Average Error: 7.9 → 3.9
Time: 19.9s
Precision: 64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -6.997492932900559610807387797805298705116 \cdot 10^{193}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}}\right) - \left(t \cdot \frac{z}{a}\right) \cdot 4.5\\ \mathbf{elif}\;x \cdot y \le 1.144427839264607317163410286721664133085 \cdot 10^{212}:\\ \;\;\;\;\frac{\left(x \cdot y\right) \cdot 0.5 - \left(z \cdot t\right) \cdot 4.5}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}}\right) - \left(t \cdot \frac{z}{a}\right) \cdot 4.5\\ \end{array}\]
\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y \le -6.997492932900559610807387797805298705116 \cdot 10^{193}:\\
\;\;\;\;0.5 \cdot \left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}}\right) - \left(t \cdot \frac{z}{a}\right) \cdot 4.5\\

\mathbf{elif}\;x \cdot y \le 1.144427839264607317163410286721664133085 \cdot 10^{212}:\\
\;\;\;\;\frac{\left(x \cdot y\right) \cdot 0.5 - \left(z \cdot t\right) \cdot 4.5}{a}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}}\right) - \left(t \cdot \frac{z}{a}\right) \cdot 4.5\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r35516923 = x;
        double r35516924 = y;
        double r35516925 = r35516923 * r35516924;
        double r35516926 = z;
        double r35516927 = 9.0;
        double r35516928 = r35516926 * r35516927;
        double r35516929 = t;
        double r35516930 = r35516928 * r35516929;
        double r35516931 = r35516925 - r35516930;
        double r35516932 = a;
        double r35516933 = 2.0;
        double r35516934 = r35516932 * r35516933;
        double r35516935 = r35516931 / r35516934;
        return r35516935;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r35516936 = x;
        double r35516937 = y;
        double r35516938 = r35516936 * r35516937;
        double r35516939 = -6.99749293290056e+193;
        bool r35516940 = r35516938 <= r35516939;
        double r35516941 = 0.5;
        double r35516942 = a;
        double r35516943 = cbrt(r35516942);
        double r35516944 = r35516943 * r35516943;
        double r35516945 = r35516936 / r35516944;
        double r35516946 = r35516937 / r35516943;
        double r35516947 = r35516945 * r35516946;
        double r35516948 = r35516941 * r35516947;
        double r35516949 = t;
        double r35516950 = z;
        double r35516951 = r35516950 / r35516942;
        double r35516952 = r35516949 * r35516951;
        double r35516953 = 4.5;
        double r35516954 = r35516952 * r35516953;
        double r35516955 = r35516948 - r35516954;
        double r35516956 = 1.1444278392646073e+212;
        bool r35516957 = r35516938 <= r35516956;
        double r35516958 = r35516938 * r35516941;
        double r35516959 = r35516950 * r35516949;
        double r35516960 = r35516959 * r35516953;
        double r35516961 = r35516958 - r35516960;
        double r35516962 = r35516961 / r35516942;
        double r35516963 = r35516957 ? r35516962 : r35516955;
        double r35516964 = r35516940 ? r35516955 : r35516963;
        return r35516964;
}

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

Original7.9
Target5.7
Herbie3.9
\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709043451944897028999329376 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.144030707833976090627817222818061808815 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* x y) < -6.99749293290056e+193 or 1.1444278392646073e+212 < (* x y)

    1. Initial program 31.1

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

    2. Taylor expanded around 0 31.1

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

    4. Applied add-cube-cbrt31.7

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

    5. Applied times-frac7.5

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

    7. Applied *-un-lft-identity7.5

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

    8. Applied times-frac2.0

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

    9. Simplified2.0

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

    if -6.99749293290056e+193 < (* x y) < 1.1444278392646073e+212

    1. Initial program 4.2

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

    2. Taylor expanded around 0 4.1

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

    4. Applied associate-*r/4.1

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

    5. Applied associate-*r/4.1

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

    6. Applied sub-div4.1

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

  4. Final simplification3.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -6.997492932900559610807387797805298705116 \cdot 10^{193}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}}\right) - \left(t \cdot \frac{z}{a}\right) \cdot 4.5\\ \mathbf{elif}\;x \cdot y \le 1.144427839264607317163410286721664133085 \cdot 10^{212}:\\ \;\;\;\;\frac{\left(x \cdot y\right) \cdot 0.5 - \left(z \cdot t\right) \cdot 4.5}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}}\right) - \left(t \cdot \frac{z}{a}\right) \cdot 4.5\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(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)))