Average Error: 2.1 → 1.5
Time: 14.9s
Precision: 64
\[\frac{x}{y} \cdot \left(z - t\right) + t\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \le -3.43872422120508929 \cdot 10^{-224} \lor \neg \left(\frac{x}{y} \le 7.5728756396099922 \cdot 10^{-286}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z - t}{y} \cdot x + t\\ \end{array}\]
\frac{x}{y} \cdot \left(z - t\right) + t
↓
\begin{array}{l}
\mathbf{if}\;\frac{x}{y} \le -3.43872422120508929 \cdot 10^{-224} \lor \neg \left(\frac{x}{y} \le 7.5728756396099922 \cdot 10^{-286}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{z - t}{y} \cdot x + t\\

\end{array}
double f(double x, double y, double z, double t) {
        double r290288 = x;
        double r290289 = y;
        double r290290 = r290288 / r290289;
        double r290291 = z;
        double r290292 = t;
        double r290293 = r290291 - r290292;
        double r290294 = r290290 * r290293;
        double r290295 = r290294 + r290292;
        return r290295;
}
↓
double f(double x, double y, double z, double t) {
        double r290296 = x;
        double r290297 = y;
        double r290298 = r290296 / r290297;
        double r290299 = -3.4387242212050893e-224;
        bool r290300 = r290298 <= r290299;
        double r290301 = 7.572875639609992e-286;
        bool r290302 = r290298 <= r290301;
        double r290303 = !r290302;
        bool r290304 = r290300 || r290303;
        double r290305 = z;
        double r290306 = t;
        double r290307 = r290305 - r290306;
        double r290308 = fma(r290298, r290307, r290306);
        double r290309 = r290307 / r290297;
        double r290310 = r290309 * r290296;
        double r290311 = r290310 + r290306;
        double r290312 = r290304 ? r290308 : r290311;
        return r290312;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original2.1
Target2.4
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;z \lt 2.7594565545626922 \cdot 10^{-282}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \mathbf{elif}\;z \lt 2.326994450874436 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ x y) < -3.4387242212050893e-224 or 7.572875639609992e-286 < (/ x y)

    1. Initial program 1.9

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

    2. Simplified1.9

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

    if -3.4387242212050893e-224 < (/ x y) < 7.572875639609992e-286

    1. Initial program 2.6

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

    2. Simplified2.6

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

    4. Applied fma-udef2.6

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

    5. Simplified0.2

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \le -3.43872422120508929 \cdot 10^{-224} \lor \neg \left(\frac{x}{y} \le 7.5728756396099922 \cdot 10^{-286}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z - t}{y} \cdot x + t\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.Signal.Multichannel:$cget from hsignal-0.2.7.1"

  :herbie-target
  (if (< z 2.759456554562692e-282) (+ (* (/ x y) (- z t)) t) (if (< z 2.326994450874436e-110) (+ (* x (/ (- z t) y)) t) (+ (* (/ x y) (- z t)) t)))

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