Average Error: 7.7 → 0.8
Time: 10.6s
Precision: 64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t = -\infty \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 1.2344983160605906 \cdot 10^{285}\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \left(\frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \left(x \cdot y\right) - 4.5 \cdot \left(t \cdot z\right)}{a}\\ \end{array}\]
\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t = -\infty \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 1.2344983160605906 \cdot 10^{285}\right):\\
\;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \left(\frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{0.5 \cdot \left(x \cdot y\right) - 4.5 \cdot \left(t \cdot z\right)}{a}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r414432 = x;
        double r414433 = y;
        double r414434 = r414432 * r414433;
        double r414435 = z;
        double r414436 = 9.0;
        double r414437 = r414435 * r414436;
        double r414438 = t;
        double r414439 = r414437 * r414438;
        double r414440 = r414434 - r414439;
        double r414441 = a;
        double r414442 = 2.0;
        double r414443 = r414441 * r414442;
        double r414444 = r414440 / r414443;
        return r414444;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r414445 = x;
        double r414446 = y;
        double r414447 = r414445 * r414446;
        double r414448 = z;
        double r414449 = 9.0;
        double r414450 = r414448 * r414449;
        double r414451 = t;
        double r414452 = r414450 * r414451;
        double r414453 = r414447 - r414452;
        double r414454 = -inf.0;
        bool r414455 = r414453 <= r414454;
        double r414456 = 1.2344983160605906e+285;
        bool r414457 = r414453 <= r414456;
        double r414458 = !r414457;
        bool r414459 = r414455 || r414458;
        double r414460 = 0.5;
        double r414461 = a;
        double r414462 = r414446 / r414461;
        double r414463 = r414445 * r414462;
        double r414464 = r414460 * r414463;
        double r414465 = 4.5;
        double r414466 = cbrt(r414461);
        double r414467 = r414466 * r414466;
        double r414468 = r414451 / r414467;
        double r414469 = r414448 / r414466;
        double r414470 = r414468 * r414469;
        double r414471 = r414465 * r414470;
        double r414472 = r414464 - r414471;
        double r414473 = r414460 * r414447;
        double r414474 = r414451 * r414448;
        double r414475 = r414465 * r414474;
        double r414476 = r414473 - r414475;
        double r414477 = r414476 / r414461;
        double r414478 = r414459 ? r414472 : r414477;
        return r414478;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original7.7
Target5.8
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.14403070783397609 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- (* x y) (* (* z 9.0) t)) < -inf.0 or 1.2344983160605906e+285 < (- (* x y) (* (* z 9.0) t))

    1. Initial program 57.1

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

    2. Taylor expanded around 0 57.0

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt57.0

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}\]

    5. Applied times-frac31.3

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \color{blue}{\left(\frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}\right)}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity31.3

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{\color{blue}{1 \cdot a}} - 4.5 \cdot \left(\frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}\right)\]

    8. Applied times-frac0.9

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{1} \cdot \frac{y}{a}\right)} - 4.5 \cdot \left(\frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}\right)\]

    9. Simplified0.9

      \[\leadsto 0.5 \cdot \left(\color{blue}{x} \cdot \frac{y}{a}\right) - 4.5 \cdot \left(\frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}\right)\]

    if -inf.0 < (- (* x y) (* (* z 9.0) t)) < 1.2344983160605906e+285

    1. Initial program 0.9

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

    2. Taylor expanded around 0 0.9

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}}\]
    3. Using strategy rm

    4. Applied associate-*r/0.9

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - \color{blue}{\frac{4.5 \cdot \left(t \cdot z\right)}{a}}\]

    5. Applied associate-*r/0.8

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(x \cdot y\right)}{a}} - \frac{4.5 \cdot \left(t \cdot z\right)}{a}\]

    6. Applied sub-div0.8

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(x \cdot y\right) - 4.5 \cdot \left(t \cdot z\right)}{a}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t = -\infty \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 1.2344983160605906 \cdot 10^{285}\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \left(\frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \left(x \cdot y\right) - 4.5 \cdot \left(t \cdot z\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

  :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 t))) (* a 2)) (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5)))))

  (/ (- (* x y) (* (* z 9) t)) (* a 2)))