Average Error: 14.6 → 1.7
Time: 29.8s
Precision: 64
\[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -2.2665860626305797 \cdot 10^{-139}:\\ \;\;\;\;\frac{\sqrt[3]{y}}{\sqrt[3]{z}} \cdot \frac{x}{\frac{\sqrt[3]{z}}{\sqrt[3]{y}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{y}}}\\ \mathbf{elif}\;\frac{y}{z} \le 8.578702620784637 \cdot 10^{-293}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{y}}{\sqrt[3]{z}} \cdot \frac{x}{\frac{\sqrt[3]{z}}{\sqrt[3]{y}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{y}}}\\ \end{array}\]
double f(double x, double y, double z, double t) {
        double r9610966 = x;
        double r9610967 = y;
        double r9610968 = z;
        double r9610969 = r9610967 / r9610968;
        double r9610970 = t;
        double r9610971 = r9610969 * r9610970;
        double r9610972 = r9610971 / r9610970;
        double r9610973 = r9610966 * r9610972;
        return r9610973;
}
↓
double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r9610974 = y;
        double r9610975 = z;
        double r9610976 = r9610974 / r9610975;
        double r9610977 = -2.2665860626305797e-139;
        bool r9610978 = r9610976 <= r9610977;
        double r9610979 = cbrt(r9610974);
        double r9610980 = cbrt(r9610975);
        double r9610981 = r9610979 / r9610980;
        double r9610982 = x;
        double r9610983 = r9610980 / r9610979;
        double r9610984 = r9610983 * r9610983;
        double r9610985 = r9610982 / r9610984;
        double r9610986 = r9610981 * r9610985;
        double r9610987 = 8.578702620784637e-293;
        bool r9610988 = r9610976 <= r9610987;
        double r9610989 = r9610982 * r9610974;
        double r9610990 = r9610989 / r9610975;
        double r9610991 = r9610988 ? r9610990 : r9610986;
        double r9610992 = r9610978 ? r9610986 : r9610991;
        return r9610992;
}


x \cdot \frac{\frac{y}{z} \cdot t}{t}
↓
\begin{array}{l}
\mathbf{if}\;\frac{y}{z} \le -2.2665860626305797 \cdot 10^{-139}:\\
\;\;\;\;\frac{\sqrt[3]{y}}{\sqrt[3]{z}} \cdot \frac{x}{\frac{\sqrt[3]{z}}{\sqrt[3]{y}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{y}}}\\

\mathbf{elif}\;\frac{y}{z} \le 8.578702620784637 \cdot 10^{-293}:\\
\;\;\;\;\frac{x \cdot y}{z}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt[3]{y}}{\sqrt[3]{z}} \cdot \frac{x}{\frac{\sqrt[3]{z}}{\sqrt[3]{y}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{y}}}\\

\end{array}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Split input into 2 regimes

  2. if (/ y z) < -2.2665860626305797e-139 or 8.578702620784637e-293 < (/ y z)

    1. Initial program 13.3

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

    2. Simplified4.3

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt5.2

      \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}\]

    5. Applied add-cube-cbrt5.5

      \[\leadsto x \cdot \frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}\]

    6. Applied times-frac5.5

      \[\leadsto x \cdot \color{blue}{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\right)}\]

    7. Applied associate-*r*2.0

      \[\leadsto \color{blue}{\left(x \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}}\]
    8. Using strategy rm

    9. Applied pow12.0

      \[\leadsto \left(x \cdot \color{blue}{{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right)}^{1}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\]

    10. Applied pow12.0

      \[\leadsto \left(\color{blue}{{x}^{1}} \cdot {\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right)}^{1}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\]

    11. Applied pow-prod-down2.0

      \[\leadsto \color{blue}{{\left(x \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right)}^{1}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\]

    12. Simplified1.9

      \[\leadsto {\color{blue}{\left(\frac{x}{\frac{\sqrt[3]{z}}{\sqrt[3]{y}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{y}}}\right)}}^{1} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\]

    if -2.2665860626305797e-139 < (/ y z) < 8.578702620784637e-293

    1. Initial program 18.1

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

    2. Simplified11.0

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}}\]

    3. Taylor expanded around inf 1.2

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -2.2665860626305797 \cdot 10^{-139}:\\ \;\;\;\;\frac{\sqrt[3]{y}}{\sqrt[3]{z}} \cdot \frac{x}{\frac{\sqrt[3]{z}}{\sqrt[3]{y}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{y}}}\\ \mathbf{elif}\;\frac{y}{z} \le 8.578702620784637 \cdot 10^{-293}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{y}}{\sqrt[3]{z}} \cdot \frac{x}{\frac{\sqrt[3]{z}}{\sqrt[3]{y}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{y}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019101 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1"
  (* x (/ (* (/ y z) t) t)))