lor \neg \left(t \le 3.266284109707386469243141298434361497441 \cdot 10^{-36}\right):\\
\;\;\;\;\mathsf{fma}\left(2, x, \left(\left(\left(27 \cdot b\right) \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a} - t \cdot \left(\left(9 \cdot y\right) \cdot z\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(2, x, \left(27 \cdot b\right) \cdot a - y \cdot \left(t \cdot \left(9 \cdot z\right)\right)\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r479268 = x;
        double r479269 = 2.0;
        double r479270 = r479268 * r479269;
        double r479271 = y;
        double r479272 = 9.0;
        double r479273 = r479271 * r479272;
        double r479274 = z;
        double r479275 = r479273 * r479274;
        double r479276 = t;
        double r479277 = r479275 * r479276;
        double r479278 = r479270 - r479277;
        double r479279 = a;
        double r479280 = 27.0;
        double r479281 = r479279 * r479280;
        double r479282 = b;
        double r479283 = r479281 * r479282;
        double r479284 = r479278 + r479283;
        return r479284;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r479285 = t;
        double r479286 = -5.710514206256463e-143;
        bool r479287 = r479285 <= r479286;
        double r479288 = 3.2662841097073865e-36;
        bool r479289 = r479285 <= r479288;
        double r479290 = !r479289;
        bool r479291 = r479287 || r479290;
        double r479292 = 2.0;
        double r479293 = x;
        double r479294 = 27.0;
        double r479295 = b;
        double r479296 = r479294 * r479295;
        double r479297 = a;
        double r479298 = cbrt(r479297);
        double r479299 = r479296 * r479298;
        double r479300 = r479299 * r479298;
        double r479301 = r479300 * r479298;
        double r479302 = 9.0;
        double r479303 = y;
        double r479304 = r479302 * r479303;
        double r479305 = z;
        double r479306 = r479304 * r479305;
        double r479307 = r479285 * r479306;
        double r479308 = r479301 - r479307;
        double r479309 = fma(r479292, r479293, r479308);
        double r479310 = r479296 * r479297;
        double r479311 = r479302 * r479305;
        double r479312 = r479285 * r479311;
        double r479313 = r479303 * r479312;
        double r479314 = r479310 - r479313;
        double r479315 = fma(r479292, r479293, r479314);
        double r479316 = r479291 ? r479309 : r479315;
        return r479316;
}

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

Target

Original3.8
Target2.8
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;y \lt 7.590524218811188954625810696587370427881 \cdot 10^{-161}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - 9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -5.710514206256463e-143 or 3.2662841097073865e-36 < t

    1. Initial program 1.3

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]

    2. Simplified1.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, \left(27 \cdot b\right) \cdot a - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt1.6

      \[\leadsto \mathsf{fma}\left(2, x, \left(27 \cdot b\right) \cdot \color{blue}{\left(\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}\right)} - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\]

    5. Applied associate-*r*1.6

      \[\leadsto \mathsf{fma}\left(2, x, \color{blue}{\left(\left(27 \cdot b\right) \cdot \left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right)\right) \cdot \sqrt[3]{a}} - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\]

    6. Simplified1.6

      \[\leadsto \mathsf{fma}\left(2, x, \color{blue}{\left(\sqrt[3]{a} \cdot \left(\sqrt[3]{a} \cdot \left(27 \cdot b\right)\right)\right)} \cdot \sqrt[3]{a} - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\]

    if -5.710514206256463e-143 < t < 3.2662841097073865e-36

    1. Initial program 7.4

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]

    2. Simplified7.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, \left(27 \cdot b\right) \cdot a - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)}\]
    3. Using strategy rm

    4. Applied associate-*l*0.7

      \[\leadsto \mathsf{fma}\left(2, x, \left(27 \cdot b\right) \cdot a - \color{blue}{\left(y \cdot 9\right) \cdot \left(z \cdot t\right)}\right)\]
    5. Using strategy rm

    6. Applied associate-*l*0.5

      \[\leadsto \mathsf{fma}\left(2, x, \left(27 \cdot b\right) \cdot a - \color{blue}{y \cdot \left(9 \cdot \left(z \cdot t\right)\right)}\right)\]

    7. Simplified0.6

      \[\leadsto \mathsf{fma}\left(2, x, \left(27 \cdot b\right) \cdot a - y \cdot \color{blue}{\left(\left(z \cdot 9\right) \cdot t\right)}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -5.710514206256463262912944743704297947195 \cdot 10^{-143} \lor \neg \left(t \le 3.266284109707386469243141298434361497441 \cdot 10^{-36}\right):\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(\left(\left(27 \cdot b\right) \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a} - t \cdot \left(\left(9 \cdot y\right) \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(27 \cdot b\right) \cdot a - y \cdot \left(t \cdot \left(9 \cdot z\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, A"

  :herbie-target
  (if (< y 7.590524218811189e-161) (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* a (* 27.0 b))) (+ (- (* x 2.0) (* 9.0 (* y (* t z)))) (* (* a 27.0) b)))

  (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* (* a 27.0) b)))