Average Error: 7.5 → 4.9
Time: 22.1s
Precision: 64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -1.925359839423540310793648918342674435965 \cdot 10^{180} \lor \neg \left(a \le 1.301701073128679922027601299899353444017 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{x}{2} \cdot \frac{y}{a} - t \cdot \frac{9 \cdot \frac{z}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(z, 9 \cdot \left(-t\right), y \cdot x\right)}{a}}{2}\\ \end{array}\]
\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
\begin{array}{l}
\mathbf{if}\;a \le -1.925359839423540310793648918342674435965 \cdot 10^{180} \lor \neg \left(a \le 1.301701073128679922027601299899353444017 \cdot 10^{-25}\right):\\
\;\;\;\;\frac{x}{2} \cdot \frac{y}{a} - t \cdot \frac{9 \cdot \frac{z}{2}}{a}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r688515 = x;
        double r688516 = y;
        double r688517 = r688515 * r688516;
        double r688518 = z;
        double r688519 = 9.0;
        double r688520 = r688518 * r688519;
        double r688521 = t;
        double r688522 = r688520 * r688521;
        double r688523 = r688517 - r688522;
        double r688524 = a;
        double r688525 = 2.0;
        double r688526 = r688524 * r688525;
        double r688527 = r688523 / r688526;
        return r688527;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r688528 = a;
        double r688529 = -1.9253598394235403e+180;
        bool r688530 = r688528 <= r688529;
        double r688531 = 1.30170107312868e-25;
        bool r688532 = r688528 <= r688531;
        double r688533 = !r688532;
        bool r688534 = r688530 || r688533;
        double r688535 = x;
        double r688536 = 2.0;
        double r688537 = r688535 / r688536;
        double r688538 = y;
        double r688539 = r688538 / r688528;
        double r688540 = r688537 * r688539;
        double r688541 = t;
        double r688542 = 9.0;
        double r688543 = z;
        double r688544 = r688543 / r688536;
        double r688545 = r688542 * r688544;
        double r688546 = r688545 / r688528;
        double r688547 = r688541 * r688546;
        double r688548 = r688540 - r688547;
        double r688549 = -r688541;
        double r688550 = r688542 * r688549;
        double r688551 = r688538 * r688535;
        double r688552 = fma(r688543, r688550, r688551);
        double r688553 = r688552 / r688528;
        double r688554 = r688553 / r688536;
        double r688555 = r688534 ? r688548 : r688554;
        return r688555;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original7.5
Target5.5
Herbie4.9
\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709043451944897028999329376 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.144030707833976090627817222818061808815 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if a < -1.9253598394235403e+180 or 1.30170107312868e-25 < a

    1. Initial program 11.4

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
    2. Using strategy rm

    3. Applied div-sub11.4

      \[\leadsto \color{blue}{\frac{x \cdot y}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}}\]

    4. Simplified9.2

      \[\leadsto \color{blue}{\frac{y}{a} \cdot \frac{x}{2}} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}\]

    5. Simplified5.8

      \[\leadsto \frac{y}{a} \cdot \frac{x}{2} - \color{blue}{\frac{t \cdot 9}{a} \cdot \frac{z}{2}}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity5.8

      \[\leadsto \frac{y}{a} \cdot \frac{x}{2} - \frac{t \cdot 9}{\color{blue}{1 \cdot a}} \cdot \frac{z}{2}\]

    8. Applied times-frac5.7

      \[\leadsto \frac{y}{a} \cdot \frac{x}{2} - \color{blue}{\left(\frac{t}{1} \cdot \frac{9}{a}\right)} \cdot \frac{z}{2}\]

    9. Applied associate-*l*6.1

      \[\leadsto \frac{y}{a} \cdot \frac{x}{2} - \color{blue}{\frac{t}{1} \cdot \left(\frac{9}{a} \cdot \frac{z}{2}\right)}\]

    10. Simplified6.2

      \[\leadsto \frac{y}{a} \cdot \frac{x}{2} - \frac{t}{1} \cdot \color{blue}{\frac{\frac{z}{2} \cdot 9}{a}}\]

    if -1.9253598394235403e+180 < a < 1.30170107312868e-25

    1. Initial program 3.7

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
    2. Using strategy rm

    3. Applied associate-/r*3.7

      \[\leadsto \color{blue}{\frac{\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a}}{2}}\]

    4. Simplified3.7

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

  4. Final simplification4.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.925359839423540310793648918342674435965 \cdot 10^{180} \lor \neg \left(a \le 1.301701073128679922027601299899353444017 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{x}{2} \cdot \frac{y}{a} - t \cdot \frac{9 \cdot \frac{z}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(z, 9 \cdot \left(-t\right), y \cdot x\right)}{a}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"

  :herbie-target
  (if (< a -2.090464557976709e+86) (- (* 0.5 (/ (* y x) a)) (* 4.5 (/ t (/ a z)))) (if (< a 2.144030707833976e+99) (/ (- (* x y) (* z (* 9.0 t))) (* a 2.0)) (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5)))))

  (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))