Average Error: 3.6 → 0.7
Time: 16.6s
Precision: 64
\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot 3 \le -73562257687642543292240068825919258624:\\ \;\;\;\;\left(x - \frac{1}{\frac{z \cdot 3}{y}}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{elif}\;z \cdot 3 \le 2.357897429842402930998768322142604435202 \cdot 10^{78}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{y}}{z \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, x, \frac{y}{3} \cdot \frac{-1}{z}\right) + \mathsf{fma}\left(\frac{1}{z}, \left(-\frac{y}{3}\right) + \frac{y}{3}, \frac{\frac{\frac{t}{3}}{z}}{y}\right)\\ \end{array}\]
\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;z \cdot 3 \le -73562257687642543292240068825919258624:\\
\;\;\;\;\left(x - \frac{1}{\frac{z \cdot 3}{y}}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\

\mathbf{elif}\;z \cdot 3 \le 2.357897429842402930998768322142604435202 \cdot 10^{78}:\\
\;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{y}}{z \cdot 3}\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r391818 = x;
        double r391819 = y;
        double r391820 = z;
        double r391821 = 3.0;
        double r391822 = r391820 * r391821;
        double r391823 = r391819 / r391822;
        double r391824 = r391818 - r391823;
        double r391825 = t;
        double r391826 = r391822 * r391819;
        double r391827 = r391825 / r391826;
        double r391828 = r391824 + r391827;
        return r391828;
}
↓
double f(double x, double y, double z, double t) {
        double r391829 = z;
        double r391830 = 3.0;
        double r391831 = r391829 * r391830;
        double r391832 = -7.356225768764254e+37;
        bool r391833 = r391831 <= r391832;
        double r391834 = x;
        double r391835 = 1.0;
        double r391836 = y;
        double r391837 = r391831 / r391836;
        double r391838 = r391835 / r391837;
        double r391839 = r391834 - r391838;
        double r391840 = t;
        double r391841 = r391831 * r391836;
        double r391842 = r391840 / r391841;
        double r391843 = r391839 + r391842;
        double r391844 = 2.357897429842403e+78;
        bool r391845 = r391831 <= r391844;
        double r391846 = r391836 / r391831;
        double r391847 = r391834 - r391846;
        double r391848 = r391840 / r391836;
        double r391849 = r391848 / r391831;
        double r391850 = r391847 + r391849;
        double r391851 = r391836 / r391830;
        double r391852 = -1.0;
        double r391853 = r391852 / r391829;
        double r391854 = r391851 * r391853;
        double r391855 = fma(r391835, r391834, r391854);
        double r391856 = r391835 / r391829;
        double r391857 = -r391851;
        double r391858 = r391857 + r391851;
        double r391859 = r391840 / r391830;
        double r391860 = r391859 / r391829;
        double r391861 = r391860 / r391836;
        double r391862 = fma(r391856, r391858, r391861);
        double r391863 = r391855 + r391862;
        double r391864 = r391845 ? r391850 : r391863;
        double r391865 = r391833 ? r391843 : r391864;
        return r391865;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original3.6
Target1.9
Herbie0.7
\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y}\]

Derivation

  1. Split input into 3 regimes

  2. if (* z 3.0) < -7.356225768764254e+37

    1. Initial program 0.4

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

    3. Applied clear-num0.4

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

    if -7.356225768764254e+37 < (* z 3.0) < 2.357897429842403e+78

    1. Initial program 7.8

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

    3. Applied add-cube-cbrt8.0

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

    4. Applied times-frac0.9

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

    6. Applied associate-*l/0.9

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

    7. Simplified0.6

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

    if 2.357897429842403e+78 < (* z 3.0)

    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*1.2

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

    5. Applied *-un-lft-identity1.2

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

    6. Applied times-frac1.2

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

    8. Applied *-un-lft-identity1.2

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

    9. Applied times-frac1.2

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

    10. Applied *-un-lft-identity1.2

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

    11. Applied prod-diff1.2

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

    12. Applied associate-+l+1.2

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

    13. Simplified1.2

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 3 \le -73562257687642543292240068825919258624:\\ \;\;\;\;\left(x - \frac{1}{\frac{z \cdot 3}{y}}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{elif}\;z \cdot 3 \le 2.357897429842402930998768322142604435202 \cdot 10^{78}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{y}}{z \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, x, \frac{y}{3} \cdot \frac{-1}{z}\right) + \mathsf{fma}\left(\frac{1}{z}, \left(-\frac{y}{3}\right) + \frac{y}{3}, \frac{\frac{\frac{t}{3}}{z}}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 +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))))