Average Error: 6.5 → 1.6
Time: 17.0s
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 r446176 = x;
        double r446177 = y;
        double r446178 = z;
        double r446179 = r446178 - r446176;
        double r446180 = r446177 * r446179;
        double r446181 = t;
        double r446182 = r446180 / r446181;
        double r446183 = r446176 + r446182;
        return r446183;
}
↓
double f(double x, double y, double z, double t) {
        double r446184 = t;
        double r446185 = -3.1568940568519223e+78;
        bool r446186 = r446184 <= r446185;
        double r446187 = 3.179604535548228e-75;
        bool r446188 = r446184 <= r446187;
        double r446189 = !r446188;
        bool r446190 = r446186 || r446189;
        double r446191 = y;
        double r446192 = r446191 / r446184;
        double r446193 = z;
        double r446194 = x;
        double r446195 = r446193 - r446194;
        double r446196 = fma(r446192, r446195, r446194);
        double r446197 = r446193 * r446191;
        double r446198 = r446197 / r446184;
        double r446199 = r446194 * r446191;
        double r446200 = r446199 / r446184;
        double r446201 = r446198 - r446200;
        double r446202 = r446201 + r446194;
        double r446203 = r446190 ? r446196 : r446202;
        return r446203;
}

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)))