Average Error: 3.7 → 0.8
Time: 25.9s
Precision: 64
\[\left(x \cdot 2.0 - \left(\left(y \cdot 9.0\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27.0\right) \cdot b\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(y \cdot 9.0\right) \cdot z \le -5.677468110041499 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(27.0 \cdot a, b, x \cdot 2.0\right) - \left(\sqrt[3]{9.0} \cdot \left(\left(\left(t \cdot y\right) \cdot \sqrt[3]{9.0}\right) \cdot \sqrt[3]{9.0}\right)\right) \cdot z\\ \mathbf{elif}\;\left(y \cdot 9.0\right) \cdot z \le 3.2463793104744963 \cdot 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(27.0 \cdot a, b, x \cdot 2.0\right) - \left(t \cdot \left(z \cdot y\right)\right) \cdot 9.0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 2.0, a \cdot \left(27.0 \cdot b\right)\right) - \left(\left(\left(t \cdot y\right) \cdot \left(\sqrt[3]{9.0} \cdot \sqrt[3]{9.0}\right)\right) \cdot \sqrt[3]{9.0}\right) \cdot z\\ \end{array}\]
\left(x \cdot 2.0 - \left(\left(y \cdot 9.0\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27.0\right) \cdot b
↓
\begin{array}{l}
\mathbf{if}\;\left(y \cdot 9.0\right) \cdot z \le -5.677468110041499 \cdot 10^{+60}:\\
\;\;\;\;\mathsf{fma}\left(27.0 \cdot a, b, x \cdot 2.0\right) - \left(\sqrt[3]{9.0} \cdot \left(\left(\left(t \cdot y\right) \cdot \sqrt[3]{9.0}\right) \cdot \sqrt[3]{9.0}\right)\right) \cdot z\\

\mathbf{elif}\;\left(y \cdot 9.0\right) \cdot z \le 3.2463793104744963 \cdot 10^{+302}:\\
\;\;\;\;\mathsf{fma}\left(27.0 \cdot a, b, x \cdot 2.0\right) - \left(t \cdot \left(z \cdot y\right)\right) \cdot 9.0\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(x, 2.0, a \cdot \left(27.0 \cdot b\right)\right) - \left(\left(\left(t \cdot y\right) \cdot \left(\sqrt[3]{9.0} \cdot \sqrt[3]{9.0}\right)\right) \cdot \sqrt[3]{9.0}\right) \cdot z\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r35288258 = x;
        double r35288259 = 2.0;
        double r35288260 = r35288258 * r35288259;
        double r35288261 = y;
        double r35288262 = 9.0;
        double r35288263 = r35288261 * r35288262;
        double r35288264 = z;
        double r35288265 = r35288263 * r35288264;
        double r35288266 = t;
        double r35288267 = r35288265 * r35288266;
        double r35288268 = r35288260 - r35288267;
        double r35288269 = a;
        double r35288270 = 27.0;
        double r35288271 = r35288269 * r35288270;
        double r35288272 = b;
        double r35288273 = r35288271 * r35288272;
        double r35288274 = r35288268 + r35288273;
        return r35288274;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r35288275 = y;
        double r35288276 = 9.0;
        double r35288277 = r35288275 * r35288276;
        double r35288278 = z;
        double r35288279 = r35288277 * r35288278;
        double r35288280 = -5.677468110041499e+60;
        bool r35288281 = r35288279 <= r35288280;
        double r35288282 = 27.0;
        double r35288283 = a;
        double r35288284 = r35288282 * r35288283;
        double r35288285 = b;
        double r35288286 = x;
        double r35288287 = 2.0;
        double r35288288 = r35288286 * r35288287;
        double r35288289 = fma(r35288284, r35288285, r35288288);
        double r35288290 = cbrt(r35288276);
        double r35288291 = t;
        double r35288292 = r35288291 * r35288275;
        double r35288293 = r35288292 * r35288290;
        double r35288294 = r35288293 * r35288290;
        double r35288295 = r35288290 * r35288294;
        double r35288296 = r35288295 * r35288278;
        double r35288297 = r35288289 - r35288296;
        double r35288298 = 3.2463793104744963e+302;
        bool r35288299 = r35288279 <= r35288298;
        double r35288300 = r35288278 * r35288275;
        double r35288301 = r35288291 * r35288300;
        double r35288302 = r35288301 * r35288276;
        double r35288303 = r35288289 - r35288302;
        double r35288304 = r35288282 * r35288285;
        double r35288305 = r35288283 * r35288304;
        double r35288306 = fma(r35288286, r35288287, r35288305);
        double r35288307 = r35288290 * r35288290;
        double r35288308 = r35288292 * r35288307;
        double r35288309 = r35288308 * r35288290;
        double r35288310 = r35288309 * r35288278;
        double r35288311 = r35288306 - r35288310;
        double r35288312 = r35288299 ? r35288303 : r35288311;
        double r35288313 = r35288281 ? r35288297 : r35288312;
        return r35288313;
}

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

Original3.7
Target2.6
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;y \lt 7.590524218811189 \cdot 10^{-161}:\\ \;\;\;\;\left(x \cdot 2.0 - \left(\left(y \cdot 9.0\right) \cdot z\right) \cdot t\right) + a \cdot \left(27.0 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2.0 - 9.0 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27.0\right) \cdot b\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* (* y 9.0) z) < -5.677468110041499e+60

    1. Initial program 10.8

      \[\left(x \cdot 2.0 - \left(\left(y \cdot 9.0\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27.0\right) \cdot b\]

    2. Simplified3.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(27.0 \cdot a, b, x \cdot 2.0\right) - z \cdot \left(\left(t \cdot y\right) \cdot 9.0\right)}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt3.1

      \[\leadsto \mathsf{fma}\left(27.0 \cdot a, b, x \cdot 2.0\right) - z \cdot \left(\left(t \cdot y\right) \cdot \color{blue}{\left(\left(\sqrt[3]{9.0} \cdot \sqrt[3]{9.0}\right) \cdot \sqrt[3]{9.0}\right)}\right)\]

    5. Applied associate-*r*3.1

      \[\leadsto \mathsf{fma}\left(27.0 \cdot a, b, x \cdot 2.0\right) - z \cdot \color{blue}{\left(\left(\left(t \cdot y\right) \cdot \left(\sqrt[3]{9.0} \cdot \sqrt[3]{9.0}\right)\right) \cdot \sqrt[3]{9.0}\right)}\]
    6. Using strategy rm

    7. Applied associate-*r*3.1

      \[\leadsto \mathsf{fma}\left(27.0 \cdot a, b, x \cdot 2.0\right) - z \cdot \left(\color{blue}{\left(\left(\left(t \cdot y\right) \cdot \sqrt[3]{9.0}\right) \cdot \sqrt[3]{9.0}\right)} \cdot \sqrt[3]{9.0}\right)\]

    if -5.677468110041499e+60 < (* (* y 9.0) z) < 3.2463793104744963e+302

    1. Initial program 0.4

      \[\left(x \cdot 2.0 - \left(\left(y \cdot 9.0\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27.0\right) \cdot b\]

    2. Simplified3.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(27.0 \cdot a, b, x \cdot 2.0\right) - z \cdot \left(\left(t \cdot y\right) \cdot 9.0\right)}\]

    3. Taylor expanded around inf 0.4

      \[\leadsto \mathsf{fma}\left(27.0 \cdot a, b, x \cdot 2.0\right) - \color{blue}{9.0 \cdot \left(t \cdot \left(z \cdot y\right)\right)}\]

    if 3.2463793104744963e+302 < (* (* y 9.0) z)

    1. Initial program 58.0

      \[\left(x \cdot 2.0 - \left(\left(y \cdot 9.0\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27.0\right) \cdot b\]

    2. Simplified0.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(27.0 \cdot a, b, x \cdot 2.0\right) - z \cdot \left(\left(t \cdot y\right) \cdot 9.0\right)}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt0.8

      \[\leadsto \mathsf{fma}\left(27.0 \cdot a, b, x \cdot 2.0\right) - z \cdot \left(\left(t \cdot y\right) \cdot \color{blue}{\left(\left(\sqrt[3]{9.0} \cdot \sqrt[3]{9.0}\right) \cdot \sqrt[3]{9.0}\right)}\right)\]

    5. Applied associate-*r*0.9

      \[\leadsto \mathsf{fma}\left(27.0 \cdot a, b, x \cdot 2.0\right) - z \cdot \color{blue}{\left(\left(\left(t \cdot y\right) \cdot \left(\sqrt[3]{9.0} \cdot \sqrt[3]{9.0}\right)\right) \cdot \sqrt[3]{9.0}\right)}\]

    6. Taylor expanded around 0 0.6

      \[\leadsto \color{blue}{\left(2.0 \cdot x + 27.0 \cdot \left(a \cdot b\right)\right)} - z \cdot \left(\left(\left(t \cdot y\right) \cdot \left(\sqrt[3]{9.0} \cdot \sqrt[3]{9.0}\right)\right) \cdot \sqrt[3]{9.0}\right)\]

    7. Simplified0.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 2.0, a \cdot \left(b \cdot 27.0\right)\right)} - z \cdot \left(\left(\left(t \cdot y\right) \cdot \left(\sqrt[3]{9.0} \cdot \sqrt[3]{9.0}\right)\right) \cdot \sqrt[3]{9.0}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot 9.0\right) \cdot z \le -5.677468110041499 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(27.0 \cdot a, b, x \cdot 2.0\right) - \left(\sqrt[3]{9.0} \cdot \left(\left(\left(t \cdot y\right) \cdot \sqrt[3]{9.0}\right) \cdot \sqrt[3]{9.0}\right)\right) \cdot z\\ \mathbf{elif}\;\left(y \cdot 9.0\right) \cdot z \le 3.2463793104744963 \cdot 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(27.0 \cdot a, b, x \cdot 2.0\right) - \left(t \cdot \left(z \cdot y\right)\right) \cdot 9.0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 2.0, a \cdot \left(27.0 \cdot b\right)\right) - \left(\left(\left(t \cdot y\right) \cdot \left(\sqrt[3]{9.0} \cdot \sqrt[3]{9.0}\right)\right) \cdot \sqrt[3]{9.0}\right) \cdot z\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +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)))