Average Error: 3.7 → 0.5
Time: 4.7s
Precision: 64
\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.982808370584645728390571403324870832662 \cdot 10^{-59} \lor \neg \left(z \le 1.637385439772020732742719657482536703395 \cdot 10^{-52}\right):\\ \;\;\;\;\left(x - \frac{\frac{y}{z}}{3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x + \left(\mathsf{fma}\left(\frac{1}{z}, \left(-\frac{y}{3}\right) + \frac{y}{3}, \frac{1}{z} \cdot \frac{\frac{t}{3}}{y}\right) - \frac{\frac{y}{3}}{z}\right)\\ \end{array}\]
\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;z \le -2.982808370584645728390571403324870832662 \cdot 10^{-59} \lor \neg \left(z \le 1.637385439772020732742719657482536703395 \cdot 10^{-52}\right):\\
\;\;\;\;\left(x - \frac{\frac{y}{z}}{3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot x + \left(\mathsf{fma}\left(\frac{1}{z}, \left(-\frac{y}{3}\right) + \frac{y}{3}, \frac{1}{z} \cdot \frac{\frac{t}{3}}{y}\right) - \frac{\frac{y}{3}}{z}\right)\\

\end{array}
double f(double x, double y, double z, double t) {
        double r746824 = x;
        double r746825 = y;
        double r746826 = z;
        double r746827 = 3.0;
        double r746828 = r746826 * r746827;
        double r746829 = r746825 / r746828;
        double r746830 = r746824 - r746829;
        double r746831 = t;
        double r746832 = r746828 * r746825;
        double r746833 = r746831 / r746832;
        double r746834 = r746830 + r746833;
        return r746834;
}
↓
double f(double x, double y, double z, double t) {
        double r746835 = z;
        double r746836 = -2.9828083705846457e-59;
        bool r746837 = r746835 <= r746836;
        double r746838 = 1.6373854397720207e-52;
        bool r746839 = r746835 <= r746838;
        double r746840 = !r746839;
        bool r746841 = r746837 || r746840;
        double r746842 = x;
        double r746843 = y;
        double r746844 = r746843 / r746835;
        double r746845 = 3.0;
        double r746846 = r746844 / r746845;
        double r746847 = r746842 - r746846;
        double r746848 = t;
        double r746849 = r746835 * r746845;
        double r746850 = r746849 * r746843;
        double r746851 = r746848 / r746850;
        double r746852 = r746847 + r746851;
        double r746853 = 1.0;
        double r746854 = r746853 * r746842;
        double r746855 = r746853 / r746835;
        double r746856 = r746843 / r746845;
        double r746857 = -r746856;
        double r746858 = r746857 + r746856;
        double r746859 = r746848 / r746845;
        double r746860 = r746859 / r746843;
        double r746861 = r746855 * r746860;
        double r746862 = fma(r746855, r746858, r746861);
        double r746863 = r746856 / r746835;
        double r746864 = r746862 - r746863;
        double r746865 = r746854 + r746864;
        double r746866 = r746841 ? r746852 : r746865;
        return r746866;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original3.7
Target1.6
Herbie0.5
\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -2.9828083705846457e-59 or 1.6373854397720207e-52 < z

    1. Initial program 0.5

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\]
    2. Using strategy rm

    3. Applied associate-/r*0.5

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

    if -2.9828083705846457e-59 < z < 1.6373854397720207e-52

    1. Initial program 13.7

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\]
    2. Using strategy rm

    3. Applied associate-/r*3.6

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{\frac{t}{z \cdot 3}}{y}}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity3.6

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

    6. Applied times-frac3.6

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

    7. Applied *-un-lft-identity3.6

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

    8. Applied prod-diff3.6

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(1, x, -\frac{y}{3} \cdot \frac{1}{z}\right) + \mathsf{fma}\left(-\frac{y}{3}, \frac{1}{z}, \frac{y}{3} \cdot \frac{1}{z}\right)\right)} + \frac{\frac{t}{z \cdot 3}}{y}\]

    9. Applied associate-+l+3.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, x, -\frac{y}{3} \cdot \frac{1}{z}\right) + \left(\mathsf{fma}\left(-\frac{y}{3}, \frac{1}{z}, \frac{y}{3} \cdot \frac{1}{z}\right) + \frac{\frac{t}{z \cdot 3}}{y}\right)}\]

    10. Simplified3.6

      \[\leadsto \mathsf{fma}\left(1, x, -\frac{y}{3} \cdot \frac{1}{z}\right) + \color{blue}{\mathsf{fma}\left(\frac{1}{z}, \left(-\frac{y}{3}\right) + \frac{y}{3}, \frac{\frac{t}{z \cdot 3}}{y}\right)}\]
    11. Using strategy rm

    12. Applied fma-udef3.6

      \[\leadsto \color{blue}{\left(1 \cdot x + \left(-\frac{y}{3} \cdot \frac{1}{z}\right)\right)} + \mathsf{fma}\left(\frac{1}{z}, \left(-\frac{y}{3}\right) + \frac{y}{3}, \frac{\frac{t}{z \cdot 3}}{y}\right)\]

    13. Applied associate-+l+3.6

      \[\leadsto \color{blue}{1 \cdot x + \left(\left(-\frac{y}{3} \cdot \frac{1}{z}\right) + \mathsf{fma}\left(\frac{1}{z}, \left(-\frac{y}{3}\right) + \frac{y}{3}, \frac{\frac{t}{z \cdot 3}}{y}\right)\right)}\]

    14. Simplified3.6

      \[\leadsto 1 \cdot x + \color{blue}{\left(\mathsf{fma}\left(\frac{1}{z}, \left(-\frac{y}{3}\right) + \frac{y}{3}, \frac{\frac{t}{z \cdot 3}}{y}\right) - \frac{\frac{y}{3}}{z}\right)}\]
    15. Using strategy rm

    16. Applied *-un-lft-identity3.6

      \[\leadsto 1 \cdot x + \left(\mathsf{fma}\left(\frac{1}{z}, \left(-\frac{y}{3}\right) + \frac{y}{3}, \frac{\frac{t}{z \cdot 3}}{\color{blue}{1 \cdot y}}\right) - \frac{\frac{y}{3}}{z}\right)\]

    17. Applied *-un-lft-identity3.6

      \[\leadsto 1 \cdot x + \left(\mathsf{fma}\left(\frac{1}{z}, \left(-\frac{y}{3}\right) + \frac{y}{3}, \frac{\frac{\color{blue}{1 \cdot t}}{z \cdot 3}}{1 \cdot y}\right) - \frac{\frac{y}{3}}{z}\right)\]

    18. Applied times-frac3.6

      \[\leadsto 1 \cdot x + \left(\mathsf{fma}\left(\frac{1}{z}, \left(-\frac{y}{3}\right) + \frac{y}{3}, \frac{\color{blue}{\frac{1}{z} \cdot \frac{t}{3}}}{1 \cdot y}\right) - \frac{\frac{y}{3}}{z}\right)\]

    19. Applied times-frac0.3

      \[\leadsto 1 \cdot x + \left(\mathsf{fma}\left(\frac{1}{z}, \left(-\frac{y}{3}\right) + \frac{y}{3}, \color{blue}{\frac{\frac{1}{z}}{1} \cdot \frac{\frac{t}{3}}{y}}\right) - \frac{\frac{y}{3}}{z}\right)\]

    20. Simplified0.3

      \[\leadsto 1 \cdot x + \left(\mathsf{fma}\left(\frac{1}{z}, \left(-\frac{y}{3}\right) + \frac{y}{3}, \color{blue}{\frac{1}{z}} \cdot \frac{\frac{t}{3}}{y}\right) - \frac{\frac{y}{3}}{z}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.982808370584645728390571403324870832662 \cdot 10^{-59} \lor \neg \left(z \le 1.637385439772020732742719657482536703395 \cdot 10^{-52}\right):\\ \;\;\;\;\left(x - \frac{\frac{y}{z}}{3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x + \left(\mathsf{fma}\left(\frac{1}{z}, \left(-\frac{y}{3}\right) + \frac{y}{3}, \frac{1}{z} \cdot \frac{\frac{t}{3}}{y}\right) - \frac{\frac{y}{3}}{z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, H"
  :precision binary64

  :herbie-target
  (+ (- x (/ y (* z 3))) (/ (/ t (* z 3)) y))

  (+ (- x (/ y (* z 3))) (/ t (* (* z 3) y))))