Average Error: 2.2 → 1.6
Time: 3.5s
Precision: 64
\[\frac{x}{y} \cdot \left(z - t\right) + t\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -7.2999266771936822 \cdot 10^{-146} \lor \neg \left(y \le 1.5849138783300118 \cdot 10^{35}\right):\\ \;\;\;\;\mathsf{fma}\left(1, z, -\sqrt[3]{t} \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right) \cdot \frac{x}{y} + \mathsf{fma}\left(\mathsf{fma}\left(-t, 1, t\right), \frac{x}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(z - t\right)}{y} + t\\ \end{array}\]
\frac{x}{y} \cdot \left(z - t\right) + t
↓
\begin{array}{l}
\mathbf{if}\;y \le -7.2999266771936822 \cdot 10^{-146} \lor \neg \left(y \le 1.5849138783300118 \cdot 10^{35}\right):\\
\;\;\;\;\mathsf{fma}\left(1, z, -\sqrt[3]{t} \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right) \cdot \frac{x}{y} + \mathsf{fma}\left(\mathsf{fma}\left(-t, 1, t\right), \frac{x}{y}, t\right)\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r488611 = x;
        double r488612 = y;
        double r488613 = r488611 / r488612;
        double r488614 = z;
        double r488615 = t;
        double r488616 = r488614 - r488615;
        double r488617 = r488613 * r488616;
        double r488618 = r488617 + r488615;
        return r488618;
}
↓
double f(double x, double y, double z, double t) {
        double r488619 = y;
        double r488620 = -7.299926677193682e-146;
        bool r488621 = r488619 <= r488620;
        double r488622 = 1.5849138783300118e+35;
        bool r488623 = r488619 <= r488622;
        double r488624 = !r488623;
        bool r488625 = r488621 || r488624;
        double r488626 = 1.0;
        double r488627 = z;
        double r488628 = t;
        double r488629 = cbrt(r488628);
        double r488630 = r488629 * r488629;
        double r488631 = r488629 * r488630;
        double r488632 = -r488631;
        double r488633 = fma(r488626, r488627, r488632);
        double r488634 = x;
        double r488635 = r488634 / r488619;
        double r488636 = r488633 * r488635;
        double r488637 = -r488628;
        double r488638 = fma(r488637, r488626, r488628);
        double r488639 = fma(r488638, r488635, r488628);
        double r488640 = r488636 + r488639;
        double r488641 = r488627 - r488628;
        double r488642 = r488634 * r488641;
        double r488643 = r488642 / r488619;
        double r488644 = r488643 + r488628;
        double r488645 = r488625 ? r488640 : r488644;
        return r488645;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original2.2
Target2.4
Herbie1.6
\[\begin{array}{l} \mathbf{if}\;z \lt 2.7594565545626922 \cdot 10^{-282}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \mathbf{elif}\;z \lt 2.326994450874436 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -7.299926677193682e-146 or 1.5849138783300118e+35 < y

    1. Initial program 1.3

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

    3. Applied add-cube-cbrt1.4

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

    4. Applied *-un-lft-identity1.4

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

    5. Applied prod-diff1.4

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

    6. Applied distribute-rgt-in1.4

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

    7. Applied associate-+l+1.4

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

    8. Simplified1.4

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

    if -7.299926677193682e-146 < y < 1.5849138783300118e+35

    1. Initial program 4.3

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

    3. Applied associate-*l/2.2

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

  4. Final simplification1.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -7.2999266771936822 \cdot 10^{-146} \lor \neg \left(y \le 1.5849138783300118 \cdot 10^{35}\right):\\ \;\;\;\;\mathsf{fma}\left(1, z, -\sqrt[3]{t} \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right) \cdot \frac{x}{y} + \mathsf{fma}\left(\mathsf{fma}\left(-t, 1, t\right), \frac{x}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(z - t\right)}{y} + t\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.Signal.Multichannel:$cget from hsignal-0.2.7.1"
  :precision binary64

  :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))