Average Error: 6.5 → 1.6
Time: 20.8s
Precision: 64
\[x + \frac{y \cdot \left(z - x\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -3.1568940568519223 \cdot 10^{78} \lor \neg \left(t \le 3.1796045355482281 \cdot 10^{-75}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z \cdot y}{t} - \frac{x \cdot y}{t}\right) + x\\ \end{array}\]
x + \frac{y \cdot \left(z - x\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -3.1568940568519223 \cdot 10^{78} \lor \neg \left(t \le 3.1796045355482281 \cdot 10^{-75}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r404876 = x;
        double r404877 = y;
        double r404878 = z;
        double r404879 = r404878 - r404876;
        double r404880 = r404877 * r404879;
        double r404881 = t;
        double r404882 = r404880 / r404881;
        double r404883 = r404876 + r404882;
        return r404883;
}
↓
double f(double x, double y, double z, double t) {
        double r404884 = t;
        double r404885 = -3.1568940568519223e+78;
        bool r404886 = r404884 <= r404885;
        double r404887 = 3.179604535548228e-75;
        bool r404888 = r404884 <= r404887;
        double r404889 = !r404888;
        bool r404890 = r404886 || r404889;
        double r404891 = y;
        double r404892 = r404891 / r404884;
        double r404893 = z;
        double r404894 = x;
        double r404895 = r404893 - r404894;
        double r404896 = fma(r404892, r404895, r404894);
        double r404897 = r404893 * r404891;
        double r404898 = r404897 / r404884;
        double r404899 = r404894 * r404891;
        double r404900 = r404899 / r404884;
        double r404901 = r404898 - r404900;
        double r404902 = r404901 + r404894;
        double r404903 = r404890 ? r404896 : r404902;
        return r404903;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.5
Target2.2
Herbie1.6
\[x - \left(x \cdot \frac{y}{t} + \left(-z\right) \cdot \frac{y}{t}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if t < -3.1568940568519223e+78 or 3.179604535548228e-75 < t

    1. Initial program 9.4

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

    2. Simplified1.1

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

    if -3.1568940568519223e+78 < t < 3.179604535548228e-75

    1. Initial program 2.2

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

    2. Simplified3.9

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

    4. Applied fma-udef3.9

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

    5. Simplified12.7

      \[\leadsto \color{blue}{\frac{y}{\frac{t}{z - x}}} + x\]

    6. Taylor expanded around 0 2.2

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

  4. Final simplification1.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.1568940568519223 \cdot 10^{78} \lor \neg \left(t \le 3.1796045355482281 \cdot 10^{-75}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z \cdot y}{t} - \frac{x \cdot y}{t}\right) + x\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D"
  :precision binary64

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

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