Average Error: 3.5 → 0.5
Time: 21.0s
Precision: 64
\[\left(x - \frac{y}{z \cdot 3.0}\right) + \frac{t}{\left(z \cdot 3.0\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.055519881012369 \cdot 10^{-07}:\\ \;\;\;\;\frac{t}{y \cdot \left(z \cdot 3.0\right)} + \left(x - \frac{\frac{y}{z}}{3.0}\right)\\ \mathbf{elif}\;z \le 6.041072736026979 \cdot 10^{-09}:\\ \;\;\;\;\frac{\frac{t}{3.0}}{y} \cdot \frac{1}{z} + \left(\left(x - \frac{\frac{y}{3.0}}{z}\right) + \mathsf{fma}\left(\frac{\frac{y}{3.0}}{z}, -1, \frac{\frac{y}{3.0}}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t}{z}}{3.0 \cdot y} + \left(x - \frac{y}{z \cdot 3.0}\right)\\ \end{array}\]
\left(x - \frac{y}{z \cdot 3.0}\right) + \frac{t}{\left(z \cdot 3.0\right) \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;z \le -2.055519881012369 \cdot 10^{-07}:\\
\;\;\;\;\frac{t}{y \cdot \left(z \cdot 3.0\right)} + \left(x - \frac{\frac{y}{z}}{3.0}\right)\\

\mathbf{elif}\;z \le 6.041072736026979 \cdot 10^{-09}:\\
\;\;\;\;\frac{\frac{t}{3.0}}{y} \cdot \frac{1}{z} + \left(\left(x - \frac{\frac{y}{3.0}}{z}\right) + \mathsf{fma}\left(\frac{\frac{y}{3.0}}{z}, -1, \frac{\frac{y}{3.0}}{z}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{t}{z}}{3.0 \cdot y} + \left(x - \frac{y}{z \cdot 3.0}\right)\\

\end{array}
double f(double x, double y, double z, double t) {
        double r30651074 = x;
        double r30651075 = y;
        double r30651076 = z;
        double r30651077 = 3.0;
        double r30651078 = r30651076 * r30651077;
        double r30651079 = r30651075 / r30651078;
        double r30651080 = r30651074 - r30651079;
        double r30651081 = t;
        double r30651082 = r30651078 * r30651075;
        double r30651083 = r30651081 / r30651082;
        double r30651084 = r30651080 + r30651083;
        return r30651084;
}
↓
double f(double x, double y, double z, double t) {
        double r30651085 = z;
        double r30651086 = -2.055519881012369e-07;
        bool r30651087 = r30651085 <= r30651086;
        double r30651088 = t;
        double r30651089 = y;
        double r30651090 = 3.0;
        double r30651091 = r30651085 * r30651090;
        double r30651092 = r30651089 * r30651091;
        double r30651093 = r30651088 / r30651092;
        double r30651094 = x;
        double r30651095 = r30651089 / r30651085;
        double r30651096 = r30651095 / r30651090;
        double r30651097 = r30651094 - r30651096;
        double r30651098 = r30651093 + r30651097;
        double r30651099 = 6.041072736026979e-09;
        bool r30651100 = r30651085 <= r30651099;
        double r30651101 = r30651088 / r30651090;
        double r30651102 = r30651101 / r30651089;
        double r30651103 = 1.0;
        double r30651104 = r30651103 / r30651085;
        double r30651105 = r30651102 * r30651104;
        double r30651106 = r30651089 / r30651090;
        double r30651107 = r30651106 / r30651085;
        double r30651108 = r30651094 - r30651107;
        double r30651109 = -1.0;
        double r30651110 = fma(r30651107, r30651109, r30651107);
        double r30651111 = r30651108 + r30651110;
        double r30651112 = r30651105 + r30651111;
        double r30651113 = r30651088 / r30651085;
        double r30651114 = r30651090 * r30651089;
        double r30651115 = r30651113 / r30651114;
        double r30651116 = r30651089 / r30651091;
        double r30651117 = r30651094 - r30651116;
        double r30651118 = r30651115 + r30651117;
        double r30651119 = r30651100 ? r30651112 : r30651118;
        double r30651120 = r30651087 ? r30651098 : r30651119;
        return r30651120;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original3.5
Target1.8
Herbie0.5
\[\left(x - \frac{y}{z \cdot 3.0}\right) + \frac{\frac{t}{z \cdot 3.0}}{y}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -2.055519881012369e-07

    1. Initial program 0.3

      \[\left(x - \frac{y}{z \cdot 3.0}\right) + \frac{t}{\left(z \cdot 3.0\right) \cdot y}\]
    2. Using strategy rm

    3. Applied associate-/r*0.3

      \[\leadsto \left(x - \color{blue}{\frac{\frac{y}{z}}{3.0}}\right) + \frac{t}{\left(z \cdot 3.0\right) \cdot y}\]

    if -2.055519881012369e-07 < z < 6.041072736026979e-09

    1. Initial program 10.2

      \[\left(x - \frac{y}{z \cdot 3.0}\right) + \frac{t}{\left(z \cdot 3.0\right) \cdot y}\]
    2. Using strategy rm

    3. Applied associate-/r*3.0

      \[\leadsto \left(x - \frac{y}{z \cdot 3.0}\right) + \color{blue}{\frac{\frac{t}{z \cdot 3.0}}{y}}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt3.4

      \[\leadsto \left(x - \color{blue}{\left(\sqrt[3]{\frac{y}{z \cdot 3.0}} \cdot \sqrt[3]{\frac{y}{z \cdot 3.0}}\right) \cdot \sqrt[3]{\frac{y}{z \cdot 3.0}}}\right) + \frac{\frac{t}{z \cdot 3.0}}{y}\]

    6. Applied add-sqr-sqrt32.3

      \[\leadsto \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \left(\sqrt[3]{\frac{y}{z \cdot 3.0}} \cdot \sqrt[3]{\frac{y}{z \cdot 3.0}}\right) \cdot \sqrt[3]{\frac{y}{z \cdot 3.0}}\right) + \frac{\frac{t}{z \cdot 3.0}}{y}\]

    7. Applied prod-diff32.3

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\sqrt{x}, \sqrt{x}, -\sqrt[3]{\frac{y}{z \cdot 3.0}} \cdot \left(\sqrt[3]{\frac{y}{z \cdot 3.0}} \cdot \sqrt[3]{\frac{y}{z \cdot 3.0}}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\frac{y}{z \cdot 3.0}}, \sqrt[3]{\frac{y}{z \cdot 3.0}} \cdot \sqrt[3]{\frac{y}{z \cdot 3.0}}, \sqrt[3]{\frac{y}{z \cdot 3.0}} \cdot \left(\sqrt[3]{\frac{y}{z \cdot 3.0}} \cdot \sqrt[3]{\frac{y}{z \cdot 3.0}}\right)\right)\right)} + \frac{\frac{t}{z \cdot 3.0}}{y}\]

    8. Simplified3.0

      \[\leadsto \left(\color{blue}{\left(x - \frac{\frac{y}{3.0}}{z}\right)} + \mathsf{fma}\left(-\sqrt[3]{\frac{y}{z \cdot 3.0}}, \sqrt[3]{\frac{y}{z \cdot 3.0}} \cdot \sqrt[3]{\frac{y}{z \cdot 3.0}}, \sqrt[3]{\frac{y}{z \cdot 3.0}} \cdot \left(\sqrt[3]{\frac{y}{z \cdot 3.0}} \cdot \sqrt[3]{\frac{y}{z \cdot 3.0}}\right)\right)\right) + \frac{\frac{t}{z \cdot 3.0}}{y}\]

    9. Simplified3.0

      \[\leadsto \left(\left(x - \frac{\frac{y}{3.0}}{z}\right) + \color{blue}{\mathsf{fma}\left(\frac{\frac{y}{3.0}}{z}, -1, \frac{\frac{y}{3.0}}{z}\right)}\right) + \frac{\frac{t}{z \cdot 3.0}}{y}\]
    10. Using strategy rm

    11. Applied *-un-lft-identity3.0

      \[\leadsto \left(\left(x - \frac{\frac{y}{3.0}}{z}\right) + \mathsf{fma}\left(\frac{\frac{y}{3.0}}{z}, -1, \frac{\frac{y}{3.0}}{z}\right)\right) + \frac{\frac{t}{z \cdot 3.0}}{\color{blue}{1 \cdot y}}\]

    12. Applied *-un-lft-identity3.0

      \[\leadsto \left(\left(x - \frac{\frac{y}{3.0}}{z}\right) + \mathsf{fma}\left(\frac{\frac{y}{3.0}}{z}, -1, \frac{\frac{y}{3.0}}{z}\right)\right) + \frac{\frac{\color{blue}{1 \cdot t}}{z \cdot 3.0}}{1 \cdot y}\]

    13. Applied times-frac3.0

      \[\leadsto \left(\left(x - \frac{\frac{y}{3.0}}{z}\right) + \mathsf{fma}\left(\frac{\frac{y}{3.0}}{z}, -1, \frac{\frac{y}{3.0}}{z}\right)\right) + \frac{\color{blue}{\frac{1}{z} \cdot \frac{t}{3.0}}}{1 \cdot y}\]

    14. Applied times-frac0.3

      \[\leadsto \left(\left(x - \frac{\frac{y}{3.0}}{z}\right) + \mathsf{fma}\left(\frac{\frac{y}{3.0}}{z}, -1, \frac{\frac{y}{3.0}}{z}\right)\right) + \color{blue}{\frac{\frac{1}{z}}{1} \cdot \frac{\frac{t}{3.0}}{y}}\]

    15. Simplified0.3

      \[\leadsto \left(\left(x - \frac{\frac{y}{3.0}}{z}\right) + \mathsf{fma}\left(\frac{\frac{y}{3.0}}{z}, -1, \frac{\frac{y}{3.0}}{z}\right)\right) + \color{blue}{\frac{1}{z}} \cdot \frac{\frac{t}{3.0}}{y}\]

    if 6.041072736026979e-09 < z

    1. Initial program 0.3

      \[\left(x - \frac{y}{z \cdot 3.0}\right) + \frac{t}{\left(z \cdot 3.0\right) \cdot y}\]
    2. Using strategy rm

    3. Applied associate-/r*1.1

      \[\leadsto \left(x - \frac{y}{z \cdot 3.0}\right) + \color{blue}{\frac{\frac{t}{z \cdot 3.0}}{y}}\]
    4. Using strategy rm

    5. Applied associate-/r*1.1

      \[\leadsto \left(x - \frac{y}{z \cdot 3.0}\right) + \frac{\color{blue}{\frac{\frac{t}{z}}{3.0}}}{y}\]
    6. Using strategy rm

    7. Applied associate-/l/1.1

      \[\leadsto \left(x - \frac{y}{z \cdot 3.0}\right) + \color{blue}{\frac{\frac{t}{z}}{y \cdot 3.0}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.055519881012369 \cdot 10^{-07}:\\ \;\;\;\;\frac{t}{y \cdot \left(z \cdot 3.0\right)} + \left(x - \frac{\frac{y}{z}}{3.0}\right)\\ \mathbf{elif}\;z \le 6.041072736026979 \cdot 10^{-09}:\\ \;\;\;\;\frac{\frac{t}{3.0}}{y} \cdot \frac{1}{z} + \left(\left(x - \frac{\frac{y}{3.0}}{z}\right) + \mathsf{fma}\left(\frac{\frac{y}{3.0}}{z}, -1, \frac{\frac{y}{3.0}}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t}{z}}{3.0 \cdot y} + \left(x - \frac{y}{z \cdot 3.0}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, H"

  :herbie-target
  (+ (- x (/ y (* z 3.0))) (/ (/ t (* z 3.0)) y))

  (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))