Average Error: 7.6 → 0.8
Time: 17.1s
Precision: 64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t = -\infty \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 2.321426214984744761253195196318689903382 \cdot 10^{264}\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - \left(4.5 \cdot \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right) \cdot \frac{z}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \left(x \cdot y\right) - 4.5 \cdot \left(t \cdot z\right)}{a}\\ \end{array}\]
\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 2.321426214984744761253195196318689903382 \cdot 10^{264}\right):\\
\;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - \left(4.5 \cdot \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right) \cdot \frac{z}{\sqrt[3]{a}}\\

\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 r455985 = x;
        double r455986 = y;
        double r455987 = r455985 * r455986;
        double r455988 = z;
        double r455989 = 9.0;
        double r455990 = r455988 * r455989;
        double r455991 = t;
        double r455992 = r455990 * r455991;
        double r455993 = r455987 - r455992;
        double r455994 = a;
        double r455995 = 2.0;
        double r455996 = r455994 * r455995;
        double r455997 = r455993 / r455996;
        return r455997;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r455998 = x;
        double r455999 = y;
        double r456000 = r455998 * r455999;
        double r456001 = z;
        double r456002 = 9.0;
        double r456003 = r456001 * r456002;
        double r456004 = t;
        double r456005 = r456003 * r456004;
        double r456006 = r456000 - r456005;
        double r456007 = -inf.0;
        bool r456008 = r456006 <= r456007;
        double r456009 = 2.3214262149847448e+264;
        bool r456010 = r456006 <= r456009;
        double r456011 = !r456010;
        bool r456012 = r456008 || r456011;
        double r456013 = 0.5;
        double r456014 = a;
        double r456015 = r455999 / r456014;
        double r456016 = r455998 * r456015;
        double r456017 = r456013 * r456016;
        double r456018 = 4.5;
        double r456019 = cbrt(r456014);
        double r456020 = r456019 * r456019;
        double r456021 = r456004 / r456020;
        double r456022 = r456018 * r456021;
        double r456023 = r456001 / r456019;
        double r456024 = r456022 * r456023;
        double r456025 = r456017 - r456024;
        double r456026 = r456013 * r456000;
        double r456027 = r456004 * r456001;
        double r456028 = r456018 * r456027;
        double r456029 = r456026 - r456028;
        double r456030 = r456029 / r456014;
        double r456031 = r456012 ? r456025 : r456030;
        return r456031;
}

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

Original7.6
Target5.6
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709043451944897028999329376 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.144030707833976090627817222818061808815 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- (* x y) (* (* z 9.0) t)) < -inf.0 or 2.3214262149847448e+264 < (- (* x y) (* (* z 9.0) t))

    1. Initial program 53.1

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

    2. Taylor expanded around 0 52.6

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

    4. Applied add-cube-cbrt52.7

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

    5. Applied times-frac27.9

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

    6. Applied associate-*r*27.9

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

    8. Applied *-un-lft-identity27.9

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

    9. Applied times-frac1.1

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

    10. Simplified1.1

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

    if -inf.0 < (- (* x y) (* (* z 9.0) t)) < 2.3214262149847448e+264

    1. Initial program 0.8

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

    2. Taylor expanded around 0 0.8

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

    4. Applied associate-*r/0.8

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t = -\infty \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 2.321426214984744761253195196318689903382 \cdot 10^{264}\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - \left(4.5 \cdot \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right) \cdot \frac{z}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \left(x \cdot y\right) - 4.5 \cdot \left(t \cdot z\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +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)))